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# First Measurement of the Muon Anti-Neutrino Charged Current Quasielastic Double-Differential Cross Section

## Material Information

Title:
First Measurement of the Muon Anti-Neutrino Charged Current Quasielastic Double-Differential Cross Section
Physical Description:
1 online resource (211 p.)
Language:
english
Creator:
Grange, Joseph M
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

## Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Ray, Heather
Committee Members:
Furic, Ivan Kresimir
Woodard, Richard P
Saab, Tarek
Wagener, Kenneth B

## Subjects

Subjects / Keywords:
antineutrino -- neutrino
Physics -- Dissertations, Academic -- UF
Genre:
Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract:
This dissertation presents the first measurement of the muon anti-neutrino charged current quasi-elastic double-differential cross section. These data significantly extend the knowledge of neutrino and anti-neutrino interactions in the GeV range, a region that has recently come under scrutiny due to a number of conflicting experimental results. To facilitate this measurement, a set of novel techniques to measure the neutrino component ofthe dominantly anti-neutrino beam were opportunistically developed and executed. Perhaps more importantly, these data help to understand signal and background processes that will be observed in the coming decades in long-baseline neutrino experiments that aim to measure the ordering of the neutrino masses and a process that may well explain why our universe is almost entirely matter-dominated. The confirmation of the latter process would mean the discovery of a physical mechanism that explains the existence of our universe.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Joseph M Grange.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-11-30

## Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045262:00001

Full Text

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FIRST MEASUREMENT OF THE MUON ANTI-NEUTRINO CHAR GED CURRENT QUASIELASTIC DOUBLE-DIFFERENTIAL CR OSS SECTION By JOSEPH M. GRANGE DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2013 1

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c r 2013 Joseph M. Grange 2

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T o all the lo v ely p eople who ha v e help ed me along the w a y 3

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A CKNO WLEDGEMENTS So man y folks had a hand in guiding m y progress that it w ould b e disingen uous to try and form a complete list here. But I certainly could not ha v e accomplished what I did without the guidance, insigh t, and con tagious en th usiasm of m y advisor Heather Ra y and those at F ermilab most inruen tial to m y analysis: Sam Zeller, T epp ei Katori, and Chris P olly I am so luc ky to call y ou m y men tors, and I hop e to liv e up to y our exp ectations in m y p ostgraduate w ork. 4

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T ABLE OF CONTENTS page A CKNO WLEDGEMENTS . . . . . . . 4 LIST OF T ABLES . . . . . . . . 8 LIST OF FIGURES . . . . . . . . 9 ABSTRA CT . . . . . . . . . 13 CHAPTER 1 INTR ODUCTION . . . . . . . 14 2 NEUTRINO O VER VIEW . . . . . . . 15 2.1 Disco v ery . . . . . . . . 15 2.2 In teraction and Propagation States . . . . . 17 2.3 Fla v ors of Neutrinos . . . . . . . 19 2.4 Chiralit y . . . . . . . . 20 2.5 Magnetic Momen t . . . . . . . 22 2.6 Absolute Mass . . . . . . . 23 2.7 Neutrino Sources . . . . . . . 23 3 NEUTRINO OSCILLA TIONS . . . . . . 25 3.1 F ormalism . . . . . . . . 25 3.1.1 Three-Neutrino Mixing . . . . . . 27 3.1.2 Tw o-Neutrino Mixing . . . . . . 28 3.2 Exp erimen tal Evidence for Neutrino Oscillations . . . 30 3.2.1 Solar Oscillations . . . . . . 30 3.2.2 A tmospheric Oscillations . . . . . 33 3.2.3 13 and Oscillations . . . . . . 35 3.2.4 Hin ts for m 2 1 eV 2 . . . . . . 38 3.3 Summary and Outstanding Questions . . . . 40 4 NEUTRINO INTERA CTIONS IN MINIBOONE . . . . 44 4.1 Ov erview . . . . . . . . 44 4.2 CCQE . . . . . . . . 46 4.3 CC . . . . . . . . 50 4.4 Nuclear Eects . . . . . . . 50 4.4.1 Nuclear Mo deling . . . . . . 50 5

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4.4.2 Final-State In teractions . . . . . . 54 5 THE MINIBOONE EXPERIMENT . . . . . 56 5.1 Ov erview . . . . . . . . 56 5.2 The Bo oster Neutrino Beamline . . . . . 56 5.2.1 The Primary Proton Beam . . . . . 56 5.2.2 Beryllium T arget and Magnetic F o cusing Horn . . . 60 5.2.3 Meson Deca y Region . . . . . . 62 5.2.4 Neutrino Flux Calculation . . . . . 63 5.3 Detector . . . . . . . . 69 5.3.1 Ph ysical La y out . . . . . . 69 5.3.2 Mineral Oil and its Prop erties . . . . . 70 5.3.3 Photom ultiplier T ub es . . . . . . 73 5.3.4 Calibration Systems . . . . . . 73 5.3.5 Analysis T o ols . . . . . . . 75 6 INTR ODUCTION TO THE CR OSS-SECTION MEASUREMENT . 81 6.1 Ov erview . . . . . . . . 81 6.2 Ev en t Selection . . . . . . . 81 7 BA CK GR OUND MEASUREMENTS AND CONSTRAINTS . . 91 7.1 Measuremen ts of the Bac kground . . . . . 91 7.1.1 Motiv ation . . . . . . . 91 7.1.2 General Strategy . . . . . . 93 7.1.3 Flux Measuremen t Using CC + Ev en ts . . . 96 Implemen tation of the CC + Cross Section . . . 97 The Selected Sample . . . . . . 99 Flux Measuremen t Using CC + . . . . 101 7.1.4 Flux Measuremen t Using Nuclear Capture . . 103 Implemen tation of the CC Cross Sections . . . 104 Muon Capture Mo del and Ev en t Selection . . . 106 Calibrations and Stabilit y Chec ks Using the Neutrino-Mo de Data 113 Flux Measuremen t Using Capture . . . . 117 Systematic Errors . . . . . . 119 7.1.5 Flux Measuremen t Using the cos Distribution . . 122 Ov erview . . . . . . . 122 Sample Selection . . . . . . 123 Measuremen t Execution . . . . . . 124 The Co v ariance Matrix . . . . . . 127 Results and Systematic Errors . . . . . 130 7.1.6 Summary of Flux Measuremen ts . . . . 136 7.2 The CC Bac kground . . . . . . 139 6

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7.3 All Other Bac kgrounds . . . . . . 143 7.4 Bac kground Constrain t Summary . . . . . 143 8 CCQE CR OSS-SECTION MEASUREMENT . . . . 146 8.1 Data Stabilit y . . . . . . . 146 8.2 CCQE Sim ulation . . . . . . 149 8.3 Cross-Section Calculation . . . . . . 152 8.3.1 Unsmearing to T rue Quan tities . . . . . 154 8.3.2 Eciency Correction . . . . . . 161 8.3.3 Flux and In teraction T argets . . . . . 163 8.3.4 Statistical Uncertain t y . . . . . . 164 8.4 Systematic Uncertain t y . . . . . . 165 8.4.1 Bac kground Uncertain ties . . . . . 165 8.4.2 Signal Errors . . . . . . . 167 8.4.3 Uncertain t y Summary for CCQE on Mineral Oil . . 170 8.4.4 Uncertain t y Summary for CCQE on Carb on . . . 173 8.5 Results . . . . . . . . 176 8.5.1 Results Inciden t on Mineral Oil . . . . . 176 8.5.2 Results Inciden t on Carb on . . . . . 182 9 COMBINED AND CCQE MEASUREMENTS . . . 190 9.1 Correlated Measuremen ts . . . . . . 190 9.2 Com bined and CCQE Measuremen ts . . . . 192 10 CONCLUSION . . . . . . . . 200 LIST OF REFERENCES . . . . . . . 201 BIOGRAPHICAL SKETCH . . . . . . . 211 7

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LIST OF T ABLES T able page 5-1 P article Cerenk o v thresholds in the MiniBo oNE detector oil . . 71 6-1 Purit y and detection eciency for CCQE . . . . 85 6-2 CCQE sample summary . . . . . . 86 7-1 Data set b y absorb er p erio d used in the CC + and cos analyses . 97 7-2 Selection eciency and purit y for the CC + sample . . . 101 7-3 Predicted in teraction comp osition of the an tineutrino-mo de CC + sample 102 7-4 Summary of the CC + rux measuremen t . . . . 104 7-5 and CC purit y for the capture samples . . . 112 7-6 Predicted in teraction con tributions to the capture samples . 113 7-7 Summary of the neutrino-mo de capture samples . . . 114 7-8 Calibration summary for the capture and Mic hel detection mo dels . 115 7-9 Neutrino-mo de sub ev en t ratios . . . . . 116 7-10 Calibration eects on the rux measuremen t using capture . 119 7-11 Uncertain t y summary for the capture-based measuremen t of the rux 121 7-12 Summary of the rux measuremen t using capture . . 122 7-13 Summary of uncertain ties for the rux measuremen t using capture 130 7-14 Summary of cross-section uncertain ties for the analysis of cos ev en ts 131 7-15 Summary of the rux measuremen t using the cos distribution . 135 7-16 and con tributions to the CCQE sample b efore and after the cos t 136 7-17 Summary of bac kgrounds in the CCQE sample . . . 145 8-1 Summary of data groups input to stabilit y tests . . . 147 8-2 Normalization uncertain ties for the cross sections on mineral oil . 176 8-3 Normalization uncertain ties for cross sections on carb on . . 177 8

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LIST OF FIGURES Figure page 2-1 Detection sc hematic for the rst observ ation of the neutrino . . 17 2-2 Measured width of the Z 0 b oson . . . . . 20 2-3 Neutrino sources . . . . . . . 24 3-1 SNO solar neutrino results . . . . . . 32 3-2 KamLAND e disapp earance results . . . . . 33 3-3 Solar oscillation parameters . . . . . . 34 3-4 Evidence of atmospheric oscillations from Sup er-Kamiok ande . . 35 3-5 A tmospheric oscillation parameters . . . . . 36 3-6 Observ ation of e disapp earance at Da y a Ba y . . . . 37 3-7 Curren t con train ts on and the mass hierarc h y . . . 38 3-8 Com bined e and e app earance results from MiniBo oNE . . 39 3-9 Degeneracy in the ordering of neutrino mass states . . . 42 4-1 Comparison of exp erimen tal data in and CC in teractions . 45 4-2 Decomp osition of the CCQE cross section . . . . 49 4-3 Binding energy and F ermi momen tum in electron scattering data . 52 4-4 Pion rein teraction cross sections . . . . . 55 5-1 Carto on of the rst stage of the F ermilab accelerator net w ork . . 57 5-2 Injection in to the Bo oster sync hrotron . . . . 58 5-3 Timing structure of the BNB proton spills . . . . 59 5-4 The Bo oster Neutrino Beamline b eryllium target . . . 61 5-5 Magnetic eld at the BNB horn . . . . . 62 5-6 Primary hadropro duction mo dels at MiniBo oNE . . . 64 5-7 Double-dieren tial pro duction at the BNB . . . 66 5-8 Con tributions to neutrino b eams from hadronic rein teractions . . 67 5-9 BNB rux at MiniBo oNE for neutrino and an tineutrino mo des . 68 9

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5-10 Ov erview of the MiniBo oNE detector housing . . . 69 5-11 MiniBo oNE detector cuta w a y and v eto region photo . . . 70 5-12 Extinction rate sp ectrum in the MiniBo oNE oil . . . 72 5-13 Quan tum eciency for MiniBo oNE PMTs . . . . 73 5-14 Laser calibration system sc hematic . . . . . 74 5-15 Carto on of the m uon calibration system and calibration summary . 76 5-16 T ypical CCQE PMT signature . . . . . 77 5-17 T op ology and timing of t ypical electron and m uon ev en ts in MiniBo oNE 78 5-18 Energy and angular resolutions of the MiniBo oNE reconstruction . 80 6-1 V eto tank hit distributions . . . . . . 82 6-2 Basic CCQE selection gures . . . . . 84 6-3 Spatial correlation b et w een the sub ev en ts CCQE sample . . 89 6-4 A generic t-c hannel exc hange . . . . . 89 6-5 Kinematic distributions of the CCQE sample . . . 90 7-1 Pion pro duction phase space at the BNB . . . . 92 7-2 Accepted sp ectra in neutrino and an tineutrino run mo des . . 95 7-3 MiniBo oNE CC + total cross section . . . . 98 7-4 CC + fractional uncertain t y con v ersion . . . . 100 7-5 Cross-section correction to the sim ulated CCQE pro cess . . 105 7-6 Log-lik eliho o d =e particle-ID in the single-sub ev en t sample . . 107 7-7 F our-momen tum transfer cut on for 1 and 2 sub ev en ts . . 109 7-8 Predicted m uon stopping radius for 1 and 2 sub ev en ts . . 110 7-9 Timing dierence b et w een the rst and second sub ev en ts . . 111 7-10 T ank hit distribution for the Mic hel sub ev en t in an tineutrino mo de . 111 7-11 T ank hit distribution for the Mic hel sub ev en t in neutrino mo de . 115 7-12 Stabilit y c hec k of the timing systems with the neutrino-mo de data . 117 7-13 Before-t and cos distributions . . . . 122 10

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7-14 Tw o options for spatially correlating the m uon and its Mic hel . . 125 7-15 V ariations in the cos distribution to due cross-section uncertain ties . 127 7-16 Correlation matrix b et w een and ev en ts in the cos distribution 129 7-17 Angular t results in the CCQE sample . . . . 134 7-18 Summary of rux measuremen ts . . . . . 138 7-19 CC + correction based on MiniBo oNE neutrino-mo de data . . 140 7-20 Comparison of v arious mo dels to MiniBo oNE CC + data . . 141 7-21 Comparison of v arious mo dels to the prediction of CC ev en ts . 142 8-1 Kolmogoro v-Smirno test for the CCQE sample . . . 147 8-2 Data stabilit y for E QE in the CCQE sample . . . 148 8-3 Normalization stabilit y for the CCQE sample . . . 148 8-4 Generator-lev el comparison of the sim ulate CCQE cross section on carb on 150 8-5 Generator-lev el cross-section correction for CCQE in teractions on carb on 151 8-6 Generator-lev el correction for CCQE on h ydrogen . . . 152 8-7 Generator-lev el comparisons of the sim ulated CC pro cesses . . 153 8-8 Reconstructed-truth lev el comparisons for the CCQE distributions . 154 8-9 Unsmearing matrices for the CCQE distributions . . . 158 8-10 Comparison of b efore and after the unsmearing pro cess . . 159 8-11 Comparison of the Ba y esian and matrix in v ersion metho ds of unsmearing 160 8-12 Detection eciency for the CCQE distributions . . . 162 8-13 paren t phase space at the BNB in an tineutrino mo de . . 163 8-14 rux prediction . . . . . . . 164 8-15 F ractional uncertain t y for the CCQE cross sections on mineral oil . 171 8-16 More fractional uncertain ties for the CCQE cross sections on mineral oil 172 8-17 F ractional uncertain t y for the CCQE cross sections on carb on . 174 8-18 More fractional uncertain ties for the CCQE cross sections on mineral oil 175 8-19 CCQE cross sections on mineral oil . . . . 179 11

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8-20 Pro jections of the CCQE d 2 dT d cos cross section on mineral oil . 180 8-21 Shap e comparison of the cross section on mineral oil with MC . 181 8-22 CCQE cross sections on carb on . . . . . 183 8-23 One-dimensional views of the double-dieren tial cross section on carb on 184 8-24 Shap e comparison of the CCQE cross section on carb on to MC . 185 8-25 Shap e comparison of the single-dieren tial cross section on mineral oil 186 8-26 Comparison b et w een the and data from MiniBo oNE and NOMAD 186 8-27 T otal cross section for CCQE on carb on . . . . 188 8-28 T otal cross section for and CCQE on carb on . . . 189 9-1 and CCQE cross sections used in the correlation analysis . 194 9-2 Ov erall correlation co ecien ts b et w een the and CCQE cross sections 196 9-3 Correlation co ecien ts b y uncertain t y t yp e . . . . 197 9-4 Dierence measuremen ts of the and CCQE cross sections . 198 9-5 Ratio measuremen ts of the and CCQE cross sections . . 199 12

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Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y Of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y FIRST MEASUREMENT OF THE MUON ANTI-NEUTRINO CHAR GED CURRENT QUASIELASTIC DOUBLE-DIFFERENTIAL CR OSS SECTION By Joseph M. Grange Ma y 2013 Chair: Heather Ra y Ma jor: Ph ysics This dissertation presen ts the rst measuremen t of the m uon an tineutrino c harged curren t quasi-elastic double-dieren tial cross section. These data signican tly extend the kno wledge of neutrino and an tineutrino in teractions in the GeV range, a region that has recen tly come under scrutin y due to a n um b er of conricting exp erimen tal results. T o maximize the precision of this measuremen t, three no v el tec hniques w ere emplo y ed to measure the neutrino bac kground comp onen t of the data set. Represen ting the rst measuremen ts of the neutrino con tribution to an accelerator-based an tineutrino b eam in the absence of a magnetic eld, the successful execution of these tec hniques carry implications for curren t and future neutrino exp erimen ts. Finally com bined measuremen ts of these an tineutrino and the previously-published neutrino cross section data using the same apparatus maximize the extracted information from these results b y exploiting correlated systematic uncertain ties. The results of this analysis will help to understand signal and bac kground pro cesses in presen t and future long-baseline neutrino exp erimen ts, the principle goal of whic h is to measure the ordering of the neutrino masses and a pro cess that ma y ultimately explain the origin of our matter-dominated univ erse. 13

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CHAPTER 1 INTR ODUCTION The w ork presen ted here represen ts a ma jor step forw ard in exp erimen tally understanding the b eha vior of m uon neutrinos and an tineutrinos. Apart from pro viding a w orld's-rst measuremen t of these in teractions in a mostly-unexplored energy region, these data adv ance the neutrino comm unit y's preparedness to searc h for an asymmetry b et w een matter and an ti-matter that ma y w ell pro vide the ph ysical mec hanism for the existence of our univ erse. The details of these measuremen ts are preceded b y brief summaries of the history of the neutrino (Chapter 2 ), the phenomenon of neutrino oscillations (Chapter 3 ), and a description of their in teractions (Chapter 4 ). Details of the exp erimen tal setup for the measuremen ts are giv en in Chapter 5 Chapter 6 in tro duces the m uon an tineutrino cross-section measuremen t and motiv ates the need for dedicated, in situ bac kground constrain ts. The w orld's rst measuremen ts of the neutrino comp onen t of an an tineutrino b eam using a non-magnetized detector, as w ell as other crucial bac kground constrain ts, are presen ted in Chapter 7 The m uon an tineutrino cross-section measuremen t is giv en in Chapter 8 By exploiting correlated systematic uncertain ties, com bined measuremen ts of the m uon neutrino and an tineutrino cross sections describ ed in Chapter 9 maximize the precision of the extracted information from b oth results. Finally the results are summarized in Chapter 10 14

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CHAPTER 2 NEUTRINO O VER VIEW This c hapter touc hes on the v arious milestone measuremen ts in the history of the neutrino and outlines man y of their basic prop erties. As p erhaps the most imp ortan t dev elopmen t in its y oung history a more complete dev elopmen t and review of neutrino oscillations is sa v ed for Chapter 3 2.1 Disco v ery A t the b eginning of the t w en tieth cen tury m uc h of the ph ysics comm unit y w ere con ten t to b eliev e the univ erse w as fundamen tally comp osed of electrons, photons and, in the literal sense, atoms. An example of historically bad advice giv en b y an advisor to their studen t came when the sup ervisor of Max Planc k suggested that \in this eld, almost ev erything is already disco v ered, and all that remains is to ll a few unimp ortan t holes" [ 1 ]. Planc k w en t on to rev olutionize the eld with his description of quan tum mec hanics and ultimately help ed exp ose h uman kno wledge to a litan y of new particles and phenomena. And so, lik e man y of the particles disco v ered in the t w en tieth cen tury the neutrino came as a surprise. The rst hin ts came through observ ations of so-called deca y in the 1920's, where a neutron inside a n ucleus sp on taneously deca ys. Though the picture of the proton and neutron structure of the n ucleus w as not y et clear, b y energy and spin conserv ation the deca y w as b eliev ed to b e a t w o-b o dy pro cess: n p + e ; E e = m 2n + m 2e m 2p 2 m n ; (2.1) where E e is the energy of the ejected electron in the rest frame of the neutron and m n ; m p and m e are the neutron, proton, and electron masses, resp ectiv ely As the neutrons housed in n uclei are on a v erage at rest, the observ ed electron sp ectrum ough t to b e nearly mono energetic. Multiple exp erimen ts using a v ariet y of -deca y sources conclusiv ely rejected this h yp othesis [ 2 ]. 15

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Of the man y alternate explanations for the observ ed electron sp ectrum, W olfgang P auli prop osed in 1931 the ultimately pro v en h yp othesis: the pro ducts of deca y include a third, electrically neutral particle of mass far less than the electron and whose in teractions are rare enough to ha v e escap ed direct detection. The deca y reaction is no w describ ed as: n p + e + ; E e 2 ( m e ; m 2n + m 2e m 2p + m 2 2 m n ) ; (2.2) where the presence of the neutrino kinematically allo ws the electron to ha v e a con tin uous energy sp ectrum. Regarding the prop osed feeble in teraction rate of the neutrino, P auli famously quipp ed \I ha v e done a terrible thing. I ha v e p ostulated a particle that cannot b e detected" [ 3 ]. F ortunately it to ok only a few decades for exp erimen tal tec hnology and tec hniques to reac h the lev el of precision necessary to directly observ e the neutrino. If P auli's in terpretation of the deca y sp ectrum and Enrico F ermi's extended description of the particle [ 4 5 ] w ere correct, t w o prolic sources of neutrinos in the 1950's w ere a v ailable in atomic explosions and n uclear reactors. One of the early plans to detect the neutrino in v olv ed a retrosp ectiv ely comical prop osal to detonate a dedicated atomic b om b while a neutrino detector w as sim ultaneously dropp ed do wn a nearb y mineshaft [ 6 ]. They ev en tually pro ceeded with a more pacic design, aiming to observ e neutrinos pro duced in a n uclear reactor. If Eq. 2.2 is the correct description of deca y the in v erted pro cess induced b y a neutrino should also b e allo w ed: + p n + e + In a tank of liquid scin tillator dop ed with cadmium, the p ositron pro duced in this in v erted deca y reaction will annihilate with an in-medium electron, pro ducing t w o prompt gamma ra ys emitted in opp osing directions ( e + + e 2 r r r = ). The neutron has a large probabilit y for b eing captured on the cadmium n uclei, and the c haracteristic de-excitation photons follo wing this pro cess 16

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Figure 2-1. The detection sc heme for the rst conclusiv e demonstration of the existence of the neutrino. Image from Ref. [ 7 ]. pro vides a clean neutron signature. The detection sc hematic is sho wn in Figure 2-1 Photom ultiplier tub es collect the photons from the annihilation and capture reactions, and data from a detector using these principles yielded the rst denitiv e detection of the neutrino in 1956 [ 8 ]. 2.2 In teraction and Propagation States An y in telligen t discussion of the nature of the neutrino m ust b e built on an understanding of its p eculiarly misaligned in teraction and propagation eigenstates. Generically an eigenstate is a v ector returned b y the action of a particular op erator. Neutrinos only in teract through the w eak force, so the eigenstates a v ailable up on action with the w eak op erator (or, more in tuitiv ely up on an in teraction with a W or Z b oson) are the w eakly-participating leptonic \ra v ors" e; and The other eigenstate that go v erns the b eha vior of the neutrino is its Hamiltonian state. This describ es the ph ysical propagation of the neutrino in time with a denite v alue of mass. Throughout this w ork, the in teraction eigenstate is often referred to as the w eak, or ra v or, state, just as the Hamiltonian state is equated with the propagation, or mass, eigenstate. F or most fundamen tal particles, their in teraction and Hamiltonian eigenstates are 17

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indistinguishable. Ho w ev er, nothing demands this b e the case, and a div ergence b et w een the states has b een observ ed in t w o leptonic sectors: quarks and neutrinos. A helpful w a y to describ e the relationship b et w een the ra v or and Hamiltonian eigenstates is a unitary mixing matrix U that connects the arbitrary ra v or state j i to the mass states j k i : j i = P U k j k i Precision measuremen ts of a v ariet y of bary onic w eak in teractions yield the follo wing appro ximate relationship b et w een the quark Hamiltonian and ra v or states [ 9 ]: 0BBBB@ d 0 s 0 b 0 1CCCCA = 0BBBB@ 0 : 974 0 : 225 0 : 003 0 : 225 0 : 973 0 : 041 0 : 009 0 : 040 0 : 999 1CCCCA 0BBBB@ d s b 1CCCCA ; (2.3) where d 0 s 0 and b 0 refer to the quarks of ra v or do wn, strange, and b ottom, resp ectiv ely and the con v en tion of using d s and b for the Hamiltonian states is used. Note that, due to the unitary nature of U this description of the mixing in terms of the do wn-t yp e quarks instead of the up-t yp e quarks is arbitrary The same relationship for neutrino mixing, the formalism for whic h and whose measuremen ts are describ ed in Chapter 3 is giv en b y: 0BBBB@ e 1CCCCA = 0BBBB@ 0 : 822 0 : 547 0 : 150 0 : 356 0 : 704 0 : 614 0 : 442 0 : 452 0 : 774 1CCCCA 0BBBB@ 1 2 3 1CCCCA ; (2.4) where the v alues sho wn assume the mass hierarc h y to b e normal and the CP-violating phase to b e zero (see Chapter 3 for details of b oth quan tities). Eqs. 2.3 and 2.4 clearly sho w the ra v or and mass eigenstates to b e distinct for neutrinos and quarks, but also that the details of this mixing dier greatly b et w een the t w o sp ecies. The origin of these mixing 18

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parameters is not understo o d, and is one of the outstanding issues prev en ting a fundamen tal understanding of the w eak in teraction. 2.3 Fla v ors of Neutrinos W e no w kno w that the creation or annihilation of a c harged lepton ( e; ; ) m ust also in v olv e either its o wn c harged an tiparticle or a neutrino (or an tineutrino) of the same ra v or ( e ; ; ). This certainly did not need to b e the case, and the prediction of this lepton conserv ation symmetry in the Standard Mo del (SM) did not arriv e un til decades after the disco v ery of the neutrino. F ollo wing the disco v ery of the neutrino, man y exp erimen ts con tributed to the quic kly-gro wing b o dy of kno wledge regarding its prop erties. Lik ely the most signican t in this p erio d, b oth in terms of the engineering that w ould b ecome the future of the eld and the milestone it represen ted in the emerging picture of the neutrino, came from the rst observ ation of neutrinos from an accelerator-based b eam [ 10 ]. The exp erimen ts, led b y L. Lederman, observ ed an o-axis rux of neutrinos dominan tly created from + deca y the reaction of whic h w as kno wn to pro ceed via + + + Ho w ev er, unlik e in the rst observ ation of the neutrino using a n uclear reactor source, the c harged particles that emerged in the detector w ere negativ ely c harged m uons, not p ositrons. This w as particularly comp elling b ecause it could ha v e b een the case that m uons w ere not created in reactor neutrino in teractions simply b ecause b ecause their pro duction w ould not b e energetically allo w ed: the energy of the source ( < 10 MeV) is m uc h less than the m uon mass ( 105 MeV). That no p ositrons or electrons w ere observ ed in the accelerator-based exp erimen t conclusiv ely demonstrated the reactor neutrinos to b e distinct from those created in + deca ys. Through the dev elopmen t of the SM, these so on b ecame to b e kno wn as electron and m uon ra v ored neutrinos e and With the disco v ery of the particle in 1975 [ 11 ], a third fundamen tal neutrino w as presumed to accompan y it. Ho w ev er, the large mass of the ( m 1.8 GeV) and its rapid deca y (with a lifetime of O (10 13 ) s) with a n um b er of b oth hadronic and leptonic mo des 19

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0 10 20 30 86 88 90 92 94 E cm M GeV Nshad M nb N 3 n 2 n 4 n average measurements, error bars increased by factor 10 ALEPH DELPHI L3 OPAL Figure 2-2. Results from a com bined analysis of precision measuremen ts of the Z 0 width. The n um b er of ligh t, activ e neutrino sp ecies is determined to b e 2.9840 0.0082, consisten t with direct observ ations of the e and neutrinos. Figure from Ref. [ 13 ]. a v ailable made the direct observ ation of the particularly c hallenging. It w asn't un til the y ear 2000 that the particle w as exp erimen tally conrmed, when a team observ ed four candidate ev en ts on a calculated bac kground of 0.34 from an accelerator-based b eam of 100 GeV 's [ 12 ]. Since then, not more than t w en t y total ev en ts ha v e b een observ ed, making it one of the least exp erimen tally-prob ed SM mem b ers. Finally consisten t with the direct observ ations discussed in this section, precision measuremen ts of the Z 0 b oson width has denitiv ely concluded that there exist exactly three neutrino ra v ors with eectiv e mass less than half the Z 0 mass and whom also participate in the w eak in teraction [ 13 ]. Figure 2-2 presen ts the measuremen t of the Z 0 width. 2.4 Chiralit y In the same y ear the neutrino w as disco v ered, another paramoun t exp erimen tal result rev ealed an unexp ected asp ect of the w eak in teraction. Using a magnetic eld to p olarize the spin of a collection of unstable 60 Co atoms, the observ ed direction of the emitted electron in the -deca y reaction w as nearly alw a ys opp osite to the direction of the 20

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aligned spin of the paren t n uclei [ 14 ]. The nearly-p erfect an ti-correlation b et w een the n uclei spins and the electron direction indicated that the mirror-symmetry of parit y is violated in w eak in teractions. It had b een suggested previously that the parit y symmetry in the w eak in teraction need not b e strictly conserv ed [ 15 ], but the exp erimen tal evidence concluded that parit y violation w as maximal The implication for the b eha vior of observ able neutrinos w as clear: to balance the spin and momen tum of the reaction, the observ ed kinematics of the electron dictate that it m ust b e accompanied b y an an tineutrino with a denite alignmen t b et w een its momen tum and spin v ectors. These data w ere am biguous b et w een the t w o v ectors b eing aligned or an ti-aligned for the neutrino, but a few y ears later a clev er tec hnique w as executed to measure this correlation: observ ations of the p olarization of de-excitation photons follo wing orbital electron capture on n uclei determined the spin and momen tum v ectors for the neutrino to b e an ti-aligned [ 16 ]. The inner pro duct b et w een a particle's spin and momen tum at an y instan t is kno wn as it's helicit y and the natural preference for helicit y v alues (if an y) is the more fundamen tal quan tit y of c hiralit y Th us, using the mec hanical analogy of ordinary screws, the c hiralit y of leptons in the w eak in teraction is left-handed, while (via the CPT theorem) an ti-leptons are righ t-handed. One of the direct consequences of neutrino oscillations is the implication of non-zero neutrino mass. With non-zero neutrino mass, it is in principle p ossible to b o ost to a frame with v elo cit y v suc h that v < v < c in whic h a neutrino (an tineutrino) w ould ha v e p ositiv e (negativ e) helicit y W orth noting, the correlation b et w een c hiralit y and helicit y is p erfect for massless particles and decreases sharply for those of non-zero mass, and so the pro duction of p ositiv e helicit y e is m uc h more allo w ed relativ e the emission of negativ e-helicit y an tineutrinos. A consequence of this suppression that is crucial to the exp erimen tal neutrino program is that the electronic deca y of pions ( e + e ) relativ e to the m uonic deca y ( + ) is suppressed b y appro ximately ( m e =m ) 2 m 2 m 2e m 2 m 2 2 1 : 2 10 4 This allo ws for high-purit y sources of and from the deca y of pions in accelerator 21

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en vironmen ts, and the con trol of the b eam energy and propagation distance aorded b y these articial sources has made this the standard for probing neutrino oscillation ph ysics. 2.5 Magnetic Momen t Non-zero neutrino mass allo ws for the p ossibilit y of a magnetic momen t. Though electrically neutral, electromagnetic prop erties of neutrinos ma y b e accessed through magnetic couplings with photons in lo op diagrams. Muc h lik e neutrino mass, the magnetic momen t w ould b e a prop ert y instrinsic to the Hamiltonian eigenstate, and therefore observ ations of magnetic momen ts through w eak in teractions prob e sup erp ositions of the true quan tities. This relationship can b e describ ed as [ 17 ]: 2 = X j X k U k e iE k L j k 2 ; (2.5) where j and k index the Hamiltonian eigenstates, U connects the Hamiltonian eigenstates to the ra v or state E and L are the energy and tra v el distance of the neutrino, resp ectiv ely and j k describ es the coupling of the mass eigenstates to the electromagnetic eld. As a calculable SM pro cess, the neutrino-electron elastic scattering c hannel + e + e is t ypically used to searc h for the neutrino's magnetic momen t. Evidence of an electromagnetic coupling b et w een the neutrino and electron w ould presen t itself as ev en ts in excess of the predicted cross section or a distortion in the recoiling electron sp ectrum. Man y searc hes for the neutrino magnetic momen t ha v e b een executed using astroph ysical data [ 18 19 20 ] and more direct observ ations of neutrinos from solar [ 21 ], accelerator [ 22 ], reactor [ 23 ], and sup erno v a [ 24 ] sources. The curren t b est limit on the eectiv e magnetic momen t for an y neutrino sp ecies comes from observ ations of reactor e 's, 22

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where it w as found e < 7 : 4 10 11 B where B = e= 2 m e (using natural units) is the Bohr magneton, at 90% condence lev el (C.L.) [ 25 ]. 2.6 Absolute Mass F rom neutrino oscillation observ ations, the mass of at least t w o of the neutrino states is kno wn to b e non-zero, but the sensitivit y of v arious exp erimen tal tests ha v e not y et reac hed the lev el of precision required to measure their v alues. Also kno wn from neutrino oscillations, the most massiv e state is greater or equal to p m 232 0 : 05 eV. Mean while, observ ations of cosmological radiation set an upp er limit on the sum of activ e neutrino masses of P m < 0 : 2 0 : 4 eV at 95% C.L., where the limit dep ends on assumptions used to analyze the Lymandata [ 26 ]. One of the curren t prosp ects in probing lo w er mass regions in v olv es the v ery in teraction that led to the neutrino's disco v ery: deca y As suggested b y Eq. 2.2 the endp oin t of the deca y sp ectrum is sensitiv e to the mass of the e This measuremen t w ould giv e the eectiv e mass of the e whic h is a sup erp osition of the true mass states according to their coupling with the electron-ra v or neutrino. Curren tly the b est limit based on observ ations of the endp oin t of the deca y sp ectrum is m e < 2 : 3 eV (95% C.L.) [ 27 ], while the next-generation exp erimen ts aim to ac hiev e sub-eV precision [ 28 29 ]. 2.7 Neutrino Sources Though the neutrino remains rather p o orly understo o d, man y natural and articial sources spanning an immense energy range are a v ailable to con tin ue to prob e its nature. Figure 2-3 sho ws the v arious neutrino sources and their appro ximate sp ectral con tributions. Generally tec hnology for direct neutrino detection is most eectiv e for neutrinos of energy 10 6 10 11 eV. While w e ha v e learned a great deal from observ ations of this energy range, Figure 2-3 suggests there are man y opp ortunities to expand this kno wledge using freely a v ailable neutrinos. It ma y b e p ossible to learn more not only ab out the neutrino, but there ma y b e ric h ph ysics in their pro duction mec hanisms as w ell. 23

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Figure 2-3. Summary of the v arious prolic neutrino sources in the univ erse. The cross section for e + e e + e is sho wn for comparison. Figure from Ref. [ 30 ]. As an example, a preliminary analysis of t w o neutrino ev en ts of energy > 10 12 eV w as recen tly rep orted [ 31 ]. These neutrinos ha v e the highest energy ev er recorded, and their origin is not clear [ 32 33 ]. A t the lo w end of the sp ectrum, neutrino remnan ts from the Big Bang are predicted to stil l p ermeate the univ erse at a densit y of 100 cm 3 Figure 2-3 sho ws the cross section for these neutrinos are man y orders of magnitude b elo w the curren tly-accessible range; ho w ev er, if these neutrinos could b e observ ed, it w ould b e a fan tastic addition to the b o dy of evidence for the birth of our univ erse. 24

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CHAPTER 3 NEUTRINO OSCILLA TIONS The phenomenon of neutrino oscillations and its immediate consequence of non-zero neutrino mass is one of the v ery few particle ph ysics results not predicted b y the SM. This c hapter describ es the disco v ery of this pro cess, the state of kno wledge as of to da y and it's phenomenological implications. 3.1 F ormalism Cen tral to an y discussion of neutrino oscillations is the div ergence b et w een its in teraction and Hamiltonian eigenstates, an in tro duction to whic h is giv en in Section 2.2 The state of the neutrino accessible b y exp erimen ts is the in teraction eigenstate, whic h is t ypically determined b y the ra v or of c harged lepton pro duced as a result of c harged curren t (CC) in teractions ( l + X l + X 0 ; l = e; ; ). If the mec hanism through whic h the neutrino is created is kno wn precisely then the w eak eigenstate at the time of creation is also kno wn. T o c haracterize neutrino oscillations w e m ust dynamically describ e the connection b et w een creation and detection. Since the neutrino propagates in its Hamiltonian eigenstate, w e b egin there. The neutrino with mass eigenstate i will ev olv e in time according to the time-dep enden t Sc hr oedinger equation: i @ @ t j i ( t ) i = ^ H j i ( t ) i = q ( p 2i + m 2i ) j i ( t ) i = p i s 1 + m 2i p 2i j i ( t ) i E + m 2i 2 E j i ( t ) i (3.1) where natural units of ~ = c = 1 are used, p is the neutrino momen tum and m its mass. The neutrino is assumed to tra v el through free space, and as its mass is m uc h smaller than its momen tum for all practical applications, terms of order t w o and higher in ( m 2i =p 2i ) in the expansion are ignored. This also implies the neutrino energy E p i for eac h mass state i A solution to this rst-order dieren tial equation is immediately apparen t: 25

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j i ( t ) i = e i ( E + m 2i = 2 E ) t j i (0) i : (3.2) This form is particularly con v enien t b ecause the only time a neutrino's propagation eigenstate corresp onds exactly to a single ra v or eigenstate o ccurs in coincidence with its creation. Cho osing t = 0 as the time of a w eak in teraction to create a neutrino of ra v or eigenstate the propagation state i can b e written as: j i (0) i = j i = X k U k j k i (3.3) where the arbitrary unitary matrix U describ es the coupling b et w een the propagation and in teraction eigenstates. If the propagation and in teraction eigenstates w ere iden tically equal, U w ould simply b e the iden tit y matrix and neutrinos w ould not oscillate. Substituting Equation 3.3 in to Equation 3.2 and again exploiting the assumption of negligible neutrino mass compared to its energy so that t L where L is the distance propagated in time t w e nd the probabilit y densit y of a neutrino created in w eak eigenstate after tra v eling a distance L to b e: j ( t = L ) i = X X k U k e i ( E + m 2i = 2 E ) L j k i U k j i ; (3.4) where also indexes the w eak eigenstates. No w w e can write the probabilit y for a neutrino created in w eak eigenstate to b e detected in state as a function of only its energy and the distance tra v ersed: 26

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P ( ) = j h j ( L ) ij 2 = X k U k U k e i ( E + m 2i = 2 E ) L 2 = X k j U k U k U j U j exp i m 2k j L 2 E = 4 X k >j Re U k U j U k U j sin 2 m 2k j L 4 E +2 X k >j Im U k U j U k U j sin m 2k j L 2 E (3.5) where m 2k j m 2k m 2j is referred to as the \mass splitting" b et w een the Hamiltonian eigenstates k and j and the unitary nature of the matrix U is used in the last step. Eq. 3.5 is v alid for an y n um b er of neutrino sp ecies; in the next t w o sections w e consider the case of three and t w o sp ecies. Three neutrino sp ecies is consisten t with direct observ ations of neutrino ra v ors (Section 2.3 ), and the study of t w o sp ecies is particularly instructiv e to understand oscillatory b eha vior and also giv es an excellen t appro ximation for most exp erimen tal prob es of the phenomenon. 3.1.1 Three-Neutrino Mixing With three observ ed ra v ors of neutrinos ( = e ; ; ), it is natural to assume there also exist three Hamiltonian eigenstates ( i = 1 ; 2 ; 3 ). This is analogous to the observ ed mixing in the quark sector. Though man y parametrizations of the mixing matrix U are p ossible, the canonical c hoice follo ws the form of the quark-mixing matrix. Under this c hoice it is referred to as the P on tecorv o-Maki-Nak aga w a-Sak ata (PMNS) matrix. Using the PMNS matrix, the ra v or states are related to the Hamiltonian states b y: 0BBBB@ e 1CCCCA = 0BBBB@ c 12 c 13 s 12 c 13 s 13 e i s 12 c 23 c 12 s 23 s 13 e i c 12 c 23 s 12 s 23 s 13 e i s 23 c 13 s 12 s 23 c 12 c 23 s 13 e i c 12 s 23 s 12 c 23 s 13 e i c 23 c 13 1CCCCA 0BBBB@ 1 2 3 1CCCCA ; (3.6) 27

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where c ij cos ( ij ) and s ij sin ( ij ) are trigonometric functions of the amplitude for mixing b et w een the Hamiltonian eigenstates i and j and is an arbitrary phase that allo ws for neutrinos to oscillate dieren tly than an tineutrinos [ 34 ]. Under the assumptions of m 3 > m 1 and = 0, the v alues of the PMNS matrix are giv en in Eq. 2.4 An adv an tage of the U PMNS matrix is that it ma y b e factored to isolate the eects of eac h mixing angle ij : U PMNS = 0BBBB@ 1 0 0 0 cos ( 23 ) sin ( 23 ) 0 sin ( 23 ) cos( 23 ) 1CCCCA 0BBBB@ cos( 13 ) 0 sin ( 13 ) e i 0 1 0 sin( 13 ) e i 0 cos ( 13 ) 1CCCCA 0BBBB@ cos ( 12 ) sin ( 12 ) 0 sin ( 12 ) cos( 12 ) 0 0 0 1 1CCCCA : (3.7) As will b e describ ed further, most exp erimen tal data are consisten t with the existence of exactly three Hamiltonian eigenstates and three w eakly-in teracting neutrinos whose eectiv e mass state lies w ell b elo w the Z 0 b oson mass of 91 GeV. Under these conditions, the PMNS matrix fully describ es the phenomenon of neutrino oscillations. Ho w ev er, some exp erimen tal evidence supp orts the existence of additional Hamiltonian eigenstates. The strongest hin ts come from the LSND [ 35 ] and MiniBo oNE [ 36 ] exp erimen ts. If these signals are conrmed, the PMNS matrix w ould ha v e to b e signican tly extended to accommo date the additional degrees of freedom [ 37 ]. Data from these exp erimen ts and their implications are further discussed in Section 3.2.4 3.1.2 Tw o-Neutrino Mixing In the case of only t w o neutrino ra v or ( ) and Hamiltonian (1 ; 2) eigenstates, the matrix U can b e expressed in terms of a single mixing angle : 28

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0B@ 1CA = 0B@ cos sin sin cos 1CA 0B@ 1 2 1CA ; (3.8) With this simplied mixing matrix, the arbitrary oscillation probabilit y (Eq. 3.5 ) reduces to: P ( ) = sin 2 2 sin 2 1 : 267 m 2 L E ; (3.9) where the units of m 2 = m 21 m 22 ; L and E are eV 2 km, and GeV, resp ectiv ely and the factor of 1.267 incorp orates n umerical constan ts including the factors of ~ and c ignored previously Ev en a cursory examination of Eq. 3.9 sho ws ho w the oscillation parameters m 2 and aect the exp erimen tal signature L and E : the amplitude of the oscillation probabilit y is prop ortional to while m 2 sets the frequency for oscillation as a function of the ratio L=E The consequences of Eq. 3.9 are w orth a few more remarks: for a giv en mixing amplitude the oscillation probabilit y is maximized for L=E ( m 2 ) 1 This informs exp erimen talists ho w to c ho ose the parameters L and E to gain sensitivit y to a certain region of mass splitting m 2 It follo ws that if the emplo y ed L and E are suc h that m 2 L=E 1, the eect of m 2 on observ ables will b e minimal. as sin 2 is an ev en function in its argumen t, neutrino oscillations are only sensitiv e to the absolute v alue of m 2 that is, the more massiv e state b et w een the t w o participating Hamiltonian eigenstates cannot b e determined from oscillation observ ations alone. if m 2 = 0, P ( ) = and neutrinos do not oscillate. This w ould imply the 29

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Hamiltonian eigenstates prob ed ha v e the same mass, whether zero or non-zero. if = 0, again P ( ) = and neutrinos do not oscillate. This w ould imply U reduces to the iden tit y matrix and the neutrino in teraction and Hamiltonian eigenstates are iden tically equal. It follo ws from the last t w o observ ations that an immediate consequence of the conrmation of neutrino oscillations is that there exist at least as man y neutrino Hamiltonian states with non-zero mass as the observ ed n um b er of mass splittings m 2 and the w eak and Hamiltonian eigenstates mix. Though the ab o v e conditions are most readily recognized with the neutrino oscillation probabilit y under the assumption of only t w o participating sp ecies, they apply equally to the arbitrary case of Eq. 3.5 As men tioned previously neutrino oscillations w ere not predicted in the SM and so the scales of m 2 v alues w ere completely unconstrained. F ortunately nature has pro vided us with t w o sources of organic neutrinos whose energy E and distance from creation to Earthly detection L is suc h that their ratio L=E prob es t w o indep enden t neutrino mass splittings. Exp erimen ts using articial neutrino sources suc h as accelerator-based b eams and neutrinos emitted from n uclear reactors ha v e conrmed and rened measuremen ts of these oscillation parameters. Neutrinos from n uclear reactors ha v e v ery recen tly also pro vided measuremen ts of the mixing angle 13 The exp erimen tal evidence for eac h follo ws. 3.2 Exp erimen tal Evidence for Neutrino Oscillations Assuming three Hamiltonian eigenstates ( 1 ; 2 and 3 ) and three ra v or eigenstates ( e ; and ), there exist t w o indep enden t mass splittings m 212 and m 223 (since j m 213 j = j m 212 m 223 j ) that mix with the w eak eigenstates through three indep enden t mixing angles and one CP-violating phase. The follo wing presen ts their curren t measuremen ts or constrain ts. 3.2.1 Solar Oscillations Often referred to collo quially as solar neutrino oscillations, the rst exp erimen tal hin ts of an y oscillation signature w ere caused b y the m 212 mass splitting and w ere 30

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observ ed in 1968 [ 38 ]. These hin ts remained a puzzle for more than three decades, when the SNO collab oration [ 39 ] pro vided observ ations of the en tire rux of neutrinos created in solar pro cesses ha ving transm uted in to a dieren t ra v or comp osition from creation to detection. Solar neutrinos are dominan tly pro duced as e 's in the n uclear fusion reaction p + p 2 H + e + + e Because of the lo w energy of these neutrinos ( < 10 MeV), only electrons are energetically allo w ed to b e pro duced in CC in teractions. Therefore, if solar neutrinos w ere oscillating in to the and w eak eigenstates, their en tire rux as seen on Earth could only b e observ ed using neutral curren t (NC) in teractions. SNO used the no v el idea of emplo ying hea vy w ater ( 2 H 2 O) as the detection medium to exploit neutron capture on deuterium and observ e b oth NC and CC ev en ts: e + 2 H p + p + e (CC) (3.10) + 2 H p + n + (NC) + 2 H 3 H + r (3.11) + e + e (ES) (3.12) where the particles observ ed to determine the reaction are in b old. The analysis of these three reactions is summarized in Figure 3-1 A global t to these data sho w they are compatible with e ; oscillations with parameters m 2solar 10 4 eV 2 and solar 34 [ 40 ]. The KamLAND exp erimen t pro vided an in v aluable conrmation of the ( m 212 12 ) v alues rep orted b y SNO using an articial neutrino source [ 41 ]. P erhaps more comp elling than the conrmation of the oscillation parameters, their data pro vided the rst clear observ ation of the sin usoidal nature of neutrino oscillations as a function of L=E Observing a rux of e from 53 n uclear p o w er reactors in Japan, KamLAND measured the probabilit y for e disapp earance using the in v erse deca y reaction 31

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Figure 3-1. Solar and rux v ersus e rux observ ed b y the SNO detector. The Standard Solar Mo del (SSM) exp ectation is sho wn b y the band b et w een the dashed lines and sho ws go o d agreemen t with the b est-t of the SNO data, represen ted b y the solid p oin t. Surrounding the b est t mark are 68%, 95% and 99% C.L. con tours. Figure tak en from Ref. [ 39 ]. e + p e + + n The strong correlation b et w een the p ositron and the inciden t an tineutrino energy and direction allo w ed for a measuremen t of the oscillation probabilit y with the ratio L=E clearly supp orting the trigonometric form of Eq. 3.9 This is sho wn in Figure 3-2 and the results of a t for the oscillation parameters including these and the SNO data are presen ted in Figure 3-3 Consistency in the observ ed oscillatory b eha vior b et w een the e 's at KamLAND and the solar e 's observ ed at SNO supp orts the CPT theorem. The t nds m 2solar = 7 : 9 +0 : 6 0 : 5 10 5 eV 2 and solar = 32 : 3 +3 : 0 2 : 4 [ 41 ]. A up date to this analysis using additional data from b oth exp erimen ts yields the most sensitiv e measuremen ts of the solar oscillation parameters to date: m 2solar = 7 : 59 +0 : 20 0 : 21 10 5 eV 2 and solar = 34 : 06 +1 : 16 0 : 84 [ 42 ]. Notice w e are b eing careful to refer to these oscillation parameters as \solar" instead of as the mixing b et w een t w o mass eigenstates. In principle, ev ery observ ed oscillation is aected b y all oscillation mo des, and so a single set of observ ed oscillation parameters are highly degenerate in in terpretations of the mass splittings and mixing angles c hosen b y nature. Ho w ev er, w e will see in the follo wing sections that the conrmed 32

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Figure 3-2. KamLAND e disapp earance results. The ratio sho wn is the observ ed data relativ e to the no-oscillation h yp othesis, and the distribution clearly fa v ors the sin usoidal form of neutrino oscillation o v er alternativ es. Figure tak en from Ref. [ 41 ]. mass splittings are sucien tly separated to ev en tually refer to these solar parameters as the mixing b et w een only t w o Hamiltonian states to excellen t appro ximation. 3.2.2 A tmospheric Oscillations Cosmically-pro duced high energy protons, electrons and stable n uclei collide with Earth's upp er atmosphere and pro duce a rux of neutrinos through pion and m uon deca y sucien tly in tense to b e studied on Earth. The Kamiok ande detector in Japan [ 43 ] w as originally designed to searc h for proton deca y but the secondary ph ysics goal of atmospheric neutrino studies pro v ed m uc h more in teresting. As in the puzzle of solar neutrino rux discussed in the previous section, exp ectations of the neutrino con ten t w ere not met: the observ ed ratio = e w as signican tly lo w er than predicted [ 43 ]. Sup er-Kamiok ande succeeded Kamiok ande and featured upgrades that allo w ed for CC measuremen ts of and e in teractions as a function of neutrino tra v el distance. Kno wledge of the neutrino propagation length w as p ossible through the strong correlation b et w een the direction of the observ ed c harged lepton in CC in teractions and the origin of the inciden t neutrino. 33

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Figure 3-3. Results from a t to the KamLAND and solar neutrino data to the oscillation parameters m 2solar and solar Figure tak en from [ 41 ]. The ratio of observ ed CC ev en ts relativ e to the no-oscillation h yp othesis from Sup er-Kamiok ande detector as a function of L=E is sho wn in Figure 3-4 A t to the t w o-neutrino oscillation h yp othesis using these data yields 1 : 9 10 3 < m 2atm < 3 : 0 10 3 and sin 2 2 atm > 0 : 90 at 90% C.L [ 44 ]. Indep enden t conrmation of these oscillation parameters come from the K2K [ 45 ] and MINOS [ 46 ] exp erimen ts, b y observing ruxes of accelerator-based neutrino b eams at m ultiple p ositions along the line of neutrino tra v el. Motiv ated b y the Sup er-Kamiok ande observ ations, the neutrino b eam energies and detector p ositions w ere c hosen suc h that L and E w ere distinct b et w een the t w o exp erimen ts but the ratio L=E for b oth K2K and MINOS aorded sensitivit y to m 2 10 3 eV 2 With suc h con trol o v er the oscillation region explored b y this exp erimen tal setup, this metho d for searc hing for neutrino oscillations has since b ecome the comm unit y standard. Both exp erimen ts observ ed decits in the observ ed rux of consisten t with m 2 10 3 eV 2 and the C.L. regions from K2K, MINOS, and Sup er-Kamiok ande are sho wn in Figure 3-5 A more recen t t to w orld 34

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Figure 3-4. Data from the Sup er-Kamiok ande exp erimen t clearly sho wing an L=E dep endence of the observ ed rux relativ e to the prediction assuming no oscillations. Figure tak en from Ref. [ 44 ]. data sensitiv e to this mass splitting giv es j m 2atm j = 2 : 43 +0 : 06 0 : 10 10 3 eV 2 and sin 2 atm = 0 : 386 +0 : 024 0 : 014 [ 47 ]. 3.2.3 13 and Oscillations As men tioned previously the magnitude of the mass splitting m 213 is constrained b y j m 213 j = j m 212 m 223 j where the states either add or subtract dep ending on the unkno wn mass hierarc h y (discussed in Section 3.3 ). Considering the separation in the v alues m 2sol 10 5 eV 2 and m 2atm 10 3 eV 2 j m 213 j m 2atm to go o d appro ximation. Ho w ev er, the mixing angle b et w een the (1,3) Hamiltonian eigenstates is unconstrained b y the other oscillation parameters and m ust b e indep enden tly determined. The 13 mixing angle is the most recen tly measured and conrmed oscillation parameter, its measuremen t coming last mostly b ecause its dimin utiv e size leads to more subtle eects compared to the other mixing amplitudes. While the accelerator-based exp erimen ts MINOS [ 48 ] and T2K [ 49 ] pro vided indications that its v alue is non-zero through e con v ersions, it w as observ ations of reactor e disapp earance with the Da y a 35

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q 2 2 sin 0.5 0.6 0.7 0.8 0.9 1 )2 4| (eV /c2m D | 0.001 0.002 0.003 0.004 MINOS data, 90% C.L. Best-fit point Super-K 90% C.L. K2K 90% C.L. Figure 3-5. Summary of oscillation ts to data sensitiv e to m 2 10 3 eV 2 Figure tak en from Ref. [ 46 ]. Ba y [ 50 ] and Reno [ 51 ] exp erimen ts that pro vided the rst measuremen ts of 13 The Da y a Ba y exp erimen t uses an impressiv e n um b er of nearly-iden tical detectors to measure the reactor e rux at a v ariet y of distances, and as sho wn in Figure 3-6 clearly observ es e disapp earance. This measuremen t is the most precise to date and nds sin 2 2 13 = 0 : 089 0 : 011 [ 52 ]. The neutrino transitions aected b y and the sign of m 213 are most readily exp erimen tally accessible through a comparison of P ( e ) with P ( e ). These probabilities cannot b e reasonably appro ximated b y the t w o-neutrino case, as all three mixing angles and mass splittings con tribute signican tly to the pro cess [ 53 ]. Though this indicates Eq. 3.9 is less helpful here, the exp erimen tal sensitivit y to these transitions are still go v erned principally b y the appropriate ratio of L=E and in this case is O 10m 10 3 eV T o allo w for reasonable pro duction phase-space for the observ ation of the m uon in CC in teractions, E m ust b e O (1 GeV), setting L of O (10 9 m). This distance is roughly an 36

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Weighted Baseline [km] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 expected / N detected N 0.9 0.95 1 1.05 1.1 1.15 EH1 EH2 EH3 13 q 2 2 sin 0 0.05 0.1 0.15 2 c 0 10 20 30 40 50 60 70 s 1 s 3 s 5 Figure 3-6. Observ ed e rux at the v arious Da y a Ba y exp erimen tal halls (EH) as a function of distance from the e source relativ e to the prediction assuming no oscillations. The inset sho ws the compatibilit y of the data for v arious v alues of sin 2 2 13 clearly ruling out 13 = 0 at greater than 5 C.L.. Figure tak en from Ref. [ 52 ]. order of magnitude longer than an y previous observ ations of articicial sources. As the neutrino rux is roughly prop ortional to 1 =L 2 at large distances from the source, it will b e enormously c hallenging to ac hiev e the b eam p o w er and detection precision required to prob e v alues of Curren tly the only constrain ts on come from the MINOS [ 48 ] exp erimen t, and are sho wn in Figure 3-7 It can b e seen that no v alue of for either sign of m 213 (referred to in the gure simply as m 2 ) is strongly preferred o v er others. The determination of the sign of m 213 and precision measuremen ts of is curren tly at the forefron t of to da y's exp erimen tal neutrino program, and ma y dominate the high-energy ph ysics landscap e in the US for decades. Curren t exp erimen ts NO A [ 54 ] and T2K, and later LBNE [ 55 ], will lead the searc h b y comparing P ( e ) with P ( e ) using few-GeV b eams of and One of the c hallenges that m ust b e met b efore a clean measuremen t of is p ossible is a high-precision understanding of the fundamen tal con tributing and in teractions at this energy range. The w ork presen ted in this dissertation pro vides a rst measuremen t of cross sections b elo w 1 GeV and th us signican tly adv ances the comm unit y's preparedness to searc h for CP violation with 37

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Figure 3-7. Constrain ts on for v arious assumptions on the sign of m 2 and the v alue of 23 from the MINOS exp erimen t. It will b e sho wn that the 23 is nearly iden tical to atm Figure tak en from Ref. [ 48 ]. neutrinos.3.2.4 Hin ts for m 2 1 eV 2 One of the ma jor outstanding questions in neutrino ph ysics is the existence of another mass splitting in the range of m 2 1 eV 2 As with the other mixing parameters, this h yp othesis is en tirely exp erimen tally-driv en. The rst indication came from the LSND exp erimen t, where an excess of e ev en ts w ere observ ed from a stopp ed-pion source [ 35 ]. As with the accelerator-based conrmations of the solar mixing describ ed in Section 3.2.1 the MiniBo oNE exp erimen t w as designed to pro vide an indep enden t c hec k of this splitting b y probing the same ratio L=E while L and E w ere themselv es distinct from the v alues used at LSND. An indication of b oth e and e oscillations w ere observ ed in the MiniBo oNE data as w ell [ 36 ], ho w ev er neither of the signals from the t w o exp erimen ts exclude the no-oscillation h yp othesis at greater than 4 The allo w ed ( m 2 ; ) regions from b oth exp erimen ts are sho wn in Figure 3-8 Less signican t signals indicating m 2 1 eV 2 come from cosmological observ ations [ 57 ], radioactiv e source exp erimen ts [ 58 ], and from reactor an tineutrino 38

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q 2 2 sin 3 10 2 10 1 10 1 ) 2 (eV 2 m D 2 10 1 10 1 10 2 10 LSND 90% CL LSND 99% CL 68% CL 90% CL 95% CL 99% CL CL s 3 KARMEN2 90% CL Figure 3-8. Results of a t w o-neutrino oscillation t to the com bined e and e app earance data from MiniBo oNE. Also sho wn are the LNSD allo w ed regions and limits from the KARMEN exp erimen t [ 56 ]. Figure tak en from Ref. [ 36 ]. 39

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data [ 59 ]. P articularly in ligh t of the implications of suc h a mass splitting as discussed in the next section, this signal m ust b e rigorously tested in the near future. Prop osed exp erimen ts to do so include OscSNS [ 60 ], n uSTORM [ 61 ], and a searc h using deca y-at-rest k aons [ 62 ]. 3.3 Summary and Outstanding Questions Noting that the m 2 scales discussed in the previous section dier b y orders of magnitude, w e can drop their conserv ativ e solar and atmospheric lab els and refer to them as the gen uine splitting b et w een neutrino Hamiltonian eigenstates to excellen t appro ximation 1 Under the suspicion of symmetry b et w een the ordering of the neutrino mass states and that of the other leptons, w e refer to the smaller mass splitting as m 212 and the larger splitting as m 223 The conrmed v alues for the neutrino mixing parameters are: m 212 = 7 : 59 +0 : 20 0 : 21 10 5 eV 2 12 = 34 : 06 +1 : 16 0 : 84 j m 232 j = 2 : 43 +0 : 06 0 : 10 10 3 eV 2 sin 2 23 = 0 : 386 +0 : 024 0 : 014 m 213 m 232 13 = 9 : 0 +0 : 4 0 : 5 (3.13) With t w o indep enden t mass splittings, as least t w o neutrino mass eigenstates m ust b e non-zero. One of the most imp ortan t questions ab out the nature of the neutrino concerns ho w these masses ma y b e in tegrated in to the SM. This issue is fundamen tally tied to whether the neutrino is its o wn an ti-particle, indicating a Ma jorana nature; if the neutrino and an tineutrino are distinct, neutrinos are Dirac particles. F or Dirac particles the extension of neutrino mass in to the SM is quite simple in that, lik e other massiv e particles, the masses are generated b y the Higgs eld and b oth leftand righ t-handed neutrinos and an tineutrinos exist. Neutrinos with opp osing c hiralit y to the observ ed states 1 Note also this is also true if there exists a mass splitting near 1 eV 2 40

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are then not exp erimen tally accessible not b ecuase of the nature of the neutrino, but b ecause they only in teract through the maximally parit y-violating w eak in teraction. A p opular mo del of the alternativ e of Ma jorana neutrinos is equally viable and oers an explanation of the dimin utiv e scale of the neutrino mass compared to the other fermions. In this mo del the masses of the ligh t and activ e neutrinos are accompanied b y some n um b er of p ossibly non-w eakly in teracting neutrinos N suc h that the pro duct of the t w o neutrino family masses are related to the scale of the quark q or c harged lepton l families: m m N m 2q ;l [ 63 ]. In this w a y the large mass of the N neutrino pro vides a coun ter-balance for the observ ed neutrinos to b e arbitrarily ligh t, and this mo del is therefore referred to as the See-Sa w mec hanism. Curren tly the b est exp erimen tal prob e to determine whether the neutrino is Ma jorana or Dirac in v olv es the neutrino-less double-b eta deca y reaction ( n + n p + p + e + e ). The deca y w ould in v olv e the emission of an e at one v ertex of ordinary deca y and it's immediate absorption at a second deca y v ertex pla ying the role of e Consequen tly this pro cess is allo w ed for Ma jorana neutrinos but is forbidden if neutrinos are Dirac particles. One exp erimen t has claimed to ha v e observ ed evidence of this pro cess [ 64 ], but this remains unconrmed. Another curren tly-degenerate fundamen tal prop ert y related to the neutrino mass is their hierarc hical ordering. F rom the Mikhey ev-Smirno v-W olfenstein (MSW) eect in solar oscillations, it is kno wn that m 2 > m 1 while the ordinal lab el b et w een the third mass state and the others is arbitrary As sho wn in Section 3.1 observ ations of splittings sensitiv e to only t w o mass eigenstates rev eal only the absolute v alue of the splitting, and the curren t neutrino oscillation data are degenerate b et w een the smaller mass splitting separating the t w o ligh test states and the same splitting separating the most massiv e states. Figure 3-9 pictorially sho ws this degeneracy in the mass hierarc h y along with the appro ximate mixing amplitudes b et w een eac h ra v or and mass state. As men tioned in Section 3.2.3 the mass ordering will b e addressed in the curren t and next round of exp erimen ts sim ultaneously 41

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Figure 3-9. The t w o p ossible neutrino mass orderings, sho wn with their appro ximate couplings to the ra v or states. searc hing for the CP-violating phase It can b e seen in Eq. 3.7 that the CP-violating phase is inextricably tied to the mixing angle 13 and therefore the sign of m 213 The observ ation of a non-zero v alue for 13 oers the opp ortunit y to searc h for CP-violation in the lepton sector, whic h is curren tly one of the b est h yp otheses for explaining the bary on asymmetry in the univ erse [ 65 ]. It is w orth noting that a sucien tly large v alue of m ust b e accompanied b y at least one more sp ecies of neutrino, m uc h more massiv e than the kno wn t yp es, to explain the observ ed bary on asymmetry Nev ertheless, as the observ ed CP-violation in the quark sector is far to o meager to accoun t for the ev olution of our matter-dominated univ erse [ 9 ], searc hes for the origin of the asymmetry using neutrinos are w ell-motiv ated and comp elling. Finally if conrmed, the exp erimen tal hin ts of another mass splitting presen ted in Section 3.2.4 w ould imply a fourth neutrino mass state, with a fundamen tally dieren t coupling to the w eak in teraction. If there exists a mass splitting near 1 eV 2 from the precision constrain t of the Z 0 width (sho wn in Figure 2-2 ), it m ust not directly couple to the w eak in teraction. F rom the disconnect b et w een its mass and the in teraction states, this 42

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h yp othesized extra neutrino is kno wn as sterile. F urthermore, argumen ts based on symmetry b et w een neutrinos and the other fermions w ould suggest these ough t to exist in sets of three, if an y and analyses to global data do mildly prefer the addition of more than a single sterile mass state [ 37 ]. This w ould in tro duce a litan y of extra degrees of freedom in neutrinos oscillations in the form of mass splittings and mixing angles, the signals from whic h are almost en tirely degenerate in curren t exp erimen ts. It is clear b y no w a quan tum mec hanical pro cess not predicted b y the standard mo del is real and ma y b e a consequence of some deep er la ws of ph ysics w e do not y et appreciate. Though there are man y unkno wns in neutrino ph ysics that will presumably lead to a more fundamen tal understanding of the w eak in teraction and ho w it ts in to nature, the concrete observ ation of neutrino oscillations rev eal t w o pieces of information crucial to this quest: that the neutrino mass is non-zero, and that lepton n um b er is not a strictly conserv ed quan tit y As a nal note, it w as en tirely fortuitous that the comm unit y realized the phenomenon of neutrino oscillations. If the nature of oscillations w ere suc h that solar and atmospheric neutrinos w ere unaected, our ignorance of this pro cess w ould ha v e p ersisted for at least man y more decades. Therefore, though it app ears the comm unit y ma y fully p opulate the PMNS matrix (Eq. 3.6 ) with precision measuremen ts in the coming decades, it seems unlik ely that this will complete our fundamen tal understanding of neutrino oscillations. 43

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CHAPTER 4 NEUTRINO INTERA CTIONS IN MINIBOONE 4.1 Ov erview MiniBo oNE uses the nuance neutrino ev en t generator [ 66 ] to predict and sim ulate neutrino in teractions in the detector. nuance includes a comprehensiv e cross-section mo del whic h considers kno wn in teractions in the neutrino energy range from 100 MeV to 1 T eV. Ninet y-nine reactions are mo deled separately and com bined with n uclear mo dels describing b ound n ucleon states and nal-state in teractions to predict ev en t rates and kinematics. Figure 4-1 sho ws the exp ectation and exp erimen tal data for and CC in teractions across a wide range of energies. As the MiniBo oNE ruxes of and are p eak ed near 700 MeV (Figure 5-9 ), the c harged-curren t quasi-elastic (CCQE) and c harged-curren t single pion (CC ) in teractions are the most abundan t in teractions in the MiniBo oNE data sets. F or this reason, in this c hapter w e concen trate on the exp ectations and exp erimen tal evidence asso ciated with these pro cesses. A w ealth of information is summarized in Figure 4-1 and it is imp ortan t to p oin t out the o v erall structure of the cross sections and the features most relev an t to the measuremen ts executed in this dissertation. When the neutrino energy is large enough to resolv e individual quarks, the CC cross section is appro ximately linear with energy This b eha vior is a conrmation of the quark parton mo del [ 67 ], where higher energy prob es gain sensitivit y to more scattering in teractions through the quark sea. This appro ximation, of course, breaks do wn at lo w er energies where elastic in teractions are dominan t. Exp erimen tally Figure 4-1 sho ws that these in teractions at lo w er energies feature total error on the order of tens of p ercen t. This is mostly due to exp erimen tal diculties in separating the v arious con tributing pro cesses, a c hallenge that is unique to the v arious detector tec hnologies and usually includes dep endence on assumptions ab out the con tributing signal and bac kground pro cesses. Finally the an tineutrino cross sections are 44

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Figure 4-1. Comparison of exp ectations and exp erimen tal data across (top) and (b ottom) CC in teractions. The \QE" and \RES" lab els here are referred to as CCQE and CC in teractions in the text. Figure from Ref. [ 30 ]. 45

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exp erimen tally kno wn less accurately compared to the analogous neutrino pro cesses, and in particular there are no an tineutrino CC cross-section measuremen ts b elo w 1 GeV. The lo w er precision and more sparse an tineutrino data is due in general to a n um b er of eects, most notably relativ ely larger bac kgrounds and lo w statistics. The measuremen ts in this dissertation break signican t ground on b oth exp erimen tal c hallenges: Chapter 7 presen ts a rst demonstration of a set of tec hniques to statistically measure a bac kground t ypical of articial b eams of an tineutrinos, and Chapter 8 presen ts an analysis of an tineutrino in teractions with more than an order of magnitude of higher statistics compared to all other previously-published an tineutrino cross-section measuremen ts c ombine d In addition, the a v erage an tineutrino energy for these measuremen ts is 650 MeV, and so these data are sensitiv e to an almost en tirely unprob ed energy region. 4.2 CCQE The CCQE pro cess ( l + n l + p for neutrinos, and l + p l + + n for an tineutrinos) is the most abundan t in teraction at the MiniBo oNE energy range, accoun ting for 40% of in teractions in the detector. It is t ypically used as the signal pro cess in neutrino oscillation measuremen ts due to its simple m ultiplicit y and also the abilit y to reconstruct the inciden t neutrino energy under a few imp ortan t assumptions, based solely on observ ations of the c harged lepton (Chapter 6 ). T ypically credited to Llew ellyn-Smith [ 68 ], the dieren tial cross section for this pro cess assuming the exc hange of a single W b oson as a function of the momen tum transfer Q 2 is: d dQ 2 = M 2 G 2F j V ud j 2 8 E 2 A Q 2 B Q 2 s u M 2 + C Q 2 s u M 2 2 # ; (4.1) where the p ositiv e (negativ e) sign refers to neutrino (an tineutrino) scattering, G F is the F ermi coupling constan t, V ud is the Cabbib o coupling b et w een do wn and up quarks, m is 46

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the mass of the c harged lepton, M the mass of the target n ucleon, and s u are the usual Mandelstam v ariables. F or a deriv ation of this expression, see Ref. [ 69 ]. Note the terms are organized in p o w ers of s u M 2 = 4 M E Q 2 m M 2 and the in terference b et w een the axial and v ector curren ts that go v erns the dierence in scattering amplitudes b et w een neutrinos and an tineutrinos is en tirely con tained in the B ( Q 2 ) term. This in terference is a consequence of the V-A nature of the w eak in teraction. The auxiliary functions A ( Q 2 ) ; B ( Q 2 ) ; C ( Q 2 ) are parameterized in terms of v ector, axial and pseudoscalar form factors: A Q 2 = ( m 2 + Q 2 ) M 2 [(1 + ) F 2 A (1 ) F 2 1 + (1 ) F 2 2 + 4 F 1 F 2 m 2 4 M 2 ( F 1 + F 2 ) 2 + ( F A + 2 F P ) 2 4 F 2 P (1 + )] (4.2) B Q 2 = Q 2 M 2 F A ( F 1 + F 2 ) (4.3) C Q 2 = 1 4 F 2 A + F 2 1 + F 2 2 ; (4.4) where = Q 2 4 M 2 F 1 and F 2 are v ector form factors, F A is the axial form factor, and F P is the pseudoscalar form factor. The v ector form factors are: F 1 = 1 + (1 + p n ) (1 + ) 1 + Q 2 m 2V 2 (4.5) F 2 = p n (1 + ) 1 + Q 2 m 2V 2 ; (4.6) where p ( n ) = 2.793 (-1.913) N is the proton (neutron) anomalous magnetic momen t [ 70 ], and m V is the empirically-determined \v ector mass". Using the conserv ed v ector curren t (CV C) h yp othesis, the results of the plen tiful and high-qualit y elastic electron scattering ( e + N e + N ) data can b e used to constrain these form factors. The dip ole forms of Eqs. 4.5 and 4.6 are adequately describ ed with a v ector mass of 47

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m 2V = 0 : 71 GeV 2 but recen t ts to these data sho w a clear preference for a non-dip ole form [ 71 ]. The pseudoscalar form factor is giv en b y: F P = 2 M 2 m 2 + Q 2 F A ; (4.7) where m is the pion mass. Notice the con tribution of F P to the CCQE is suppressed b y m 2 M 2 and so its eect relativ e to the other terms is small. Finally and most imp ortan tly for the measuremen ts of this dissertation, the axial form factor is: F A = g A 1 + Q 2 M 2 A 2 ; (4.8) where g A and M A are empirical inputs, and the dip ole form is again assumed. Lik e the v ector form factors tak en from electron-scattering data, g A is also constrained b y external information: deca y measuremen ts giv e g A = F A ( Q 2 = 0) = 1 : 267 0 : 002 [ 72 ]. This lea v es the axial mass M A as the only free parameter in the CCQE cross section. F or decades, this parameter w as measured with observ ations of b oth the total observ ed CCQE cross section and its shap e as a function of the momen tum transfer. A com bined analysis of the w orld data through the t w en tieth cen tury yields M A = 1 : 026 0 : 021 GeV [ 73 ]. Imp ortan t to p oin t out, most of these measuremen ts w ere p erformed with bubble-c ham b er detectors housing mostly h ydrogen and deuterium media. More recen t results from exp erimen ts emplo ying larger n uclei in order to more easily gain the statistics needed for oscillation exp erimen ts ha v e found tension with these data, and as a result the mo del for n uclear eects t ypically used b y exp erimen t has come under scrutin y F urther discussion of 48

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Figure 4-2. Decomp osition of the dieren tial CCQE cross section for 700 MeV 's. The ordinate axis is prop ortional to d =dQ 2 Figure from Ref. [ 74 ]. this mo del and its implications is found in Section 4.4 As a nal remark, the in terference term prop ortional to B ( Q 2 ) in Eq. 4.2 giv es rise to stark kinematic dierences in the b eha vior of neutrinos compared to an tineutrinos in CCQE in teractions. In Figure 4-2 the dieren tial cross section for CCQE scattering is separated in to terms arising from the v ector and axial curren ts, as w ell as the in terference b et w een the t w o. As the in terference term is constructiv e for neutrino scattering and destructiv e for an tineutrinos, is clear that the div ergence of their amplitudes gro ws with momen tum transfer. Momen tum transfer of the in teraction is closely related to the pro duction angle of the c harged lepton relativ e to the neutrino direction, and this dierence is exploited in Section 7.1.5 to measure the and con ten t of the MiniBo oNE an tineutrino-mo de data. 49

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4.3 CC Neutrinos with energy 400 MeV and ab o v e can pro duce pions through the excitation and subsequen t deca y of bary onic resonances. Resonances of Delta () particles are most imp ortan t for the neutrinos observ ed b y MiniBo oNE, and their deca ys are dominated b y N [ 72 ]. The formalism to describ e these in teractions is tak en from the Rein-Sehgal mo del [ 75 ], where the relativistic harmonic oscillator quark mo del is assumed [ 76 ] and the pion angular distribution due to the spin structure of the resonances is considered. Eigh teen resonances are mo deled, though the (1232) is dominan t in the energy range spanned b y MiniBo oNE. Multi-pion pro duction mec hanisms are also mo deled, though their con tribution is predicted to b e small. As the primary in teraction for the CC pro cesses ( l + N l + ) is closely related to CCQE in teractions, the formalism also includes a single tunable axial mass parameter M 1 A The axial masses in the resonance c hannels are set sim ultaneously to repro duce inclusiv e non-MiniBo oNE c harged-curren t data [ 77 ]. The extracted v alues are M 1 A = 1 : 10 0 : 27 GeV and M m ulti A = 1 : 30 0 : 52 GeV (m ulti-pion pro duction). V arious lev els of discrepancy b et w een this mo del and the MiniBo oNE single pro duction results spanning normalization dierences of up to 60% ha v e b een observ ed [ 78 79 80 ], and these dierences con tin ue to p ersist in more mo dern singlepro duction calculations [ 81 ]. F or these reasons, whenev er p ossible, the v arious MiniBo oNE cross-section and oscillations measuremen ts rely on direct constrain ts from the v arious MiniBo oNE singlepro duction samples. 4.4 Nuclear Eects 4.4.1 Nuclear Mo deling The MiniBo oNE detector is lled with mineral oil, and as a h ydro carb on material, the bare neutrino-n ucleon in teraction amplitudes m ust b e com bined with eects arising from the n uclear en vironmen t for in teractions with material b ound in carb on. MiniBo oNE uses the Relativistic F ermi Gas (RF G) mo del [ 82 ] to describ e this connection. Broadly it 50

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com bines the free-n ucleon cross sections with a p oten tial w ell in the form of binding energy as w ell as P auli blo c king to restrict the a v ailable kinematics of struc k n ucleons. The binding energy E B increases the threshold for the reaction to o ccur, while the eects of P auli blo c king are more subtle. The phenomenon of P auli blo c king arises from the exclusion principle, whic h dictates that no t w o fermions ma y share the iden tical set of quan tum eigenstates. The RF G mo del sim ulates b ound n ucleons as a \gas" of particles, with a uniform momen tum distribution from the lo w est state up to an empirically-determined maxim um k F These mo dications to the CCQE amplitudes are implemen ted b y in tegrating, with resp ect to the initial n ucleon momen tum ~ k the free-n ucleon cross sections scaled b y a factor prop ortional to: ( k F j ~ k j ) ( j ~ k + ~ q j k F ) ( E ~ k E ~ k + ~ q E B + ) ; (4.9) where = E E l is the energy transfer, ~ q = ~ p ~ p l is the three-momen tum transfer, and E ~ k and E ~ k + ~ q are the energies of the initial and struc k n ucleon, resp ectiv ely The rst term requires the n ucleon participating in the in teraction to ha v e momen tum b elo w k F the second enforces P auli blo c king, and the third assures energy conserv ation. The second term is appropriate only to n uclear transitions in v olving n $p so that the struc k n ucleon is required to b e ab o v e the F ermi momen tum of the other fully-p opulated n ucleon F ermi sea. In the case of carb on-12, where Z = N = 6, a single momen tum k F sp ecies the maxim um of b oth the proton and neutron F ermi lev els. The energy of the lo w est-allo w ed struc k n ucleon momen tum state is closely related to lo w v alues of the squared momen tum transfer distribution Q 2 q 2 where q 2 here is the four-momen tum transfer. This region in the MiniBo oNE CCQE data w as insucien tly describ ed b y this RF G mo del [ 83 ], and ev en after a more rigorous ev aluation 51 PAGE 52 Figure 4-3. The observ ed dieren tial cross section for electron scattering e + N e 0 + N on carb on-12 in terms of the energy transfer = E e E e 0 The binding energy is determined b y the p osition of the observ ed p eak, while the F ermi momen tum is found from the p eak's width. Figure from Ref. [ 85 ]. of the bac kgrounds it w as found that a mild scaling (\ ") of this energy lev el: E lo w = q k 2 F + M 2 + E B (4.10) w as preferred b y the data at the lev el of a 1% mo dication to E lo w [ 84 ]. The v alues for the binding energy and F ermi momen tum of carb on are informed b y electron scattering data. Sho wn in Figure 4-3 the p eak of quasielastic electron-scattering ( e + N e + N ) data is w ell-describ ed b y a F ermi gas mo del and E B ( p F ) = 25 (221) MeV, where natural units are used. The F ermi momen tum p F = 221 MeV is directly implemen ted in to the RF G. Ho w ev er, for CCQE scattering where n$ p the binding energy m ust b e mo died from the determination from electron scattering data, where the initial and outgoing n ucleons are of the same t yp e. Additional coulom b repulsion for n p transitions (appropriate to CCQE in teractions) adds to the eectiv e binding energy of the system. The asymmetry 52

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term in the semi-empirical mass form ula [ 86 ] estimates this adds 9 MeV to the energy for this transition, resulting in an eectiv e binding energy of 34 MeV for CCQE in teractions on carb on-12. P articularly in the con text of this dissertation, it is imp ortan t to note the RF G assumes all n ucleons b eha v e en tirely indep enden tly of one another. Recen t deviations from RF G exp ectations in the measuremen ts of the CCQE in teraction with relativ ely hea vy n uclear targets ha v e cast suspicion on this assumption. While measuremen ts of M A using mostly ligh t n uclear material (discussed in Section 4.2 ) agree fairly w ell, data from exp erimen ts using relativ ely hea vy n uclei and higher-precision detectors ha v e extracted v alues of M A systematically higher than 1.026 GeV [ 84 87 88 89 ]. Adding complexit y the mo dern hea vy n uclear target exp erimen t NOMAD has measured v alues of M A consisten t with the ligh t-target analyses [ 90 ], while preliminary shap e results from the MINER A exp erimen t seem to also fa v or M A 1 GeV [ 91 ]. An essen tial rst step to understanding this apparen t discrepancy is to recognize the particulars of the mo del dep endence in tro duced b y comparing v alues of M A b et w een the man y exp erimen ts. Imp ortan t exp erimen tal dierences that ma y con tribute to the discrepancy include disparate neutrino sp ectra, dieren t neutrino detection tec hnologies and the size of the n uclear media emplo y ed. Among the lib erties tak en to compare M A v alues across these scattering exp erimen ts include the dip ole form of F A v arious exp ectations of hadronic activit y consisten t with single-n ucleon ejection and the previously-men tioned indep enden t n ucleon assumption implicit in b oth the formalism and in the inference of the Q 2 distribution. A p ossible reconciliation b et w een the data sets has b een prop osed b y oering a mec hanism resulting in in tra-n uclear correlations of greater strength than previously exp ected [ 92 93 94 95 96 97 98 99 100 ]. Suc h a mec hanism is consisten t with electron scattering data [ 101 102 ]. If this pro cess is conrmed for w eak in teractions via neutrino scattering, its detailed understanding will signican tly expand kno wledge of in tra-n uclear b eha vior, and some neutrino oscillation results ma y need to b e 53

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revisited [ 103 104 ]. The b est c hance to denitiv ely resolv e this crucial am biguit y lies in the comm unit y's abilit y and willingness to pro duce and compare mo del-indep enden t information in b oth the leptonic and hadronic in teraction sectors b et w een exp erimen tal data and theoretical calculations. The results of this dissertation oer a rst lo ok at an tineutrino CCQE in teractions b elo w 1 GeV and th us signican tly expand the b o dy of exp erimen tal data con tributing to this picture. In recognizing the p ossible deciencies of the RF G, the main result of this w ork is the double-dieren tial CCQE cross section d 2 dT d cos on mineral oil, where no assumptions ab out the underlying pro cess are necessary 4.4.2 Final-State In teractions An imp ortan t connection b et w een fundamen tal neutrino-n ucleus in teractions and what is observ ed in the detector are the p ossible strong in teractions b et w een the struc k bary on and its n uclear en vironmen t. F or neutrino in teractions with a n ucleon b ound in carb on, nuance propagates the outgoing hadrons including n ucleons, mesons and bary onic resonances, and sim ulates their re-in teraction as they exit the n ucleus. The initial in teraction mo del emplo ys the impulse appro ximation whic h assumes an instan taneous exc hange with indep enden t n ucleons. Subsequen t to the initial neutrino in teraction, particles pro duced inside the n ucleus are propagated step-wise in 0.3 fm incremen ts un til they emerge from the 2.5 fm radius sphere. In termitten tly the probabilit y for hadronic re-in teraction is calculated using a radially-dep enden t n ucleon densit y distribution [ 105 ] along with external N ; N N cross-section measuremen ts [ 106 ]. F or re-in teractions ( + N N + N ), an energy-indep enden t probabilit y of 20% (10%) is tak en for + + N 0 + N ( ++ + N ; + N ) based on K2K data [ 77 ] and is assigned 100% uncertain t y Out of all hadronic rein teraction pro cesses, pion absorption ( + X X 0 ) and c harge exc hange ( + X $0 + X 0 ) are the most relev an t in predicting the comp osition of the samples studied in the analyses of this dissertation. Sho wn in Figure 4-4 54 PAGE 55 Figure 4-4. Comparison of the nuance prediction for + absorption (top) and c hargeexc hange (b ottom) to relev an t data [ 107 ]. Figure from Ref. [ 84 ]. in tran uclear fractional uncertain ties on pion absorption (c harge-exc hange) are set to 25% (30%) based on comparisons b et w een external data [ 107 ] and the nuance prediction. 55 PAGE 56 CHAPTER 5 THE MINIBOONE EXPERIMENT 5.1 Ov erview The MiniBo oNE exp erimen t w as designed to optimize the searc h for the app earance of e ev en ts in a b eam of Accordingly man y design c hoices w ere made and auxiliary systems implemen ted to maximize detection eciency for and e CC ev en ts sensitiv e to mass splittings of m 2 1 eV 2 while main taining discrimination p o w er b et w een the t w o neutrino sp ecies. This c hapter describ es the ph ysical la y out of the exp erimen t and the detector subsystems most crucial to the measuremen t of m uon kinematics. An expanded description of the b eamline and neutrino rux calculation can b e found in Ref. [ 108 ], while the o v erall design and p erformance of the detector is discussed in more detail in Ref. [ 109 ]. 5.2 The Bo oster Neutrino Beamline The Bo oster Neutrino Beamline (BNB) collides 8.9 GeV/c momen tum protons on to a b eryllium target, and a magnetic horn is used to sign-select and fo cus the secondary meson b eam in the direction of the detector. Dep ending on the p olarit y of the magnetic eld, the selected meson deca y mo des yield an enhanced or b eam. This section steps through the imp ortan t instrumen ts in this pro cess, nally arriving at the calculation of the neutrino rux observ ed b y the detector. 5.2.1 The Primary Proton Beam The F ermilab accelerator c hain b egins with a b ottle of h ydrogen gas and a v oltage m ultiplier system rst demonstrated in 1913 [ 110 ]. This Co c kroft-W alton system generates a large DC v oltage from a small A C input with a ladder net w ork of capacitors and dio des. A t eac h successiv e stage, the c harge on eac h capacitor is doubled b y sim ultaneously collecting c harge stored in the previous capacitor and the A C input. The F ermilab Co c kroft W alton mac hine applies a v oltage dierence of 750 kV across an ionization c ham b er, the negativ e p oten tial side of whic h is coated with cesium metal, and the other w all is partially op en in the direction of the F ermilab linear accelerator 56 PAGE 57 Figure 5-1. Carto on of the F ermilab pre-accelerator stage. Figure tak en from Ref. [ 111 ]. (linac). Hydrogen atoms that drift in to this c ham b er will ionize, and the bare protons will collide with the cesium metal. Cesium has a relativ ely lo w w ork function, and some of these collisions result in the transfer of t w o v alence electrons to the proton, forming an H ion. These 750 k eV kinetic energy negativ ely-c harged atoms drift to the w all of p ositiv e p oten tial, and ma y pass through the op ening and con tin ue to the next accelerator stage. A carto on of this pro cess is sho wn in Figure 5-1 The H atoms then en ter the linac, where an alternately p olarized electric eld accelerates the ions b et w een gaps of F arada y cage drift tub es. Beam bunc hes are formed with pulses roughly 5 ns apart, and the 130 m long linac terminates with H batc hes of 400 MeV kinetic energy As sho wn in Figure 5-2 these bunc hes are injected in to the Bo oster sync hrotron via a system featuring a stripping foil placed b et w een a series of dip ole magnets in a \dogleg" conguration. The foil strips the H ions of their electrons, and the subsequen t magnets steer the bare protons on to the Bo oster orbit. The dogleg dip ole magnet conguration has the eect of a fo cusing and defo cusing (F ODO) quadrup ole system, where the injected H atoms and the Bo oster protons con v erge to a single b eam. 57 PAGE 58 Figure 5-2. Injection diagram for H ions on to the Bo oster orbit as bare protons. Figure tak en from Ref. [ 112 ]. T o a v oid unnecessary b eam div ergences in the Bo oster, the dogleg dip oles are only pulsed when b eam is injected from the linac. The Bo oster sync hrotron accelerates the 400 MeV kinetic energy protons up to 8 GeV through 17 radio-frequency (RF) stages and is k ept on-orbit b y 24 p erio ds of F OFDOOD cells. This acceleration tak es roughly 33 ms and 20,000 turns around the 150 m diameter ring. The harmonic n um b er of the Bo oster is 84, though t ypically 3 buc k ets are not used. These 81 bunc hes, eac h separated b y 19 ns, of 8 GeV kinetic energy protons are extracted from the Bo oster in a 1.6 s spill. Sho wn in Figure 5-3 this structure is clearly visible in the arriv al of neutrino ev en ts at the MiniBo oNE detector. These spills t ypically con tain 5 10 12 protons and are deliv ered to the MiniBo oNE target and horn system at a maxim um rate of 5 Hz. F ull details of the Bo oster sync hrotron is a v ailable in Ref. [ 112 ]. 58 PAGE 59 arrival time (ns) n 500 0 500 1000 1500 Events 0 100 200 300 400 500 600 Figure 5-3. Timing structure of the BNB proton spills through the arriv al time of neutrino ev en ts at MiniBo oNE. The represen ted data is the neutrino-mo de CCQE sample, and a time-of-righ t correction based on the observ ed in teraction v ertex along the b eam direction has b een applied. The 500 ns oset b et w een the arriv al and recorded times is due to an oset in the timing instrumen ts. 59 PAGE 60 5.2.2 Beryllium T arget and Magnetic F o cusing Horn The next stage in the BNB con v erts the proton spill in to a fo cused b eam of mesons. The proton b eam strik es a 71.1 cm long target, comp osed of sev en 10.2 cm long and 0.5 cm radius cylindrical b eryllium slugs. The proton-b eryllium in teractions dep osit 600 W under normal running conditions, and so an air-co oling system is implemen ted to reduce radiation damage to the system and the surrounding en vironmen t. The b eryllium target is separated from its housing using three supp orting \ns", also made of b eryllium, and allo ws for air to b e circulated along its en tire length. The air ro w rate is 8 10 3 m 3 /s and, due to a heat exc hanger system, ro ws con tin uously during normal running conditions. Engineering designs for the b eryllium target and its installation inside the magnetic horn are sho wn in Figure 5-4 The proton-b eryllium in teractions create a spra y of secondary particles, including man y neutrino-paren t mesons. A set of connected inner and outer conductors form a horn system, and an electric curren t of 174 kA pulsed through these conductors in time with the BNB proton spill creates a toroidal magnetic eld as sho wn in Figure 5-5 This eld sim ultaneously fo cuses particles with p ositiv e or negativ e c harge, while defo cusing the other. In this w a y the p olarit y of this system denes the running mo de fo cusing p ositiv ely-c harged mesons yields an enhanced b eam (dominan tly via + + ) while selecting negativ e mesons creates a -enhanced b eam (via ). The magnetic horn sim ultaneously con trols the neutrino comp osition of the BNB b eam and substan tially increases the neutrino rux. In neutrino-mo de running, the horn increases the observ ed rate of neutrino in teractions b y roughly a factor of six. As with the b eryllium targets, the magnetic horn m ust also b e co oled to protect against radiation damage. A closed w ater system k eeps the system exceptionally stable. The rst BNB horn pulsed 96 million times b efore failing due to corrosion, while the second horn is still op erational and has b een pulsed a w orld's record 397 million times as of Marc h 2013. 60 PAGE 61 Figure 5-4. The BNB b eryllium target. Sho wn is an expanded view of the segmen ted target (top) and its place inside the magnetic fo cusing horn (b ottom). The proton b eam strik es the target from the left. 61 PAGE 62 Figure 5-5. A comparison of the azim uthal comp onen t of the magnetic eld relativ e to the input curren t b et w een data (in p oin ts) and the exp ectation (solid curv e) of 1/r dep endence. The v ertical line iden ties the inside edge of the outer conductor. Figure tak en from Ref. [ 113 ]. 5.2.3 Meson Deca y Region The mesons accepted in to the neutrino b eam are collimated immediately do wnstream through a 60 cm op ening in a concrete slab and subsequen tly en ter a 50 m long, air-lled deca y v olume. The mesons deca y in righ t to pro duce c harged leptons and neutrinos, or are absorb ed b y a concrete w all at the end of the deca y v olume. Protons that do not strik e the b eryllium target ma y in teract with the air molecules in the deca y region b efore terminating at the b eam dump. These in teractions ma y also pro duce mesons b o osted to w ards the detector, and these pro cesses con tribute 5% of the neutrino rux at MiniBo oNE. T en 25-ton steel absorb er b eams are housed ab o v e the middle of the deca y region, and could b e deplo y ed in the hall to systematically alter the normalization and energy sp ectrum of the neutrino b eam. Sp ecically the shortened deca y region w ould remo v e higher-energy neutrinos, including an appreciable amoun t of the instrinsic e and e from 62 PAGE 63 the deca y c hain. Mean while, the o v erall and rux w ould b e reduced b y roughly 10% p er deplo y ed absorb ed. So far, these absorb ers ha v e not b een in ten tionally deplo y ed; ho w ev er, in an early p erio d of an tineutrino-mo de running, one and then another absorb er fell in to the b eamline. A total of 5 : 69 (6 : 12) 10 19 POT w as collected in an tineutrino-mo de with one (t w o) absorb ers presen t in the deca y hall. Details of the systematic eects caused b y these blo c ks w as implemen ted in to sim ulation, and consistency b et w een the observ ed and predicted rate and kinematics suggest the mo deling is adequate. As the MiniBo oNE e oscillation searc h is limited b y statistics to date [ 36 ], these data are included in the oscillation analysis, as is the case with an early determination of the con tribution to the an tineutrino-mo de b eam presen ted in Section 7.1.5 Ho w ev er, the double-dieren tial cross section d 2 dT d cos for CCQE in teractions is limited b y statistics only in small regions of the distribution tails, and so these absorb er-do wn data are not used in the main result of this dissertation. 5.2.4 Neutrino Flux Calculation The most imp ortan t piece of an absolute neutrino rux calculation is the pro duction of the neutrino and an tineutrino paren t + and created in proton-b eryllium in teractions at the target. It is common to rely on a com bination of hadropro duction mo dels and data-based extrap olations to meet this goal. Ho w ev er, Figure 5-6 sho ws mo dern mo dels [ 114 115 116 117 ] for primary hadropro duction ( p + Be + X ) at 8.9 GeV proton b eam momen tum dramatically disagree. Clearly precision neutrino and an tineutrino cross section measuremen ts cannot b e made with information from hadropro duction mo dels alone. A m uc h more clean and direct metho d for constraining the neutrino rux w as fortunately a v ailable to MiniBo oNE: the HARP hadropro duction exp erimen t at CERN collected dedicated data using the same proton momen tum and target material as in the BNB. Double-dieren tial cross sections in terms of pion kinematics w as measured for b oth + [ 118 ] and [ 119 ], allo wing for a 63 PAGE 64 Figure 5-6. Pro duction of primary p er POT for 8.9 GeV momen tum protons inciden t on b eryllium for v arious hadropro duction mo dels as a function of momen tum (left) and op ening angle with resp ect to the incoming proton b eam (righ t). Figure tak en from Ref. [ 119 ]. minimally mo del-dep enden t determination of primary pro duction at the BNB for b oth the neutrino-mo de and an tineutrino mo de run congurations. Ho w ev er, ev en with dedicated data appropriate to the exp erimen tal setup of MiniBo oNE, there remain small regions of phase space relev an t to the an tineutrino-mo de b eam not co v ered b y the HARP measuremen ts. As will b e expanded and directly addressed in Chapter 7 of particular imp ortance to this w ork is the pro duction of v ery forw ard pions with resp ect to the direction of the incoming proton b eam. In the HARP exp erimen t, this same angular region suers from re-in teractions in the target and a sev ere proton bac kground, prev en ting a clean measuremen t of the pion pro duction cross section. F or these reasons, pion cross sections in the < 30 mrad region, where is the angle the outgoing pion mak es with resp ect to incoming protons, are not co v ered b y the HARP data. Instead, the nominal primary pro duction cross section for this region in the MiniBo oNE rux calculation is extrap olated from the existing HARP data using a Sanford-W ang [ 120 ] paramaterization. More suitable for extrap olating uncertain ties, errors on primary pro duction come from the piecewise p olynomial spline in terp olation [ 121 ]. This extrap olation is only one of man y p ossible c hoices, and is therefore sub ject to large 64 PAGE 65 uncertain ties. Figure 5-7 sho ws the HARP data, the Sanford-W ang parametrization, and the pro duction uncertain t y from the spline pro cedure for primary pro duction. The HARP data w as tak en on a thin v ersion (5% proton in teraction length) of the full-sized (170%) MiniBo oNE b eryllium target, and so these data do not include p ossible hadronic re-in teractions inside the target. The total cross section for these secondary in teractions are calculated with the Glaub er mo del [ 122 ], and this calculation is v eried with comparisons to data wherev er p ossible. Based on the agreemen t b et w een this mo del and the a v ailable data, uncertain ties on the most imp ortan t pro cesses con tributing pions to the b eam are set around 20% and higher [ 108 123 ]. F ortunately while some details of this calculation are mo del-dep enden t, Figure 5-8 sho ws the o v erall con tribution of these pro cesses to the o v erall neutrino rux is rather mild, at the lev el of 10%. Moreo v er, the same gure also suggests the con tribution from tertiary pions presen t in the long MiniBo oNE target but not in the thin target data from HARP is small. Therefore, with the exception of the v ery forw ard-going angular region, the HARP data allo ws for a minimally mo del-dep enden t determination of the pro duction of neutrino and an tineutrino paren t pions at the BNB. A geant 4-based pac k age [ 124 ] is used to calculate the neutrino rux observ ed at MiniBo oNE. The sim ulation tak es as input the previously-describ ed meson pro duction and considers the b eamline geometry proton tra v el to the target, p-Be in teractions in the target, magnetic horn fo cusing, particle propagation, meson deca y and nally neutrino and an tineutrino transp ort to the detector. F or b oth neutrino and an tineutrino mo de run congurations, the uncertain t y on pion pro duction and the set of all other b eamline uncertain ties con tribute roughly equally to the 9% total uncertain t y on the absolute rux prediction for the selected neutrino sp ecies. Figure 5-9 sho ws the predicted rux of e and e observ ed b y the MiniBo oNE detector for b oth neutrino and an tineutrino run mo des. 65 PAGE 66 Figure 5-7. Double-dieren tial cross section d 2 = dp d n for p + Be + X in units of m b / (GeV sr). The angular axes ha v e units of radians, and the momen tum pro jections are in units of GeV. The blue curv e is the Sanford-W ang parametrization based on the red HARP data p oin ts, and the blac k histogram with uncertain ties is the spline in terp olation. Figure tak en from Ref. [ 125 ]. 66 PAGE 67 Figure 5-8. Fluk a [ 126 ] calculations of the tertiary + yield from rein teractions in a graphite target. Giv en as a function of inciden t proton b eam momen tum p 0 the + fraction is giv en for the indicated thresholds on the longitudinal comp onen t of the + momen tum (left), and also for targets of 0.5, 1.0, and 2.0 in teraction lengths (righ t). The primary proton b eam at the BNB has momen tum 8.9 GeV/c. Figure tak en from Ref. [ 127 ]. 67 PAGE 68 (GeV) n E 0 0.5 1 1.5 2 2.5 3 3.5 4 /POT/50 MeV)2 (1/cm F -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 (a) mode n (GeV) n E 0 0.5 1 1.5 2 2.5 3 3.5 4 /POT/50 MeV)2 (1/cm F -16 10 -15 10 -14 10 -13 10 -12 10 -11 10 m n m n e n e n (b) mode n Figure 5-9. The MiniBo oNE rux prediction for (a) neutrino mo de and (b) an tineutrino mo de. Data tak en from Ref. [ 108 ]. 68 PAGE 69 Figure 5-10. Ov erview of the MiniBo oNE detector housing. Image tak en from Ref. [ 128 ]. 5.3 Detector 5.3.1 Ph ysical La y out Sho wn sc hematically in Figure 5-10 the detector is a 12.2 m diameter sphere housed in a 13.7 m underground cylindrical v ault suc h that the top of the tank sits roughly at ground lev el. The detector shap e w as motiv ated b y maximizing the v olume to surface area ratio, aording greater photo catho de co v erage for the same n um b er of PMTs. The simple spherical geometry also allo ws for globally symmetric reconstruction algorithms and th us equal sensitivit y to particle kinematics across all scattering angles. An earth o v erburden of 3 m reduces the rate of cosmic-ra y m uons en tering the detector to 10 kHz. Bet w een the detector and the o v erburden is an access ro om housing the main electronics, including the m uon calibration system crucial to the measuremen t of this dissertation discussed in Section 5.3.4 The tank is lled with 818 tons of undop ed mineral oil, optically segregated in to an inner signal region of radius 575 m and an outer v eto shell of 35 cm thic kness. Ligh t pro duced in the detector is collected b y 1520 8-inc h Hamamatsu photom ultiplier tub es (PMTs), 1280 of whic h face in to the signal region (11.3% co v erage) while 240 are inside the outer shell. Figure 5-11 sho ws a carto on of the MiniBo oNE detector partially cut a w a y to 69 PAGE 70 Figure 5-11. On the left, cuta w a y dra wing of the MiniBo oNE detector sho wing PMTs disp ersed in the inner and outer regions. The optical barrier separating the outer v eto region (pain ted white) from the inner signal region (pain ted blac k) is sho wn on the righ t. sho w the inner comp onen ts as w ell as a photo of the optical barrier separating the t w o regions. Lo w activit y in the v eto region is required in ph ysics analyses to ensure con tainmen t of c harged particles pro duced b y b eam-induced neutrinos while also eliminating con tamination from c harged particles en tering the tank. T o encourage photon rescattering and th us maximize detection eciency for c harged particles tra v ersing the v eto region, the surfaces are pain ted white. In con trast, to impro v e the kinematical resolution of signal ev en ts, photon rescattering is minimized with a blac k surface for the inner region. 5.3.2 Mineral Oil and its Prop erties A common c hoice for the detection medium in Cerenk o v-based exp erimen ts is w ater. In the case of MiniBo oNE, mineral oil w as selected o v er w ater for a v ariet y of reasons: the increased index of refraction yields a lo w er momen tum threshold on Cerenk o v ligh t pro duction for all particles, globally impro ving detection eciency n uclear capture of stopp ed is 8% in mineral oil, compared to 20% in w ater. 70 PAGE 71 T able 5-1. Momen tum threshold for pro duction of Cerenk o v radiation for four imp ortan t particle t yp es in mineral oil compared to w ater. P article Cerenk o v threshold mineral oil, n = 1.47 w ater, n = 1.33 electron 0.7 MeV/c 0.8 MeV/c m uon 144 MeV/c 160 MeV/c pion 190 MeV/c 212 MeV/c proton 1280 MeV/c 1423 MeV/c This allo ws a cleaner tagging of CC ev en ts, again impro ving detection eciency while sim ultaneously reducing its bac kground con tribution to the e CC sample. b y exploiting PMT activit y timing information, the lo w ered sp eed of ligh t in the medium impro v es in teraction v ertex resolutions Kept at 20 C, the mineral oil has a densit y of 0.845 g/cm 3 and an index of refraction of 1.47. Under these conditions, c harged particles with v elo cit y > 0 : 68 pro duce Cerenk o v radiation. The momen tum thresholds for pro duction of Cerenk o v radiation for relev an t particle sp ecies in mineral oil and w ater are compared in T able 5-1 The ab o v e b enets come at the cost of signican tly more complex mec hanisms for ligh t pro duction and propagation through the detector. Due to impurities in the oil, molecular excitations pro duce dela y ed photons with an isotropic direction and of longer w a v elength than the absorb ed particle. These are kno wn as ruorophores, or ruors, and four distinct mo des w ere observ ed in table-top measuremen ts of the MiniBo oNE oil [ 109 ]. The measured and extrap olated extinction rates of these ruors are sho wn in Figure 5-12 The presence of ruors obfuscate the top ology of the Cerenk o v signature and bias the correlation b et w een the collected Cerenk o v ligh t and the true energy of the particle. F ortunately calibration samples and systems discussed in the next section exist to measure these biases so that their eect on most analyses (including the topic of this dissertation) are minimal. Though the ruors complicate the understanding of the detector, without 71 PAGE 72 Figure 5-12. Photon extinction rate sp ectrum in the MiniBo oNE oil. As indicated, the solid curv es corresp ond to measuremen ts, while the dashed lines are based on an extrap olation of these data and are tuned to v arious in situ calibration data samples. Figure tak en from Ref. [ 109 ]. their presence scin tillation-based measuremen ts suc h as the neutral curren t elastic cross sections [ 129 130 ] w ould not b e p ossible. 72 PAGE 73 Figure 5-13. Quan tum eciency for the new er MiniBo oNE PMTs. Figure tak en from Ref. [ 132 ]. 5.3.3 Photom ultiplier T ub es Of the 1520 PMTs, 1198 are 9-stage and ha v e b een repurp osed follo wing their use in the LSND exp erimen t, while the remainder are 10-stage tub es purc hased for MiniBo oNE. T ests for c harge and time resolution, the v oltage lev el required to meet the desired gain, and the dark curren t w ere p erformed for all PMTs installed in to MiniBo oNE. The new er tub es feature a v erage timing (c harge at one photo electron) resolution of 1.1 ns (40%), while the older tub es resolv e the same quan tities at 1.7 ns (130%) [ 131 ]. The a v erage dark curren t for the new (old) tub es w as found to b e 1.0 (1.4) kHz at their op erating v oltage. Due to their sup erior p erformance, the new er PMTs are distributed uniformly in the signal region, while the LSND tub es with higher amoun ts of dark noise are used in the v eto region. The quan tum eciency for the new PMTs is giv en in Figure 5-13 5.3.4 Calibration Systems In situ measuremen ts of the PMT p erformance and the oil atten uation length o v er the lifetime of the exp erimen t is aorded b y a pulsed laser calibration system, sho wn 73 PAGE 74 Figure 5-14. Sc hematic of the laser calibration system. Image tak en from Ref. [ 109 ]. sc hematically in Figure 5-14 F our laser disp ersion rasks and a single bare optical b er are distributed throughout the detector and are pulsed at 3.33 Hz during normal data taking. Giv en the p eak eciency for the MiniBo oNE PMTs at 400 nm, these lasers are pulsed at 397 nm and signals from the disp ersion rasks illuminate all PMTs with roughly equal in tensities. In terpretations of the signals from the disp ersion rasks are somewhat degenerate b et w een eects arising from degrading oil prop erties and c hanging PMT p erformance. This degeneracy is partially brok en b y the signal from the bare optical b er, whic h illuminates a small circle of PMTs near the b ottom of the detector and is used to more directly study an y ev olution of the oil prop erties. The most imp ortan t asp ects of the PMT p erformance prob ed at 3.33 Hz under normal running conditions (though the system is v eto ed in case of coincidence with a b eam spill from the BNB) b y the laser calibration system are PMT time osets and gain calibrations. Time osets due to diering transit times for eac h readout system are obtained b y a simple comparison of the observ ed laser signal arriv al time to the kno wn 74 PAGE 75 laser pulse, while also considering tra v el time for the laser ligh t. The gain of individual PMTs can v ary in subtle but imp ortan t w a ys, and these eects are calibrated b y normalizing the resp onse of eac h PMT to a single v alue, based on the input in tensit y of the laser ligh t. The calibrated time osets are critical to the detector's abilit y to separate the Cerenk o v signatures from dieren t particles, most notably those connected b y deca y pro cesses, while the gain corrections allo w for precise measuremen ts of particle energy uniform in pro duction p osition and direction. Ev en more crucial to the study of CCQE in teractions, cosmic-ra y m uons and a dedicated calibration system allo w the m uon reconstruction algorithm to b e v eried against data. The detector resp onse to m uons is indep enden tly measured b y observ ation of the energy and direction of cosmic-ra y m uons up to 800 MeV. A scin tillator ho doscop e directly ab o v e the detector and sev en scin tillator cub es at v arious depths within the detector are used to trac k these particles. Figure 5-15 sho ws the la y out of this system in the MiniBo oNE detector. Eac h cub e is connected b y an optical b er to a PMT for readout. The direction of cosmic-ra y m uons are measured in the ho doscop e, and they ma y b e iden tied as stopping in one of the scin tillation cub es b y the observ ation of a deca y electron pro duced inside the cub e. With kno wledge of the cub e's p osition and the m uon's inciden t p osition and angle, it's energy can b e calculated based on ho w m uc h oil it crossed and the Bethe-Blo c k form ula for energy loss. In this w a y the m uon reconstruction algorithm can b e v eried against data for a v ariet y of m uon energies. After all calibration studies, the energy (angle) resolution for m uons impro v es from 12% (5.4 deg) at 100 MeV to 3.4% (1.0 deg) at 800 MeV. More details of this reconstruction are giv en in the next section. 5.3.5 Analysis T o ols This section describ es the connection b et w een the PMT signals and the analysis of CCQE in teractions. A total of 16 triggers ma y activ ate the data acquisition (D A Q) system for a total 75 PAGE 76 Cube Range Energy (MeV) 0 100 200 300 400 500 600 700 800 Visible Tank Energy (MeV) 0 100 200 300 400 500 600 700 800 Cosmic Muon Energy Data Monte Carlo Figure 5-15. Carto on of the m uon calibration system (left) and the relationship b et w een m uon energy and range in data and sim ulation subsequen t to calibrations (righ t). Only one of the sev en scin tillator cub es are sho wn in the left gure, and the image is tak en from Ref. [ 109 ]. rate of 26 Hz under normal running conditions, and are used a v ariet y of calibration purp oses and ph ysics analyses [ 133 134 ]. Up to 5 Hz are due to the primary BNB trigger, and in this case the D A Q records PMT c harge and time information from all 1520 phototub es for a total of 19.2 s b eginning 5 s b efore the 1.6 s long proton spill. Cosmic-ra y m uons stopp ed in the signal region prior to the start of the D A Q windo w ma y deca y in time with the BNB spill, so PMT activit y 5 s b efore proton deliv ery is monitored and used to minimize this con tamination. Activit y is recorded subsequen t to the b eam windo w for more than 10 s to observ e electrons from the at-rest deca y of m uons (hereafter referred to as \Mic hel" electrons) pro duced either directly or indirectly through the primary neutrino in teraction. The PMT timing information is used to asso ciate clusters of activit y with the signature of a single particle using PMT \hits"; temp oral groups of hits form \sub ev en ts". A PMT pulse passing the discriminator threshold of 0.1 photo electrons is dened as a 76 PAGE 77 Figure 5-16. T ypical PMT hit (ordinate axis) and timing signature of a CCQE ev en t. The prompt + arriv es in time with the BNB spill (from 4600 6200 ns relativ e to the b eginning of the D A Q clo c k) with 100's of MeV in kinetic energy while the Mic hel is observ ed a time c haracteristic of the m uon lifetime later with an energetic endp oin t of 53 MeV. Their signatures are easily separated with the sub ev en t denition. hit, and forms the basic unit of the observ ed signal in tensit y A group of PMT activit y with at least 10 hits within a 200 ns windo w and individual hit times less than 10 ns apart, while allo wing for at most t w o spacings of 10 20 ns, denes a sub ev en t. These sub ev en ts separate particles whose transit emits signican t amoun ts of Cerenk o v ligh t with high eciency and so are primarily used isolate the signatures and top ologies of m uons and electrons. In teractions of CCQE t ypically yield t w o sub ev en ts, the rst from the prompt + and the second from its deca y p ositron. Figure 5-16 sho ws the timing and PMT hit signature of a t ypical CCQE ev en t. The pattern, timing, and total c harge of prompt Cerenk o v radiation collected b y the PMTs in the rst sub ev en t are used to iden tify m uon kinematics, the quan tit y most imp ortan t to the main result of this dissertation. A lik eliho o d function is compared to the top ology and timing of the observ ed PMT hits: 77 PAGE 78 Figure 5-17. T ypical PMT hit top ology and timing for m uon (left) and electron (righ t) candidate ev en ts in MiniBo oNE data. PMT c harge is correlated to the size of the displa y ed hits, while timing is giv en b y the color sp ectrum, where blue hits arriv ed earliest and red hits arriv ed last. L ( x ) = Y unhit PMTs i (1 P ( i hit ; x )) Y hit PMTs i P ( i hit ; x ) f q ( q i ; x ) f t ( t i ; x ) ; (5.1) where P ( i hit ; x ) is the probabilit y for PMT i to register a hit giv en the m uon v ertex and kinematic v ector x and f q ( f t ) is a probabilit y distribution function (PDF) for the hit to return the measured c harge (time) q i ( t i ). As the energy range of particles observ ed b y MiniBo oNE is sensitiv e to the mass dierence b et w een m uons and electrons, an electron's path of tra v el in the MiniBo oNE detector is more lik ely to b e derected compared to a m uon's via the Bremstrahlung and m ultiple scattering pro cesses. Electrons ma y also create electromagnetic sho w ers, and this leads to distinct Cerenk o v top ologies and therefore dieren t f q and f t PDFs for the t w o c harged leptons. Figure 5-17 compares t ypical electron and m uon timing and c harge signatures in MiniBo oNE. The v ector x is comp osed of the particle's time, energy and p osition at creation, as w ell as its momen tum pro jections along the azim uthal and p olar angles in spherical 78 PAGE 79 co ordinates. The negativ e logarithm of the lik eliho o d function in Eq. 5.1 sim ultaneously v aries these sev en parameters while comparing to the observ ed PMT hits. The parameters from the maximized lik eliho o d function yield the reconstructed lepton kinematics. A t w o-trac k v ersion of this reconstruction w as also dev elop ed to iden tify 0 candidate ev en ts, and the angular and energy resolutions of this reconstruction to all three particle sp ecies, op erating Eq. 5.1 under the appropriate h yp othesis, are giv en in Figure 5-18 F urther details on this reconstruction can b e found in Ref. [ 135 ]. The direct and high-resolution observ ation of m uon prop erties using this reconstruction further motiv ates the c hoice of emphasizing the CCQE cross section as a function of m uon kinematics as the main result of this w ork, while the statistics of the data set also yield unpreceden ted sensitivit y to the b eha vior of the + in CCQE in teractions. 79 PAGE 80 Figure 5-18. Kinetic energy and angular resolution of the MiniBo oNE reconstruction to electrons, m uons, and neutral pions. Figure from Ref. [ 135 ]. 80 PAGE 81 CHAPTER 6 INTR ODUCTION TO THE CR OSS-SECTION MEASUREMENT 6.1 Ov erview This dissertation describ es the rst measuremen t of the m uon an tineutrino CCQE cross section with h E i < 1 GeV. Before exploring the details, it is helpful to rst describ e the o v erall strategy and iden tify the areas of the calculation deserving of the most atten tion. Generically for one to measure a dieren tial cross section in the distribution X giv en a data set d total bac kground b using a detector housing a n um b er of in teraction targets N with detection eciency and a total exp osure the form ula is rather simple: d dX i = P j U ij ( d j b j ) X i i N ; (6.1) where i indexes the region of measuremen t in the absence of detector eects, X i is the width of this region, j lab els the same region as observ ed b y the detector, and the matrix U ij connects the t w o. The other cross-section congurations measured in this w ork are simple extensions of Eq. 6.1 and will b e discussed later. Before w e pro ceed with a cross-section calculation, an analysis sample m ust b e iden tied. In describing this in the next section, it will b ecome clear that a ma jor complication of this analysis is the presence of large and nominally uncertain bac kgrounds. 6.2 Ev en t Selection Optimizing the sample to study a particular t yp e of in teraction alw a ys in v olv es a balance b et w een retaining as man y high-qualit y signal ev en ts as p ossible while minimizing the con tamination from bac kground in teractions. The selection and its ecacy for the an tineutrino-mo de CCQE sample follo ws: 1. V eto hits < 6, all sub ev en ts 81 PAGE 82 2. First sub ev en t in b eam windo w: 4000 < T(ns) < 7000, where T is the a v erage PMT hit time 3. Tw o sub ev en ts 4. Reconstructed v ertex < 500 cm from tank cen ter, rst sub ev en t 5. T > 200 MeV (kinetic energy of rst sub ev en t) 6. ln( /e) > 0.0, rst sub ev en t 7. Distance b et w een 1st and 2nd sub ev en t v ertices > 500 cm/GeV T 100 cm 8. Distance b et w een 1st and 2nd sub ev en t v ertices > 100 cm Cut 1 sim ultaneously rejects incoming c harged particles and enforces con tainmen t of c harged particles created in the tank. The upp er b ound on the acceptable n um b er of v eto hits is motiv ated in Figure 6-1 where six v eto hits accepts lo w-lev el PMT noise but rejects most exiting and en tering activit y Cut 2 requires the rst sub ev en t b e in time with the proton b eam spill. Figure 6-1. V eto hits for early neutrino-mo de data and MC for the rst t w o sub ev en ts. P oin ts are data, the dotted (dashed) blue (red) histogram is CCQE (all nonCCQE) and the solid line is total MC. All distributions are normalized to unit area. Cosmic ra ys are not sim ulated, and this is the origin of the shap e discrepancy b et w een data and MC in the rst sub ev en t. Figure from Ref. [ 136 ]. 82 PAGE 83 T o motiv ate and isolate the eects of selections 3-8, the distribution under examination is presen ted with all other requiremen ts applied. T o a v oid placing requiremen ts on sub ev en ts that ma y not exist, the sub ev en t distribution is the lone exception. Figure 6-2 sho ws the impact of cuts 3-6. Cut 3 sim ultaneously ensures there are no nal-state pions and the ev en t is consisten t with the pro duction of a con tained m uon. The selection of the sample represen ted in the sub ev en t gure is cuts 1 and 2, where the v eto hit requiremen t is applied to eac h sub ev en t presen t. The large excess in the single sub ev en t bin is dominated b y Mic hel electrons pro duced in time with the b eam from cosmic-ra y m uons, whic h are not sim ulated, en tering prior to the start of the D A Q windo w. Cut 4 a v oids a class of ev en ts that ma y b e reconstructed p o orly due to greater sensitivit y to PMT co v erage. The spik e at high radius is due to the relativ ely dense material in the optical barrier. The requiremen t of cut 5 also impro v es reconstruction reliabilit y while a v oiding a double-coincidence of the kind of Mic hel electrons men tioned earlier. Cut 6 enhances the purit y of the sample b y rejecting man y CC ev en ts where the pion is energetic enough to pro duce some Cerenk o v ligh t and cause the m uon ring to receiv e a more electron-lik e score. The l n ( =e ) v ariable is found b y comparing the m uon-lik e to the electron-lik e score of the reconstruction describ ed in Section 5.3.5 83 PAGE 84 Number of subevents 0 1 2 3 4 5 0 50 100 150 200 250 300 350 400 450 3 10 vertex radius (cm) m Reconstructed 0 100 200 300 400 500 600 0 500 1000 1500 2000 2500 3000 3500 4000 (GeV) m Reconstructed T 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1000 2000 3000 4000 5000 6000 Data w/ stat error CCQE w/ bound p m n CCQE w/ free p m n CCQE m n p CC + p CC all others /e) m ln ( 0.15 0.1 0.05 0 0.05 0.1 0.15 0 2000 4000 6000 8000 10000 12000 14000 Figure 6-2. Selection requiremen ts 3-6. Descriptions are giv en in the text, and distributions are normalized to rux. With the exception of the sub ev en t gure, all distributions ha v e ev ery cut applied except the one indicated in the gure. Cut 7 enhances the sample purit y b y requiring the distance b et w een the v ertices of the t w o sub ev en ts b e consisten t with the pro duction and subsequen t deca y of a minim um ionizing particle (MIP). Cut 8 further reduces the small neutral curren t bac kground. Cuts 7 and 8 are sho wn in Figure 6-3 Step wise purit y and detection eciency for the resultan t sample is presen ted in T able 6-1 where v alues are giv en for b oth b ound (from the carb on con tribution to mineral oil) and free (from the h ydrogen con ten t) CCQE scattering. A breakdo wn of the predicted sample comp osition is giv en in T able 6-2 F rom these data w e will extract three main cross sections: as a function of neutrino energy with resp ect to Q 2QE (reconstructed four-momen tum transfer under the assumption 84 PAGE 85 T able 6-1. Purit y and detection eciency for the sample describ ed in this section. Signal CCQE in teractions are presen ted in their b ound (\ 12 C") and free (\H 2 ") comp onen ts and also as the sum. A pre-cut of generated v ertex radius < 550 cm for the primary sub ev en t has b een applied, and these estimates rerect the measuremen ts and constrain ts describ ed in Chapter 7 Cut Description Purit y (%) Eciency (%) 12 C H 2 T otal 12 C H 2 T otal No cuts 22.6 9.0 32.3 100 100 100 V eto hits < 6, all sub ev en ts 19.5 7.7 27.6 50.9 50.4 50.8 First sub ev en t in b eam windo w 19.6 7.7 27.7 50.5 49.9 50.3 T > 200 MeV 25.9 10.1 36.9 44.3 43.4 44.0 Tw o sub ev en ts 33.8 13.2 48.4 39.1 38.3 38.8 Reconstructed radius within 500 cm 34.4 13.5 49.2 32.8 32.1 32.6 e dist. > 500 cm/GeV T 100 cm 38.2 15.0 54.3 30.8 30.3 30.6 e dist. > 100 cm ln ( / e ) > 0 43.2 17.1 61.0 29.6 29.3 29.5 of CCQE in teractions), and the minimally mo del-dep enden t double-dieren tial cross section as a function of kinematics. Reconstructing the inciden t neutrino energy and the squared four momen tum transfer on an ev en t-b y-ev en t basis is p ossible only b y assuming the observ ed in teraction is CCQE on carb on, and also that it o ccurs on a single indep enden t n ucleon at rest. T o b egin these calculations, w e start with an arbitrary t-c hannel exc hange b et w een t w o particles 1 + 2 3 + 4 sho wn in Figure 6-4 The momen tum transfer q 2 in this in teraction is giv en b y: q 2 = ( p 1 p 3 ) 2 = ( p 2 p 4 ) 2 ; (6.2) where p is the particle's four-momen tum. Using the notation of four-v ectors and fo cusing on the momen tum transfer b et w een the 1 and 3 particles, 85 PAGE 86 T able 6-2. Predicted comp osition of the sample describ ed in this section. These estimates rerect the measuremen ts and constrain ts describ ed in the follo wing c hapters. in tegrated POT 10.1 10 20 mean energy 665 MeV energy-in tegrated rux 2.93 10 11 / cm 2 CCQE candidate ev en ts 71176 CCQE eciency ( R < 550 cm) 29.5% In teraction Con tribution (%) + p (bnd) + + n 43.2 + p (free) + + n 17.1 + n + p 16.6 + n + + n + 7.9 + A + + A + 3.3 + p + p + + 3.1 + p + + p + 2.5 . . . . . . . . . . + p + + 0 2.0 + n + + + p + + 0 . . . . . . . . . . + p + + n + 0 2.0 + n + n + + 0.7 + n + p + 0 0.8 + A + A + + 0.2 All other 0.5 q 2 = ( E 1 E 3 ; ~ p 1 ~ p 3 ) = ( E 1 E 3 ) 2 j ~ p 1 ~ p 3 j 2 = E 2 1 + E 2 3 2 E 1 E 3 j ~ p 1 j 2 + j ~ p 3 j 2 2 j ~ p 1 jj ~ p 3 j cos 1 3 ; (6.3) where ~ p is the three-momen tum v ector, and 1 3 is the scattering angle of particle 3 with resp ect to the direction of particle 1. In the case of a neutrino for particle 1, j ~ p 1 j = E 1 to excellen t appro ximation. F or a or CC in teraction, particle 3 is a m uon, and 86 PAGE 87 Q 2 = q 2 = 2 E QE ( E j ~ p j cos ) m 2 ; (6.4) where, for con v enience, Q 2 is dened to b e a p ositiv e quan tit y The form of Eq. 6.4 is particularly useful for accelerator-based neutrino measuremen ts, as the scattering angle is simply the observ ed angle of the m uon relativ e to the b eam direction, hereafter simply referred to as Ho w ev er, these neutrino sources t ypically feature a broad range of neutrino energies, and so additional information is needed to nd Q 2 T o reconstruct the inciden t neutrino energy w e emplo y the same assumptions as b efore and also in tro duce an at-rest proton target and an outgoing neutron in the hadronic v ertex, appropriate to CCQE. Note that the kinematic assumption on the proton is wildly inaccurate; ho w ev er, as the momen tum distribution ab out an y spatial direction m ust b e cen tered around zero, with enough statistics the bias is small and acceptable. Neglecting the small binding energy for in teractions with b ound n ucleons, four-momen tum conserv ation giv es ( E QE + m p ; ~ p ) = ( E + E n ; ~ p + ~ p n ) : (6.5) Note the neutrino energy is lab eled to explicitly recognize its assumption of a CCQE in teraction. Eliminating the neutron kinematics and again neglecting the neutrino mass giv es: m 2p + E 2 + 2( m p E QE E E QE m p E ) = m 2n + j ~ p j 2 2 E QE j ~ p j cos : (6.6) Rearranging Eq. 6.6 yields a determination of the neutrino energy solely in terms of m uon kinematics: 87 PAGE 88 E QE = m 2n m 2p m 2 2 m p E 2 ( m p E + j ~ p j cos ) : (6.7) W e can use this quan tit y in nding the four-momen tum transfer, carrying o v er the \ QE lab el to again recognize the propagated CCQE assumption: Q 2QE = 2 E QE ( E p cos ) m 2 : (6.8) As man y theoretical groups predict a sizable con tribution from an unexp ected bac kground to the MiniBo oNE CCQE sample (Section 4.4.1 ), the assumption of CCQE in teractions em b edded in E QE and Q 2QE is particularly troubling. This is the primary motiv ation for highligh ting the double-dieren tial cross section d 2 dT d cos as the main result of this w ork. Nev ertheless, pro ducing cross sections in E QE and Q 2QE can b e helpful to facilitate historical comparisons. The full MiniBo oNE CCQE sample in E QE Q 2QE and the kinematics of the m uon is sho wn in Figure 6-5 as w ell as the t w o-dimensional m uon kinematical ratio of data to the prediction. T able 6-2 estimates the CCQE sample features a purit y of 60%. With a signal:bac kground rate approac hing 1:1, it is crucial to ev aluate ho w w ell these bac kgrounds are understo o d b efore they can b e reliably subtracted from the data to pro duce CCQE cross sections. The next c hapter is dedicated to the v arious measuremen ts and constrain ts obtained for the dominan t bac kgrounds. 88 PAGE 89 (GeV) m T 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 -e vertex distance (cm) m 0 50 100 150 200 250 300 350 400 450 500 CCQE (GeV) m T 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 -e vertex distance (cm) m 0 50 100 150 200 250 300 350 400 450 500 p CC m n (GeV) m T 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 -e vertex distance (cm) m 0 50 100 150 200 250 300 350 400 450 500 p CC m n (GeV) m T 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 -e vertex distance (cm) m 0 50 100 150 200 250 300 350 400 450 500 other (GeV) m T 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 -e vertex distance (cm) m 0 50 100 150 200 250 300 350 400 450 500 All MC (GeV) m T 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 -e vertex distance (cm) m 0 50 100 150 200 250 300 350 400 450 500 data Figure 6-3. T dep enden t range cut for dieren t c hannels and samples, as lab eled. Ev en ts are plotted on a logarithmic scale. p 2 p 4 p 1 p 3 Figure 6-4. An arbitrary t-c hannel exc hange. 89 PAGE 90 m q Reconstructed cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 2000 4000 6000 8000 10000 12000 14000 (GeV) m Reconstructed T 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1000 2000 3000 4000 5000 6000 Data w/ stat error CCQE w/ bound p m n CCQE w/ free p m n CCQE m n p CC + p CC all others ) 2 (GeV QE 2 Reconstructed Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2000 4000 6000 8000 10000 12000 (GeV) QE n Reconstructed E 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1000 2000 3000 4000 5000 0.8 0.9 1 1.1 1.2 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 data / MC data / MC Figure 6-5. The four v ariables cross sections will b e rep orted in. Distributions are normalized to rux. 90 PAGE 91 CHAPTER 7 BA CK GR OUND MEASUREMENTS AND CONSTRAINTS With bac kgrounds accoun ting for nearly half of the total sample studied, their precise con tribution and kinematics m ust b e rigorously v eried b efore a reliable bac kground subtraction can b e made. This section presen ts v arious measuremen ts and constrain ts on these pro cesses. As and CC in teractions are dominan t, particular atten tion is paid to understanding their con tribution. 7.1 Measuremen ts of the Bac kground 7.1.1 Motiv ation In teractions induced b y ev en ts form the largest single bac kground to the CCQE sample, accoun ting for 20% of the selected ev en ts, and half of the total bac kground. Giv en the high-qualit y pion pro duction data from the HARP exp erimen t and the litan y of cross-section measuremen ts from the MiniBo oNE neutrino-mo de data [ 84 78 137 79 138 80 139 ], one migh t assume the con tribution to the an tineutrino-mo de data is w ell-constrained. Ho w ev er, as Figure 7-1 sho ws, the ma jorit y of the ev en ts con tributing to the an tineutrino-mo de data are pro duced in a kinematic region of the paren t + that is not constrained b y the HARP data. The particulars of Figure 7-1 w arran t a few more remarks: b oth paren t pion distributions leading to the \wrong-sign" con tribution (neutrinos in an tineutrino mo de and vic e versa ) p eak at the lo w est op ening angles. This sho ws ho w these ev en ts con tribute to the b eam: their transv erse momen tum is insucien t to b e signican tly altered b y the magnetic eld, and so their path is m uc h less derected compared to pions created at larger the an tineutrino con tribution to the neutrino-mo de data is min uscule in comparison to the con v erse. This is due to a con v olution of rux and cross-section eects that sim ultaneously serv e to enhance the neutrino comp onen t while the an tineutrino con tribution is suppressed: the leading-particle eect at the b eryllium target (the 91 PAGE 92 (rad) p q 0 0.05 0.1 0.15 0.2 0.25 Predicted Events 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 m n + p p+Be m n p p+Be mode n (rad) p q 0 0.05 0.1 0.15 0.2 0.25 Predicted Events 0 500 1000 1500 2000 2500 m n p p+Be m n + p p+Be mode n < 0.9 GeV n : E m n > 0.9 GeV n : E m n Figure 7-1. Predicted angular distributions of pions with resp ect to the inciden t proton b eam ( ) pro ducing and in neutrino (left) and an tineutrino (righ t) mo des. Only pions leading to and ev en ts in the detector are sho wn, and all distributions are normalized to 10.1 10 20 protons on target. Arro ws indicate the region where HARP data [ 118 ] are a v ailable. p + Be initial state has a net p ositiv e c harge) naturally leads to the creation of roughly t wice as man y + as and neutrino cross sections are t ypically around three times as large as an tineutrino cross sections around 1 GeV. the ab o v e observ ation explains wh y this is a complication unique to an tineutrino mo de: the wrong-sign comp onen t in neutrino-mo de data is small enough so that ev en for large fractional uncertain t y on this bac kground, the resultan t error on the cross-section measuremen ts are negligible compared to other systematic uncertain ties. as seen in the an tineutrino-mo de distribution, high-energy 's are strongly correlated with the deca y of + created at v ery small op ening angles. This indicates their rux is more p o orly constrained b y the HARP data compared to lo w er-energy 's. So, not only is the o v erall rux in an tineutrino mo de highly uncertain, the accuracy of the extrap olated rux prediction ma y b e a function of neutrino energy The ab o v e observ ations motiv ate dedicated studies of the con tribution to the an tineutrino-mo de b eam, and in as man y exclusiv e regions of neutrino energy as is allo w ed b y statistics to examine the rux sp ectrum. 92 PAGE 93 Man y other exp erimen ts deal with this bac kground in a m uc h more direct w a y: they emplo y a magnetic eld to determine the sign of the outgoing lepton. This pro vides / discrimination for CC in teractions on an ev en t-b y-ev en t basis. Mo dern examples of magnetized neutrino oscillation exp erimen ts include MINOS [ 46 ], NOMAD [ 140 ], and the T2K near detector [ 141 ]. The analyses in this section pro vide a rst demonstration that, in the absence of a magnetic eld, and con ten t of an y mixed neutrino rux can b e mo destly separated using statistical metho ds. These metho ds could also aid curren t and future neutrino exp erimen ts that will test for CP violation in the lepton sector using large unmagnetized detectors. This includes exp erimen ts suc h as NO A [ 54 ], the T2K far detector, LBNE [ 55 ], LA GUNA [ 142 ], and Hyp er-K [ 143 ]. Also, it has b een argued that the separation of c harged-curren t neutrino and an tineutrino ev en ts aorded b y these kinds of analyses ma y b e sucien t to meet the loft y ph ysics goals of neutrino factories [ 144 ]. Finally the MINER A [ 145 ] neutrino cross-section exp erimen t could gain crucial kinematic and statistical sensitivit y b y using these kinds of tec hniques to analyze CC ev en ts not accepted in to their magnetized m uon calorimeter. The follo wing sections presen t the rst measuremen t of the comp onen t of an an tineutrino-mo de b eam observ ed b y a non-magnetized detector. Three statistical tec hniques are used to constrain this bac kground to a sub-dominan t uncertain t y in the extraction of the CCQE cross sections. Tw o of these tec hniques are published in Ph ysical Review D [ 146 ], while the third is curren tly under review [ 147 ]. 7.1.2 General Strategy T o statistically measure the wrong-sign bac kground, w e m ust exploit asymmetries in the w a y neutrinos, an tineutrinos, and their b ypro ducts in teract in the detector. Analyzing the v arious samples giv es a direct handle on their con tribution to the data, whic h is the only kno wledge necessary for p erforming the bac kground subtraction. Ho w ev er, with the v aluable cross-section data from MiniBo oNE's neutrino-mo de run, w e can extract 93 PAGE 94 information ab out the rux as w ell, whic h can b e used to test the accuracy of the extrap olation of the HARP data (describ ed in Section 5.2.4 ) in to the lo w-angle region. The c hannels con tributing to the ph ysics samples w e will analyze are dominated b y CCQE and CC + in teractions, and results from their cross-section analyses in the neutrino-mo de data [ 84 78 ] are applied to the an tineutrino-mo de sim ulation. With accurate cross sections implemen ted in to the Mon te Carlo (MC) sim ulation, dierences of the observ ed rates and the exp ectation from sim ulation rerect the accuracy of the mo del-dep enden t rux prediction. The relev ance of the measured cross sections to the an tineutrino-mo de b eam dep ends on the relativ e o v erlap in the sp ectra b et w een the t w o running mo des. As the + -paren t particles are sign-selected in neutrino mo de and feature a fo cusing p eak, while their acceptance in an tineutrino mo de is mostly due to lo w-angle and high energy pro duction, it is reasonable to exp ect the relativ e sp ectra to b e drastically dieren t. Figure 5-9 sho ws this to b e the case, where the pr o duc e d rux sp ectrum in an tineutrino mo de is signican tly harder compared to the 's in neutrino-mo de running. If the an tineutrino-mo de data w ere sensitiv e to the details of the diering sp ectra, this w ould indicate the relev ance of the observ ed cross sections is only marginal, and in terpretations of these analyses as rux measuremen ts w ould b e inaccurate. F ortunately Figure 7-2 sho ws that the ac c epte d sp ectrum of 's in the CCQE samples across b oth run mo des is v ery similar. This is mostly due to the rejection of high-energy 's via the m uon con tainmen t requiremen t. The large o v erlap b et w een these sp ectra allo ws the observ ed in teractions in the an tineutrino-mo de b eam to b e tigh tly constrained b y the neutrino-mo de measuremen ts. In principle, the extracted rux information from these analyses could b e used to re-analyze the neutrino-mo de data with m uc h stronger constrain ts on the lo w-angle region of the rux prediction. Figure 7-1 sho ws this region con tributes roughly 10% of the rux in neutrino-mo de running. Notice that, due to the small o v erlap b et w een the paren t + phase space across the t w o running mo des, some circularit y w ould b e presen t in suc h an 94 PAGE 95 (GeV) n gen. E 0 0.5 1 1.5 2 2.5 3 arbitrary 0 0.02 0.04 0.06 0.08 0.1 in neutrino mode m n in antineutrino mode m n Figure 7-2. Generated energy distributions accepted in to the neutrino and an tineutrino CCQE samples. The substan tial o v erlap indicates the 's bac kground to the CCQE sample are kinematically co v ered b y the cross sections observ ed in the neutrino-mo de data.analysis. Nonetheless, this last adv an tage exp oses a unique feature of this tec hnique: it will b e sho wn that the rux measuremen t in an tineutrino mo de is dominated b y uncertain ties on the MiniBo oNE cross-section measuremen ts, whic h are in turn dominated b y the + HARP errors through the rux uncertain t y So the tec hniques presen ted here eectiv ely constrain regions of hadropro duction phase-space not co v ered b y the HARP data to the lev el of precision of the regions that ar e co v ered. A nal adv an tage of determining a rux with this strategy is the cancellation of systematic uncertain ties that aect the pro cesses in the same w a y across b oth run mo de congurations. These fully-correlated errors are mostly detector-related; in particular, a unique feature of this measuremen t of the rux is its indep endence of man y nal-state in teraction pro cesses. 95 PAGE 96 7.1.3 Flux Measuremen t Using CC + Ev en ts The rst and most direct measuremen t of the bac kground is a simple rate analysis of the three sub ev en t sample. In the neutrino-mo de data, this sample is dominated b y CC + pro duction, mostly through the (1232) resonance. The three sub ev en ts arise from the prompt and t w o deca y electrons, one eac h from the and + : 1 : + p ( n ) + p ( n ) + + + + 2 = 3 : e + e + 2 = 3 : e + + e + : (7.1) The mono-energetic + from deca y-at-rest + is b elo w Cerenk o v threshold, and regardless the quic k deca y of the + w ould mak e the + not separable from the prompt using timing alone. Also, due to the fast deca y of the + it is eectiv ely random whic h deca y electron yields the second or third sub ev en t. F ew other pro cesses in the MiniBo oNE detector yield this signal, and the neutrino-mo de CC + sample has a purit y of 90% [ 78 ]. F rom simple electric c harge and lepton n um b er conserv ation, the analogous c harged-curren t single pion mec hanism induced b y an tineutrinos yields a As stopp edexp eriences n uclear capture on 12 C at nearly 100% [ 148 ], its deca y is not observ ed and therefore it mostly yields t w o sub ev en ts: 1 : + p ( n ) + + p ( n ) + + 12 C X 2 : e + + e + (7.2) where the remnan ts of n uclear capture X t ypically do not yield observ able ligh t in the detector. While the n uclear capture mec hanism v acates CC ev en ts from the three sub ev en t sample and so allo ws for the presen t measuremen t of the the rux, one can 96 PAGE 97 readily recognize the sample CC ev en ts do p opulate is the main study of this dissertation, that of CCQE. This bac kground is addressed in Section 7.2 Some CC ev en ts do yield a third sub ev en t, mostly when the deca ys in righ t. Ev en with this additional bac kground, the simple requiremen ts outlined in the next section giv e a high-purit y sample of CC + ev en ts with whic h w e can use to mak e a p o w erful measuremen t of the rux in the an tineutrino-mo de b eam. A t the time of the analysis of CC + ev en ts, only a subset of the full 1.0 10 20 POT tak en in an tineutrino mo de w as a v ailable. Since few er data w ere a v ailable at this time, some less reliable runs w ere used in whic h absorb er blo c ks acciden tally fell in to the deca y tunnel at the BNB (describ ed in Section 5.2.3 ). These blo c ks preferen tially absorb high-energy 's and 's, reducing the con tribution of high-energy and to the b eam. Since this measuremen t is not limited b y statistics, the analysis w as not up dated as more POT b ecame a v ailable. This is also the case for the analysis of the cos distribution (Section 7.1.5 ). T able 7-1 sho ws the con tribution of these absorb er-do wn runs to the total amoun t analyzed. T able 7-1. Summary of data p erio ds used in the analyses of CC + ev en ts and the cos distribution (Section 7.1.5 ). P erio d POT (e20) 0 absorb ers 4.480 1 absorb er 0.569 2 absorb ers 0.612 T otal 5.661 Implemen tation of the CC + Cross Section An imp ortan t distinction in this analysis is the denition of CC + ev en ts treated as signal. T o a v oid dep endence on nal-state in teractions, the MiniBo oNE neutrino-mo de CC + cross section w as rep orted as an observ able quan tit y: sp ecically the nal state studied consisted of one one + and an y n um b er of n ucleons [ 78 ]. This nal state w as 97 PAGE 98 (MeV) n E 600 800 1000 1200 1400 1600 1800 2000 )2) (cmn(E s 0 0.02 0.04 0.06 0.08 0.1 0.12 -36 10 Figure 7-3. The CC + total cross section. Blac k p oin ts with green error bands are MiniBo oNE data, and red is the MC exp ectation. Figure repro duced from Ref. [ 78 ]. not corrected for nal-state in teractions. Some of the more imp ortan t implications are that some amoun t of n ucleon-lev el CC 0 ( + N + 0 + N 0 ) con tribute to this sample, while some CC + ev en ts are not presen t due to the pion c harge-exc hange ( + X$ 0 + X 0 ) and absorption ( + X X 0 ) pro cesses. While this in tro duces a lev el of am biguit y in in terpretations b et w een the n ucleon-lev el CC + pro cess and nal-state in teractions, it is an exp erimen tally clean signature and ma y b e used to rigorously test the CC + pro cess when nal-state in teractions are b etter understo o d. As w e use this neutrino-mo de CC + result in this w ork, observ able CC + ev en ts are also treated as signal here. A n um b er of single and double-dieren tial MiniBo oNE CC + cross sections are published in kinematics of the and + Ho w ev er, since the presen t study is a simple rate measuremen t, it is sucien t to simply implemen t the total cross section as a function of neutrino energy Figure 7-3 compares the data to the sim ulation exp ectation. 98

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F unctionally the CC + cross section data is implemen ted in to this analysis through correcting the an tineutrino-mo de exp ectation of observ able CC + The ratio data/MC is measured in regions of generated neutrino energy according to the bin delimitations. Note this is only p ossible b ecause the exact set of underlying ph ysics parameters, most imp ortan tly the single-pion axial masses and the nal-state in teraction mo del, are identic al b et w een the neutrino-mo de sim ulation used to calculate the exp ectation sho wn in Figure 7-3 and the MC used in the presen t an tineutrino-mo de analysis. With the observ ed CC + cross-section data implemen ted in this analysis, the rate measuremen t presen ted in this section is also a measuremen t of the rux con tribution to the an tineutrino-mo de b eam. As will b e sho wn, the uncertain t y on this measuremen t is dominated b y error on the CC + cross section. T o con v ert the uncertain t y from the original measuremen t to the binning optimized for this analysis, a p olynomial of order 4 is t to the fractional CC + systematic uncertain t y and the v alues of this function ev aluated in the cen ter for the bins c hosen in this analysis are tak en as the CC + uncertain t y Figure 7-4 sho ws the p olynomial function and the t w o fractional error distributions. The largest con tribution to the uncertain t y on the CC + cross section comes from the neutrino-mo de rux uncertain t y whic h is the only systematic error asso ciated with the cross-section measuremen t that is also indep enden t of the measuremen t made here. Because the other CC + uncertain ties are treated as uncorrelated b et w een the neutrino-mo de and the an tineutrino-mo de data, a partial cancellation of errors is ignored in the presen t rux measuremen t. The Selected Sample As men tioned in the previous section, the main requiremen t to select a CC + sample is the observ ation of 3 sub ev en ts. The full selection set is: 1. Three sub ev en ts 2. First sub ev en t in b eam windo w: 4000 < T(ns) < 7000, where T is the a v erage PMT 99

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(MeV) n E 600 800 1000 1200 1400 1600 1800 2000 2200 2400 Fractional Error 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 order polynomial fit th 4 original binning converted for this analysis Figure 7-4. The CC + fractional uncertain t y con v ersion. hit time 3. All sub ev en ts: reconstructed v ertex < 500 cm from tank cen ter 4. 1st sub ev en t: tank hits > 200 5. 2nd and 3rd sub ev en ts: tank hits < 200 6. All sub ev en ts: v eto hits < 6 7. Distance b et w een calculated end of 1st sub ev en t and nearest deca y electron v ertex < 150 cm Cut 1 requires the ev en t b e consisten t with the pro duction of three leptons ab o v e Cerenk o v threshold, and cut 2 assures it b e asso ciated with the proton b eam spill. Cuts 3 and 4 enhance the reliabilit y of the reconstruction used, and cut 5 requires the nal t w o sub ev en ts b e consisten t with a Mic hel electron, whose energetic endp oin t of 53 MeV leads to roughly 175 tank hits. Cut 6 ensures the leptons b e con tained and that no c harged 100

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T able 7-2. Summary of selection cuts in the CC + sample. Purit y and eciency n um b ers are sequen tial and are calculated for the \observ able CC + ev en t signature: 1 1 + Cut # Description Eciency Purit y (%) (%) 0 No cuts 100 10 1 Three sub ev en ts 30 29 2 1st sub ev en t in ev en t time windo w 28 34 4000 < T(ns) < 7000 3 All sub ev en ts: reconstructed 23 36 v ertex < 500 cm from tank cen ter 4 1st sub ev en t: tank hits > 200 22 39 5 2nd and 3rd sub ev en ts: 19 65 tank hits < 200 6 All sub ev en ts: v eto hits < 6 16 78 7 Distance b et w een reconstructed 12 82 end of 1st sub ev en t and nearest Mic hel electron v ertex < 150 cm particles en tered the detector. Finally cut 7 enforces spatial correlation b et w een one of the Mic hel electrons and the end of the calculated m uon path. As men tioned previously due to the fast deca y of the + timing alone cannot determine the origin of the deca y electrons. Using this selection, T able 7-2 presen ts the detection eciency and purit y for CC + ev en ts, and T able 7-3 summarizes the sample comp osition. Flux Measuremen t Using CC + The purit y of the CC + sample is sucien tly high to p erform a simple bac kground-subtracted rate measuremen t to test the rux. With the notation A for data, B for the exp ected con tributions, C for signal CC + and D for non-CC + ev en ts, w e calculate the rux measuremen t as = A B C + D : (7.3) The assigned uncertain ties on these quan tities are as follo ws: A: statistical uncertain t y on the data. F ollo wing gaussian statistics, the uncertain t y is 101

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T able 7-3. Summary of the CC + sample in an tineutrino mo de, including the n ucleon-lev el comp osition. in tegrated POT 5.66 10 20 CC + candidate ev en ts 3268 observ able CC + eciency ( R < 550 cm) 12.0% In teraction Con tribution (%) N + N (resonan t) 64 A + A (coheren t) 7 N + N 0 (resonan t) 6 n p 6 n 0 p 2 p + 0 n 1 Other (mostly DIS) 14 \Observ able CC + 82 (1 1 + ) tak en as p N where N is the n um b er of observ ed ev en ts. B: bac kground. This accoun ts for 14% of the sample, mostly deep inelastic scattering (DIS) and CC ev en ts in whic h the deca y ed in righ t. An o v erall uncertain t y of 30% is assigned. C: signal observ able CC + P er Figure 7-3 fractional uncertain t y on this pro cess v aries with neutrino energy and is at a minim um of 10% around 800 MeV. D: non-signal ev en ts. This accoun ts for 6% of the sample, and is dominated b y DIS. An o v erall uncertain t y of 30% is assigned. These fractional uncertain ties are propagated on to the rux measuremen t with a simple quadrature sum of the uncorrelated uncertain ties due to the pro cesses A; B ; C and D : 102

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= s A A 2 + B B 2 + C C 2 + D D 2 (7.4) = 1 C + D s ( A ) 2 + ( B ) 2 + A B C + D 2 [( C ) 2 + ( D ) 2 ] (7.5) T o test the accuracy of the sim ulated rux sp ectrum, the rux measuremen t is p erformed in exclusiv e regions of reconstructed energy E where E follo ws the deriv ation of Eq. 6.7 appropriate to CC + ev en ts: E = m 2 m 2p m 2 2 m p E 2 ( m p E + j ~ p j cos ) : (7.6) This reconstruction assumes a quasi-elastic in teraction + N + N for all ev en ts. While this is a mo del-dep enden t v aluation of the neutrino energy separating the sample in to exclusiv e regions of E nev ertheless aords statistical sensitivit y to the accuracy of the sim ulated rux sp ectrum. The rux measuremen t in the an tineutrino-mo de b eam using CC + ev en ts is summarized in T able 7-4 7.1.4 Flux Measuremen t Using Nuclear Capture Another opp ortunit y to measure the rux using n uclear capture is a v ailable through the An y CC ev en t from and will pro duce a m uon, the 's of whic h will pro duce few er Mic hel electrons due to 8% n uclear capture on carb on [ 148 ]. An adv an tage of this analysis o v er the determination of the rux in an tineutrino mo de using the CC + sample is the natural sensitivit y to lo w er energies. The dominan t mec hanism for CC + pro duction in v olv es the (1232) resonance, and the examination of the rux using these in teractions tests energies greater than 900 MeV. F ortunately the presen t 103

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T able 7-4. An tineutrino-mo de CC1 + sample details and rux comp onen t measuremen t. \Mean Gen. E is the a v erage generated neutrino energy in eac h reconstructed energy bin. E Range Mean Gen. Ev en ts Exp ected Flux (MeV) E (MeV) in Data Scale 600 700 961 465 556 104 0.65 0.10 700 800 1072 643 666 118 0.79 0.10 800 900 1181 573 586 97 0.81 0.10 900 1000 1285 495 474 78 0.88 0.11 1000 1200 1426 571 646 92 0.74 0.10 1200 2400 1685 521 614 74 0.73 0.15 Inclusiv e 1266 3268 3542 563 0.76 0.11 analysis can reac h further do wn in neutrino energy to directly test the rux sp ectrum of the 's that are bac kground to the main analysis of CCQE in teractions. A complication of this measuremen t is the substan tial comp onen t of CCQE ev en ts presen t in the analysis samples, and so it is critical to ev aluate the bias caused b y the assumptions used to predict their con tribution. If this bias w ere signican t and the measuremen t w ere used to subtract the bac kground from the data, the nal CCQE cross section w ould ha v e an appreciable dep endence on the CCQE in teraction mo del. It will b e sho wn that this is the case for the angular analysis of CCQE ev en ts presen ted in Section 7.1.5 and so its results are ignored in subtracting the bac kground from the CCQE sample. Mean while, it will b e sho wn that the bias caused b y assumptions on the CCQE cross section in the presen t capture analysis is small and negligible compared to other uncertain ties. Implemen tation of the CC Cross Sections The MiniBo oNE CCQE analysis found the shap e of the kinematics in data to b e describ ed w ell b y the RF G assuming a few empirical parameter adjustmen ts: M A = 1 : 35 0 : 17 GeV and = 1 : 007 0 : 012 [ 84 ]. While the observ ed normalization is also consisten t with this mo del within uncertain ties, the data lies 8% high. Therefore, to implemen t the CCQE cross section in to the presen t rux measuremen t, the RF G 104

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(GeV) n E 0.5 1 1.5 2 2.5 ) 2 (cm s 2 4 6 8 10 12 14 -39 10 MB data = 1.007 k = 1.35 GeV, A MC: M same MC weighted to data Figure 7-5. Summary of the normalization correction to the CCQE cross section in the an tineutrino-mo de sim ulation. The ratio of the data to the red MC histogram is applied binb y-bin to the sim ulated CCQE ev en ts. Also visible is the eect of lo w-precision sampling with the nuance generator at high energies. This is discussed further in Section 8.2 mo del with M A = 1 : 35 GeV and = 1 : 007 is assumed b y sim ulation and the mild normalization discrepancy is accoun ted for b y rew eigh ting ev en ts. The rew eigh ting v alues are found b y a generator-lev el comparison b et w een the observ ed MiniBo oNE CCQE total cross section and the RF G with the previously-men tioned parameter adjustmen ts. Figure 7-5 compares the unfolded MC distributions b efore and after the correction. The CC + in teractions also con tribute signican tly to the selected capture samples. Their in teraction rate and kinematics are implemen ted through the Q 2QE -based measuremen t in the neutrino-mo de data [ 84 ]. This function and its origin are explained further in Section 7.2 It will b e sho wn that the CCQE and CC + in teractions represen t more than 94% of the c hannels con tributing to the capture samples. With b oth the kinematics and the normalization of these in teractions implemen ted in to the an tineutrino-mo de MC, the rate measuremen t of the con tribution to the capture analysis samples ma y b e cleanly 105

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in terpreted as measuremen t of the rux comp onen t of the an tineutrino-mo de b eam. Muon Capture Mo del and Ev en t Selection In mineral oil, stopp ed are captured on carb on n uclei with a probabilit y of 7.78 0.07% [ 148 ]. F or suc h capture ev en ts in MiniBo oNE, t ypically little or no extra activit y is observ ed in the detector. Ho w ev er, the lo w-energy neutron and photons from the primary capture reaction as w ell as de-excitations of the b oron isotop e ma y b e energetic enough to pro duce a Mic hel-lik e ev en t. The sim ulated pro duction of these particles is based on the measuremen ts of Refs. [ 149 150 151 152 153 154 ], and the mo del that propagates these particles and p ossible re-in teractions through the MiniBo oNE detector estimates 6.60% of capture ev en ts lead to activit y similar to a lo w-energy Mic hel. Th us, the apparen t n uclear capture probabilit y in the detector is predicted to b e 7.78 (1 6.60%) = 7.26 0.20%, where the uncertain t y is substan tially increased to recognize the mo del dep endence of the rate to regain Mic hel-lik e ev en ts follo wing capture. This rate is partially constrained b y the calibration pro cedure describ ed in Section 7.1.4 and it will b e sho wn that the assigned uncertain t y on eectiv e n uclear capture has a negligible impact on the nal measuremen ts. Sensitivit y to the con ten t of the data is obtained b y sim ultaneously analyzing t w o samples: those with only a m uon candidate ev en t, and ev en ts consisten t with the observ ation of a m uon and its deca y electron. Therefore, this analysis tak es as signal and CC ev en ts. Apart from the requiremen t of either one or t w o sub ev en ts, the ev en t selection for this analysis closely follo ws that describ ed in Section 6.2 with a few c hanges appropriate to dieren t bac kgrounds and a higher sensitivit y to Mic hel detection eciency T able 7-5 details the and c harged-curren t purit y of the t w o samples after eac h cut. The primary samples of this analysis are separated b y cut 1, where CC ev en ts ha v e an enhanced con tribution in the single sub ev en t sample due to capture. Cuts 2-5 are common to the analysis presen ted in the main b o dy of this w ork and are motiv ated in Section 6.2 Cuts 6 and 8 reduce the NC bac kground in the single-sub ev en t sample. 106

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Figure 7-6 sho ws NC single ev en ts are largely rejected b y the requiremen t on the =e log-lik eliho o d v ariable, while Figure 7-7 sho ws cut 8 further reduces their con tribution. /e) m log L( -0.15 -0.1 -0.05 0 0.05 0.1 0.15 Events 0 100 200 300 400 500 600 700 800 900 Data Total MC Charged current p Neutralcurrent 1 Other neutral current Figure 7-6. The log-lik eliho o d =e particle-ID v ariable in the single-sub ev en t sample. All other selection requiremen ts ha v e b een applied. The sim ulation is normalized to rux, and errors sho wn on data are statistical only Cut 7 eliminates ev en ts in whic h the Mic hel electron is pro duced near the optical barrier, where mo deling of the electron detection eciency ma y b e less reliable. In this region man y more Mic hels are lost due to the minim um requiremen t of 10 PMT tank hits to form a sub ev en t, while some are missed due to Mic hels en tering the v eto region. T o explore these eects, w e b egin with a prediction of where the Mic hel ough t to b e pro duced, assuming it is the deca y pro duct of the prompt m uon. Calculating the stopping radius of the m uon based on its observ ed v ertex, direction and energy w e nd: stopping radius = q ( V x + dE =dX 1 T U x ) 2 + [ same for y and z directions ] ; (7.7) where V x and U x are the reconstructed m uon v ertex and direction in the x-direction resp ectiv ely and T is the m uon kinetic energy dE =dx is the a v erage energy dep osited p er 107

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unit of distance tra v eled for m uons in mineral oil at MiniBo oNE energies. Fitting dE/dX data for stopping p o w er of mineral oil to a linear function nds dE =dx = 1.9 MeV/cm. Figure 7-8 sho ws adequate agreemen t b et w een data and sim ulation at high radius in the 2SE sample, where a Mic hel is b oth pro duced and detected, while the agreemen t is w orse in the single-sub ev en t sample. Regardless, the ratio of single/t w o sub ev en t ev en ts as a function of the m uon endp oin t presen ted in the same gure sho ws this ratio clearly increases with radius at large v alues and so is quite sensitiv e to the details of Mic hel detection near the optical barrier. Aw a y from this barrier, where Mic hel detection is not a function of p osition, most Mic hel electrons not detected are missed due to the timing cut used to separate sub ev en ts and not the requiremen t of at least 10 tank PMT hits within the temp oral windo w. Figure 7-9 sho ws the dierence in the timing distributions for the 2SE sample, while Figure 7-10 presen ts the tank hit distribution for the second sub ev en t. Less than 0.5% of ev en ts are rejected b y the 10 PMT hit requiremen t, while 8% of Mic hels are pro duced to o close in time with the m uon to b e temp orally separated. Cut 7 also enhances purit y due to kinematic dierences b et w een and CCQE, where the more forw ard-going nature of the + from in teractions preferen tially stop at high radius in the do wnstream region of the detector. With the full selection, n ucleon-lev el in teraction con tributions to the sub ev en t samples are giv en in T able 7-6 108

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) 2 (GeV 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 100 200 300 400 500 600 700 1 subevent Total MC All CCQE WS CCQE NCE p NC p CC other data 1 subevent ) 2 (GeV 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 500 1000 1500 2000 2500 3000 2 subevents Total MC All CCQE WS CCQE NCE p NC p CC other data 2 subevents Figure 7-7. Q 2QE for the single-sub ev en t (top) and t w o-sub ev en t (b ottom) samples. All other selection cuts ha v e b een applied. Ev en ts with Q 2 > 0.2 GeV 2 are selected to reject some NC ev en ts, particularly in the single-sub ev en t sample. Distributions are normalized to rux. 109

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stopping radius (cm) m Predicted 0 100 200 300 400 500 600 0 50 100 150 200 250 1 subevent Total MC data 1 subevent stopping radius (cm) m Predicted 0 100 200 300 400 500 600 0 200 400 600 800 1000 2 subevents 2 subevents endpoint (cm) m Predicted 0 100 200 300 400 500 600 1SE/2SE 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 7-8. Predicted m uon endp oin t radius for the single-sub ev en t (top left), t w o-sub ev en t (top righ t) samples and the ratio single/t w o sub ev en ts (b ottom). All other selection cuts ha v e b een applied. Data-MC discrepancy is only presen t at high radius, presumably due to diculties in mo deling Mic hel detection close to the optical barrier. The p eak ab o v e 500 cm in the single-sub ev en t and the ratio distribution is due to the lo w er Mic hel detection eciency in this region. There are zero ev en ts in the rst bin of the single-sub ev en t data. Distributions are relativ ely normalized to data. 110

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Michel avg. time Muon avg. time (ns) 0 2000 4000 6000 8000 10000 12000 14000 0 200 400 600 800 1000 1200 1400 Total MC data Figure 7-9. Dierence b et w een a v erage hit times for the Mic hel-lik e sub ev en t and the m uonlik e sub ev en t in an tineutrino mo de. The distributions deviate from an exp onen tial form at lo w timing dierences due to the temp oral denition of a sub ev en t. Sim ulation is relativ ely normalized to data. Michel tank hits 0 50 100 150 200 250 0 200 400 600 800 1000 1200 Total MC data Figure 7-10. T ank hit distributions for the second sub ev en t in an tineutrino mo de. Sim ulation is normalized to data. 111

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T able 7-5. An tineutrino-mo de purit y in % for all and c harged-curren t ev en ts in the oneand t w o-sub ev en t samples. A pre-cut of generated radius < 550 cm is applied. Cut # Description One sub ev en t Tw o sub ev en ts CC CC CC CC 1 Sub ev en t cut 18 33 26 57 2 V eto hits < 6 for all sub ev en ts 9 11 30 65 3 First sub ev en t in b eam windo w: 4000 < T (ns) < 7000 9 11 29 65 4 Reconstructed v ertex radius < 500 cm for rst sub ev en t 8 11 29 65 5 Kinetic energy > 200 MeV for the rst sub ev en t 20 27 29 68 6 =e log-lik eliho o d ratio > 0 : 02 for rst sub ev en t 36 54 27 72 7 Predicted stopping radius < 500 cm 39 46 28 71 8 Q2QE> 0.2 GeV257 36 43 56 112

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T able 7-6. Summary of predicted n ucleon-lev el in teractions in the an tineutrino-mo de sub ev en t samples. The small con tribution from neutral curren t pro cesses are presen ted as the sum of the and in teractions. Pro cess Con tribution (%) to One sub ev en t Tw o sub ev en ts + p + + n 31 49 + n + p 48 36 + N + + N + 3 5 + N + N + + 7 7 ( ) + N ( ) + N 1 0 ( ) + N ( ) + N + 0 3 0 ( ) + N ( ) + N + 4 0 other 3 3 All 58 43 All 42 57 Calibrations and Stabilit y Chec ks Using the Neutrino-Mo de Data The success of this analysis is dep enden t up on b eing able to in terpret dierences b et w een the oneand t w o-sub ev en t an tineutrino-mo de data and MC samples as b eing due to the amoun t of in the b eam. In principle, an y dierence disco v ered b et w een data and the sim ulation is am biguous b et w een the con ten t and inadequate mo deling of the total migration rate b et w een the sub ev en t samples. F ortunately the neutrino-mo de data oers an opp ortunit y to calibrate the migration rate b et w een the sub ev en t samples for CC ev en ts. Due to the con v olution of rux and cross-section eects discussed in Section 7.1.1 the neutrino-mo de sub ev en t samples are dominan tly due to CC in teractions. T able 7-7 sho ws the predicted neutrino sp ecies and in teraction t yp e con tributions to the neutrino-mo de sub ev en t samples with the same selection describ ed in the previous section. With a high-purit y CC sample, the accuracy of Mic hel detection and eectiv e capture in sim ulation can b e tested. F or CC ev en ts without nal-state pions (\ CC "), the n um b er of ev en ts in the neutrino-mo de one-sub ev en t (\1SE ") and t w o-sub ev en t (\2SE ") samples are giv en b y: 113

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T able 7-7. A brief description of the neutrino-mo de sub ev en t samples for the same selection describ ed in the previous section. Pro cess Con tribution (%) to One sub ev en t Tw o sub ev en ts All CC 95.4 99.0 All 0.4 0.7 All NC 4.3 0.3 1SE = CC ( + (1 )) + N 1 (7.8) 2SE = CC (1 (1 )) + N 2 (7.9) where N 1 (N 2 ) is the NC con tribution to the 1SE (2SE) sample, is the Mic hel detection ineciency and is the eectiv e capture rate describ ed previously The rate for Mic hel non-detection can b e solv ed in terms of the eectiv e capture rate and the small NC con tribution: = 1SE N 1 1SE +2SE (N 1 +N 2 ) 1 (7.10) Noting the symmetry in Equations 7.8 and 7.9 b et w een and Equation 7.10 can also express the eectiv e capture rate in terms of Mic hel detection with $T able 7-8 giv es v alues of and from sim ulation and data based on the observ ed or predicted ev en t rates in the 1SE and 2SE samples. As the c harged-curren t migration rate to the single-sub ev en t sample is due to a con v olution of Mic hel detection and eectiv e capture, the pro cesses cannot b e sim ultaneously calibrated with the neutrino-mo de data that is, for example, the calibration of assumes the MC v aluation of is correct. F uture exp erimen ts ma y b e able 114 PAGE 115 T able 7-8. Calibration summary for Mic hel detection ineciency ( ) and the rate of eectiv e n uclear capture ( ). Note that b oth pro cesses cannot b e sim ultaneously constrained. Pro cess data MC data/MC 0.073 0.074 0.98 0.071 0.073 0.98 Michel Tank Hits 0 50 100 150 200 250 0 500 1000 1500 2000 2500 3000 3500 4000 4500 mode n Total MC data mode n Figure 7-11. T ank hit distributions for the second sub ev en t in neutrino mo de. Ev en ts from capture are exp ected to only con tribute in the lo w est bins, and the consistency b et w een data and sim ulation giv es condence that the capture mo del is adequate. MC is relativ ely normalized to data. to break this degeneracy b y examining the lo w-energy region of the Mic hel sp ectrum across b oth neutrino and an tineutrino mo des, where the con tribution from activit y follo wing capture is enhanced. In the case of MiniBo oNE, the Mic hel sp ectrum in an tineutrino-mo de is giv en in Figure 7-10 and the neutrino-mo de analogue is sho wn in Figure 7-11 While consistency in the lo w-energy region b et w een data and sim ulation indicate the capture mo del is not grossly wrong, the statistics of the an tineutrino sample prev en t a rigorous test of the Mic hel-lik e con tributions follo wing capture. As the calibration results sho wn in T able 7-8 are quite mild and within systematic 115 PAGE 116 uncertain ties, this pro cedure giv es condence in the abilit y to unam biguously measure the con ten t of the an tineutrino-mo de data using capture. The substan tially higher ev en t rate in neutrino-mo de compared to an tineutrino-mo de also oers the opp ortunit y for a robust stabilit y c hec k of the Mic hel detection eciency If there w ere some v ariation or degradation of the electronics during neutrino-mo de running that w ould aect the detection of Mic hels, it should app ear as dierences in some suitable v ariables b et w een temp oral bins. The neutrino-mo de data is separated in to four c hronologically sequen tial groups of data with roughly equal POT con tributions. The rst v ariable to lo ok at is the ratio 1SE/2SE. T able 7-9 oers ev en t coun ts in the 1 and 2SE samples and their ratio with statistical error for the four data groups. Within statistical uncertain t y the sub ev en t ratios are consisten t and w e nd no evidence of systematic v ariations aecting Mic hel detection. T able 7-9. 1SE and 2SE ev en t details in four sequen tial and roughly equally sized neutrinomo de data groups. The 1SE/2SE ratios are consisten t within one standard deviation. Run n um b ers 1SE ev en ts 2SE ev en ts 1SE/2SE 3539 7999 3658 21318 0.172 0.003 8000 10999 4413 26380 0.167 0.003 11000 11999 2355 13933 0.169 0.003 12000 12842, 3112 18576 0.168 0.003 15833 17160 A nal c hec k on Mic hel detection stabilit y can b e made b y lo oking at the v ery early timing distribution of the 2SE sample. Figure 7-12 presen ts a 0 800 ns windo w of the a v erage time separating the t w o sub ev en ts for the four sets of neutrino-mo de data. No evidence of a time-dep enden t shift b et w een the data runs is observ ed. W e conclude that in the sample most sensitiv e to an y pathological ev olution of Mic hel detection in time, none are observ ed. The statistics of the single-sub ev en t sample in an tineutrino mo de prohibit the execution of the same tests using the primary analysis samples. 116 PAGE 117 Michel avg. time Muon avg. time (ns) 0 100 200 300 400 500 600 700 800 scaled to unit area 0 0.01 0.02 0.03 0.04 0.05 MC Runs 3539 7999 Runs 8000 10999 Runs 11000 11999 Runs 12000 12842, 15833 17160 Figure 7-12. Early separation b et w een the t w o sub ev en ts in neutrino mo de. No signican t shap e dierence is observ ed b et w een the four c hronologically sequen tial groups of data. Distributions are scaled to unit area. Flux Measuremen t Using Capture The rux in the an tineutrino-mo de b eam is measured b y adjusting the MC prediction of the and con ten t to matc h the data in regions of reconstructed energy for the sub ev en t samples. F ollo wing the con v en tions of Eqs. 7.8 and 7.9 and in tro ducing CC for the c harged-curren t con ten t, the predicted and con tributions to the sub ev en t samples in an tineutrino mo de are dened as 1SE MC = CC ( + (1 )) (7.11) 2SE MC = CC (1 (1 )) (7.12) 1SE MC = CC (7.13) 2SE MC = CC (1 ) (7.14) 117 PAGE 118 Then the single(\1SE ") and t w o-sub ev en t (\2SE ") data samples in an tineutrino mo de are giv en b y 1SE = 1SE MC + 1SE MC + N 1 (7.15) 2SE = 2SE MC + 2SE MC + N 2 (7.16) where and are scale factors for the and c harged-curren t con ten t, resp ectiv ely to b e measured in this analysis. The NC con ten t (N 2 and N 1 ) include con tributions from b oth and Equations 7.15 and 7.16 can b e solv ed for and : = (1SE N 1 ) 2SE MC (2SE N 2 ) 1SE MC 2SE MC 1SE MC 1SE MC 2SE MC (7.17) = (1SE N 1 ) 2SE MC (2SE N 2 ) 1SE MC 2SE MC 1SE MC 1SE MC 2SE MC (7.18) T o c hec k the mo deling of the rux sp ectrum, this measuremen t is p erformed in three regions of reconstructed energy E QE (Eq. 6.7 ): ab o v e and b elo w 900 MeV, and an inclusiv e energy sample. As describ ed in the previous section, the calibration from the neutrino-mo de data is am biguous b et w een Mic hel detection and the eectiv e capture mo del. As these eects c hange the exp ectations for 1SE MC ; 2SE MC ; 1SE MC and 2SE MC in dieren t w a ys, the measuremen t of and is, in principle, sensitiv e to whic h rate is calibrated. In the absence of a comp elling reason to c ho ose one o v er the other, the nal ev aluations for and are tak en to b e the a v erage of the t w o calculations assuming eac h rate is calibrated. A calibration uncertain t y spanning the dierence in the t w o measuremen ts is added to the systematic errors discussed next. The cen tral v alues for and are presen ted in T able 7-10 118 PAGE 119 T able 7-10. Results for scale factors relativ e to the exp ectation for the and c hargedcurren t con ten t of the an tineutrino-mo de data. P arameter Calibrated E QE range (GeV) pro cess < 0.9 0.9 All 0.78 0.79 0.78 0.78 0.79 0.78 Av erage 0.78 0.79 0.78 1.16 1.15 1.16 1.16 1.15 1.16 Av erage 1.16 1.15 1.16 Systematic Errors Systematic uncertain ties on and are ev aluated b y assigning relev an t errors to the ph ysics pro cesses con tributing to the sub ev en t samples and observing ho w the measuremen t c hanges as the c hannels are v aried within their uncertain t y These uncertain ties are treated as uncorrelated, so the uncertain t y on for example, due to ph ysics pro cesses P 1 ; ; P N is simply 2 = N X i =1 @ @ P i P i 2 (7.19) T able 7-11 sho ws the errors assigned to the v arious con tributing pro cesses and their propagated uncertain t y on to and The most imp ortan t pro cess for extracting the rux measuremen t is the CCQE in teraction, and its cross section and assigned uncertain t y rerect the measuremen t and accuracy of the MiniBo oNE result [ 84 ]. The same is true for the and neutral-curren t single 0 c hannels [ 155 ]; ho w ev er the error is increased to recognize a p ossible rate dierence in these in teractions b et w een the cross-section measuremen ts and this analysis due to using the opp osite side of the log-lik eliho o d v ariable sho wn in Figure 7-6 The and c harged-curren t single c harged c hannels are adjusted to rerect the measuremen t [ 84 ] and their uncertain t y is increased 119 PAGE 120 to recognize the extrap olation to the pro cesses. T reating the uncertain ties on the pro cesses constrained b y MiniBo oNE data as uncorrelated ignores a common dep endence on the neutrino-mo de rux uncertain ties, and a small cancellation of errors that could b e propagated on to and is ignored. The neutral-curren t elastic pro cess is also constrained b y MiniBo oNE data [ 156 ], while the neutral-curren t c harged-pion pro duction pro cesses are completely unconstrained and so the assigned uncertain t y is large. Preliminary results for the CCQE pro cess [ 157 ] informs the c hoice of a 20% uncertain t y relativ e to the RF G mo del with M A = 1.35 GeV. With these systematic uncertain t y assumptions, as seen in T able 7-11 the uncertain t y on the main result of this w ork is dominated b y statistics and the CCQE cross section. As the CCQE pro cess is directly constrained b y MiniBo oNE data, the measuremen t of the rux scale features negligible mo del dep endence. T able 7-12 summarizes the measuremen ts of and As the cross sections for the dominan t pro cesses ha v e b een applied to sim ulation, the deviation from unit y for represen ts the accuracy of the rux prediction in an tineutrino mo de. As the bulk of the rux prediction is constrained b y the HARP data, the scale factor is represen tativ e of the lev el of cross-section agreemen t b et w een data and the RF G with M A = 1.35 GeV for the CCQE pro cess. 120 PAGE 121 T able 7-11. Uncertain t y summary for the capture-based measuremen t of the rux. Included are the assumed error on ph ysics pro cesses and their con tribution to the total error in and in the regions of reconstructed neutrino energy studied. The regions of EQE are giv en in GeV. The statistics of the -mo de data en ter the uncertain t y from the calibration pro cedure describ ed in Section 7.1.4 Note the 20% uncertain t y on the CCQE pro cess con tributes an uncertain t y 0.04 or less to the measuremen t of This assures the rux measuremen t is not biased b y assumptions on CCQE. Uncertain t y F rac. Uncertain t y con tribution to Uncertain t y con tribution to source error (%) EQE < 0.9 EQE 0.9 All EQE < 0.9 EQE 0.9 All + n + p 10 0.07 0.08 0.07 0.00 0.00 0.00 + p ++ n 20 0.04 0.02 0.03 0.20 0.20 0.21 ( ) + N ( +) + N + +( ) 20 0.04 0.05 0.04 0.02 0.02 0.01 ( ) + N ( ) + N 30 0.00 0.00 0.00 0.00 0.00 0.00 ( ) + N ( ) + N + 025 0.02 0.01 0.01 0.01 0.01 0.01 ( ) + N ( ) + N + 50 0.05 0.02 0.01 0.03 0.03 0.01 capture 2.8 0.00 0.00 0.00 0.00 0.00 0.00 -mo de statistics 0.10 0.11 0.08 0.08 0.08 0.06 -mo de statistics 0.04 0.05 0.04 0.03 0.03 0.03 All 0.14 0.16 0.12 0.22 0.22 0.22 121 PAGE 122 m q cos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1 Events 10 2 10 3 10 4 10 MC n MC n = 1) n a = 1, n a ( MC T data Composition n Predicted : 29 % m n : 71 % m n Figure 7-13. The cos distribution of the CCQE sample b y neutrino t yp e b efore tting. The nominal MC prediction is normalized to rux, and notations used in the legend are used in the execution of the t. T able 7-12. Summary of measuremen ts for the rux scale and the rate scale P arameter E QE range (GeV) < 0.9 0.9 All 0.78 0.14 0.79 0.16 0.78 0.12 1.16 0.22 1.15 0.22 1.16 0.22 7.1.5 Flux Measuremen t Using the cos Distribution Ov erview The nal constrain t on ev en ts comes from the observ ed m uon angular distribution cos where is the m uon scattering direction relativ e to the incoming neutrino b eam. Due to the axial-v ector in terference term (Section 4.2 ), the con tribution from ev en ts to bac kw ard-scattering m uons is predicted to b e hea vily suppressed. Figure 7-13 compares the predicted and con tributions to the m uon scattering angle with data. This large asymmetry oers the opp ortunit y to t a com bination of the and 122 PAGE 123 con ten t to the observ ed data. Ho w ev er, this asymmetry is mo del-dep enden t, as the details of CCQE scattering are not w ell kno wn, and in fact the pro cesses con tributing to the MiniBo oNE CCQE sample ma y b e m uc h more isotropic than suggested b y Figure 7-13 Of course, detailed measuremen ts of CCQE scattering is the main fo cus of this dissertation. Therefore, an y dep endence of the bac kground estimation on assumptions of CCQE m ust b e strictly a v oided. F or this reason, the results of this analysis are not used to subtract the bac kground to the CCQE sample. Ho w ev er, in the future, when the pro cesses con tributing to samples lik e these are b etter understo o d, this tec hnique could pro v e to b e p o w erful. As men tioned in Section 7.1.3 the data used in this analysis do es include the small absorb er-do wn an tineutrino-mo de runs. Sample Selection The only dierence b et w een the CCQE sample selected here and that used in the main analysis of CCQE in teractions (detailed in Section 6.2 ) is the replacemen t of the range-based cut with the requiremen t that the reconstructed v ertex of the second sub ev en t b e within a 100 cm radius of the predicted stopping p oin t. The radius b et w een the predicted m uon stopping p oin t and the Mic hel v ertex is hereafter referred to as the \Mic hel distance." T o directly see the dierence b et w een these cuts, the slop e of the range cut v ersus the Mic hel distance can b e examined. The slop e of the range cut is Rang e +100 cm T (cm/GeV), and 500 cm/GeV is the cut used in the neutrino CCQE analysis. The 500 cm/GeV cut is simply the in v erse of the standard MIP energy loss 2 MeV/cm. The range cut slop e v ersus Mic hel distance for is plotted for data, MC, and the signican t in teraction c hannels in Figure 7-14 Note the eect of the 100 cm range cut (cut 8 in Section 6.2 ) is not included in this comparison. It can b e seen that the range-based cut k eeps more signal ev en ts while rejecting around the same amoun t of bac kground. Ho w ev er, either c hoice of spatial correlation requiremen t b et w een the m uon and its deca y electron result in mostly the same 123 PAGE 124 purit y and eciency for and ev en ts. Measuremen t Execution T o measure the neutrino con ten t using the m uon angular distribution, the MC sample is separated in to t w o cos templates, one arising from all in teractions and the other from regardless of in teraction c hannel. A linear com bination of these t w o templates is then formed, T M C ( ; ) M C + M C (7.20) where T M C is the total predicted cos distribution to b e t to data, and are neutrino and an tineutrino rate scales, and M C and M C are the MC neutrino and an tineutrino scattering angular predictions, resp ectiv ely Man y bac kgrounds to the CCQE sample p eak in the most forw ard scattering region of the cos distribution. This includes pion pro duction and h ydrogen CCQE scattering while the latter is tec hnically not a bac kground, the prop er handling of the dierence in n uclear eects b et w een b ound and free targets is not straigh tforw ard. Additionally the forw ard scattering region is dominated b y in teractions, while the presen t analysis is principally in terested in -dominated bac kw ards scattering region. F or these reasons, ev en ts with cos > 0 : 91 are not included in the t to data, where is the outgoing m uon angle relativ e to the incoming neutrino b eam. Ignoring this forw ard-scattering region, the mo died sim ulation sample in Eq. 7.20 is compared to data b y forming a go o dness-of-t 2 test as a function of the rate scales: 2 ( ; ) = X i;j ( T M C ( ; ) i d i ) M 1 ij; FIT ( T M C ( ; ) j d j ) : (7.21) where i and j lab el bins of cos d is data and M FIT is the co v ariance matrix describ ed in the next section. 124 PAGE 125 1 10 2 10 3 10 Michel distance (cm) 0 50 100 150 200 250 300 350 400 450 500 (cm/GeV)m[Range + 100 cm]/T 0 200 400 600 800 1000 1200 CCQE Range cut Michel distance cut 1 10 2 10 Michel distance (cm) 0 50 100 150 200 250 300 350 400 450 500 (cm/GeV)m[Range + 100 cm]/T 0 200 400 600 800 1000 1200 p CC m n Range cut Michel distance cut 1 10 2 10 Michel distance (cm) 0 50 100 150 200 250 300 350 400 450 500 (cm/GeV)m[Range + 100 cm]/T 0 200 400 600 800 1000 1200 + p CC m n Range cut Michel distance cut -1 10 1 10 Michel distance (cm) 0 50 100 150 200 250 300 350 400 450 500 (cm/GeV)m[Range + 100 cm]/T 0 200 400 600 800 1000 1200 CCpi0 Range cut Michel distance cut -1 10 1 10 Michel distance (cm) 0 50 100 150 200 250 300 350 400 450 500 (cm/GeV)m[Range + 100 cm]/T 0 200 400 600 800 1000 1200 other Range cut Michel distance cut 1 10 2 10 3 10 Michel distance (cm) 0 50 100 150 200 250 300 350 400 450 500 (cm/GeV)m[Range + 100 cm]/T 0 200 400 600 800 1000 1200 All MC Range cut Michel distance cut 1 10 2 10 3 10 Michel distance (cm) 0 50 100 150 200 250 300 350 400 450 500 (cm/GeV)m[Range + 100 cm]/T 0 200 400 600 800 1000 1200 data Range cut Michel distance cut Figure 7-14. Mic hel distance v ersus the range slop e as describ ed in the text. Distributions are absolutely normalized. 125 PAGE 126 The linearit y of this t allo ws for an analytic solution. The scales and describ e the data b est when the 2 function in Eq. 7.21 is minimized sim ultaneously with resp ect to b oth parameters: @ 2 @ = @ 2 @ = 0 : (7.22) By exploiting the symmetric nature of the error matrix, w e can simplify and arriv e at a unique solution for the t parameters and in terms of the data d and MC distributions of (\ M C ") and (\ M C "): 264 BF BF 375 = 26664 X i;j M C i M C j M 1 ij; FIT X i;j M C i M C j M 1 ij; FIT X i;j M C i M C j M 1 ij; FIT X i;j M C i M C j M 1 ij; FIT 37775 1 26664 X i;j M C i d j M 1 ij; FIT X i;j M C i d j M 1 ij; FIT 37775 ; (7.23) where BF and BF are the b est-t scales for the neutrino and an tineutrino distributions, resp ectiv ely The uncertain t y on and is determined b y the pro jections of the 2 function (Eq. 7.21 ) for eac h parameter while holding the other xed at its b est-t v alue. The uncertain t y on the parameters is: = BF 2 ( BF ; BF ) 2 (7.24) = BF 2 ( BF ; BF ) 2 ; (7.25) where 2 for the 68% C.L. in a t w o-parameter t is 2.30 [ 158 ]. Note the uncertain ties symmetric in the t parameters assumed b y Eqs. 7.24 and 7.25 are not general, and is the case here due to the linearit y of the t. 126 PAGE 127 The Co v ariance Matrix The co v ariance matrix is used to propagate correlated uncertain ties on parameters and pro cesses to the quan tities rep orted in the analysis while accoun ting for correlations b et w een and ev en ts. It is made b y rst forming w eigh ts corresp onding to sim ulation excursions set b y Gaussian v ariations of parameters within their asso ciated error. The dierence of these w eigh ted ev en ts from the sim ulated cen tral v alue forms the error matrix. Correlations b et w een and are not considered in the generation of these excursions, and so m ust b e explicitly addressed in this analysis. The cos correlations b et w een and are treated b y rst expanding the distributions input to the co v ariance matrix to include b oth and cos templates, side-b y-side. An example of the cen tral v alue distribution and 100 instances of cross section uncertain ties related to the v arious con tributing in teractions is sho wn in Figure 7-15 0 1000 2000 3000 4000 5000 6000 Xsec throws CV [-1 RS 1][-1 W S 1] m q cos m q cos Figure 7-15. Cen tral-v alue (CV) cos prediction v ersus 100 distributions created b y cross section thro ws for righ t-sign (RS) and wrong-sign (WS) ev en ts. 127 PAGE 128 Using these distributions, the co v ariance matrix is calculated as: M ij = 1 K K P s =1 ( N s i N C V i ) ( N s j N C V j ) = ij i j ; (7.26) where K sim ulation excursions are used ( K = 100 in this analysis), N s is the re-w eigh ted n um b er of en tries corresp onding to the s th sim ulation set and N C V represen ts the sim ulation cen tral v alue. The total uncertain t y in eac h bin i is i and the correlation b et w een bins i and j is giv en b y ij In this analysis, for uncertain ties on pro cesses with correlated errors, K = 100 while K = 1 is sucien t for uncorrelated errors. This tec hnique is further describ ed in Ref. [ 159 ]. Systematic uncertain ties requiring correlated errors include the pro duction of in the proton b eam target, the connection b et w een pro duction and the fo cused b eam, optical transp ort in the detector, and nal-state in teractions. With the and input distributions separated as in Figure 7-15 this matrix con tains and -only co v ariance information on the blo c k-diagonals, while the o-diagonal pieces con tain the lev el of correlation b et w een and ev en ts. A simple rearranging of Eq. 7.26 giv es the correlation v alues: ij = M ij = ( i j ) ; (7.27) where the individual bin uncertain ties i are trivially found from M ii = 2 i since ii = 1. Figure 7-16 sho ws the lev el of correlation b et w een all bins in the and cos distributions. It will b e sho wn that the o v erall p ositiv e correlation is mostly due to the dominan t uncertain ties related to highly-correlated and cross sections. T o use the co v ariance matrix in the con text of a t, its size m ust rst b e reduced to the dimension of a single cos distribution. Since the total sample T M C in eac h bin i is a simple sum of and ev en ts, 128 PAGE 129 RS WS RS WS 0.4 0.5 0.6 0.7 0.8 0.9 1 ij r Figure 7-16. Correlation matrix for and ev en ts in the cos distribution. F rom left to righ t and b ottom to top, the domain is [-1,1] for the RS and WS distributions. The blo c k-diagonals giv e bin correlation information b et w een the ev en ts (top righ t) and (b ottom left) ev en ts, while the o-diagonals con tain the correlations. All bins are p ositiv ely correlated to eac h other due to the dominan t cross-section uncertain ties, man y of whic h aect the generation of and ev en ts in the same w a y T i;M C (1 ; 1) = M C i + M C i (7.28) Using j for another arbitrary cos bin, the co v ariance for this distribution is: T i;M C T j;M C ( T i;M C T j;M C ) = M C i M C j ( M C i M C j ) + M C i M C j ( M C i M C j ) +2 M C i M C j ( M C i M C j ) (7.29) The terms on the righ t side of Eq. 7.29 can b e recognized as en tries of the full co v ariance matrix in Eq. 7.26 Finally if the dimension of a single cos distribution is N d 129 PAGE 130 T able 7-13. Summary of systematic error con tribution to the scale parameter in the inclusiv e energy t. Individual error con tributions are found for the i th systematic error b y rst rep eating the ts with only indep enden t systematics considered. The fractional error con tributions are then found b y q ( = ) 2sy st i + stat ( = ) 2stat where is the onesigma error rep orted in T able 7-15 The statistical error is found b y considering the second term only This metho d do es not accoun t for small c hanges in the b est t parameter b et w een the ts considering v arious errors, and so the individual fractional errors do not add in quadrature to pro duce the total fractional error rep orted in T able 7-15 and in the nal column. Source of Error F ractional Uncertain t y (%) Statistical 8 Detector Mo deling 11 CC + Constrain t 4 Cross Section 26 T otal F ractional Error 35 the nal error matrix to b e used in this analysis is: M FIT i;j = M i;j + M i + N d ;j + M i;j + N d + M i + N d ;j + N d ; (7.30) where i and j 2 [0 ; N d ]. Results and Systematic Errors As the presen t analysis directly measures the neutrino comp onen t in the an tineutrino-mo de b eam, systematic errors relating to b eam geometry and meson pro duction at the target are not considered. The remaining systematic errors include those arising from detector mo deling, the single pion pro duction bac kground, and the cross section parameters in the underlying CCQE mo del. Con tributions propagated from these errors to the uncertain t y on the parameter in the inclusiv e energy sample are giv en in T able 7-13 Apart from nal-state in teraction uncertain ties leading to errors on the cross section, the error on the CC + bac kground con tributes to the systematic error through the 130 PAGE 131 T able 7-14. Summary of cross-section errors used in this analysis. The b ottom p ortion presen ts fractional uncertain ties assigned to pro cesses in addition to parameter errors. Errors giv en on pion absorption and c harge exc hange are relev an t to pion propagation in the detector medium. P arameter V alue with Error M C A (carb on target) 1.35 0.07 GeV M H A (h ydrogen target) 1.03 0.02 GeV 1.007 0.005 E B 34 9 MeV s 0.0 0.1 M 1 A 1.10 0.28 GeV M m ulti A 1.30 0.52 GeV p F 220 30 MeV Pro cess F ractional Uncertain t y (%) + Charge Exc hange 50 + Absorption 35 CCQE Normalization 10 All Normalization 10 + N N + N 100 error lab eled \CC + Constrain t" in T able 7-13 This measuremen t uncertain t y is based on a Q 2 -dep enden t shap e-only scale factor to impro v e data-sim ulation agreemen t in the neutrino-mo de CC + sample [ 160 ]. The cross section (b oth CCQE and CC + ) uncertain t y is dominan t in these ts and w arran ts further discussion. T able 7-14 oers a breakdo wn of cross section parameters and asso ciated errors. The error on carb on M C A has b een reduced from that rep orted in Ref. [ 84 ] to a v oid double-coun ting MiniBo oNE systematic errors applicable to b oth the measuremen t of M A and the measuremen t rep orted here. The 26% uncertain t y due to cross-section errors rep orted in T able 7-13 can b e expanded as the quadrature sum of 16% from the 10% normalization errors on and CCQE pro cesses, 14% from the error on M A and and 15% from the remaining pro cesses. The t is p erformed analytically in three bins of reconstructed energy and also in an inclusiv e energy sample. Results including statistical and systematic uncertain ties are presen ted in T able 7-15 and the ts to data are sho wn in Figure 7-17 As the main 131 PAGE 132 con tributions to the dominan t cross section systematic error apply to b oth and scattering, and are p ositiv ely correlated as rep orted in T able 7-15 The adjusted con tributions of and to the CCQE sample are compared to the prediction in T able 7-16 The 2 v alue for the angular t in the reconstructed energy range E QE > 900 MeV is un usually lo w at 2 = 7 for 21 degrees of freedom. This is lik ely to b e simply due to c hance, as the statistical error only t agrees with the data exceptionally w ell within the error, returning 2 = 13 for 21 degrees of freedom. As the angular template has b een corrected for the observ ed cross section p er Ref. [ 84 ], ma y b e in terpreted as a rux scale factor, and signican t deviations from unit y w ould imply a rux mismo deling. Consisten t with the results rep orted in the previous sections using CC + and capture ev en ts, ts in the an tineutrino-mo de CCQE sample indicate the true neutrino rux to b e somewhat lo w er than the sim ulation predicts. Ov er all reconstructed energies, the neutrino rux comp onen t of the an tineutrino-mo de b eam should b e scaled b y 0.65 to matc h the observ ed data. Fits in individual reconstructed energy bins indicate that the neutrino rux comp onen t shap e is w ell-mo deled. The rate scale is am biguous in in terpretation, as the cross section is y et unmeasured. The results from this tec hnique dep end on kno wing the angular distributions of neutrino and an tineutrino CCQE in teractions in the detector. While the pro cedure relies on exploiting the eect of the in terference term in the CCQE cross section, the angular distributions ma y b e somewhat altered b y n uclear eects. In this analysis the measured angular distribution of neutrino in teractions on carb on [ 84 ] is emplo y ed, but the measuremen t relies on the scattering mo del to predict an tineutrino in teractions. This mo del do es not include t w o-b o dy curren t eects whic h ma y b e larger than previously exp ected [ 161 93 95 98 99 100 ] and ma y in tro duce additional neutrino and an tineutrino angular dierences. Despite this inheren t mo del dep endence, the results presen t a demonstration of a tec hnique aimed at informing future exp erimen ts lo oking to separately 132 PAGE 133 constrain neutrino and an tineutrino ev en ts in an unmagnetized en vironmen t. By that time, the eect of additional n uclear pro cesses on the angular dep endence of an tineutrino CCQE scattering should b e b etter kno wn. 133 PAGE 134 Events 10 2 10 3 10 0.22 = 0.65 n a 0.18 = 0.98 n a (MeV) < 600 QEn E 10 2 10 3 10 0.20 = 0.61 n a 0.19 = 1.05 n a (MeV) < 900 QEn E £ 600 10 2 10 3 10 0.20 = 0.64 n a 0.21 = 1.18 n a 900 (MeV) QEn E m q cos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1 10 2 10 3 10 4 10 (a) 0.23 = 0.65 n a 0.22 = 1.00 n a QEn Inclusive E MC n n a MC n n a ) n a n a ( MC T Syst. Error Data (data MC) / MC -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 (MeV) < 600 QEn E -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 (MeV) < 900 QEn E £ 600 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 900 (MeV) QEn E m q cos -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 QEn Inclusive E MC = Prediction MC = Best Fit Frac. Syst. Error (b) Figure 7-17. Results of the m uon angular ts to the CCQE data. Sho wn are (a) the ts and (b) fractional dierences (data sim ulation) / sim ulation for b oth the unmo died prediction and the b est t. Along with an inclusiv e sample, three reconstructed energy bins are considered. The b efore-t sim ulation is absolutely normalized to 5.66 10 20 protons on target. As indicated, only ev en ts with cos < 0.91 participate in the t. 134 PAGE 135 T able 7-15. Fit results in three energy bins and an inclusiv e sample. The results are consisten t with an o v er-prediction of the con tamination of the MiniBo oNE an tineutrino-mo de CCQE sample. EQE Range Mean Generated Ev en ts t t t 2(MeV) E(MeV) in Data correlation (DOF = 21) < 600 675 15242 0.65 0.22 0.98 0.18 0.33 13 600 900 897 16598 0.61 0.20 1.05 0.19 0.49 21 > 900 1277 15626 0.64 0.20 1.18 0.21 0.45 7 Inclusiv e 950 47466 0.65 0.23 1.00 0.22 0.25 16 135 PAGE 136 T able 7-16. F ractional comp osition of the an tineutrino-mo de CCQE sample b efore and after angular ts. E QE Range Before Fit (%) After Fit (%) (MeV) < 600 25 75 18 6 82 16 600 900 26 74 17 6 83 15 > 900 35 65 23 7 77 15 Inclusiv e 29 71 21 8 79 18 7.1.6 Summary of Flux Measuremen ts The results from Sections 7.1.3 7.1.5 pro vide the rst demonstration of a set of statistical tec hniques used to measure the comp onen t of an an tineutrino-mo de b eam. Their results are summarized in Figure 7-18 where measuremen ts p erformed in exclusiv e reconstructed neutrino energy bins are giv en as a function of the mean generated neutrino energy for that region. Results from all three measuremen ts indicate the normalization of the nominal rux prediction using a Sanford-W ang-based [ 120 ] extrap olation of the HARP data (discussed in Section 5.2.4 ) requires a uniform reduction of 20-30%. This indicates the sim ulated shap e of the rux sp ectrum app ears to b e adequate. In teresting to note, giv en the results in Figure 7-18 along with the comparison of t w o p ossible extrap olations in to the lo w-angle region sho wn in Figure 5-7 the spline-based prediction app ears to more accurately describ e the data. It is helpful to men tion again that the analysis of the cos distribution is somewhat dep enden t on the mo del for CCQE in teractions assumed b y the sim ulation, and so its results are not used in the bac kground subtraction of ev en ts from the CCQE sample. Con v ersely it has b een sho wn that the analyses based on CC + and capture ev en ts are almost en tirely free from mo del dep endence. Moreo v er, that the analyzed samples are dominated b y dieren t ph ysics pro cesses indicates a lev el of indep endence b et w een the t w o measuremen ts. The results of these t w o analyses can 136 PAGE 137 therefore b e used to nd a com bined measuremen t of the rux in the an tineutrino-mo de b eam featuring a reduced uncertain t y compared to either measuremen t alone. F or t w o measuremen ts x 1 x 2 along with their asso ciated uncertain ties 1 2 and correlation the com bined measuremen t and uncertain t y can b e expressed as [ 162 ]: h x i = x 1 + 1 a 1 2 a + a 2 ( x 2 x 1 ) (7.31) h i 2 = (1 2 ) a 2 2 1 1 2 a + a 2 (7.32) where a = 2 = 1 and 2 1 Consistency of the capture and CC + measuremen ts across the observ ed energy range (Figure 7-18 ) indicates the sim ulated rux shap e to b e w ell-mo deled, and so a com bined measuremen t applied univ ersally to the bac kground ev en ts is adequate. The measuremen t from the capture measuremen t giv es 0.78 0.12, while the CC + measuremen t yields 0 : 76 0 : 11. The uncertain t y in the capture measuremen t is in roughly equal parts due to statistics and the neutrino-mo de rux errors, while the error in the CC + measuremen t is dominated b y the neutrino-mo de rux uncertain t y Based on this, the correlation co ecien t is estimated to b e 0.5. With these v alues implemen ted in to Eqs. 7.31 and 7.32 the com bined measuremen t of the rux in the an tineutrino-mo de b eam is 0.77 0.10 relativ e to the extrap olated and highly-uncertain prediction. This will b e the data-based constrain t of the uncertain t y assumed in the bac kground subtraction pro cess in nding the CCQE cross sections presen ted in Chapter 8 Notice the lev el of kno wledge necessary for bac kground subtraction is ho w man y ev en ts are presen t in the analysis sample, whic h is directly measured through the capture-based measuremen t. Therefore, using the uncertain t y on the rux results in a mild o v erestimate of the uncertain t y of the bac kground. 137 PAGE 138 (GeV) n generated E 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 neutrino flux scale 0 0.2 0.4 0.6 0.8 1 1.2 ALL + p CC1 CCQE capture m Figure 7-18. Summary of the results from three tec hniques used to measure the rux in the an tineutrino-mo de b eam. Measuremen ts p erformed in exclusiv e regions of reconstructed energy are placed here at the mean of their asso ciated distribution of generated energy Sho wn as a dotted line at unit y the measuremen ts are made relativ e to an extrap olation of HARP data in to a region where no relev an t hadropro duction data exists. 138 PAGE 139 7.2 The CC Bac kground While the high rate of stopp edn uclear capture allo ws for the p o w erful measuremen t of the rux using CC + ev en ts presen ted in Section 7.1.3 CC ev en ts migrate in to the CCQE sample and form an irreducible bac kground to the main analysis of this dissertation. F ollo wing the discussion of CCQE selection ecacy giv en in Section 6.2 T able 6-2 rep orts that these CC ev en ts are predicted to accoun t for 15% of the sample. Considering the lev el of agreemen t b et w een v arious calculations and the exp erimen tal data for single-pion in teractions discussed in Section 4.3 constrain ts and uncertain ties for CC ev en ts are based en tirely on direct comparisons with MiniBo oNE data. An indirect constrain t of CC ev en ts is obtained through an extrap olation of a MiniBo oNE CC + -based measuremen t, and a discussion of the origin of this correction is w arran ted. As suggested in Section 7.1.3 single-pion ev en ts induced b y t ypically giv e rise to Mic hel electrons through the deca y c hain + + e + of stopp ed pions, whic h can b e observ ed and used to reject these ev en ts. Ho w ev er, an appreciable n um b er of + are destro y ed in righ t through the n uclear absorption pro cess ( + + X X 0 ) and therefore formed a signican t bac kground to the neutrino-mo de CCQE sample. Measuremen ts of CC + ev en ts tagged through the observ ation of an additional Mic hel allo w ed a direct constrain t of the rate and kinematics of the CC + bac kground to the CCQE analysis. Figure 7-19 sho ws the MiniBo oNE neutrino-mo de CC + data, prediction, and the obtained constrain t. T o guaran tee the selected sample prob es the same kinematics of CC + ev en ts that en ter as bac kgrounds to the CCQE selection, sample formation w as iden tical to that for CCQE describ ed in Section 6.2 with the replacemen t of the t w o sub ev en t requiremen t with three sub ev en ts. Along the same lines, the measuremen t is based on Q 2QE (Eq. 6.8 with n$ p appropriate to CCQE scattering), whic h assumes the underlying in teraction to b e CCQE. Clearly this assumption is incorrect for this ph ysics sample, and so the comparison in Figure 7-19 cannot b e rigorously used to iden tify 139

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Events 0 1000 2000 3000 4000 5000 6000 7000 (a) 2 subevent ) 2 (GeV QE 2 Q data MC total CCQE + p CC1 others 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Events 0 500 1000 1500 2000 2500 (b) 3 subevent ) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 weight+p CC1 Figure 7-19. Summary of the MiniBo oNE CC + analysis for the bac kground measuremen t to the CCQE sample. Sho wn is (a) the neutrino-mo de CCQE sample and (b) the CC + sample b efore the application of constrain ts and parameter ts. The dashed line in (b) sho ws the ratio of prediction to data in the CC + sample, and its v alue is giv en b y the righ t ordinate axis. This measuremen t is used to indirectly constrain the rate and kinematics of the CC con tribution to the CCQE sample. Figure tak en from Ref. [ 84 ]. the lev el of agreemen t b et w een data and the underlying mo del for CC + in teractions. Ho w ev er, obtaining the constrain t in this v ariable do es allo w its direct application to CC + ev en ts bac kground to the CCQE sample. Due to n uclear capture, using an analogous pro cedure to measure the rate and kinematics of CC ev en ts is not p ossible. In the absence of suc h a measuremen t, the constrain t obtained in neutrino mo de for CC + is applied to the CC Rein-Sehgal prediction. This assumes the underlying eects observ ed in the MiniBo oNE CC + data not predicted b y the Rein-Sehgal mo del are iden tical for the CC pro cess. In the absence of additional information, a large extrap olation uncertain t y w ould b e w arran ted. F ortunately a more mo dern external calculation with success in describing w orld single-pion pro duction data is a v ailable to use in predicting the con tribution of CC 140

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) 2 (GeV QE 2 Q 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 500 1000 1500 2000 2500 3000 Data = 1.1 GeV p 1 A RS M = 1.1 GeV p 1 A ); M 2 (Q A 5 Dipole C Figure 7-20. Comparison of the nominal Rein-Sehgal mo del to a more mo dern mo del that also describ es w orld pion-pro duction data more accurately ev en ts to the CCQE sample [ 163 ]. This alternate mo del is implemen ted in nuance and is based on impro v emen ts to the Rein-Sehgal mo del originally dev elop ed in Refs. [ 164 165 166 ]. This up dated calculation includes m uon mass terms and a mo died v ector form factor to yield b etter agreemen t with w orld pion pro duction data [ 167 ]. Figure 7-20 sho ws this mo del also oers excellen t agreemen t with the MiniBo oNE CC + data for Q 2 & 0 : 1 GeV 2 This mo del is used as a second constrain t on the prediction for the con tribution of CC ev en ts to the CCQE sample. The lev el of agreemen t b et w een this calculation and the indirect constrain t based on the observ ed MiniBo oNE CC + data is sho wn in Figure 7-21 Consistency b et w een these t w o predictions for CC pro duction suggests an uncertain t y of 20% is sucien t for the CC bac kground. 141

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m q cos 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Events 0 500 1000 1500 2000 2500 3000 3500 4000 20% Central value prediction BergerSehgal extended model Nominal ReinSehgal prediction CCQE sample m n events in p CC1 Figure 7-21. Three calculations for the CC bac kground con tribution to the CCQE sample as a function of cos The \cen tral v alue" distribution corresp onds to the nominal Rein-Sehgal prediction for CC ev en ts in MiniBo oNE constrained b y the observ ed kinematics in the neutrino-mo de CC + sample. This agrees w ell with a more recen t calculation (\Berger-Sehgal extended mo del") that is based on an impro v ed v ersion of the Rein-Sehgal mo del. F or comparison, the nominal Rein-Sehgal prediction for CC ev en ts is also sho wn. Distributions are normalized to rux. 142

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7.3 All Other Bac kgrounds The analyses in Sections 7.1 and 7.2 constrain the con tribution of CCQE and CC in teractions to the CCQE sample. According to T able 6-2 the remaining c hannels accoun t for 6% of the analysis sample. Ab out half of these in teractions are from and CC 0 pro duction, and follo wing the normalization discrepancy found in the MiniBo oNE CC 0 cross-section analysis [ 80 ], the con tribution from b oth and CC 0 ev en ts is increased b y 60% of the prediction. The remaining half is dominated b y Cabibb o-suppressed quasi-elastic h yp eron pro duction, whic h ough t to b e closely related to the CCQE results, but is exp erimen tally p o orly understo o d. Tw o congurations of CCQE cross sections are pro duced in this dissertation: inciden t on mineral oil (with atomic comp osition CH 2 ) and inciden t on carb on only Man y measuremen ts of and CCQE on free or quasi-free n ucleons at a v ariet y of energies ha v e pro duced results consisten t with the RF G mo del and M A 1 GeV [ 168 ], and so the carb on-only conguration is attained b y treating the CCQE ev en ts on h ydrogen as bac kground. Its con tribution is predicted with the RF G and M A = 1 : 026 0 : 021 GeV follo wing the analysis of relev an t global data in Ref. [ 73 ]. 7.4 Bac kground Constrain t Summary The largest bac kgrounds in the CCQE sample are those from and from the CC con tributions. The dominan t in teractions are from CCQE and CC + c hannels, and their fundamen tal cross sections ha v e b een measured in the MiniBo oNE data [ 84 78 ]. The implemen tation of these direct constrain ts is explained in Section 7.1.4 The rux accepted in to the an tineutrino-mo de data is constrained b y the no v el and unique measuremen ts presen ted in Section 7.1 and p er Section 7.1.6 the com bined constrain t on the rux relativ e to the extrap olated and highly-uncertain prediction (describ ed in Section 5.2.4 ) is 0 : 77 0 : 10. No additional error is tak en on the CCQE and CC + in teractions, as the rux uncertain t y is nearly fully correlated with the CCQE and CC + cross section errors. 143

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As discussed in Section 7.2 the CC in teraction cross sections are only indirectly constrained through the measuremen t of the rate and kinematics of CC + ev en ts extrap olated to the pro cesses. Consistency b et w een this prediction for the CC con tribution to the CCQE analysis sample with an external mo del for resonance ev en ts capable of describing w orld CC pro duction data pro vides condence in our description of CC ev en ts. F ollo wing these studies, a 20% normalization uncertain t y is assigned to the CC in teractions. Finally the small con tribution from and CC 0 ev en ts are increased b y 60% of the nuance -based prediction to rerect the MiniBo oNE CC 0 results [ 80 ]. The uncertain t y on in teractions not induced b y and are non-CC in teractions is 30% of their prediction. A summary of the v arious bac kgrounds in the CCQE sample, including their uncertain ties and constrain ts, if an y is pro vided in T able 7-17 144

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T able 7-17. Summary of bac kground constrain ts and uncertain ties in the CCQE sample. Notice there is a small o v erlap b et w een the and non-CCQE, non-CC categories. Ev en ts b elonging to b oth classes are sub ject to b oth categorical uncertain ties. In teractions of CCQE on h ydrogen are only treated as a bac kground in the extraction of CCQE cross sections on carb on. Additional uncertain ties on the pro cesses en ter through the HARP pro duction data and b eamline sim ulations. In teraction Con tribution to Flux constrain t Cross-section constrain t F ractional CCQE sample (%) uncertain t y (%) All 22 This c hapter Refs. [ 84 78 80 ] 13 CCQE on h ydrogen 17 Ref. [ 119 ] Ref. [ 73 ] 2 CC 14 Ref. [ 119 ] Ref. [ 84 ] (indirect) 20 Non-CCQE, non-CC 6 This c hapter ( pro cesses) Ref. [ 80 ] 30 and Ref. [ 119 ] ( pro cesses) ( CC 0only) 145

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CHAPTER 8 CCQE CR OSS-SECTION MEASUREMENT F ollo wing the in tro duction to this measuremen t giv en in Chapter 6 and the opp ortunistic bac kground measuremen ts and constrain ts presen ted in Chapter 7 w e no w turn to the cross-section calculation and its v arious ingredien ts. This measuremen t is also describ ed in Ref. [ 147 ]. 8.1 Data Stabilit y Certainly the most imp ortan t quan tit y to ha v e condence in is the data itself. A total of 10.09 10 20 POT of an tineutrino-mo de data are used in this analysis. This corresp onds to the full MiniBo oNE an tineutrino-mo de data set through April 2012, excluding the brief p erio d in 2006 when absorb er blo c ks fell in to the meson deca y tunnel. Man y stabilit y c hec ks ha v e b een p erformed on the CCQE sample o v er the y ears and they'v e t ypically sho wn consistency within 1%. Those most directly related to this analysis are presen ted here. F or historical reasons, the data selection used in these stabilit y c hec ks dier sligh tly from that describ ed in the Section 6.2 ho w ev er the t w o selection sets result in roughly the same purit y and eciency T o test for an y eectiv e time dep endence in the distributions, p erhaps due to a systematic c hange in the exp erimen tal setup, the data is separated c hronologically in to groups describ ed in T able 8-1 The shap e compatibilit y of the four distributions w e will turn in to cross sections, Q 2QE (Eq. 6.8 ), E QE (Eq. 6.7 ) and the m uon kinematic prop erties T and cos are assessed o v er dieren t run p erio ds are assessed through the Kolmogoro v-Smirno (K-S) test [ 169 ] and are presen ted in Figure 8-1 Not indep enden t from Figure 8-1 but p erhaps more accessible is the shap e of the E QE distribution sho wn for the same run p erio ds on Figure 8-2 A direct test of the normalization of the primary analysis sample is presen ted in Figure 8-3 where the ev en ts passing selection are giv en p er POT for eac h p erio d. 146

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T able 8-1. Summary of data groups input to the stabilit y tests. Lab el Run dates POT ( 10 20 ) \jul07" Jan. 2006 Sep. 2008 2.205 \sep09" Sep. 2008 Jun. 2009 1.477 \mar10" Aug. 2009 Mar. 2010 0.798 \o ct10" Mar. 2010 Oct. 2010 1.160 \ma y11" Oct. 2010 Ma y 2011 1.763 \mar12" Ma y 2011 Apr. 2012 2.688 m q cos QE n E 2 Q m T K-S Prob. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 jul07 sep09 mar10 oct10 may11 mar12 Figure 8-1. Kolmogoro v-Smirno test for dieren t an tineutrino run p erio ds. Eac h data p oin t is the result of the K-S test of that run against the sum of the other subsamples. The data are consisten t with a uniform distribution b et w een n ull and unit y Imp ortan t to note is that E QE and Q 2QE are deriv ed quan tities from T and cos so these tests are not indep enden t. 147

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Figure 8-2. Reconstructed an tineutrino data for v arious run p erio ds. Go o d agreemen t within statistical errors indicate stable running. Included here in \jul07" and not in the main analysis is the small absorb er do wn sample. jul07 (0abs) sep09 mar10 oct10 may11 mar12 Events / POT 68 69 70 71 72 73 -18 10 Figure 8-3. Normalization stabilit y o v er v arious run p erio ds. Included uncertain ties are statistics and a 2% error on deliv ered POT. 148

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8.2 CCQE Sim ulation Data extraction is mostly insensitiv e to assumptions on the signal pro cesses, but it is imp ortan t to qualify their ev en t generation to b etter understand the nal cross-section comparisons b et w een data and sim ulation. The RF G sim ulation of CCQE in teractions in this analysis assumes an axial mass for CCQE ev en ts on protons b ound in carb on atoms (hereafter referred to as \ M C A ") of 1.35 GeV, and for ev en ts on free protons (\ M H A ") of 1.02 GeV. Signal ev en ts in v olving b ound n ucleons also receiv e the mild P auli blo c king mo dication = 1.007 (Section 4.4.1 ). These parameters are c hosen b ecause they adequately repro duce the shap e of the data in the reconstructed quan tities (Figure 6-5 ) while main taining consistency with the MiniBo oNE CCQE analysis [ 84 ] and the ligh t-target CCQE data [ 73 ]. If not for a few issues in generating the MiniBo oNE MC, this description of our signal assumptions w ould b e sucien t. The rst issue originates in the rew eigh ting sc heme for nding = 1.007. F ollo wing the 2008 CCQE analysis [ 83 ], the MC les w ere generated with M C A = 1.23 GeV and = 1.022. As is a lo w er b ound on the a v ailable outgoing n ucleon phase space, w e cannot pro duce a lo w er v alue of compared to the generated v alue using traditional rew eigh ting. In other w ords, ev en ts that do not exist cannot b e reco v ered b y rew eigh ting. F or our sim ulation to rerect = 1.007, w e rst calculate the absolute cross section p er n ucleon for signal ev en ts in v olving b ound protons from the baseline MC. W e exclude h ydrogen ev en ts here b ecause they are unaected b y This cross section is compared to the nuance -generated rate for M C A = 1.35 GeV and = 1.007 in Figure 8-4 The decit in MC due to lo w ering is clear at lo w er energies, while an indep enden t problem sho ws itself ab o v e E 1.5 GeV. When the same baseline MC w as generated, nuance w as run in the logarithmic lo w precision rux sampling mo de, meaning the rux sp ectrum w as sampled increasingly sp oradically at higher energies. As the rux rapidly decreases with E > 1 GeV or so (Figure 5-9 ), this lev el of sensitivit y to the rux shap e is sucien t for the all 149

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(GeV) n E 0.5 1 1.5 2 2.5 ) 2 / N (cm s 1 2 3 4 5 6 7 8 -39 10 = 1.007 k = 1.35 GeV, C A MC: M = 1.007 k = 1.35 GeV, C A Nuance: M Figure 8-4. Comparison b et w een generator-lev el total cross section for b ound signal ev en ts to the unfolded MC. The les input to the MC distribution w ere generated with = 1.022, and the failed attempt to reco v er = 1.007 through traditional rew eigh ting is clear at lo w er energies, while the rux sampling issue dominates the discrepancy at high energy non-cross section MiniBo oNE analyses. In the presen t analysis, the rux w as sucien tly sampled for the bulk of the distribution but the high energy tail w as not accurately explored b y nuance If not corrected, the details of the issue w ould lead to a to o-lo w detection eciency and a spuriously high cross section. As w as in tro duced to impro v e kinematic agreemen t with CCQE data, it w ould b e insucien t to reco v er it b y rew eigh ting MC to nuance based on the total cross section. Therefore w e scale MC to the generator-lev el distribution in bins of the double-dieren tial cross section in m uon kinematics d 2 dT d cos and the resultan t w eigh ts are sho wn in Figure 8-5 As the double-dieren tial cross section is rux-in tegrated, it is only mildly sensitiv e to the high-energy rux tail. A nal set of w eigh ts in the absolute cross section is applied to MC to ac hiev e generator-lev el agreemen t at the few-% lev el ev en at high energies. As men tioned, the issue is irrelev an t for signal scattering o h ydrogen, but of 150

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0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 (GeV) m T 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 = 1.007 k = 1.35 GeV, C A Nuance / MC, M = 1.007 k = 1.35 GeV, C A Nuance / MC, M Figure 8-5. W eigh ts as a function of m uon kinematics applied to MC to reco v er = 1.007. As exp ected, the w eigh ts are strongest in regions of forw ard-going, lo w-energy m uons. course the high energy issue is presen t. Figure 8-6 sho ws MC b efore and after correcting for the rux sampling problem. It is imp ortan t to note that, with the exception of the high-energy issue aecting the eciency calculation, these signal assumptions hardly aect our extracted cross sections. The normalization of the MC signal ev en ts do es not en ter the cross-section calculation, while sensitivit y to the sim ulated true shap e of signal ev en ts is quite small. It will b e sho wn later that a conserv ativ ely large span of M C A and signal v alues lead to negligible dierences on the nal cross sections. Ho w ev er, this pro cedure of rew eigh ting our sim ulation to the nuance cross sections with = 1.007 w as an imp ortan t step to ha v e a reliable eciency calculation and b e able to faithfully rep ort the mo del used in data extraction and comparisons. A nal note on the high-energy rux sampling problem in v olv es its eect on the CC in teractions. Since the Q 2QE -based correction (describ ed in Section 7.2 ) to CC + and CC ev en ts w as measured in the neutrino-mo de CC + sample, the constrain t could b e sensitiv e 151

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(GeV) n E 0.5 1 1.5 2 2.5 ) 2 (cm s 1 2 3 4 5 6 7 8 -39 10 = 1.02 GeV H A Nuance free scattering, M = 1.02 GeV H A MC free scattering, M MC weighted to nuance Figure 8-6. Summary of the high energy rate correction for CCQE in teractions on h ydrogen. to the details of the high-energy problem and its implemen tation in to the an tineutrino-mo de analyses could b e erroneous through the mildly dieren t accepted sp ectra b et w een the t w o run congurations. Ho w ev er, Figure 8-7 sho ws that the rux sampling problem has a m uc h more mild eect on CC in teractions compared to CCQE, presumably due to the dieren t shap e of the total cross section around the MiniBo oNE energy range. 8.3 Cross-Section Calculation The total cross section p er n ucleon in the i th bin is giv en b y ( E ) i = P j U ij ( d j b j ) i i N ; (8.1) where d j ( b j ) is the data (bac kground) reconstructed in the j th bin, U ij is the probabilit y for an ev en t of true quan tit y within bin i to b e reconstructed in bin j is the detection 152

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(GeV) n E 0.5 1 1.5 2 2.5 ) 2 (cm s 0 1 2 3 4 5 6 7 8 9 -39 10 hist = MC curve = nuance + p p m p m n + p n m n m n p n + m n m n p p + m p m n Figure 8-7. Comparison of nuance -generated single pion cross sections to the unfolded MC. Unlik e in the CCQE in teractions, the high energy rux sampling problem is small enough to ignore here. eciency is the rux corresp onding to the deliv ered protons on target, and N is the n um b er of n uclear targets in the v olume considered. The dieren tial expressions are similar: for Q 2QE w e ha v e d dQ 2QE i = P j U ij ( d j b j ) Q 2QE i N ; (8.2) where Q 2QE is the width of the i th bin and is no w the in tegrated rux. The double-dieren tial calculation is a trivial extension: d 2 dT d ( cos ) i = P j U ij ( d j b j ) T cos i N : (8.3) The follo wing subsections expands on eac h of these quan tities. 153

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Entries 621605 Mean 0.001558 RMS 0.09808 RFG n ) / E QE n E RFG n (E -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 3000 3500 Entries 621605 Mean 0.001558 RMS 0.09808 Entries 621605 Mean -0.004789 RMS 0.1269 QE, true 2 ) / Q QE, reco 2 Q QE, true 2 (Q -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Entries 621605 Mean -0.004789 RMS 0.1269 Entries 621605 Mean -0.001091 RMS 0.05453 true m ) / T reco m T true m (T -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 1000 2000 3000 4000 5000 Entries 621605 Mean -0.001091 RMS 0.05453 Entries 621605 Mean 0.001013 RMS 0.04088 true m q ) / cos reco m q cos true m q (cos -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 2000 4000 6000 8000 10000 12000 Entries 621605 Mean 0.001013 RMS 0.04088 Figure 8-8. F ractional dierences b et w een truth and reconstructed quan tities in the four relev an t v ariables. As in all distributions p eak ed near n ull, the a v erage fractional dierence in Q 2QE is higher than it w ould b e otherwise. 8.3.1 Unsmearing to T rue Quan tities All measuremen ts are biased at some lev el b y detectors and analysis to ols. The unsmearing pro cess remo v es this bias so that the underlying quan tities ma y b e rep orted. First w e should understand the o v erall relationship b et w een the quan tities w e are lo oking to connect. Figure 8-8 sho ws the fractional dierence b et w een the reconstructed and truth-lev el v ariables relev an t to this analysis. It is tempting to refer to the RMS v alues prin ted in Figure 8-8 as the resolution of the MiniBo oNE detector to these quan tities, but this is not the case for Q 2QE and, in particular, for the neutrino energy T o a v oid dep endence on the n uclear mo del of the RF G, reconstructed Q 2QE is unsmeared to \true" Q 2QE that is using the truth-lev el quan tities 154

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in Eq. 6.8 instead of the generator-lev el squared four-momen tum transfer. Of course this is not a p erfect solution, as an y theoretical calculation of the underlying Q 2 will ha v e to rst b e translated in to Q 2QE b efore a rigorous comparison with these data can b e made. In con trast, reconstructed E QE is corrected to the generator-lev el neutrino energy referred to here as E RF G E QE and E RF G are en tirely dieren t quan tities, and the RMS prin ted on Figure 8-8 is simply the a v erage fractional dierence b et w een the t w o. The c hoice of correcting the neutrino energy bac k to the generated v alue is made to b e able to compare with historical data pro duced with the same assumptions. It is partially for these reasons that the double-dieren tial cross section d 2 dT d cos is the main result of this w ork, as it do es not rely on a ph ysics mo del to connect the reconstructed and true quan tities. Of course the other adv an tage is that no assumptions (as are implicit in Equations 6.7 and 6.8 ) ab out the primary in teraction need b e made to reconstruct m uon kinematics. F ollo wing the other MiniBo oNE cross-section analyses [ 129 84 78 79 80 ], the primary metho d for unsmearing detector eects emplo ys the Ba y esian approac h [ 170 ]. This metho d is biased b y MC assumptions on the underlying distribution, but w e will sho w this prejudice is small. An un biased estimator will b e used to cross-c hec k the results (alb eit in less comp elling binning), and ultimately the bias is assessed b y ev aluating the results under a conserv ativ ely wide range of signal assumptions. F ortunately error due to this bias is negligible compared to rux and bac kground uncertain ties. Another motiv ation to use the Ba y esian metho d is the aim to follo w as closely as reasonable the analysis c hoices of the CCQE cross section to b etter facilitate the com bined analysis of Chapter 9 If w e refer to the underlying true data distribution as ~ and to the same distribution under the inruence of detector and reconstruction biases as ~ the t w o are connected b y the unsmearing matrix U presen t in Equations 8.1 8.3 as simply: 155

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~ = U ~ (8.4) Eac h en try U ij represen ts the probabilit y for an ev en t dra wn from the underlying distribution in bin i to b e reconstructed in the j th bin. In a p erfect detector, U w ould b e the iden tit y matrix. The reconstructed v ector ~ is readily recognized as ~ d ~ b in the cross section form ulae. T o build U w e rst p opulate a matrix with signal MC ev en ts in the reconstructed vs. true v ariables, referred to here as M The en tries of M are simply connected to U b y normalization factors U ij = M ij P k M k j (8.5) This naturally conserv es the n um b er of reconstructed ev en ts, i.e. P i U ij = 1 for all j The probabilit y matrix U is sho wn in Figure 8-9 for the four standard distributions in the binning c hosen for this analysis, as w ell as the diagonal en tries for m uon kinematic plane. In the application of this metho d to histograms whose domain ma y exclude part of the sample, underro w and o v erro w bins are included. A simple consistency c hec k, passed for all distributions, is that this unsmearing pro cedure applied to reconstructed MC signal ev en ts exactly returns the generated distribution. Figure 8-10 sho ws the eect of the U matrix to the v ector ~ d ~ b where ~ d and ~ b are the reconstructed data and bac kground, resp ectiv ely The distributions in the same gure represen t the n umerator in cross-section Eqs. 8.1 8.3 The so-called \in v ersion metho d" of connecting the reconstructed to true distributions is un biased b y a priori assumptions ab out the underlying in teractions. In this pro cedure the matrix (referred to as R ) that describ es unsmearing op erates on the true distribution ~ : 156

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~ = R ~ (8.6) A quic k comparison with Equation 8.4 sho ws R = U 1 The matrix R is also formed with M but this time b y normalizing o v er the reconstructed index: R ij = R ij P k M ik (8.7) Since R m ust b e in v erted in order to nd the true distribution, the matrix in v ersion metho d of unsmearing is exceptionally unstable. In particular, to o-ne binning giv es rise to the \Gibb's phenomenon", where the calculated true distribution oscillates wildly bin-to-bin. Anecdotally this can b e a v oided b y c ho osing the binning suc h that on-diagonal elemen ts of R are no lo w er than 0.8. This requiremen t constrains the bins to b e quite mo dest. This is particularly true in the case of the neutrino energy where the correlation b et w een the E QE and E RF G v ariables is relativ ely w eak (Figure 8-8 ). Figure 8-11 compares the results of the matrix in v ersion metho d to the Ba y esian pro cedure describ ed previously The binning has b een optimized suc h that the en tries R ii are close to 0.8, b ecause binning ner than those sho wn migh t b e sub ject to the Gibb's phenomenon. That the Ba y esian metho d giv es results consisten t with the un biased matrix in v ersion metho d giv es a qualitativ e upp er limit to its bias. Ho w ev er, since unsmearing is a shap e-only pro cedure and the binning in the un biased metho d is relativ ely conserv ativ e, this is not an esp ecially p o w erful test. More comp elling evidence that the Ba y esian metho d do es not signican tly bias the results is sho wn in Section 8.4 where the prejudice is ev aluated to b e negligible in the presence of other systematic uncertain ties. 157

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (GeV) QE n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (GeV) RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ) 2 (GeV QE 2 Reco Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 (GeV QE 2 True Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (GeV) m Reco T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (GeV) m True T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 m q Reco cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 m q True cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Diagonal entries of U Diagonal entries of U Figure 8-9. The matrix U connecting the true and reconstructed quan tities for the four v ariables of in terest. F or the t w o-dimensional distribution, in principle U is four dimensional and only its diagonal en tries are sho wn here. 158

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(GeV) n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Events 1500 2000 2500 3000 3500 4000 4500 5000 5500 QE n Reco E RFG n Unsmeared to E ) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Events 0 1000 2000 3000 4000 5000 6000 7000 8000 Reco Unsmeared to true (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Events 0 1000 2000 3000 4000 5000 6000 7000 Reco Unsmeared to true m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Events 0 2000 4000 6000 8000 10000 12000 14000 16000 Reco Unsmeared to true 0.7 0.8 0.9 1 1.1 1.2 1.3 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 unsmeared / reconstructed unsmeared / reconstructed Figure 8-10. Comparison of b efore and after the unsmearing pro cedure for signal ev en ts. The t w o-dimensional ratio includes a requiremen t that there b e at least 10 ev en ts in eac h bin of b oth the reconstructed and unsmeared distributions. 159

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(GeV) RFG n E 0.5 1 1.5 2 2.5 Events/bin width 0 10000 20000 30000 40000 50000 60000 ) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Events/bin width 0 20 40 60 80 100 120 140 160 3 10 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Events/bin width 0 10000 20000 30000 40000 50000 60000 70000 Bayesian method Inversion method m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Events/bin width 0 20 40 60 80 100 120 140 160 3 10 Figure 8-11. Comparison of the t w o unsmearing metho ds studied in this section. 160

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8.3.2 Eciency Correction T o correct for the signal ev en ts lost due to sample selection, the detection eciency is calculated and applied to data bin-b y-bin in eac h distribution. Since the unsmearing pro cedure describ ed previously has (up to some uncertain t y) returned the observ ed data to the true distribution, eciencies are measured as a function of the true v ariable. The eciency is ev aluated in a sim ulated sample of signal ev en ts in a spherical v olume of radius 550 cm, the v alue of whic h is c hosen to a v oid a p oten tial rate bias due to the iron optical barrier at 575 cm (this eect is visible in Figure 6-2 ), while a negligible n um b er of signal ev en ts ( < 0.2%) that pass selection criteria ha v e a generated radius greater than 550 cm. Figure 8-12 sho ws sequen tial eciency for eac h analysis cut in the four standard v ariables, as w ell as the total eciency for the t w o-dimensional distribution. The ma jorit y of the loss of ev en ts is caused b y requiremen ts on the kinematics of the m uon, the sim ulation for whic h has b een v etted most rigorously against cosmic ra y m uon data (Section 5.3.4 ). 161

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(GeV) RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Sequential Efficiency 0 0.2 0.4 0.6 0.8 1 ) 2 (GeV QE 2 True Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Sequential Efficiency 0 0.2 0.4 0.6 0.8 1 (GeV) m True T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Sequential Efficiency 0 0.2 0.4 0.6 0.8 1 No cuts + veto hits + beam window > 200 MeV m + T + 2 SE + in fiducial vol /e) > 0 m + ln( range cut m + m q True cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Sequential Efficiency 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (GeV) m True T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q True cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 8-12. Detection eciency for eac h cut in the relev an t distributions. Only the total eciency is sho wn for t w o-dimensional m uon kinematics. 162

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(GeV) p p 0 1 2 3 4 5 6 7 8 (rad) p q 0 0.05 0.1 0.15 0.2 0.25 0.3 87.2% m n p p+Be HARP coverage Figure 8-13. Predicted outgoing phase space for b efore horn fo cusing. Only 's fo cused b y the horn and that subsequen tly lead to an in teraction in the detector are sho wn. As prin ted on the gure, roughly 90% of the rux is co v ered b y the HARP cross-section data.8.3.3 Flux and In teraction T argets As describ ed in Section 5.2.4 the prediction for the absolute rux in an tineutrino mo de is nearly mo del indep enden t. Figure 8-13 sho ws the outgoing phase space at the BNB target con tributing to the presen t data set is w ell-constrained b y the HARP data. Com bining the HARP pro duction data with detailed Gean t4 target, horn and b eamline geometry giv es the absolute rux prediction sho wn in Figure 8-14 Since the total cross section and the rux are b oth functions of the neutrino energy the rux histogram in Figure 8-14 is rebinned to matc h that used in the analysis. In the case of the dieren tial measuremen ts, the in tegrated rux is used excluding the region E < 100 MeV due to the in teraction requiremen t of m uon pro duction. The nal elemen t in the cross-section calculations is the n um b er of n ucleon targets for the signal. This in v olv es the detector v olume corresp onding to that assumed b y the eciency correction in Section 8.3.2 the mineral oil densit y the mass densit y of relev an t 163

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(GeV) n E 0 0.5 1 1.5 2 2.5 3 3.5 4 / 50 MeV) 2 / cm m n ( F 0 2 4 6 8 10 12 14 16 -12 10 Figure 8-14. Flux prediction for in an tineutrino mo de for 10.1 10 20 POT. protons p er molecule, and Av ogadro's n um b er. This is calculated for all protons (b ound protons only) b y: N = 4 3 [550cm] 3 0 : 845 g cm 3 6 : 02214 10 23 6 : 0(8 : 06) 14 : 06 = 1 : 5134 (2 : 0330) 10 32 (8.8) The densit y of the oil is measured from a sample extracted from the detector, and the comp osition of the oil is determined to b e C n H 2 n +2 n 30. 8.3.4 Statistical Uncertain t y T o a v oid regions of statistics that w ould b e incorrectly analyzed with a Gaussian treatmen t, at least 25 ev en ts are required to app ear in eac h reconstructed bin ~ d ~ b The statistical error in the i th bin is calculated b y Stat error i = q T i data MC S i i (8.9) 164

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where T i and S i are the predicted n um b er of ev en ts in the total and signal samples resp ectiv ely \data" and \MC" are the resp ectiv e normalizations and is the (total or dieren tial) cross section in that bin. The data / MC ratio is included to correct for the observ ed statistics. It migh t b e tempting to not include the normalization ratio and instead use the n um b er of data ev en ts for T i but then the statistical error itself w ould b e sub ject to statistical ructuations. As with all MiniBo oNE cross-section measuremen ts, the statistical error will pro v e to b e negligible with the exception of the tails of the double-dieren tial cross section. 8.4 Systematic Uncertain t y Broadly systematic errors are ev aluated b y recalculating the cross sections under appropriate excursions from the assumptions input to the MC regarding parameters and pro cesses that migh t aect the extraction of the CCQE cross sections. The implemen tation of this idea v aries among the systematic errors and the details are presen ted in this section. The dierences b et w een these alternate cross sections and the one describ ed in the previous section are then used to form co v ariance matrices, and the on-diagonal elemen ts of the quadrature addition of all error matrices sets the total uncertain t y on the data. The formation of this co v ariance matrix is dev elop ed in Section 7.1.5 and is not rep eated here. 8.4.1 Bac kground Uncertain ties Uncertain t y on the bac kground is ev aluated b y the \unisim" metho d, where a single excursion from the cen tral v alue prediction is sucien t to propagate uncertain t y on to data. Appropriate to the description in Section 7.4 bac kgrounds are separated in to three categories: ev en ts from + deca y in the b eam, CC and non-CCQE, non-CC in teractions. As rep orted in T able 6-2 these bac kgrounds comprise 17%, 14%, and 6% of the sample, resp ectiv ely Note there is some o v erlap in the and the non-CCQE, non-CC categories. Section 7.4 summarizes the bac kground constrain ts and assumed uncertain ties on their con tributions to the CCQE sample. 165

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With the uncertain ties on the bac kground in teractions set, w e re-calculate the cross sections from data with the v arious bac kgrounds one standard deviation from their nominal prediction: ( E ) bkg err i = P j U ij ( d j [ b j b j ]) i i N ; d dX bkg err i = P j U ij ( d j [ b j b j ]) X i N ; (8.10) d 2 dT d ( cos ) bkg err i = P j U ij ( d j [ b j b j ]) T cos i N : where X is a single-dieren tial cross sections and b is the uncertain t y on eac h bac kground. Since the error matrix formed b y these uncertain ties in v olv es squaring dierences b et w een these alternate calculations and the cen tral v alue cross sections, it is irrelev an t whether the uncertain t y on the bac kground is added or subtracted in Equation 8.10 A separate error matrix is formed for eac h bac kground b y EM bkgij = CV i bkg i CV j bkg j ; (8.11) where refers to an y of the cross section measuremen ts. The total error in a giv en bin i for these error matrices is simply p EM ii The gross deviation of the cross sections calculated in Equation 8.10 compared to the cen tral v alue is summarized in Section 8.4.3 As men tioned in Section 8.4.2 uncertain ties on pro cesses aecting signal rates lik e rux and the optical mo del also aect bac kground lev els and so a small part of these errors are due to bac kgrounds. 166

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8.4.2 Signal Errors Uncertain ties aecting signal rates are handled in subtly dieren t w a ys, according to ho w the excursions from the cen tral v alue are practically generated. Flux, in ter-medium pion in teractions, and mo del dep endence errors are ev aluated b y the MultisimMatrix pac k age. It tak es as input a co v ariance matrix for a set of parameters and generates a set of w eigh ts corresp onding to individual thro ws against a Gaussian distribution for eac h parameter, according to its sp ecied uncertain t y and is constrained b y correlations. Eac h set of w eigh ts k are then used to calculate the range of cross sections due to the input co v ariance matrix: ( E ) ki = P j U k ij d j b kj h N acc,k N gen,CV i i i N ; d dX ki = P j U k ij d j b kj h N acc,k N gen,CV i i X N ; (8.12) d 2 dT d ( cos ) ki = P j U k ij d j b kj h N acc,k N gen,CV i i T cos N : The set of w eigh ts app ear in three terms: the unsmearing matrix U k ij the bac kground prediction b kj and in an alternate eciency calculation h N acc,k N gen,CV i i The alternate unsmearing matrix incorp orates shap e uncertain ties in the generated signal distribution. P erhaps most in tuitiv ely for example, this is where uncertain ties on the rux sp ectrum will result in shap e errors on the total cross section. The nal term N acc,k N gen,CV incorp orates uncertain ties on the signal pro cess due to the k th thro w, if an y N acc,k refers to the n um b er of signal ev en ts passing selection for the k th excursion from the cen tral v alue, while N gen,CV is the distribution of signal ev en ts b efore cuts for the cen tral v alue predition. It ma y b e non-in tuitiv e to accoun t for rux uncertain ties through the eeciency term, but it is trivial to see ho w a rux excursion from 167

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the nominal prediction w ould aect the calculated error on the cross section in an iden tical fashion if the normalization dierence originated in the term rather than the term. Note that, in principle, this could lead to a calculated eciency greater than 1, but of course these factors are related to normalization uncertain ties and not true detection rates. Since the errors discussed in this section are generated b y rew eigh ting the cen tral v alue prediction, it is crucial that the denominator in this alternate eciency calculation refer to the generated cen tral v alue prediction, and not that from the k th generated distribution. If the w eigh ted generated distribution w ere tak en, the w eigh ts in tended to b e propagated as uncertain ties on to the data w ould b e suppressed. This is also the metho d for measuring the cross-section uncertain t y due to the mo del for ligh t propagation in the detector (describ ed in Section 5.3.2 ), where 35 p ossibly correlated parameters are v aried within their uncertain t y according to external measuremen ts and calibration data. In this case, k in Equation 8.12 refers to the k th optical mo del. Additionally to remo v e the statistical error the optical mo dels w ere generated with, for the neutrino energy and Q 2QE v ariables the ratio of eac h optical mo del to the cen tral v alue prediction is smo othed to a fourth order p olynomial. Suc h attempts at smo othing for the t w o-dimensional distribution w ould b e un tenable, so to minimize the in trinsic statistical error of the optical mo dels, the size of eac h sample used is increased to a little more than t wice that of the data statistics. As statistical error in this analysis is negligible, this mild o v erestimate negligibly aects the extracted cross section. F or the optical mo del and eac h systematic uncertain t y ev aluated with MultisimMatrix, the error matrix is calculated b y: EM ij = P N k k =1 CV i k i CV j k j N k 1 ; (8.13) 168

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where N k refers to the n um b er of v ariations from the cen tral v alue used, and again refers to the v arious cross sections calculated. N k = 100 for uncertain ties on the rux, the CCQE mo del dep endence, pro cesses en tering the sample due to c harge exc hange or absorption, and, in the case of calculating the carb on-only CCQE cross sections, the CCQE h ydrogen bac kground. N k = 70 for the optical mo del v ariations. The rux errors due to pro duction at the target are tak en from a spline t to the HARP double-dieren tial cross-section uncertain ties. All other rux uncertain ties not directly related to secondary pro duction are referred to as b eam unisim errors, and the most imp ortan t of these include rein teractions in the target and uncertain ties due to magnetic fo cusing and are further describ ed in Section 5.2.4 The uncertain ties on the in ter-medium pion in teractions of c harge exc hange (absorption) are set at 50% (35%) based on comparisons b et w een sim ulation and external data [ 107 ]. The mo del dep endence of the unsmearing pro cedure is ev aluated b y forming v ariations of the unsmearing matrix U with conserv ativ e errors on the underlying mo del parameters M C A = 1.35 0.35 GeV, M H A = 1.02 0.35 GeV, and = 1.007 0.007. In this case, only the matrix U is v aried in Equation 8.12 as it is the only term sensitiv e to the underlying ph ysics mo del. Finally applicable only when the h ydrogen CCQE comp onen t is treated as bac kground, its rates are v aried according to a global t to the ligh t-target data, where M H A = 1.020 0.014 GeV. The nal systematic errors are related to uncertain ties on the PMT discriminator threshold (lab eled in gures and tables as \disc") and c hanges in rates due to c harge-time correlation (\QT corr") eects. An indep enden t MC sample is a v ailable for eac h uncertain t y and so the alternate cross sections extracted using these samples are en tirely based on their distributions: 169

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( E ) pi = P j U p ij d j b pj pi i N ; d dX pi = P j U p ij d j b pj pi X N ; (8.14) d 2 dT d ( cos ) pi = P j U p ij d j b pj pi T cos N : where p refers to eac h of the t w o distributions with alternate assumptions on the PMT b eha vior. Note that this is iden tical to Equation 8.12 with the replacemen t of the MC cen tral v alue generated distribution in the eectiv e eciency calculation b y the same quan tit y in the indep enden t samples. The error matrices asso ciated with these detector uncertain ties are calculated b y: EM pij = CV i p i CV j p j : (8.15) The v arious con tributions to the total uncertain t y for the CCQE cross sections inciden t on mineral oil and on carb on only are summarized in Sections 8.4.3 and 8.4.4 8.4.3 Uncertain t y Summary for CCQE on Mineral Oil The total error matrix is formed b y simply adding together eac h error matrix calculated in the previous subsections. As the en tries of the error matrix represen t bin-b y-bin v ariances and co v ariances, the linear addition of the en tries is equiv alen t to addition in quadrature. Then, the total uncertain t y in bin i is simply p E M tot ii The o v erall eect and relativ e imp ortance of eac h error can b e ev aluated with bin-b y-bin fractional error p E M tot ii CV i where CV is the cen tral v alue cross section for eac h uncertain t y Figure 8-15 sho ws fractional errors for eac h source of uncertain t y for the four one-dimensional distributions and their sum for the t w o-dimensional cross section. Figure 8-16 sho ws the same for those with large maxim um uncertain ties. 170

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(GeV) RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 fractional error 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Total stat OM disc QT corr prod p Beam WS p CC Other bkg abs/cex p Model dep ) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 fractional error 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 fractional error 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 fractional error 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.1 0.2 0.3 0.4 0.5 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Total fractional error Total fractional error Figure 8-15. F ractional uncertain t y con tributions to the total and dieren tial cross sections including the h ydrogen con ten t as signal. The total uncertain t y is the quadrature sum of the error sources sho wn. 171

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) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 fractional error 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Total stat OM disc QT corr prod p Beam WS p CC Other bkg abs/cex p Model dep m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 fractional error 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Total fractional error Total fractional error Figure 8-16. The full range of fractional uncertain t y for Q 2QE cos and the doubledieren tial cross sections including the h ydrogen con ten t as signal. 172

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The o v erall uncertain t y for eac h error source can b e rep orted n umerically with the total normalization error. This quan tit y is equiv alen t to the total uncertain t y if the measuremen t w ere a single n um b er (e.g., a distribution with a single bin). Using the sum rule for v ariances and co v ariances ( 2 i + j = 2 i + 2 j + 2 ij i j where [ ] refers to a total error [correlation]), the total normalization error for a giv en uncertain t y in terms of its error matrix is simply Norm. error = q P ij EM ij P i CV i (8.16) These v alues are giv en in T able 8-2 for eac h uncertain t y and eac h result. Due to the exclusion of some ev en ts whose v alue in the distribution ma y b e excluded from the c hoices in binning, care m ust b e tak en in comparing normalization uncertain ties across the dieren t distributions. Ev en ts generated with T < 0.2 GeV and T > 2 GeV are not reco v ered through the eciency calculation in the d dT and d 2 dT d cos cross sections, while only ev en ts whose true energy w ould lie in 0.4 > E RF G > 2 GeV are included for the total cross section. F or cos the en tire range of kinematics is included, while the same is almost true for Q 2QE where the eectiv e cut of Q 2QE > 2 GeV 2 excludes v ery few ev en ts. 8.4.4 Uncertain t y Summary for CCQE on Carb on F ractional uncertain t y lev els for the results treating the free scattering comp onen t of CCQE in teractions as bac kground are giv en in T able 8-3 and Figure 8-17 Figure 8-18 sho ws the full range of fractional uncertain t y for those with some v alues greater than unit y 173

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(GeV) RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 fractional error 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Total stat OM disc QT corr prod p Beam WS p CC Other bkg abs/cex p Model dep Hyd bkg ) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 fractional error 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 fractional error 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 fractional error 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.1 0.2 0.3 0.4 0.5 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Total fractional error Total fractional error Figure 8-17. F ractional uncertain t y con tributions to the total and dieren tial cross sections taking CCQE in teraction on h ydrogen as bac kground. The total uncertain t y is the quadrature sum of the error sources sho wn. 174

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) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 fractional error 0 0.5 1 1.5 2 2.5 Total stat OM disc QT corr prod p Beam WS p CC Other bkg abs/cex p Model dep Hyd bkg m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 fractional error 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Total fractional error Total fractional error Figure 8-18. The full range of fractional uncertain t y for Q 2QE cos and the doubledieren tial cross sections. 175

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T able 8-2. Normalization errors for eac h cross section and eac h source of error for CCQE ev en ts on mineral oil. Due to dierences in cross-section shap es and relativ e regional sensitivit y to eac h uncertain t y the normalization errors v ary b y a few p ercen t across the distributions. The h ydrogen con ten t is included as signal here. T cos refers to the doubledieren tial cross section. Uncertain t y source Normalization error (%) E RF G Q 2QE T cos T cos Statistics 1.1 0.7 1.0 0.7 0.8 Optical mo del 2.8 3.1 2.9 3.8 3.8 PMT discriminator thresh. 1.5 0.1 0.7 1.7 0.1 PMT c harge-time corr. 2.9 1.4 0.1 0.7 0.9 pro duction 5.1 5.3 5.4 5.2 6.4 Beam unisims 6.4 5.0 6.5 5.5 7.2 All bac kground 4.4 3.1 3.6 4.6 3.5 CC bac kground 3.8 4.3 4.0 4.4 4.0 Non-CCQE, non-CC bac kground 2.6 2.5 2.6 2.7 2.6 Unsmearing mo del dep endence 1.7 0.0 0.2 0.2 0.8 c harge exc hange + absorption 2.3 2.1 2.3 2.2 2.3 T otal 11.7 10.2 11.1 11.4 12.9 8.5 Results Results for dening the analysis signal as either all CCQE in teractions or only those b ound in carb on atoms are presen ted in this section. While the more inclusiv e measuremen t is a more precise and less mo del-dep enden t measuremen t, the assumption that the free scattering in teraction is w ell-kno wn is motiv ated b oth b y consistency among the previous ligh t-target data sets and b y theoretical calculations predicting enhancemen ts only for b ound n ucleon targets. 8.5.1 Results Inciden t on Mineral Oil Results including the h ydrogen CCQE comp onen t are presen ted in Figure 8-19 The agreemen t b et w een data and the RF G mo del under v arious assumptions in the double-dieren tial cross section is sho wn in Figure 8-20 It is clear that the MC lies somewhat lo w from data in normalization, and the lev el of agreemen t in the shap e can b e ev aluated b y forming the shap e-only error matrix. The 176

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T able 8-3. Normalization errors for eac h cross section and eac h source of error, treating CCQE ev en ts on h ydrogen as bac kground. F ractional uncertain ties are generally higher compared to those rep orted in T able 8-2 due to the signican tly lo w er purit y of the sample. Uncertain t y source Normalization error (%) E RF G Q 2QE T cos T cos Statistics 1.4 0.9 1.3 1.0 1.2 Optical mo del 3.9 4.1 4.6 5.0 4.2 PMT discriminator thresh. 2.3 0.3 0.4 2.3 1.1 PMT c harge-time corr. 4.2 1.5 1.9 1.4 2.6 pro duction 6.8 7.2 6.7 7.1 8.0 Beam unisims 8.3 6.8 9.0 7.4 9.2 All bac kground 5.8 4.1 6.2 5.4 6.8 CC bac kground 5.1 5.8 6.7 5.9 7.4 Non-CCQE, non-CC bac kground 3.4 3.3 4.8 3.6 5.5 Unsmearing mo del dep endence 2.4 0.0 1.4 0.2 2.2 Hydrogen bac kground 0.8 1.0 1.6 0.9 2.2 c harge exc hange + absorption 3.0 2.9 3.3 3.0 3.6 T otal 15.5 13.8 17.2 15.0 18.6 co v ariance matrix can b e used to separate the correlated normalization uncertain ties from the total error, lea ving information related to ho w m uc h the shap e of the observ ed data ma y v ary within the systematic errors [ 69 ]. These uncertain ties are iden tied b y rst dening a data v ector V with en tries corresp onding to the observ ed relativ e normalization of eac h bin: V i = f D 1 =D T ; D 2 =D T ; ; D n =D T ; D T g Notice this v ector has dimension n + 1, where n is the n um b er of bins measured. The co v ariance matrix Q for this new v ector V in v olv es the Jacobian matrix of partial deriv ativ es J and is giv en b y: Q k l = n X ij J k i M ij J l j = n X ij @ V k @ D i M ij @ V l @ D j : (8.17) The diagonals of the matrix Q are related to the shap e uncertain t y in eac h kinematic bin. F or en tries f 1 ; 2 ; ; n g 177

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Q k k = 1 D 2 T M k k 2 D k D T n X i M ik + N 2 k N 2 T n X ij M ij # (8.18) = ( D k ; shap e ) 2 The shap e and total error of the cross sections is compared to MC normalized to data in Figure 8-21 As v alues for M A are t ypically extracted from the Q 2 distribution, it's helpful to at least calculate the compatibilit y b et w een data and MC. Prin ted on the Q 2QE distribution is the 2 b et w een MC and data using shap e-only uncertain ties. 178

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(GeV) RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 (cm s 2 4 6 8 10 12 -39 10 ) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 /GeV 2 (cm QE 2 /dQ s d 2 4 6 8 10 12 -39 10 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /GeV) 2 (cm m /dT s d 0.5 1 1.5 2 2.5 3 3.5 4 -39 10 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ) 2 ) (cm m q /d(cos s d 2 4 6 8 10 12 14 -39 10 ) 2 CCQE data with stat, syst error (CH m n = 1.007 k ) = 1.35 (1.02) GeV, H A (M C A MC: M (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 -39 10 /GeV) 2 (cm ) m q d(cos m dT s 2 d 2 CH Figure 8-19. Extracted CCQE cross sections with total uncertain t y compared to cen tral v alue MC with the h ydrogen comp onen t not subtracted. 179

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/GeV) 2 (cm ) m q d(cos m dT s 2 d (GeV) m T (GeV) m T (GeV) m T (GeV) m T (GeV) m T 0 5 10 -39 10 < 1.0 m q 0.9 < cos 0 5 10 -39 10 < 0.9 m q 0.8 < cos 0 2 4 6 -39 10 < 0.8 m q 0.7 < cos 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 -39 10 < 0.7 m q 0.6 < cos 0 1 2 3 -39 10 < 0.6 m q 0.5 < cos 0 1 2 -39 10 < 0.5 m q 0.4 < cos 0 1 2 -39 10 < 0.4 m q 0.3 < cos 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 -39 10 < 0.3 m q 0.2 < cos 0 0.5 1 -39 10 ) 2 MB data (CH = 1.35 GeV C A RFG: M = 1.02 GeV C A RFG: M < 0.2 m q 0.1 < cos 0 0.5 1 -39 10 < 0.1 m q 0.0 < cos 0 0.5 1 -39 10 < 0.0 m q -0.1 < cos 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 -39 10 < -0.1 m q -0.2 < cos 0 0.2 0.4 -39 10 < -0.2 m q -0.3 < cos 0 0.2 0.4 -39 10 < -0.3 m q -0.4 < cos 0 0.2 0.4 -39 10 < -0.4 m q -0.5 < cos 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 -39 10 < -0.5 m q -0.6 < cos 0 0.1 0.2 0.3 -39 10 < -0.6 m q -0.7 < cos 0 0.2 -39 10 < -0.7 m q -0.8 < cos 0 0.1 0.2 -39 10 < -0.8 m q -0.9 < cos 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 -39 10 < -0.9 m q -1.0 < cos Figure 8-20. Pro jections of the double-dieren tial cross section on mineral oil with total uncertain t y F or MC, = 1.007 (1.000) for MC A= 1 : 35 (1 : 02) GeV and in b oth cases MH A= 1 : 02 GeV.180

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(GeV) RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 (cm s 2 4 6 8 10 12 -39 10 data CV shape error total error 1.10 MC NOT subtracted 2 H 11.7% total norm. error ) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 /GeV 2 (cm QE 2 /dQ s d 2 4 6 8 10 12 -39 10 data CV shape error total error /dof = 6.24 / 17 2 c 1.11; MC NOT subtracted 2 H 10.2% total norm. error (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /GeV) 2 (cm m /dT s d 0.5 1 1.5 2 2.5 3 3.5 4 -39 10 data CV shape error total error 1.15 MC NOT subtracted 2 H 11.1% total norm. error m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ) 2 ) (cm m q /d(cos s d 2 4 6 8 10 12 14 -39 10 data CV shape error total error 1.10 MC NOT subtracted 2 H 11.4% total norm. error (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 -39 10 /GeV) 2 (cm ) m q d(cos m dT s 2 d shape error total error Figure 8-21. Extracted CCQE cross sections with total and shap e uncertain t y compared to cen tral v alue MC with the h ydrogen comp onen t not subtracted. F or the dieren tial cross sections, MC is scaled to the in tegrated cross section from data ( R d dX dX ), while for the total cross section MC is scaled to data based on the discrepancy in the bin at the in teraction p eak (0.65 0.7 GeV). 181

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8.5.2 Results Inciden t on Carb on Results for all distributions treating the free scattering CCQE comp onen t as bac kground are sho wn in Figure 8-22 and Figure 8-23 presen ts a detailed view of the double-dieren tial cross section with comparisons to the RF G and three external predictions [ 98 99 96 ]. Analogous to Figure 8-21 Figure 8-24 compares shap e and total errors on data to relativ ely normalized MC. As CCQE mo del parameters are t ypically extracted from the Q 2 distribution, it's in teresting to see ho w these data compare to the historically-accepted v alues. Figure 8-25 compares the shap e of the RF G with v arious c hoices of M A and to the data. T o giv e a feel n umerically for the shap e compatibilit y of eac h distribution with the data, prin ted on the gure legend is the 2 for eac h parameter c hoice using the shap e-only uncertain ties. The 2 for the RF G with M A = 1.35 GeV, = 1.007 is surprisingly lo w at just 3.7 for 17 degrees of freedom. Along the same lines, one of the only recen t exp erimen ts using n uclear targets to measure M A 1 GeV is the NOMAD exp erimen t. Muc h sp eculation rev olving around the disparate energy regimes and detector t yp es has b een made in attempts to reconcile the discrepancy in CCQE cross section b et w een MiniBo oNE and NOMAD [ 140 ], and Figure 8-26 compares the and CCQE data sets from b oth exp erimen ts. Figure 8-27 compares the total cross sections to the a v ailable theoretical predictions [ 93 95 96 97 98 99 100 ], and Figure 8-28 also includes the lev el of agreemen t b et w een the same mo dels and the data. It is clear that the RF G mo del with canonical assumptions do es not adequately describ e these data neither in shap e nor in normalization. Consisten t with other recen t CCQE measuremen ts on n uclear material [ 84 87 88 125 ], a signican t enhancemen t in the normalization that gro ws with decreasing m uon scattering angle is observ ed compared to the exp ectation with M A = 1 GeV. As discussed in Section 4.4 these observ ations are 182

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(GeV) RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 (cm s 2 4 6 8 10 12 -39 10 ) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 /GeV 2 (cm QE 2 /dQ s d 2 4 6 8 10 12 -39 10 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /GeV) 2 (cm m /dT s d 0.5 1 1.5 2 2.5 3 3.5 4 -39 10 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ) 2 ) (cm m q /d(cos s d 2 4 6 8 10 12 14 -39 10 subt.) 2 CCQE data with stat, syst error (H m n = 1.007 k = 1.35 GeV, A MC: M (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 -39 10 /GeV) 2 (cm ) m q d(cos m dT s 2 d Figure 8-22. Extracted CCQE cross sections with total uncertain t y compared to cen tral v alue MC with the h ydrogen comp onen t subtracted. 183

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/GeV) 2 (cm ) m q d(cos m dT s 2 d (GeV) m T (GeV) m T (GeV) m T (GeV) m T (GeV) m T 0 5 10 -39 10 < 1.0 m q 0.9 < cos 0 5 10 -39 10 < 0.9 m q 0.8 < cos 0 2 4 6 -39 10 < 0.8 m q 0.7 < cos 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 -39 10 < 0.7 m q 0.6 < cos 0 1 2 3 -39 10 < 0.6 m q 0.5 < cos 0 1 2 3 -39 10 < 0.5 m q 0.4 < cos 0 1 2 -39 10 < 0.4 m q 0.3 < cos 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 -39 10 < 0.3 m q 0.2 < cos 0 0.5 1 1.5 -39 10 subt) 2 MB data (H Nieves et al. Amaro et al. Meucci et al. EDAI = 1.35 GeV A RFG: M = 1.02 GeV A RFG: M < 0.2 m q 0.1 < cos 0 0.5 1 -39 10 < 0.1 m q 0.0 < cos 0 0.5 1 -39 10 < 0.0 m q -0.1 < cos 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 -39 10 < -0.1 m q -0.2 < cos 0 0.2 0.4 -39 10 < -0.2 m q -0.3 < cos 0 0.2 0.4 -39 10 < -0.3 m q -0.4 < cos 0 0.2 0.4 -39 10 < -0.4 m q -0.5 < cos 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 -39 10 < -0.5 m q -0.6 < cos 0 0.1 0.2 0.3 -39 10 < -0.6 m q -0.7 < cos 0 0.2 -39 10 < -0.7 m q -0.8 < cos 0 0.1 0.2 -39 10 < -0.8 m q -0.9 < cos 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 -39 10 < -0.9 m q -1.0 < cos Figure 8-23. Pro jections of the double-dieren tial cross section on carb on with total uncertain t y .184

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(GeV) RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 (cm s 2 4 6 8 10 12 -39 10 data CV shape error total error 1.15 MC subtracted 2 H 15.5% total norm. error ) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 /GeV 2 (cm QE 2 /dQ s d 2 4 6 8 10 12 -39 10 data CV shape error total error 1.16 MC subtracted 2 H 13.8% total norm. error (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /GeV) 2 (cm m /dT s d 0.5 1 1.5 2 2.5 3 3.5 4 -39 10 data CV shape error total error 1.19 MC subtracted 2 H 17.2% total norm. error m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ) 2 ) (cm m q /d(cos s d 2 4 6 8 10 12 14 -39 10 data CV shape error total error 1.14 MC subtracted 2 H 15.0% total norm. error (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 -39 10 /GeV) 2 (cm ) m q d(cos m dT s 2 d subtracted 2 H shape error total error Figure 8-24. Extracted CCQE cross sections with total and shap e uncertain t y compared to cen tral v alue MC with the h ydrogen comp onen t subtr acted F or the dieren tial cross sections, MC is scaled to the in tegrated cross section from data ( R d dX dX ), while for the total cross section MC is scaled to data based on the discrepancy in the bin at the in teraction p eak (0.65 0.7 GeV). 185

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) 2 (GeV QE 2 Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 /GeV 2 (cm QE 2 /dQ s d 2 4 6 8 10 12 -39 10 subtracted) 2 data with shape error (H / dof = 90.34 / 17 2 c 1.34; = 1.000, k = 1.02 GeV, A RFG: M / dof = 3.73 / 17 2 c 1.16; = 1.007, k = 1.35 GeV, A RFG: M / dof = 22.84 / 17 2 c 1.11; = 1.000, k = 1.35 GeV, A RFG: M Figure 8-25. Shap e comparison for h ydrogen-subtracted data and the RF G under v arious parameter c hoices. The shap e of cen tral v alue MC with M A = 1.35 GeV and = 1.007 agrees with data remark ably w ell. In particular, the mild c hange in (1 : 000 1 : 007) determined from the CCQE analysis seems to b e preferred b y the data. Also prin ted on the gure is the scale for eac h prediction to matc h the data in normalization. (GeV) QE,RFG n E 1 10 2 10 ) 2 (cm s 2 4 6 8 10 12 14 16 18 20 -39 10 MiniBooNE m n NOMAD m n = 1.007 k = 1.35 GeV, eff,C A RFG: M m n = 1.000 k = 1.02 GeV, eff,C A RFG: M m n MiniBooNE (hyd subt.) m n NOMAD m n = 1.007 k = 1.35 GeV, eff,C A RFG: M m n = 1.000 k = 1.02 GeV, eff,C A RFG: M m n Figure 8-26. Comparison b et w een MiniBo oNE and NOMAD and data, as w ell as some predictions from the RF G. T ension exists across b oth and data from the t w o exp erimen ts under the assumptions of CCQE with the RF G. NOMAD data from Ref. [ 171 ]. 186

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consisten t with the presence of an in tra-n uclear mec hanism of greater imp ortance than previously exp ected, and con tributions from suc h a source are tested in comparisons b et w een v arious predictions and the data analyzed here in Figures 8-23 and 8-27 Ho w ev er, these data nd tension with the NOMAD CCQE results, whic h are describ ed b oth in shap e and normalization b y M A 1 GeV [ 90 ]. This tension is also common to the CCQE analyses from the t w o exp erimen ts. Ho w ev er, care should b e tak en in comparing mo del-dep enden t results among exp erimen ts with suc h dieren t neutrino ruxes and detector tec hnologies. A denitiv e unication of these apparen tly discrepan t data sets will require the con tin ued increase of b oth exp erimen tal and theoretical activit y surrounding this topic. F ortunately man y exp erimen ts at a v ariet y of neutrino energies capable of making high-resolution, mo del-indep enden t neutrino and an tineutrino CCQE measuremen ts with dieren t detector tec hnologies and n uclear media curren tly ha v e data or will so on. These include MINER A [ 145 ], SciBo oNE [ 172 ], MicroBo oNE [ 173 ], ArgoNeuT [ 174 ], ICAR US [ 175 ] and the T2K [ 141 ] and NO A [ 54 ] near detectors. 187

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(GeV) n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 (cm s 0 2 4 6 8 10 -39 10 subt.) with total error 2 Data (H = 1.35 GeV A RFG M = 1.02 GeV A RFG M = 1.02 GeV A Free nucleon M Martini et al. Amaro et al. Bodek et al. Nieves et al. Meucci et al. EDAI Figure 8-27. T otal cross section p er n ucleon for CCQE data with the h ydrogen con ten t remo v ed compared to v arious predictions. 188

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(GeV) n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 (cm s 2 4 6 8 10 12 14 16 18 20 22 -39 10 data m n RFG m n Martini et. al m n Amaro et. al m n Bodek et. al m n Nieves et. al m n Meucci et. al EDAI m n subt.) 2 data (H m n RFG m n Martini et. al m n Amaro et. al m n Bodek et. al m n Nieves et. al m n Meucci et. al EDAI m n Figure 8-28. T otal cross section p er n ucleon for and CCQE data compared to v arious predictions. The \RF G" distribution assumes M A = 1.35 GeV and = 1.007. T otal uncertain t y is sho wn with b oth MiniBo oNE data sets. 189

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CHAPTER 9 COMBINED AND CCQE MEASUREMENTS With the high-statistics MiniBo oNE (Ref. [ 84 ]) and (Chapter 8 and Ref. [ 147 ]) CCQE cross sections, an opp ortunit y exists to extract ev en more information out of these data sets b y exploiting correlated systematic uncertain ties b et w een the t w o measuremen ts. Simple dierence and ratio analyses b et w een the t w o results will more stringen tly test the v arious mo dels for CCQE-lik e in teractions around 1 GeV. W e b egin with a brief discussion of ho w to use correlated information to reduce uncertain ties in com bined measuremen ts. 9.1 Correlated Measuremen ts This treatmen t of systematic correlations follo ws Ref [ 176 ]. Consider t w o arbitrary results x and y that are used to calculate some com bined measuremen t q ( x; y ) As in an y quan tit y giv en the set of N measuremen ts of the quan tit y q its uncertain t y q is: 2 q = 1 N N X i ( q i q ) 2 ; (9.1) where q is the CV measuremen t of q W e are in terested in ho w the ob ject q c hanges under v ariations in x and y so w e b egin b y examining ho w individual excursions x i and y i from their resp ectiv e results ( x; y ) propagate on to q : q i = q ( x i ; y i ) (9.2) = 1 X n 1 =0 1 X n 2 =0 ( x i x ) n 1 ( y i y ) n 2 n 1 n 2 @ n 1 + n 2 @ x n 1 @ y n 2 q ( x; y ) (9.3) q ( x; y ) + ( x i x ) @ @ x q ( x; y ) + ( y i y ) @ @ y q ( x; y ) ; (9.4) where disregarding the higher order terms in the last step assumes the deviations ( x i x ) and ( y i y ) to b e small. Recognizing q ( x; y ) = q and substituting Eq. 9.4 in to Eq. 9.1 w e 190

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get: 2 q = 1 N N X i @ q @ x ( x i x ) + @ q @ y ( y i y ) 2 (9.5) = @ q @ x 2 1 N N X i ( x i x ) 2 + @ q @ y 2 1 N N X i ( y i y ) 2 +2 @ q @ x @ q @ y 1 N N X i ( x i x )( y i y ) ; (9.6) where the partial deriv aties of q are still ev aluated at the p oin t ( x ; y ). The rst t w o terms in Eq. 9.6 are readily recognized as the standard deviations x and y while the last term giv es information ab out the correlation b et w een x and y It is easy to see that if the measuremen ts of x and y w ere indep enden t of one another, the last term will approac h zero as N 1 It is con v enien t to dene a correlation co ecien t xy in terms of this information and the standard deviations of x and y : xy = P Ni ( x i x )( y i y ) q P Ni ( x i x ) 2 P Ni ( y i y ) 2 = 1 N P Ni ( x i x )( y i y ) x y : (9.7) Notice that xy 2 f 1 ; 1 g Then Eq. 9.6 b ecomes: 2 q = @ q @ x 2 2 x + @ q @ y 2 2 y + 2 @ q @ x @ q @ y xy x y : (9.8) Dep ending on the sign of the pro duct @ q @ y @ q @ y xy the uncertain t y on the measuremen t of q will either b e increased or reduced b y including the correlation information. 191

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9.2 Com bined and CCQE Measuremen ts Man y systematic uncertain ties of the MiniBo oNE and CCQE cross-section results are related to the resolution of and + kinematics in the detector, and are therefore exp ected to aect the t w o measuremen ts in a similar w a y Up on examination of Eq. 9.7 suc h an exp ectation w ould predict the correlation co ecien t to b e a p ositiv e quan tit y in most regions. Then, to form a com bined measuremen t for the and CCQE cross sections that features greater precision b y including the information ab out their correlation, the sign of @ q @ @ q @ ough t to b e negativ e. Tw o simple cases of this are dierence ( ) and ratio measuremen ts of the cross sections in the v arious distributions, most imp ortan tly the double-dieren tial cross section d 2 dT d cos It is imp ortan t to men tion that this study is incomplete: while all systematic uncertain ties inheren t to the MiniBo oNE instrumen ts are included here, p ossible correlations b et w een the + and pro duction data from the HARP exp erimen t (Section 5.2.4 ) are unkno wn. In this study the correlation b et w een the and paren t + and primary pro duction cross sections are assumed to b e uncorrelated. As the uncertain ties on these quan tities signican tly con tribute to the and CCQE cross-section errors, it will b e imp ortan t to ev en tually include this information. The goal of this study is to simply ev aluate the lev el of correlation b et w een eac h bin in the v arious and CCQE cross sections in order to use Eq. 9.7 to extract the most information p ossible from the MiniBo oNE data sets. T o more easily in terpret the results of this study as measuremen ts of n uclear eects in carb on, w e use the CCQE cross section congurations in whic h the h ydrogen CCQE comp onen t is treated as bac kground. W e b egin b y forming the co v ariance matrix to b e used in the calculation of an arbitrary com bined measuremen t q in the same w a y as presen ted in Section 7.1.5 : the v arious and CCQE cross sections and the systematic uncertain t y histograms are com bined in to a single distribution, side-b y-side. Then, as in Eq. 7.26 a co v ariance matrix is formed that no w includes the correlation information b et w een eac h and CCQE bin. 192

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The details of the calculated co v ariance matrix oer a few imp ortan t consistency c hec ks: the normalization uncertain t y (Eq. 8.16 ) of eac h systematic error when only considering the or region of the co v ariance matrix m ust b e compared to the original normalization uncertain t y ndings, and the calculated cross sections from data m ust of course matc h the results of the original analyses. In this analysis, the relev an t quan tities matc h the original and CCQE results within a few p ercen t of their v alue. Also imp ortan t to note when cross-c hec king these results, mild statistical dierences are exp ected b et w een the obtained neutrino-mo de CCQE cross sections compared to those in Ref. [ 84 ] due to the inclusion of additional data. The analysis in Ref. [ 84 ] includes a total of 5 : 6 10 20 POT, while w e use the additional neutrino-mo de data collected since then in this analysis for a total of 6 : 4 10 20 POT. Figure 9-1 sho ws the resultan t cross sections, along with the reco v ered total uncertain t y from the diagonal en tries of the co v ariance matrix. The o v erall correlation co ecien ts b et w een eac h bin in the and CCQE cross sections are ev aluated through Eq. 9.7 where the term 1 N P Ni ( x i x )( y i y ) can b e recognized as a giv en en try in the co v ariance matrix b et w een t w o arbitrary bins x and y Figure 9-2 sho ws the o v erall correlation b et w een eac h bin in the and CCQE total and single-dieren tial cross sections. Recall correlations in the HARP data are not tak en in to accoun t, so the correlations presen ted here are generally exp ected to b e a mild underestimate. Also sho wn in the same gure is the total correlation b et w een a giv en bin in the and CCQE double-dieren tial cross sections. Figure 9-2 sho ws the correlations b et w een the and CCQE cross sections to b e rather mild. This is mostly due to the presence of large and CC bac kgrounds unique to the CCQE analysis. Figure 9-3 compares the co ecien ts ; for the most imp ortan t correlated systematic uncertain ties. F rom this correlation information, w e will extract t w o quan tities: the dierence b et w een the (\A") and (\B") CCQE measuremen ts q di = A B and their ratio 193

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QE,RFG n E ) 2 (cm s 0 2 4 6 8 10 12 14 16 18 20 -39 10 m n m n QE 2 Q ) 2 /GeV 2 (cm QE 2 /dQ s d 0 2 4 6 8 10 12 14 16 18 20 -39 10 m n m n Figure 9-1. Reco v ered (left half of eac h distribution) and (righ t half of eac h distribution) CCQE cross sections for the correlation analysis. Eac h bin in the E QE ;RF G distribution (left) is 2 f 0 : 4 ; 2 : 0 g GeV and is iden tical to the bins delimited in Figure 8-27 while the Q 2QE distribution (righ t) is 2 f 0 : 0 ; 2 : 0 g GeV 2 and corresp onds to the binning in Figure 8-25 194

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q ratio = A B The application of Eq. 9.8 to q di is straigh tforw ard: 2 q di = 2 A + 2 B 2 AB A B ; (9.9) while the uncertain t y for q ratio : 2 q ratio = A B 2 + A A B 2 2 2 A B 3 AB A B (9.10) can b e written more coheren tly as a com bination of fractional uncertain ties: q ratio q ratio 2 = A A 2 + B B 2 2 AB A A B B : (9.11) Notice Eqs. 9.9 and 9.11 are symmetric under A \$ B Using these expressions for the uncertain t y and the lev el of correlation b et w een eac h bin in the and CCQE cross section distributions, the com bined measuremen ts ma y b e executed. Figure 9-4 sho ws the dierence measuremen ts, while Figure 9-5 presen ts results from the ratio analysis. There is some indep enden t information gleaned when comparing v arious predictions to the data across b oth the dierence and the ratio measuremen ts: the ratio q ratio is sensitiv e only to the absolute normalization of the and CCQE cross sections, while the dierence q di is also sensitiv e to the relativ e normalization b et w een the t w o cross sections. Up to the inclusion of the correlation of the HARP + and pro duction data, these measuremen ts represen t the extraction of the most CCQE information p ossible with the MiniBo oNE detector using only observ ations of the m uon. 195

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-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ) correlation matrix QE,RFG n (E s m n m n m n m n ) correlation matrix QE,RFG n (E s 0 0.2 0.4 0.6 0.8 1 correlation matrix QE 2 /dQ s d m n m n m n m n correlation matrix QE 2 /dQ s d 0 0.1 0.2 0.3 0.4 0.5 0.6 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 n n r CCQE total correlation m n m n n n r CCQE total correlation m n m n Figure 9-2. Correlation matrices for the v arious and cross section results. T o highligh t the imp ortan t information, only the ; co ecien ts are sho wn for eac h bin in the d 2 dT d cos distribution (b ottom). The analogous en tries corresp ond to the on-diagonal co ecien ts of the o-diagonal blo c ks in the ( E QE ;RF G ) (top left) and d =dQ 2QE (top righ t) matrices. 196

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(GeV) QE,RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 n n r -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Total correlation OM beam unisims reinteraction background p ) 2 (GeV 2 QE Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 n n r -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Total correlation OM beam unisims reinteraction background p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 n n r reinteration background p n n r reinteration background p -0.4 -0.2 0 0.2 0.4 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 n n r beam unisims n n r beam unisims -0.2 0 0.2 0.4 0.6 0.8 (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 n n r optical model n n r optical model Figure 9-3. Correlation co ecien ts b y uncertain t y t yp e for the v arious distributions. As exp ected, the correlations are mostly p ositiv e b et w een the and CCQE measuremen ts. 197

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(GeV) QE,RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 (cm s CCQE m n m n 3 4 5 6 7 8 9 -39 10 uncorrelated errors correlated errors RFG Amaro et al. Bodek et al. Martini et al. Nieves et al. Meucci et al. EDAI ) 2 (GeV 2 QE Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) 2 / GeV 2 (cm 2 QE /dQ s CCQE d m n m n 0 2 4 6 8 10 12 -39 10 uncorrelated errors correlated errors = 1.007 k = 1.35 GeV, A RFG: M = 1.000 k = 1.02 GeV, A RFG: M (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 16 18 -39 10 /GeV) 2 (cm ) m q d(cos m dT s 2 d CCQE m n m n uncorrelated error correlated error Figure 9-4. Dierence measuremen ts of the and CCQE cross sections compared to v arious predictions. The \RF G" curv e in the top left gure assumes M A = 1.35 GeV and = 1.007. The \uncorrelated uncertain t y", found b y setting ; = 0 in the uncertain t y determination, is included to appreciate the lev el of sensitivit y gained b y using the correlation information. 198

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(GeV) QE,RFG n E 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s CCQE m n / m n 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 uncorrelated errors correlated errors RFG Amaro et al. Bodek et al. Martini et al. Nieves et al. Meucci et al. EDAI ) 2 (GeV 2 QE Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 QE /dQ s CCQE d m n / m n 0 0.2 0.4 0.6 0.8 1 uncorrelated errors correlated errors = 1.007 k = 1.35 GeV, A RFG: M = 1.000 k = 1.02 GeV, A RFG: M (GeV) m T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m q cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) m q d(cos m dT s 2 d CCQE m n m n uncorrelated error correlated error Figure 9-5. Ratio measuremen ts of the and CCQE cross sections compared to v arious predictions. The \RF G" curv e in the top left gure assumes M A = 1.35 GeV and = 1.007. The \uncorrelated uncertain t y", found b y setting ; = 0 in the uncertain t y determination, is included to appreciate the lev el of sensitivit y gained b y using the correlation information. 199

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CHAPTER 10 CONCLUSION This dissertation presen ts a v ariet y of an tineutrino CCQE cross sections, including the minimally mo del-dep enden t double-dieren tial cross section. While the cross section congurations including the free scattering comp onen t of CCQE in teractions in the detector features the least mo del dep endence and is the main result of this w ork, the results ev oking an axial mass M A 1 GeV to exclude the h ydrogen comp onen t are also giv en to facilitate historical comparisons for neutrino and an tineutrino in teractions on carb on. This result is also used to test mo dern n uclear mo dels that predict ho w a bac kground arising from in tra-n uclear correlations of greater size than exp ected migh t con tribute to the analysis sample. These data are the rst an tineutrino cross-section results b elo w 1 GeV, a crucial energy region for presen t and future neutrino oscillation exp erimen ts lo oking to measure CP violation. T o facilitate this measuremen t, no v el and crucial ev aluations of the bac kground to the CCQE sample w ere dev elop ed and executed. In the absence of a magnetic eld, the analyses describ ed in Chapter 7 measure the rux of the an tineutrino mo de b eam with 15% fractional uncertain t y These tec hniques could b e used in curren t and future neutrino oscillation programs, particularly when mo dest c harge iden tication is sucien t to meet the ph ysics goals [ 144 ]. The com bined measuremen ts of the MiniBo oNE and CCQE cross sections extract the most information of the CCQE pro cesses p ossible with the MiniBo oNE detector using only observ ations of the m uon. While these measuremen ts are en tirely ignoran t of hadronic activit y these analyses a v oid mo del dep endence t ypically asso ciated with quan tities suc h as momen tum trac king thresholds and n ucleon rein teractions used to iden tify CCQE in teractions. 200

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BIOGRAPHICAL SKETCH Joseph Grange receiv ed his bac helor's degree from the Univ ersit y of Puget Sound in 2006. His degrees from the Univ ersit y of Florida include a Master of Science in 2009 and a Do ctor of Philosoph y in 2013. 211