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## Material Information- Title:
- Renormalization group analyses of the standard model and its minimal supersymmetric extension
- Creator:
- Wright, Brian D., 1963-
- Publication Date:
- 1992
- Language:
- English
- Physical Description:
- v, 101 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Average linear density ( jstor )
Fermions ( jstor ) Grand unified theory ( jstor ) Leptons ( jstor ) Mass ( jstor ) Physics ( jstor ) Quarks ( jstor ) Renormalization group ( jstor ) Scalars ( jstor ) Supersymmetry ( jstor ) Dissertations, Academic -- Physics -- UF Nuclear physics ( lcsh ) Physics thesis Ph. D - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1992.
- Bibliography:
- Includes bibliographical references (leaves 94-100).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Brian D. Wright.
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RENORMALIZATION GROUP ANALYSES OF THE STANDARD MODEL AND ITS MINIMAL SUPERSYMMETRIC EXTENSION By BRIAN D. WRIGHT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1992 CF , ACKNOWLEDGEMENTS I would like to acknowledge those who have helped me to pursue the study of particle physics. I have encountered many setbacks over the years and I am indebted to my advisor Pierre Ramond for his patience and guidance. He has taught me most of what I know of particle physics. This work would not have been possible without the contributions of my other collaborators Haukur Arason, Diego Castafio, Bettina Keszthelyi, Sam Mikaelian and Eric Piard. I would like to thank Diego and Eric for the use of the / function appendices, Sam for his thoughts on quark masses, and Haukur for discussions on the extraction of the strong coupling constant. I am especially indebted to Bettina for her collaboration in the threshold calculations. I would not have been possible to get through these years without Haukur's ice cream and cookie sessions, Dr. Gary Kleppe's good evenings in the morning and burning the midnight oil with Sam and Bettina. I am especially grateful to Bettina for her patience during the writing of this work and helping to make me more human. I must thank my family for their support during my absence from the real world. TABLE OF CONTENTS ACKNOWLEDGEMENTS ................... ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTERS 1 INTRODUCTION ...................... 2 SUPERSYMMETRY AND THE STANDARD MODEL Page ii . v . 1 . 6 .6 ~8 . . . . . 6 . . . . . 8 . . . . . 10 . . . . . 13 . . . . 17 . . . . . 18 . . . . . 24 . . . . . 29 . . . . . 32 . . . . .32 . . . . .34 . . . . . 39 . . . . . 41 . . . . .42 . . . . . 43 . . . . . 44 . . . . . 47 . . . . 47 . . . . . 51 . . . . . 55 . . . . 58 2.1 Overview and Motivation . . . . . . . . . . . . . 2.2 Supersymmetry Generalities-Superfields . . . . . . . 2.3 The Minimal Supersymmetric Standard Model . . . . 2.4 Supersymmetry Breaking . . . . . . . . . . . . . 3 SUPERSYMMETRY AND GRAND UNIFIED THEORIES 3.1 Minimal SU(5) . . . . . . . . . . . . . . . . . . 3.2 Supersymmetric SU(5) . . . . . . . . . . . . . . . 4 THE RENORMALIZATION GROUP . . . . . . . . . 5 STANDARD MODEL PARAMETERS AT MZ . . . . . 5.1 al(MZ) and a2(Mz) .. . ............. 5.2 as(M z) . . . . . . . . . . . . . . . . . . . . 5.3 Yukawa Couplings ................. 5.4 Known Quark Masses ............... 5.5 Lepton Masses ............ ...... 5.6 Higgs Boson and Top Quark Masses . . . . . . . . . 5.7 Vacuum Expectation Value of the Scalar Field . . . . 6 THRESHOLD EFFECTS . . . . . . ............. 6.1 Effective Gauge Theories . . . . . . . . . . . . . . 6.2 Gauge Coupling Thresholds in the SM . . . . . . . . 6.3 Threshold Effects in Fermion Masses (Yukawas) . . . . 6.4 Thresholds Beyond the Standard Model . . . . . . . . . . . 7 NUMERICAL TECHNIQUES . . . . . . . . . . . . 8 RG ANALYSIS OF THE STANDARD MODEL . . . . 9 RG ANALYSIS OF THE MSSM and SUSY-SU(5) . . . 10 CONCLUSIONS .................. APPENDIX I THE STANDARD MODEL p FUNCTIONS APPENDIX II THE MSSM 0 FUNCTIONS . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . BIOGRAPHICAL SKETCH ............... . . . . . 61 . . . . . 63 . . . . . 71 . . . . . 79 . . . . . 81 . . . . . 88 . . . . . 94 . . . . 101 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillmnt of the Requirements for the Degree of Doctor of Philosophy RENORMALIZATION GROUP ANALYSES OF THE STANDARD MODEL AND ITS MINIMAL SUPERSYMMETRIC EXTENSION By BRIAN D. WRIGHT December 1992 Chairman: Pierre Ramond Major Department: Physics This work presents some aspects of reno-nalization group studies in the Standard Model and its minimal supersymmetric extension with an eye toward their application to grand unified predictions. We give an overview of the Minimal Supersymmetric Standard Model (MSSM.) and grand unified models to highligh.-t relevant issues. Ve next present a comprehensive analysis of the running of all the couplings of the Standard Model to two loops, including thresholds effects. The purpose ;is twofold-to determine what the running of these parameters may indicate for the physics of the Standard Model and to provide a template for the study of i:s extensions up to Planck mass. WVe discuss in detail the subject of extract:in inital data on Standard Model parameters and numerical methods applied in the analysis. This material is meant to be a complete toolkit for any renormaHzation group study including the Standard Model in the low energy regime. We then apply these tools to run all the couplings of the MISSM, taking full acco' .nt of the Yukawa sector. WVe note the successful unification of gauge couplings in this: model given the best data available. After identifying the scale v CHAPTER 1 INTRODUCTION The Standard Model of the strong and electroweak interactions has so far successfully met all experimental challenges. The situation may soon change if the proper resolution of the solar neutrino problem [1] calls for the existence of massive neutrinos [2]. Other unresolved problems are theoretical in nature. There is no explanation of the mass spectrum, the replication of fermion families or the chiral structure of the Standard Model. Nor is there an adequate explanation of the smallness of strong CP violation or the hierarchy of scales between the electroweak scale and Planck mass. Some of these aesthetic problems find partial resolution, when the gauge group of the Standard Model is imbedded in some larger symmetry group. Certain relations between parameters of the theory arise from the higher symmetry and can be extrapolated to lower energies by means of the renormalization group (RG). Thus, one approach to seeking structure beyond the Standard Model is to build such models and attempt to deduce low energy measurable predictions which can test their viability. This has led to the early grand unified theories (GUTs) [3,4,5,6]. A second approach [7] is to use the renormalization group to extrapolate the Standard Model parameters to the unexplored scales. The purpose is to find if those parameters satisfy interesting relations at shorter distances. When used in conjunction with the former approach, this can give powerful hints of the physics expected at shorter scales. Of course, it all depends on having accurate data to input as initial conditions for the renormalization group equations, as well as a strong theoretical basis for the evolution equations themselves. The minimal SU(5) GUT is the prototype for such analyses [8]. There, properties 2 of the model at energies of order 1015 GeV are translated with the aid of the renormalization group to a prediction for proton decay that is not consistent with experiment. Data on the coupling constants is now sufficiently precise to rule out most simple GUTs, including SU(5), because of the absence of unification of the running couplings (GUT triangle) [9,10,11]. The same analysis has recently improved the feasibility of the supersymmetric (SUSY) extensions of these GUTs [9,10,11,12,13]. More recently, constraints coming from Yukawa coupling unification in supersymmetric SU(5) and SO(10) models have led to bounds on the mass of the top quark [14,15,16]. Renormalization group methods are of enduring practical importance in the attempts of high energy physicists to glean indications of more fundamental theories from radiative corrections. The purpose of this work is twofold. We first attempt to give a comprehensive guide to the renormalization group techniques necessary to study the Standard Model and its extensions (see also ref. 17). In particular we emphasize the importance of threshold corrections in renormalization group analyses. Such corrections, incorporating information about as yet unseen heavy mass spectra, are becoming increasingly important as parameters of the Standard Model become more precisely known. The possibility of probing for such spectra in the deep ultraviolet through the use of the renormalization group is an exciting propect. We discuss some threshold effects not previously described in the literature. Second, we apply these techniques to the minimal supersymmetric extension of the Standard Model (MSSM) in the context of SU(5) grand unified predictions. We find significant theoretical bounds on the mass of the top quark in order to maintain gauge coupling and Yukawa unification in this model. The 3 bounds are such that the model considered can be ruled out if a top quark near its present experimental lower bound is discovered. We view these results as a success of our approach. An outline of this work follows. We present an overview of supersymmetry and the MSSM in Chapter 2. The hierarchy problem is discussed as a primary motivation for supersymmetry. A brief introduction to the superfield formalism is given to establish our notational conventions. We then discuss the particle content of the MSSM and consider mechanisms for supersymmetry breaking. Finally we present the form of the scalar potential when soft symmetry breaking effects from a supergravity theory are included. Chapter 3 gives a short review of grand unified theories and their supersymmetric extensions. Similarities and differences in their predictions are pointed out, especially with regard to proton decay. The difference in renormalization behaviors due to the presence of superpartner fields in the supersymmetric case is discussed, translating to very different comparisons with experiment. We find that a minimal SU(5) GUT is ruled out experimentally while its supersymmetric extension has dramatic agreement with available data. An introduction to renormalization and scheme dependence is presented in Chapter 4. Chapters 5, 6 and 7 provide a toolkit for renormalization group studies involving the Standard Model as a low energy theory. A review of initial data extraction from experiments is presented in Chapter 5. Many excellent reviews may be found in the literature, e.g., Marciano's [18] or Peccei's [19]. We identify the values of the various parameters at the different scales where they are most accurately known. The determination of the gauge couplings is discussed in Sections 5.1 and 5.2. Initial data extraction of the Yukawas and the CKM angles is discussed in Section 5.3. 4 The extraction of the quark masses from data is briefly discussed in Section 5.4. In the low energy regime, we consider it necessary to include the pure QCD three-loop contribution to our analysis of the running of the quark masses. Initial data for lepton masses follows in Section 5.5. In Section 5.6 we consider the extraction of and constraints on the physical top and Higgs masses. We address the scale dependence of the renormalized scalar vacuum expectation value in Section 5.7. In Chapter 6, we discuss how threshold effects are incorporated into our analysis. It is well known that in the context of GUTs the effects of particle thresholds are of great importance in analyzing their low energy predictions [20-21], such as decreasing the naive estimate of the proton lifetime [20]. For completeness we present a detailed analysis of threshold effects in the Standard Model where these effects are numerically less important; the two-loop effects dominate the effects of the electroweak threshold. This work is of theoretical interest because the same methods are applicable to other models. We include threshold effects in the running fermion masses, an analysis absent from the present literature. Chapter 7 briefly describes the numerical methods used to solve the complicated coupled two loop / functions with threshold matching functions applied at the one loop level. A quantitative analysis of our results for the case of the Standard Model makes up Chapter 8. We contrast the effects of using one-loop versus two-loop / functions and of including a proper versus a naive treatment of thresholds. We include plots of all the running parameters over the entire range of mass scales and also use these plots to display the effects discussed. Furthermore we present some tables with actual numerical differences associated with these effects. 5 In Chapter 9 we describe the application of our methods to the MSSM. We restrict our attention to testing supersymmetric SU(5) (SUSY-SU(5)) predictions for gauge and Yukawa coupling unification and what they indicate for Standard Model parameters. We find bounds on the mass of the top quark and the Higgs scalar in a minimal supersymmetric extension of the Standard Model (MSSM) with minimal Higgs structure imbedded in SU(5). These bounds will be subject to direct experimental tests in the next few years. CHAPTER 2 SUPERSYMMETRY AND THE STANDARD MODEL 2.1 Overview and Motivation Supersymmetry was introduced in the early 70s [22] as a generalization of the Poincard group which connects bosonic to fermionic degrees of freedom. At the same time such a symmetry was being studied in the context of dual resonance models [23] and was later applied in a four dimensional field theory [24]. The generators of the Lie superalgebra include those of the usual Poincar6 group as well as spinorial generators obeying a mixture of commutation and anticommutation relations. The Lie superalgebra for N = 1 supersymmetry (N is essentially the number of sets of spinorial generators) involves the additional relations: {QaQ4} =2omPm [Pm, Qa] =[Pm, QZ = 0 (2.1) {Qa, ,Q} ={Q10& } = 0 , where Q and Q are two component spinors and Pm is the generator of spacetime translations. These relations must be supplemented by generalized Jacobi identities. Equations (2.1) have many interesting consequences. An immediate consequence is that the numbers of bosonic and fermionic degrees of freedom in any irreducible representation of the superalgebra are equal. Another is that the Hamiltonian for a supersymmetric theory is positive definite: the ground state is a state of zero energy. The perhaps most often cited motivation for supersymmetry is its softening of ultraviolet divergences in Feynman amplitudes. In grand unified models this 7 enables one to avoid the gauge hierarchy problem [25]. This problem arises in gauge theories in which there is a large separation of energy scales which is unstable in the presence of radiative corrections. In GUTs [3,4,5,6,7], gauge couplings of order g ~ 10-2 create radiative corrections to ratios of the order of MEW/MGur 10-'. In general, without extreme fine-tuning of parameters, this separation of scales will not be maintained when radiative corrections are included [25]. The philosophy of this problem is that all extremely small parameters of a theory, e.g. small ratios of mass scales, must obtain radiative corrections of the same order or smaller to avoid such fine-tuning. This is the concept of naturalness and it ensures that the properties of the theory are stable against small variations in the fundamental parameters [26]. Ordinary GUTs are not natural in this sense as Higgs scalar masses for example receive corrections of the form m2 =m + g2A2, (2.2) where m0 is bare mass and A is some physical cutoff of order MGUT coming from quadratic divergences in the one loop corrections to the scalar mass. The scalar mass is thus additively renormalized by a term of order 1013 GeV which requires incredible adjustment of the bare mass at every order of perturbation theory to keep m - 100 GeV. Supersymmetric theories, through cancellations between graphs containing boson and fermion loops, eliminate such quadratic divergences and cure the GUT hierarchy problem. This statement must be qualified by the requirement that the effective supersymmetry breaking scale Msusy not be to much larger than the electroweak scale, since this would introduce yet another hierarchy problem. 8 2.2 Supersymmetry Generalities-Superfields Supersymmetric field theories are constructed in terms of irreducible representations of the supersymmetry algebra (2.1). In N = 1 supersymmetry these representations consist of essentially a pair of boson and fermion fields known as superpartners with the same mass and internal quantum numbers. Supersymmetric Yang-Mills gauge theories are constructed with chiral or scalar supermultiplets of helicity 0 and 1/2 and vector or gauge supermultiplets of helicity 1/2 and 1. These multiplets can be represented using the well-known superfield formalism. We present below a few aspects of this formalism intended to establish some conventions. Readers seeking more information should consult the literature [27]. Superfields are functions of superspace coordinates z = (x, 0, ), where 0 and # are anticommuting Grassmann variables. The transformations properties of the component fields of a superfield are obtained by applying elements of (2.1). Infinitesimal supersymmetry transformations parametrized by a Grassmann parameters (, I are given by 644(x, 0, ) = (Q + Q) , (2.3) where t is a generic superfield and the generators Q can be represented by 8 Q = - %ia0m 9a amla (2.4) -a 9 a/ 06 It is also useful to define the supersymmetric covariant derivatives Da= + a&am 8D& Wa m, (2.5) 8 0a which anticommute with the Qs. 9 For each scalar supermultiplet containing a complex scalar field A(x) and a Weyl spinor (x), we associate a chiral superfield 4 characterized by D&O = 0. Written in terms of ym = xm + iOam , this means that q is independent of 9: 0(y, 0) = A(y) + v#(y)0 + F(y)00 , (2.6) where F is an auxiliary field of mass dimension 2 needed to preserve supersymmetry off-shell. For each vector supermultiplet containing a vector field vm(x) and a Majorana spinor field A(x), we associate a vector superfield characterized by Vt = V, given by V(x,0, 0) = -Ouamvm(x) + iOOA(x) - i-OOOA(x) + 000OD(x) , (2.7) where D is a dimension 2 auxiliary field and we have written V in the WessZumino gauge for simplicity. Supersymmetric Lagrangians are constructed by noting that the highest dimension field in a superfield tranforms under (2.3) as a total derivative. Actions constructed from the F or D terms of chiral or vector superfields, respectively, will be invariant. Imposing the conditions of gauge invariance, the Lagrangian of the most general renormalizable supersymmetric non-Abelian gauge theory is ï¿½ = Ckin + int, where ï¿½kin = Tr(WaWI F + WaW 1) + "e D (2.8) and int = (Aiqi + miji + gijkl j k + h.c.) , (2.9) where the gauge covariant chiral spinor superfield Wa is given by Wa = -1DDe-VDaeV , (2.10) 10 and I(F,D) projects out F and D terms. Eliminating the F fields from (2.9) gives Lint = i- 8 ij + h.c.) 1I2 (2.11) 3 where P, the superpotential, is given by (2.9) before projection over the F terms and where the derivatives in (2.11) are evaluated with the scalar component of 4i. Eliminating D in (2.8) and collecting terms depending only on the scalar components one obtains the scalar potential V(A) = I+ jDa 2 (2.12) a where Da =ga A TaAi , (2.13) and Ta are the generators of the gauge group with couplings ga for each semisimple or Abelian factor of the group. The Yukawa couplings from the gauge kinetic term can be combined with those in (2.11) to give the Yukawa part of the Lagrangian ï¿½y = - v : ga(Ai (Ta)'Aad + h.c.) a (2.14) 1 2P(A) (i 2 + h.c.) iAi8AJ We now turn to the application of this formalism in the construction of realistic supersymmetric models. 2.3 The Minimal Supersymmetric Standard Model The construction of the minimal supersymmetric standard model (MSSM) is straightforward. Each field of the Standard Model is assigned a superpartner and these fields are combined into supermultiplets with definite SU(3)C x 11 SU(2)L x U(1)y quantum numbers. Due to the chiral nature of fermion families in the Standard Model, these multiplets are irreducible representations of N = 1 supersymmetry. Models with N > 1 have matter supermultiplets in vectorlike representations, so that ordinary fermion families have unobserved mirror counterparts. Attempts to give large masses to these mirror fermions have not been successful. The model must be free of gauge anomalies. One possibility is that the superpartner of the Higgs field is the lepton doublet but more fields must be introduced to cancel the hypercharge anomaly and to give up-type quarks masses and one is beset with phenomenological problems arising from lepton number violation. Another possibility is that the Higgs that gives rise to uptype quark masses is part of a mirror family. This chiral completion must be modified to radiatively generate down-type quark and lepton masses and has difficulty suppressing lepton and baryon number violation. The minimal choice is to introduce a second Higgs doublet and its superpartners of opposite hypercharge from the usual Higgs field to cancel the anomaly. The supermultiplets and their component fields are listed in Table 1. The kinetic and interaction terms of the theory are obtained in terms of the superfields of Table 1 by generalizing (2.8) and (2.9) to the semisimple gauge group SU(3)C x SU(2)L x U(1)y, projecting out the D and F terms and eliminating the auxiliary fields using their equations of motion. To do this one requires the superpotential which is given by P = iuYufilc + dQ Yddc + dYec + Pidu, (2.15) where the hat denotes a chiral supermultiplet. This is the most general superpotential consistent with conservation of R-parity under which usual standard 12 model fields are assigned a value of +1 and their superpartners a value -1. This discrete symmetry is a special form of a continuous R-symmetry under which vector and chiral superfields transform as V(x,0,-) ~ V(x, Oe-a,Oeia) (2.16) 4a(x, 0, 0) -- exp (inaa)Oa(x, e-i, e() The D terms of a superfield are R-invariant while the F terms of a product of chiral superfields are invariant only if Ea na = 2. This continuous R-symmetry forbids Majorana gaugino masses and must be broken to account for them, leaving the discrete R-parity. The most important consequence of R-parity conservation is conservation of baryon and lepton numbers. It also implies that superparticles must always be produced in pairs, leading to the existence of a stable lightest supersymmetric particle (e.g. the photino). Table 1. Particle Content of the MSSM Superfield Component Superpartner Quantum Numbers Fields Names SU(3) SU(2) Y A ~A VG gi, gluino 8 1 0 VW W'iITvi wino 1 3 0 VB Bm,Bm bino 1 1 0 a _u Hu,Hu higgsino 1 2 1 d Hd,Hd 1 2 -1 d ' squark 3 2 1 c 4 U UL,UL 3 1 -3 de dL,dL 3 1 2 Sslepton 1 2 -1 r.eLeL r2 pc eLeL 1 1 2 13 2.4 Supersymmetry Breaking One major difficulty of supersymmetric theories is the lack of experimental evidence for superparticles. Superparticles must have the same masses as their ordinary partners since P2 is a Casimir of supersymmetry transformations. Thus supersymmetry must be broken to be consistent with phenomenology. It can be broken either explicitly, spontaneously or dynamically. 2.4.1 Spontaneous Breaking In gauge theories with an abelian component in the gauge group it is possible to have a term in Lagrangian proportional to the vector superfield V (since it is gauge invariant in this case), Cï¿½ = -2#, ;V D (2.17) known as the Fayet-Iliopoulos term [28]. This term induces an extra piece in the D part of the scalar potential (2.12): V(A2) ~QA! Ai + 1 2 (2.18) i where Qi are the charges of the scalars Ai. As we shall see this mechanism is not sufficient to break the MSSM and is inappropriate for theories imbedded in a simple gauge group in which case the Fayet-Iliopoulos term is absent. It is possible to break supersymmetry without gauge interactions by picking special forms for the superpotential. This mechanism, invented by O'Raifeartaigh [29], requires at least three chiral superfields A, t, C with associated scalar components. The cleverly chosen superpotential P = p1-B + A(A2 - i2)C, (2.19) has no solution which gives V(A, B, C) = 0. One of the auxiliary fields FA,B,C has nonvanishing VEV and supersymmetry is broken. This mechanism is also 14 not sufficient to break the MSSM and has the unaesthetic property of requiring many extra supermultiplets. The MSSM scalar potential derived from (2.12) and (2.15) is given by V = IYjeigj + pHI2 + P2IHdl2 + IYeiHd 2 + IY iHd 2 +12(H - H2dHd i 4 m )2 (2.20) +9gl(HuHu - Hd+ ++ 2(H, + HdfHd i)2 +9g2(HutHu + Ht d+ ,lii where the squark contribution has been omitted since we forbid SU(3) breaking. Here mFI comes from the Abelian Fayet-Iliopoulos term. A solution to V = 0 exists for a nonzero slepton VEV although this minimum breaks lepton number. By adding more fields to the minimal model one may be able to induce supersymmetry breaking spontaneously; however, for any anomaly free theory broken via D or F type breaking the following constraint on the mass spectrum holds [30]: E(-1)2J(2J + 1) Tr{MJ)} = 0, (2.21) J where the trace is over squared particle masses M2 of spin J. Such a mass relation is phenomenologically unacceptable as it requires the existence of an unobserved massive squark which is lighter than the lightest quark [31]. Thus we shall consider soft explicit breaking of supersymmetry in the low energy form of the model. The most promising mechanisms for inducing such terms are through local supersymmetry or dynamical breaking via gaugino condensation. We shall limit our discussion to the former case. The low energy theory derived from the latter can be parametrized in a similar way. 2.4.2 Induced Breaking from Supergravity An alternative to the troublesome scenarios for breaking supersymmetry spontaneously in the low-energy theory is simply to introduce soft supersymmetry breaking parameters into the effective theory which do not spoil the nice 15 renormalization properties of the theory. In particular, one introduces terms which do not introduce quadratic divergences into the theory so as to maintain an attractive solution to the hierarchy problem. Not all mass dimension 2 and 3 terms are allowed by this criterion. Girardello and Grisaru [32] showed that explicit masses for the spin 1/2 components of chiral superfields are not allowed, nor are simple cubic terms in a single scalar field. Such soft supersymmetry breaking parameters are regarded as originating from some unknown higher theory. It is attractive to think that if supersymmetry occurs in nature that it appears as a local symmetry. Such a theory must contain the local extension of the Poincard group, namely general coordinate invariance, and therefore include Einstein gravity. One obvious motivation for such a supergravity theory is that it can solve the gauge hierarchy problem when extended to the Planck scale. It also arises naturally as a low energy theory from superstrings. The attractiveness of supergravity is that supersymmetry breaking can occur in some massive hidden sector of a theory which couples to the usual sector only through gravity [33]. Due to the weakness of the gravitational interactions the symmetry can be broken at a high scale but appear in the low energy effective theory at a much lower scale. The supergravity theory must be broken spontaneously to preserve the Lorentz invariance of the theory. We consider the low energy effective theory arising from an N = 1 supergravity model interacting with a hidden sector in which supersymmetry is spontaneously broken and a lower energy sector that contains the fields of the MSSM and possibly heavier grand-unified fields. The general scalar potential of globally symmetric Yang-Mills theory in (2.12) is modified in the presence of supergravity. Because supergravity theories, like their nonsupersymmetric 16 counterparts, are not renormalizable, the superpotential can contain arbitrarily high powers of chiral superfields. This leads to certain arbitrary functions of the complex scalar components Ai of the chiral superfields in the expression for the supergravity-induced scalar potential. Remarkably, in the case of supergravity interacting with a single chiral superfield, only one arbitrary function appears [34] while in the case including Yang-Mills and an arbitrary number of chiral superfields, two such functions are present [35]. Soft supersymmetry breaking effects in the MSSM are given by additional terms in the Lagrangian: Vsoft = m A Ai + By(4u'd + h.c.) + Z( AijYu'ii4iuj + A/YdiJdidOj +A YeiJji4dLj + h.c.), i,j 3 ï¿½gaugino = - M lXA + h.c. 1=1 (2.22) where Vsot must be added to Eq. (2.20) and we have omitted a possible FayetIliopoulos term. Here the sum over i includes all scalar fields and Au,d,e, B and mA, are of order mg = m3/2 where mg is the gravitino mass. We generically denote the scalar mass terms by the scale m0. The gaugino masses are also approximately equal at Mpl to a common mass ml/2. Since the breaking of supersymmetry is flavor blind in this scenario, one also has Au = Ad = Ae at the Planck scale which suppresses scalar induced flavor-changing processes in these models. Of course these relations no longer hold at the electroweak scale when the potential is renormalization group improved through the use of the ,3 functions of these parameters given in Appendix II. For further analysis and recent reviews see refs. 36,37,38 and 39. CHAPTER 3 SUPERSYMMETRY AND GRAND UNIFIED THEORIES The subject of grand unified theories and their supersymmetric counterparts is vast and here we give only the essential details. The curious reader should consult one of the many excellent reviews in the subject (see refs. 40 and 41 for reviews of GUTs and refs. 41 and 42 for reviews of SUSY-GUTs). Grand unified theories originated as another step in theorists' search for higher symmetries in particle physics. Such theories are describable by a single simple gauge group which must contain as a subgroup the SU(3)C x SU(2)L x U(1)y symmetry group of the Standard Model. The minimal group is SU(5), the most predictive of all grand unified models [4] and we will base most of the discussion on this model, although models based on SO(10), and E6 [5,6] have similar features and predictions. Besides the aesthestics of the larger symmetry manifest in grand unified theories, there are other motivations to study them. One is that GUTs provide simple relations among the parameters of the theory. The penultimate prediction of GUTs is coupling constant unification at the grand-unified scale as is required by imbedding the Standard Model group in a simple group. This decreases the number of gauge parameters in the theory although many more parameters must be introduced in the scalar sector. Typically GUTs also predict definite mass relations between quarks and leptons since they appear together in grand unified multiplets. Grand unified models generically violate CP and baryon number, giving a possible explanation of the cosmological baryon asymmetry [43]. Finally, especially in light of recent observations of 18 solar neutrinos [44,45,46], certain GUTs such as SO(10) give a simple explanation of the solar neutrino problem [1] by predicting neutrino masses via the so-called seesaw mechanism [47] which give rise to matter induced neutrino oscillations [2] with numerically consistent values for the oscillation parameters [48]. 3.1 Minimal SU(5) The minimal SU(5) GUT of Georgi and Glashow [4] contains only the fermions of the Standard Model. These fall into the 5 and 10 representations of SU(5), which decompose under SU(3)C x SU(2)L as 5 = (,1) + (1,2) (3.1) 10 = (3,2) + (E, 1) + (1, 1) . The quarks and leptons appear in these multiplets as follows. For the 5 representation we have 5 = a e ,(3.2) and for the 10 = (5 x 5)A we have 0 U3 -U2 ul d1 1 -u3 0 Ul u2 d2 X u2 -U 0 u3 d3 (3.3) 7= -u1 -u2 -u3 0 e+ -dl -d2 -d3 -e+ 0 L where the indices on the quark fields are color indices. The gauge bosons lie in the adjoint 24 representation which decomposes as 24 = (3, 2) + (3, 2) + (8, 1) + (1, 3)+ (1, 1), (3.4) and contains the Standard Model gauge bosons and two leptoquark gauge bosons X and Y of charge - and 1 respectively which mediate baryon and lepton number violating processes although B - L is conserved. 19 Already we can use the simplicity of the gauge group to make a prediction for the Weinberg angle, Ow at the GUT scale [7]. Let us write the Standard Model SU(2)L x U(1)y couplings as g2 and g' = gl/C where C appears in the formula for the U(1)em charge generator Q in terms of the third generator of weak isospin and the hypercharge generator: Q = T3 - CTO. (3.5) Normalizing the generators of the group in a representation R by Tr(TATB) NR6AB we have Tr(Q2) = (1 + C2) Tr(T3), (3.6) for any representation of the gauge group. Performing the traces for the 5 representation gives C2 = 5/3 and sin o 1 g3 (3.7) g2 +2 1 +C2 = In fact this prediction also holds for SO(10) and most E6 models. Note that this prediction holds only at the GUT scale and one must use the renormalization group to find its value at the electroweak scale. The SU(5) symmetry must be broken at a high energy scale, MGUT, to account for the lack of experimental evidence of GUT particles and to make its predictions viable. Masses for the quarks and leptons arise via spontaneous symmetry breaking of SU(2)L x U(1)y to U(1)em. The SU(5) symmetry breaking is achieved spontaneously by including an adjoint 24 of scalar fields, .EA The fermion mass terms transform as S10= 5 + 4-5, (3.8) 10 x 10= 5+45+50, 20 so we minimally require a 5 Higgs multiplet H which contains the usual Higgs doublet. With this minimal set of scalar fields both symmetries can be broken and masses given to the fermions leading immediately to the mass relations md = me, ms = mp, mb = mr , (3.9) at the GUT scale. We shall see that only the last relation is pheomenologically feasible. One can replace these extra mass relations by improved ones if more scalar multiplets including 45s are added and if various global symmetries are imposed on the scalar potential. For examples of such models see ref. 49 and in the context of SO(10) see ref. 50. For recent re-examinations of the renormalization group predictions of these models see refs. 51,52,53 and 54. Perhaps the most exciting prediction of GUTs is the instability of the proton. X and Y leptoquarks mediate baryon number violating processes such as those depicted in Figure 1. u e+ u e U e+ U 1Ld dd U Figure 1. Some baryon number violating diagrams in minimal SU(5). At low energies these processes give rise to effective dimension 6 operators in the low energy Lagrangian lrBI 2 g2 1eAB=1 201+ 8202, (3.10) w eff qMu2 8M2mixin where, omitting quark mixing, 01 =(fijk cL u)( LdL k (3.11) 02 =(ik!,u -[ -+ + " )--d 21 and c stands for charge conjugate spinor. Using MX - My MaGUT we can naively estimate the proton lifetime to be of order M5 rp oc (3.12) P on dimensional grounds [7]. However various effects modify the coefficient of the expression. Using renormalization group techniques one can compute the anomalous dimensions of O1 and 02 coming from virtual gluon, W and Z boson and photon exchange. Large logarithms of order logn(Mx/mp) can be summed, giving rise to enhancement factor for these operators. The factor from gluonic corrections is [55] A3 C3( )]-, (3.13) la3(Mx) where p - 1 GeV is a typical scale appropriate for the proton decay process and ng is the number of fermion generations. The actual decay rate requires knowledge of the hadronic matrix elements of Eqs. (3.11), which can be estimated by current algebra or bag model techniques. These estimates indicate a favored decay mode of p --+ e+7ro. The best theoretical estimate of this rate, using data from the early 80s is [41] rp/B(p + e+r0) = 4.5 x 1029ï¿½1.7 yr, (3.14) where the range is due to uncertainty in the QCD scale, A-. In the determination of MX in this rate one must also take care to properly treat mass thresholds effects. We discuss this issue in Chapter 6. The operators of Eqs. (3.11) are not the only dimension 6 operators that can mediate proton decay. One can also have decays mediated by color triplet Higgs scalars. Since these appear in the same SU(5) representation as the ordinary light Higgs doublet parameters of the potential must be fine tuned to 22 incredible accuracy to insure that they obtain large masses of order MGUarT. This is another manifestation of the gauge hierarchy problem. The scalar contribution is largest for processes involving heavy quarks; it is usually suppressed by the ratio of quark or lepton masses to the W boson mass. Huge underground water tanks have been set up around the world to search for this process and have found no signal.1 The current limit from the Kamiokande detector in Japan [56] is rp/B(p -- e+r') 9 x 10" 32yr, (3.15) ruling out the minimal SU(5) model. The situation is made worse for SU(5) when one uses the renormalization group to obtain low energy predictions from GUT scale coupling constant and mass relations. In Figure 2 we see that using the two loop 0 functions for the gauge couplings (see Appendix I) and inserting their known values at Mz with errors, minimal SU(5) is inconsistent with coupling unification [11,17] This result holds for any GUT model with the Standard Model as the effective theory below MGUT. The situation for the only viable mass relation, mb = m, is not improved. By using the renormalization group equations of Appendix I, we can determine the prediction for this ratio at low energies [55]. In Figure 3 we see that they meet at a scale far below the required GUT scale, although it might indicate some ununified intermediate scale theory with this relation which breaks around 108 GeV. These issues will be discussed further in Chapter 8. 1 However these same experiments may have discovered massive neutrinos from SN1987A - an SO(10) grand unified prediction! 60 50 V4 40 T" 3 0 -- GUT TRIANGI! - - as 0 g" - 4 820 - 46 - 44 " 10 - 40 aIIIIL Ij ....l. a1 13 14 16 17 0 "1 " " 1" 1 "1i 0 5 10 15 20 o1gto(G) Figure 2. Running of the Standard Model inverse gauge couplings to two loops using their propagated experimental errors. The picture for minimal grand unified models-those whose low energy theory below the GUT scale is the Standard Model-is bleak. This does not, however, rule out models with intermediate scales of symmetry breaking and those with additional intermediate mass fields beyond the Standard Model. One example includes nonminimal SU(5) models with 45s of scalars [49] where asymptotic freedom of the gauge coupling is lost due to the enormous Higgs structure. Another involves analogous SO(10) models with intermediate breaking scales (for example one can have an intermediate theory with the color group being 24 SU(4) of Pati and Salam [3]) [50]. These theories also predict many interesting mass relations (see for example ref. 51) although they arise from special forms of yukawa and scalar interactions and a plethora of scalar multiplets which make it difficult to take seriously such models as fundamental theories. 0 5 10 15 lo go4) 20 Figure 3. Intermediate quark and lepton masses in the Standard Model for Mt = 100 GeV and MH = 100 GeV. 3.2 Supersymmetric SU(5) The basic predictions of minimal SU(5) also hold for its supersymmetric extension [31,57,58]. However due to the presence of superpartners for all standard particles, the renormalization group equations are modified [59,60] in a manner that gives much better agreement with experimental data. One 25 of the most exciting recent results of improved data on the strong coupling, as and sin OW, was the striking gauge unification of Figure 4 in the context of supersymmetric theories [9,10,11,12,13]. So far this is the best indirect evidence for the existence of such theories. MsusY = 1 TeV 20 0 0 5 10 15 log o10( ) 20 Figure 4. Running of the MSSM inverse gauge couplings to two loops using their propagated experimental errors. The window depicts a blow-up of the area around the unification point. Note the small region where all three couplings intersect. As seen in Figure 5 the mass relation mb = mr also is not in perfect agreement for Mt = 100 GeV and is sensitive to this mass. Agreement can only be obtained for a certain range of top quark masses. We shall discuss these renormalization group results and exploit this mass relation in Chapter 9. 6 2 Loop Mt-100 GeV ME=100 GeV bMb 0 5 10 15 20 loglo0(p) Figure 5. Intermediate quark and lepton masses in the MSSM for Mt = 100 GeV and MH = 100 GeV. One major distinction between SUSY-GUTs and ordinary GUTs is the mechanism for proton decay. In SUSY-GUTs, in addition to the usual dimension 6 operators, one has dimension 5 operators produced by exchange of color triplet Higgs (Hx) and its superpartner sHiggs field (Hx) [61]. These arise from the diagrams of Figure 6. Proton decay follows from a one loop effect when wino, bino or gluino fields are exchanged between the sfermions as in Figure 7. "% o f f f % bf ' XH1 - + tH f f / f Figure 6. Diagrams inducing dimension 5 baryon number violating operators in SUSY-SU(5). Here Hx is a colored Higgs triplet and f is fermion. A tilde is used to denote the superpartners of these fields. These contributions are on the order of [61] 1 ( 2) y2 8 3 n y2 12 ( ) H or(g 3 H (3.16) 16r2. 2 mvmHx 3 msqmHx where YH is is a typical Yukawa coupling and the latter expression holds for squark masses, msq > m. W, B, or! /f Figure 7. Diagrams for colored Higgs and sHiggs mediated proton decay. Anomalous dimensions for the relevant dimension 5 operators give rise to supression factors rather than enhancement factors found in the minimal SU(5) case. This increases the proton lifetime associated with these processes from what is naively expected. The supersymmetric case is clearly less predictive than the minimal one. For sufficiently heavy wino, squark and Higgs and sHiggs color triplet masses, the proton decay rates obtained cannot be ruled out by experiment. However this assumes that the Higgs triplet fields are superheavy. 28 This is not the case in models in which supersymmetry is spontaneously broken in the GUT model as they will always produce light scalar color triplet fields as a consequence of the mass relation (2.21) [31]. We shall assume that supersymmetry is broken by a hidden sector coupled to ordinary and GUT matter via supergravity. We therefore avoid this mass relation as discussed in the previous chapter. Because of the dependence on the Yukawa couplings the preferred decay modes will go into heavier quarks than in gauge boson mediated decay. After taking into account suppression from CKM angles, one finds the dominant decays are strangeness changing: p --+ p+KO and p --+ IK+ [61]. The current limits on the latter decay are rp/B(p -+ -FOK+) Z 1032 years [56]. In order to perform detailed analyses of GUT and SUSY-GUT predictions, it is essential to properly evaluate the corrections to tree level relations through the use of the renormalization group. This requires knowledge of the f functions of the parameters of the theory and their behavior when mass thresholds are crossed. We have developed essentially a toolkit for such analyses that includes extraction of the measureable parameters of the low energy theory, incorporation of threshold effects that insure decoupling of heavy fields from the low energy effective theory and numerical evaluation of the renormalization group equations according to the hierarchy of theories under study. We therefore devote the next three chapters to these procedures. We first turn to a review of the renormalization group. CHAPTER 4 THE RENORMALIZATION GROUP Renormalization is a procedure by which finite physical observables are extracted from typically divergent Feynman amplitudes in quantum field theory. These divergences arise from large quantum fluctuations of the theory when probed at short distances. As the infinities appear in each order in perturbation theory, they are systematically subtracted out via infinite counterterms added to the Lagrangian of the theory. The renormalized Lagrangian Iren = Co + ,ct (4.1) yields finite results for physical quantities at every order. This procedure only works when the counterterms are finite in number and typically of the same form as terms present in the bare tree-level Lagrangian Co. In this case the theory is considered renormalizable. The renormalized Lagrangian is the same as the bare one but with its parameters redefined by infinite amounts from the counterterms: g = go + 6g, (4.2) where go is the bare parameter, g is the renormalized parameter, and bg is the counterterm. There is of course arbitrariness in this prescription as the finite parts of the counterterms are undetermined. The renormalization group equations for a renormalizable theory state that the theory is invariant under reparametrizations of the theory obtained by choosing different finite parts of 6 g. 1 1 However, quantities computed at finite order in perturbation theory exhibit a dependence on the scheme by which the renormalized parameters are defined. In QCD where the strong coupling as is large, significant uncertainties arise from this scheme dependence. 30 We shall work in the modified minimal subtraction scheme (MS) commonly used in the literature. The minimal subtraction (MS) prescription [621 is defined by fixing the counterterms by requiring them to consist only of the infinite terms needed to render the theory finite. In this scheme divergent integrals are regularized by continuing them to lower dimensions and an arbitrary mass scale p is introduced in order to keep couplings dimensionless. A family of MS schemes can be defined via go(O)-' = g + g , (4.3) where o is a constant parametrizing the arbitrariness in the finite parts of divergent integrals in dimensional regularization, e = (4 - d)/2 and d is the space-time dimension. The MS scheme is given by e = 1, while the so-called modified minimal subtraction (MS) prescription [63] is given by g2 -= e7E /4r, where yE is the Euler-Mascheroni constant. The extra finite piece is typically present in dimensionally regulated integrals and it is mere convenience to subtract it out. The renormalization scheme independence of a physical quantity, P, when expressed in terms of p and the running parameters of the theory, {gi(P)}, is given by the renormalization group equation, -P({gi(p)}, Y) = (P-- + #i )P = 0, (4.4) dp ap 94 where the Pi are the # functions which determine the evolution of the renormalized parameters with scale. These running parameters are not in general equal to their corresponding physical values (consequently, for the masses, we adopt a convention wherein upper case M refers to physical values and lower case m denotes MS values). This is to be contrasted with the on-shell renormalization 31 scheme in which, for example, the renormalized masses equal their physical values and the renormalized electromagnetic coupling equals the fine structure constant. Also y cannot be identified with any specific momentum scale such as are encountered in experiments. It is difficult to attach an approximate physical meaning to p and the uncertainty in the appropriate scale to use to describe a certain process is a significant one in extracting MS quantities from data (see e.g. Section 5.2). Another problem is that the decoupling of massive states in lower energy processes in such mass independent prescriptions is not manifest. This apparent contradiction of the decoupling theorem [64] is solved though the effective gauge theory formalism applied at the thresholds of such massive particles (see Chapter 6). Despite the above drawbacks, the MS schemes have the attractive characteristic that the # functions are p independent and therefore particularly simple to integrate. The two-loop # functions of the Standard Model and the MSSM in the MS scheme have been collected in Appendices I and II. To solve these first order differential equation, we require the values of the parameter at one scale, which we take to be MZ CHAPTER 5 STANDARD MODEL PARAMETERS AT MZ Below we give a brief description of the determination of the parameters of the Standard Model. Details can be found in the cited literature. 5.1 ai(Mr) and a,(M7) The determination of the SU(2)L x U(1)y couplings proceeds from the Standard Model relations: ( g2() C2 a(p) 4r cos2 OW() ' (5.1) 4r sin2 w() where a(y) = e2(p)/47r and C2, defined in (3.5), equals 1 for the Standard Model and 5 when the Standard Model is incorporated in grand unified theories of the SU(N) and SO(N) type [7]. These couplings can be specified through the MS values of a(p) and sin2 Ow(p). The electromagnetic fine structure constant (a-1 zt 137.036) is extrapolated from zero momentum scale to a scale p equal to MZ in our case. The renormalized coupling a(y) is related to the fine structure constant aem as follows: aem = ,() (5.2) 1 + II(0) where II(0) is the photon vacuum polarization function at zero momentum and includes radiative effects from charged gauge bosons and fermions as well as hadronic contributions. Threshold effects are treated by including the contributions of charged particles only at scales p above their masses. This procedure gives [65] a- 1(Mz) = 127.9 ï¿½ 0.3. (5.3) 32 33 Many definitions of the renormalized Weinberg angle exist in the literature and many methods are used to extract it from data. We give a brief overview of definitions in current use. The process independent, renormalized weak mixing angle sin2 OW of the on-shell scheme is defined to be sin 20w 1 - M (5.4) M2 z where MW and MZ are the physical masses of the W and Z gauge bosons. One can compute sin OW by accurately measuring MW and MZ or one can use the bare relation involving the low energy Fermi constant measured in muon decay and the W boson mass 0o _e . (5.5) v 8 sin2 0woM W2 This may be corrected to order a and rewritten [66,67] Mw = MzcosOw = ($j 1 , (5.6) G sin 9w(1 - Ar)2 with (iraem/v'2Gp)2 = 37.281 GeV and Ar is a parameter containing order a radiative corrections and which depends on the mass of the top and Higgs. We can absorb the radiative effects using the renormalization group by replacing Gp and aem with corresponding running parameters at MZ. For large values of Mt and MH (Mt, MH > Mz) one finds [66,68] Ar1 aem 3aem M2 + 11aem nM2H (5.7) a(Mz) 167r sin4 W M2 487r sin2 Ow M2 An estimation of the first two terms using (5.3) gives Ar a 0.07. (5.8) A third way of extracting sin2 OW is from neutral current experiments, among which deep inelastic neutrino scattering appears to provide the best determination. 34 A running sin2 OW(P) may be defined in MS and differs from the above sin2 BW by order a corrections. The MS running W boson mass mw(p) and the corresponding physical mass MW, identified as the simple pole at q2 = M2 of the W propagator, are related as follows A 2T 2 MW = m2(p) + Aww(My,4 P) (5.9) where AT is the transverse part of the W self-energy. A similar relation holds for the Z boson. In MS renormalization, the following relation defines the running sin2 4W(t) m2 sin2 OW(p) = 1- W() (5.10) Equation (5.9) and its Z analog may be combined with Eq. (5.10) to give sin2 OW(p) cos2 WAZZ(M 2,) A (M ) 1A- (w ') (5.11) sin2 Ow sin2 O M2 M2 An explicit expression relating sin2 Ow and sin2 8W(MW) is given in ref. 69. Other relations for sin2 Ow(p) may be arrived at directly linking it to MZ [70] or Mw [71]. A fit to all neutral current data gives sin2 Ow(Mz) = 0.2324 ï¿½ 0.0011 , (5.12) for arbitrary Mt [72]. Using these values of a(Mz) and sin2 Ow(Mz) yields al(Mz) = 0.01698 ï¿½ 0.00009, (5.13) a2(MZ) = 0.03364 ï¿½ 0.0002. 5.2 as(M) The value of the strong coupling is known with less precision than most of the parameters of the Standard Model. This is due to large theoretical uncertainties arising from the nonperturbative nature of low energy QCD and the 35 slow convergence of perturbation series in high energy QCD. The determination of the strong coupling constant as, is most critical to attempts to probe grand unified structure. The strong coupling constant is measured in a wide variety of processes at various energy scales. Early measurements of as came from deep inelastic electron and muon scattering and W + jet production at pji colliders. This data has been combined in recent years with results from LEP including jet rates, energy-energy correlations and global event shapes. One must also include new results in deep inelastic neutrino scattering, J/', and T decays, the hadronic decay width of the Z boson, b quark production in pp collisions and the r lepton hadronic width. A recent summary of these results can be found in ref. 73 with details found in the references therein. In Table 2 we give a list of the results for each type of as measurement. Note that the higher energy LEP results tend to give larger values of as than the extrapolated lower energy results. Table 2. Values of as at MZ. Process as(Mz) a(e+e- -- hadrons) 0.135 ï¿½ 0.015 e+e- -* hadrons (shapes) 0.119 ï¿½ 0.014 J/I, T decay 0.113+0007 ï¿½ _0.005 Deep inelastic scattering (v, p) 0.112 ï¿½ 0.004 pj-4 bbX 0.108+0.015 - _0.014 p -* W + jets 0.121 ï¿½ 0.026 Rr (world avg.) .118+0.004 '0.006 I(Z0 --+ had.) 0.133 ï¿½ 0.012 Z0 jets and ev. shapes 0.121 ï¿½ 0.004 36 As an example of the uncertainties encountered in measurements of as consider the determination of as from jet rates (see the reviews of refs. 74,75,76 and 77 for details). One of the dominant uncertainties is theoretical - the renormalization scale dependence of the rate. All the processes from which the strong coupling is extracted have such theoretical errors. Depending on the structure of the basic partonic process, the cascades of hadrons produced in an e+e- collision may form clusters strongly peaked in energy and constrained in solid angle known as jets. The basic quarkantiquark-gluon (qqg) process can give rise to a 3-jet event if the final state gluon radiation is hard and acollinear with the quark pair. The analysis of jets gives the best indication of the underlying partonic process. The differential cross-section for this process can be written [77] 1 d2a 2a z +2 S- (5.14) a dxrdx2 37r (1 - x1)(1 - x2)' where 2Ei xi = 2E (5.15) are the center-of-mass energy fractions of the final quarks. In fragmentation models parton structure and hadronic structure can be related by a scale factor (local parton-hadron duality) with excellent experimental agreement. Once this relation is established one can compute the dependence of the process on a8. A further difficulty is the problem of deciding experimentally what constitutes a jet. One must work backwards to recombine hadrons into fundamental partons. One algorithm for recombination of two final state particles into one is to take all pairs of particles and compute their weighted invariant mass squared (Pi + py)2 SEE Yij EP)2 2 2 (1 - cos Oij), (5.16) cm cm 37 where the second equality holds for massless particles (mi < Ecm). Then find the pair with smallest Yij. If ij < ycut where yeuJt is a chosen jet resolution parameter then combine this pair into a single pseudoparticle. Continue until all Yij > ycut and one is left with a fundamental partonic process. The number of partons remaining corresponds well with the number of jets. The ratio for the n-jet to the total cross-section depends on the jet resolution parameter y and is given by Rn(y, p) - an-jet (5.17) UT where for example R2(y, p) =1 + C2,1(y)as(p) + C2,2(y,f)a s(p) + ... , (5.18a) R3(y, P) =C3,1(y)as(p) + C3,2(U,)() + ... . (5.18b) Here f is an unknown scale factor presumably of order 1 relating the renormaliation scale p to the center of mass energy: p = fEcm. (5.19) The appearance of the renormalization scale in Rn is disturbing as physical quantities should be independent of this arbitrary scale. However, one works in practice only to finite order in perturbation theory and a dependence on p is introduced. A physical quantity C(a((p), p) expressed in terms of p and the running parameters of the theory, a(p), must be p independent: P C(a(p), P) = (P ( + a )c = 0, (5.20) dp '9P Do where the the # function is given by d+( (p) = p - 0a2 + la +... (5.21) dp 38 Now write C as a perturbative expansion in a C = Co(pl) + Cl(p)a(p) + C2(P)a2(p) +... + Cn(P)an(p) +... (5.22) The independence of C from p implies that C2() - 2() = -0oCiln ) (5.23) We denote by C(n)(p) and C(n)'(P') the physical quantity C computed to nth order in perturbation theory. The difference between these quantities is of order an+1. The effect of changing renormalization scale in the estimation of a physical quantity is to introduce higher-order corrections and modify the coefficients in the perturbative expansion. If we write tt' = fu then (5.23) gives C2 = C2 - /30Cllnf. Thus at finite order in perturbation theory a dependence on renormalization scale is introduced starting at second order in the coupling, justifying Eqs. (5.18). By varying the jet resolution parameter y, a given event can appear as a two or three jet event. Too avoid a given event contributing to different classes of jets, one often defines the differential 2-jet rate for the distribution of y values for a transition from a three to two jet event: D2(y) - R2(y) - R2(y- Ay) Ay (5.24) Another method used is to measure R3 at a fixed value of y. The LEP average for as from jet rates is included in Table 2. The errors reflect a weighted sum of the errors due to statistics, experimental systematics, errors from hadronization and parton shower models, recombination scheme dependence, next order QCD uncertainties and renormalization scale dependence. The largest sources of error are typically those coming from the renormalization scale and the energy 39 scale for transition from parton shower to hadronization. The former error comes from fitting experimental results with f free and with f = 1 and taking the difference of these extreme values. Values of f from such fits range from 0.008 - 0.25. A global average of the results of Table 2 which we use is as = 0.117 ï¿½ 0.004, (5.25) where the error can be as large as ï¿½0.011 depending upon the determination of theoretical uncertainties. 5.3 Yukawa Couplings To take full account of the Yukawa sector in running all the couplings, initial values for the Yukawa couplings are necessary. They must be extracted from physical data such as quark masses and CKM mixing angles. Furthermore, the interesting parameters to be plotted must be determined step by step in the process of running to Planck mass. These two procedures are not unrelated and require the diagonalization of the up-type, down-type, and leptonic Yukawa matrices. We use Machacek and Vaughn's [781 convention where the interaction Lagrangian for the Yukawa sector is ï¿½ = QLYUtuR + QL YdtdR + -LY eteR + h.c.. (5.26) The Yukawa couplings are given in terms of 3x3 complex matrices. After electroweak symmetry breaking, these translate into the quark and lepton masses r(me 0 0 Ye =v2 0 my 0 V O 0 mr) Yd ,2 (md 0 0 Yd= - (0 m ) , (5.27) v 0 0 mb mV ( 0MU 0 0 YU= 0 mC 0 V, v O 0 mt where V is the CKM matrix which appears in the charged current j+ Uir)yVdL . (5.28) It is a unitary 3 x 3 matrix often parametrized as follows: ( cl 81C3 Sls3 = -sic2 clc2c3 - s2s3ei Clc2s3 + 82c3e6 , (5.29) -81S2 ClS2C3 + c2s3e6 ClS2S3 - c2c3e6 where si = sin 0i and ci = cos 0i, i = 1,2,3. The entries of the parametrized CKM matrix can be related simply to the experimentally known CKM entries. Using limits on the magnitudes of these entries from the particle data book [72] (assuming unitarity) we find the following bounds for si, i = 1, 2, 3. 0.2188 < sin 01 < 0.2235, 0.0216 < sin 92 < 0.0543, (5.30) 0.0045 < sin 03 < 0.0290. These limits do not constrain sinb. A set of angles {01,02,03,6} was chosen that falls within the ranges quoted above. The initial data needed to run the Yukawa elements is extracted from the CKM matrix and the quark masses. A problem arises though for the mixing angles, which was solved for the quark masses (see Section 5.4), in that it is not clear at what scale the chosen initial 41 values for these angles should be considered known. However, we have observed that for the whole range of initial values, the running of the mixing angles is quite flat, with a perceptible increase in 02 between MW and the Planck scale for higher top masses. This is in accordance with the angles being related to ratios of quark masses, and therefore, the exact knowledge of that scale is not critical. 5.4 Known Quark Masses As QCD is assumed to imply quark confinement, extraction of quark masses from experiment is problematic. On-shell renormalization will not have the same physical significance as it would if quarks comprised the observed particle spectrum. The quark masses then may simply be considered as additional couplings of the model. The light quark masses are the ones least accurately known. They are determined by a combination of chiral perturbation techniques [79] and QCD spectral sum rules (QSSR) [80,81,82,83]. These techniques yield [17] mu(1 GeV) = 5.2 ï¿½ 0.5 MeV , md(1 GeV) = 9.2 ï¿½ 0.5 MeV, (5.31) ms(1 GeV) = 194 ï¿½ 4 MeV . For the heavier quarks, charm and bottom, one can make a more precise prediction. Here, the nonrelativistic bound state approximation may be applied. The physical mass M(q2 = M2) appearing in the Balmer series may be identified with the gauge and renormalization scheme invariant pole of the quark propagator S(q) = z(q)[y . q - M(q2)]-1 42 The running mass is determined from the pole mass to three loops via m(q2= M2) = M(q2 = M2)5.32) 1 a (M + K(aM))2 (5.32) where K = 13.3 for the charm and K = 12.4 for the bottom quarks [84]. The charm and bottom pole masses have been determined from J/ and T sum rules [85] and recently from CUSB and CLEO II by analysis of the heavy-light B and B* and D and D* meson masses and the semileptonic B and D decays [861. A weighted average of these results yields Mc(q2 = M2) = 1.53 ï¿½ 0.04 GeV (5.33) Mb(q2 - M2) - 4.89 ï¿½ 0.04 GeV (5.33) The running masses at the corresponding pole masses then follow from Eq. (5.32) mc(Mc) = 1.22 ï¿½ 0.06 GeV, (5.34) mb(Mb) = 4.32 ï¿½ 0.06 GeV. It should be stressed that at the low scales under consideration the three-loop as corrections included in the mass and strong coupling #3 functions are often comparable to the two-loop ones and hence affect the accuracy of the final values noticeably [17]. These heavy quark masses can also be determined by studying nonperturbative potential models (see ref. 87) which predict appreciably higher values of the heavy quark masses than the ones quoted here. These models, however, are not as fundamental as the approach considered here, and their connection to field theory is rather problematic. 5.5 Lepton Masses The physical (pole) masses of the leptons are very well known [72] Me = 0.51099906 ï¿½ 0.00000015 MeV, Mp = 105.658387 ï¿½ 0.000034 MeV, (5.35) Mr = 1.7841+0027 GeV ...-0.0036 43 We use these values to determine initial data for the running masses. Some authors neglect QED corrections and use the physical values for the running values at - MZ, which introduces only a small error. By calculating the oneloop self-energy corrections, one arrives at a QED relation between the running MS masses and the corresponding physical masses 3a(p) p2 4 ml(p) = Mt[1 - 3a(In 2 + -)]. (5.36) 47r m 3 Choosing p = 1 GeV as in the quark mass case and using Eqs. (5.36) yields the running lepton masses (taking mi = M in the log term above is an appropriate approximation to order a) me(1 GeV) = 0.4960 MeV , mp(1 GeV) = 104.57 MeV , (5.37) mr(1 GeV) = 1.7835 GeV. 5.6 Higgs Boson and Top Quark Masses The Higgs boson and top quark masses have not been measured directly at present, however their values affect radiative corrections such as Ar. Consistency with experimental data on sin2 0W requires Mt < 197 GeV for MH = 1 TeV at 99% CL assuming no physics beyond the Standard Model [9]. Precision measurements of the Z mass and its decay properties combined with low energy neutral current data have been used to set stringent bounds on the top quark mass within the minimal Standard Model. A global analysis of this data yields Mt = 122+41 GeV, for all allowed values of MH [88]. Recent -32 direct search results set the experimental lower bound Mt Z 91 GeV. As for the Higgs, the analysis of ref. 88 gives the restrictive bound, MH ; 600 GeV, if Mt < 120 GeV, and MH < 6 TeV, for all allowed Mr. Since perturbation 44 theory breaks down for MH Z 1 TeV, the latter bound on the Higgs boson mass is not necessarily meaningful. LEP data set a lower bound on the Higgs boson mass of 48 GeV [89]. In our analysis, initial values of the MS running top quark mass mt and of the scalar quartic coupling A at MZ are chosen arbitrarily (consistent with the bounds quoted above). As noted earlier in Chapter 1, these running parameters are not equal to their physical counterparts. However, any reasonable prediction for the masses of the top quark and of the Higgs boson that may come from our analysis should be that of experimentally relevant, physical masses. Therefore, formulas similar to Eq. (5.36) relating MS running parameters to physical masses are needed. To calculate the physical or pole mass of the top quark, we use Eq. (5.32) in its general form M 1ï¿½4 a +(M,) 1- M a( ) A 4 1 + - + '1611 - 1.04 (1 - M)(as(Mt))2 , (5.38) mi(Mt) 3 7r M 7r i= 1 where Mi, i = 1,..., 5, represent the masses of the five lighter quarks. Likewise the physical mass of the Higgs boson can be extracted from the following relation: A(p) = G 2 M'(1 + b(p)) , (5.39) where 6(p) contains the radiative corrections and is given in ref. 90. Equations (5.38) and (5.39) are highly nonlinear functions of Mt and MH, respectively. Their solution requires numerical routines that are described in Chapter 7. 5.7 Vacuum Expectation Value of the Scalar Field The vacuum expectation value (vev) of the scalar field may be extracted from the well known lowest order relation v = ( G"Gy)- = 246.22 GeV. (5.40) 45 From the very well measured value of the muon lifetime, rp = 2.197035 .000040 x 10-6 s [72], the Fermi constant can be extracted using the following formula [91] 1 G2 m2 3 m2 )[1 a(mp) (25 it = 7 ~2)] , (5.41) 1921r m2 5m2 21r 4 P W where f(x) = 1 - 8x + 8x3 - x - 12x21 nx, (5.42) giving G/ = 1.16637 ï¿½ 0.00002 x 10-5 GeV2 . (5.43) This parameter may be viewed as the coefficient of the effective four-fermion operator for muon decay in an effective low energy theory [ e.(1 - 75)e][Wyp(1 - 75)vu] . (5.44) A direct calculation (e.g., in the Landau gauge) of the electromagnetic corrections yields that the operator is finitely renormalized (i.e., GP does not run) [67,92]. Another way to see this is by using a Fierz transformation to rewrite the above expression - [Te_(1 - 5)vu][v_~(1 - 75)e] . (5.45) The neutrino current does not couple to the photon field, and the e - p current is conserved and is hence not multiplicatively renormalized. We need an initial value for the running vacuum expectation value at some scale p. Wheater and Llewellyn Smith [93] consider muon decay to order a in the context of the full electroweak theory and derive an equation relating an MS running Gp to the experimentally measured value. From this formula we 46 can extract a value for v(Mz). However, the formula is derived in the 't HooftFeynman gauge, and the evolution equation Eq. (I.18) of Appendix I for the vev is valid only in the Landau gauge. Nevertheless, motivated by the discussion of the previous paragraph, we choose the initial condition for the vev to be: v(Mw) = 246.22 GeV. Using the initialization algorithm (see Chapter 7), we arrive at v(Mz). We find that this procedure leads to no significant correction, and we therefore take, ab initio, v(Mz) = 246.22 GeV. CHAPTER 6 THRESHOLD EFFECTS 6.1 Effective Gauge Theories We are using the MS renormalization scheme to determine the running of the Standard Model couplings. From the Appelquist-Carrazzone decoupling theorem [64] we expect the physics at energies below a given mass scale to be independent of the particles with masses higher than this threshold. However, such minimal subtraction schemes are not physical in the sense that they are scale dependent and mass independent so that the decoupling theorem is not manifest. As described in refs. 94,95 and 96, we have to formulate a low energy effective theory by integrating out the heavy fields to one loop and find matching functions in order to take care of the threshold effects. We give a brief review of threshold effects in the running of the gauge couplings for a spontaneously broken GUT and for the Standard Model. Details can be found in the appendices of ref. 95. We next discuss thresholds in running fermion masses, a subject which to our knowledge has not been adequately treated in the literature. In chapter 7 we give a quantitative description of the effect of thresholds in the running parameters. The starting point for treating thresholds in MS and MS schemes is the construction of low energy effective gauge theories [94-96]. The basic idea is to integrate out the heavy fields in such a way the the remaining effective action is gauge invariant under the residual gauge group. Let the simple gauge group G be broken to G, and let t and 0 be a set of heavy and light fields, respectively. 48 Then the action I[] of the effective field theory is obtained from the action I[0, #] of the full theory by functional integration over the heavy fields: ei] =[d]el[,] . (6.1) Because there are no superheavy fields in the effective theory the decoupling theorem is not needed. However, there is a difficulty having to do with gauge invariance. Namely, in order to integrate out the heavy fields one has to add a gauge fixing term, and such a term usually spoils the gauge invariance of the low energy theory. The usual R gauge fixing action for the full gauge group is 18f Ig.f. = fJ f(x;,), (6.2) where, for example, fa = (-/[8yAu + ig~vtasS] , (6.3) where a runs over all generators taS of G, Al are the gauge fields, and S is a column of Hermitian scalar fields with vacuum expectation value v. However, when the heavy gauge fields are integrated out with the natural choice of the gauge fixing functional as in (6.3) but summed only over the index of the broken group generators (given in uppercase Latin letters), one obtains an effective theory that is not invariant under G. We correct this by changing the derivative in (6.3) into a G covariant derivative so that f becomes fA = +-1/2[OAA 9gCABaAu Aap + ig~vtAsS] , (6.4) where the lowercase Latin indices run over the generators of the unbroken gauge group and Cap3 are the completely antisymmetrized structure constants of G. We now apply this gauge fixing to the calculation of thresholds for the gauge couplings in the MS scheme. Although these calculations have been 49 presented in previous works [94-96], for completeness we include some of the details here. We begin with the case of a general gauge theory in which some simple group G is broken to a product of simple or Abelian groups, Gi. Let 1'a"Fa~+~ PMF)+DP) 14-( )L =_F1 Fqu+(J-MF)+D D -V()y+g.f.+ï¿½gh (6.5) be the Lagrangian of the full, G invariant theory where the gauge fixing term Cg.f. is to be chosen as in ref. 94. The full gauge group, G is broken to G= r Gi x U(1) x ... at some scale. Below this scale the Lagrangian for the effective Gi subgroup will be: =ij"Fai ai w + ,j aiFp'FalV + --- (6.6) The second term can be obtained perturbatively by calculating the contribution of heavy fields to the vacuum polarization of light fields. To one loop there are only the two diagrams of Figure 8. H H V V a~ ~ 4- at a, ' Figure 8. Light gauge field vacuum polarization diagrams. Here H represents a generic heavy field. The second diagram is not present in the case that H is a heavy fermion. Because of gauge invariance, it is enough to evaluate the contribution from the first diagram, because the second one does not give terms proportional to k2, where k is the momentum of the light external gauge field. The resulting vacuum polarization is given by: IHap;a:v(k2) =aia(1 - li)(k2g - kpk" + ... ) 2. (6.7) =g2(Ai - ) + O(g4), where in the case = 1, 82i = (Tr(tiVtiV) - 21 Tr(tivtivln MV) + Tr(tistisAlnMS) + 8Tr(tiFtiFlnMI)), (6.8) 1 I - = 8 (-21 Tr(tivtiv) + Tr(tistiSA) + 8 Tr(tiFiF)) . Here A projects onto non-Goldstone boson scalar fields, tiS, tiF and tiV are the representation matrices of di for the heavy scalars, fermions and vectors fields respectively. MSF,V are their mass matrices. Because of the presence of a nonzero 1i in Eq. (6.6) the kinetic term of the effective theory is not canonically normalized. It can be normalized by introducing canonical bare fields and bare coupling constants Aai = (1 - li)/2Aaig, (6.9) and gi = (1 - 1)-1/2g. (6.10) Having related the bare coupling constants of the full and effective theories, we consider the MS renormalized coupling constants gi(p) and g(p): bG93( + gP =g(p) + + ...(, 2 + .(6.11) big3(p) gip =gip) + + 277 where bG and bi are the lowest-order coefficients of the / functions. Substituting these equations for g and gi into (6.10) and equating the finite terms we obtain a relationship between the renormalized coupling constants gi(p) = (1 - li)1/2g(#) = g(#) + I Oi(P)g3(p) , (6.12) where i is the finite part of Ii. This is the desired boundary condition. 51 This formula can also be used to determine the effect of integrating out a heavy quark in lower energy QCD. Here one need only include the heavy fermion part of the integration to determine the low energy gauge coupling in terms of the coupling of the full theory above the heavy quark threshold. Note that equation (6.12) only holds in the neighborhood of p - M, M being the heavy scale, and as such provides an initial condition for the running of the effective couplings for p << M. In application to the Standard Model we will integrate out the heavy fields using matching functions as described above. As we are running couplings to two loops, it is sufficient to integrate out these fields to one loop. The heavy gauge bosons, their ghosts, and the top quark are integrated out near MW, the other fermions at their physical masses Mj. One may integrate out different mass particles at one scale as long as the two-loop contribution to coupling constant renormalization between two threshold scales is negligible. The errors arising from not integrating out fields at a scale p exactly equal to their physical mass M is of order al,2,31n(M/p), which is negligible within the perturbative regime. 6.2 Gauge Coupling Thresholds in the SM At the electroweak threshold, the point at which W and Z bosons, their associated Nambu-Goldstone bosons, ghosts, and the top quark are integrated out, one imposes matching conditions similar to (6.12). Above the threshold the theory has the SU(3)C x SU(2)L x U(1)y gauge symmetry of the Standard Model and a SU(3)C x U(1)em effective symmetry below. Following refs. 94 and 95, we gauge fix the Standard Model in such a way that the low energy 52 theory is SU(3)C x U(1)em invariant after the SU(2) gauge fields are integrated out. The gauge fixing part of the Lagrangian is -g.f. = --((Op Wj' + ig2 riV(l)ij~j +92sW2"Ap)2 +(OpW2' + ig2vi(r2)ijbj + g2sWj"Ap)2 (6.13) +(OpZ" + ig2c-1vi(r3)ijj)2) , where c = cos Ow, s = sin Ow, and Oi is the shifted Higgs field with vi its vacuum expectation value. The heavy gauge bosons, Higgs and Nambu-Goldstone bosons, and ghosts are integrated out to one loop by evaluating their contribution to photon vacuum polarization. As in Eq. (6.6) the contribution of the photon vacuum polarization diagrams to the effective action gives rise to a noncanonically normalized kinetic term. ï¿½SM = Gla B +. (6.14) becomes Leff= - FpF "(1 - 1) +... (6.15) where I contains loop effects, and GA,, Bv and Fov are the SU(2)L, U(1)y and U(1)em field strengths, respectively. The low energy effective charge e is related to the Standard Model couplings by normalizing the kinetic term as in equations (6.9) and (6.10) : Ap Aj(1 _- 1)1/2, e = g2s(1 - )1/2, (6.16) where the last relation is a relation between bare couplings. Here 2 - (6.17) l=1 where 2 - m 72(1 - 211n( )),(6.18) fli) =4 72( (6.18) 87 To get the running relation between e, s and 92 we use the high and low energy 3 functions which can be derived from the following: + 1 be3 () f S= j(g(p) + ) + 92,bl! (P) + +29 (P, s = s(P) ){1 + (1 - s2(p))( 2 )} 2e Inserting these expressions into (6.16) yields 1 1 1 --- -2 4r(p) , (6.20) a(p) a2(P)s2 p)' with the running Weinberg angle s(p) defined in (5.1). Using the GUT normalization for al, one can also write 1 3 1 - s2() 3 5( ( ) + -(1 - (p))4rt(p), i 5 s2(p) (P +S)2 (6.21) 1 s2 ) a2 a(p) where the first term on the right side of each equation gives the usual tree level relation. One can always impose these tree level relations by fixing p so that the matching function Q vanishes. Here this occurs for f = 0.95 MW. When the heavy top quark is included in the analysis one may integrate it out separately in which case one has an effective Standard Model without a top quark between MW and Mr. Alternatively one can integrate it out at the same scale as the massive gauge bosons in which case the top quark loop in the photon propagator contributes to the matching function above Q(p) -- P(p) + 2b)EDln( Mt) , (6.22) QED 54 where b(t)ED = 1/97r2 is the contribution of the top quark to the coefficient of QED r e3 in the QED fP function. The perturbativity of QED at MW and the small mass difference between the top quark and gauge bosons (so that aln(Mt/Mw) is small) means that this is a reliable approximation. We incorporate this matching condition which includes the top quark into our numerical integration of the electroweak # functions. The strong QCD coupling has thresholds near quark masses. The effect of integrating out a quark is to produce a matching condition of the form 1 1 1 - - 4r$3(p) (6.23) Oas(P) c3(Wu where S13(p) = 2b(q)ln(M ) , (6.24) and b) = 1/2472 appears as the contribution of each quark to the one-loop QCD p function: 43 = b3g3. The top quark is integrated out at p = j so that the strong coupling does not match continuously across the threshold. For the other heavy quark thresholds however, we choose the renormalization scale for matching to be the physical quark masses so that the matching functions vanish there. The only effect of these thresholds is a step in the number of flavors nft as each quark threshold is passed [97]. Light quarks require greater care as the strong coupling becomes nonperturbative at these mass scales and the evaluation of hadronic contributions to vacuum polarization must be evaluated. In this analysis however we remain in the perturbative region. Likewise the QED coupling in the low energy theory has thresholds at charged particle masses, the matching condition having the form 1 1 em - -(p) 47rtQED(p) . (6.25) 55 Here aem is evaluated in the zero frequency limit and 1 QED(8 Q 2Fln( !) + E Q2ln(MS) . (6.26) QQED( = ZT F S The F and S subscripts refer to charged fermions and scalars, respectively. Again the matching function vanishes at each particle mass so the effect of the threshold is simply to produce a step in the number of fermions or scalars of a given charge. 6.3 Threshold Effects in Fermion Masses (Yukawas) It is important to realize, that the running fermion masses also experience threshold effects near physical particle masses. To our knowledge there is no mention of these effects in the literature although they are potentially important in analyses of mass relations predicted in many grand unified models and in the full two-loop running of gauge couplings. Since the Higgs field is integrated out at the electroweak scale the Yukawa couplings appear in the low energy theory through particle masses and various nonrenormalizable (nr.) interactions: ï¿½SM = - YijVakii j- YjSa ik j+(62 (6.27) 'low= - mjbkj- + nr. terms + .... We evaluate the matching conditions in MS running fermion masses at the electroweak threshold by integrating out Higgs1, Nambu-Goldstone and gauge bosons, as well as ghosts and the top quark. The diagrams corresponding to integrating out these fields that contribute to one-loop renormalization of quark and lepton two-point Greens functions in the Standard Model are depicted in Figure 9. 1 The Higgs field does not contribute at one-loop to the above matching functions as it has no SU(3)C x U(1)em couplings. 56 z w* *1 I I 02 s 1 s , I lI I Figure 9. One-loop corrections to fermion two-point functions. As in (6.13), we work in a specific gauge required for gauge invariance of the low energy effective Lagrangian under SU(3)C x U(1)em. As for the gauge bosons the finite piece of the contributions of these diagrams gives the matching function and the divergent piece gives the one-loop 3 function (see Appendix I). In terms of bare parameters and fields the relevant parts of the Standard Model Lagrangian for this calculation are 1SM = iLi PiL + TiRz PkiR (6.28) - YijVa iL jR - Yijka iLjR + h.c. ... where Oi can be a quark or lepton and i is a family index. When the heavy fields are integrated out we generate the following low energy effective Lagrangian, low = (1 + KiL)/iLi PiL + (1 + KiR)VbiRi PkiR (6.29) - (mij + bmij)(biLtjR + h.c.)+... , where mi= yVa. The KL,R contain wavefunction renormalization contributions of the left and right handed fermion fields along with finite parts, and 6mij is the fermion self-energy contribution. 57 We must rescale the bare fermion fields in the low energy theory so they have the canonically normalized kinetic term, iL,R = (1 + KiL,R)1I/2iL,R. (6.30) The relation between the Standard Model and low energy bare masses becomes m = .) = (m ) + m~j)(1 + KiL)-1/2(1 + KjR)-1/2 . (6.31) Note that the left and right handed fermion fields are differently renormalized due to their different gauge couplings, so KL # KR . Also note that in the case of quarks the self-energy corrections Smij introduce additional nondiagonal contributions to the mass matrix. However, in the limit Mq We determine the matching functions for the fermion masses in the Standard Model in terms of their electroweak quantum numbers. Since we work in the limit where the self-energy contributions are diagonal, we write 6miy = bijmi(1 + Kim). The functions, KiL, KiR, and Kim are 2 ii KiL=,2iK KiL = 256r2c2 L) ) 2 it KiR - 256 2c2- - , (6.32) Kim = -6-m-) 2 if =i 92 K M Km 64-,2c2 t' where 1 1 - = - + ln47r - yE (6.33) We work in ( = 1 gauge. The coefficients of the divergent parts, if = (g + g)2 + 8c2 S= (g - g )2, (6.34) sti2 i2 m = gv - 9A) 58 give the one-loop p functions. Those of the finite parts, M22 ' M2 - 8c2(ln__) + 0) = (g- g)2(InM - 1) (6.35) ï¿½ . M2 i, = (g - g )(ln - ) give the matching functions. Here gViA = 2(g g), where T LR gV,A = 2(g' g' g), where g' --L, s2Qi and T LR and Qi are the third component of weak isospin and the electric charge, respectively, for a given handedness of the ith fermion. For the different quark and lepton charge sectors one has A= 1 V= 1 g = -1 ge = -1 + 4s2 g= 1 gV= 1-s (6.36) d = 1 gd = 1+ 4 2. Inserting the # functions into (6.31) we obtain the relation between the diagonalized, renormalized masses in the Standard Model and the low energy effective theory, lo~w) (S) gg)i m (low))=m (SM)(p) 1 + 22 (P) + -(,L(Y) + K-(p))) . (6.37) We next use the results of the preceding sections to analyze the importance of threshold conditions in the study of theories beyond the Standard Model. 6.4 Thresholds Beyond the Standard Model The difficulty with applying the general formula in (6.8) and (6.12) to higher energy theories such as GUTs is that little is known of the particle spectrum of such theories to which the thresholds are sensitive. Fortunately some reasonable assumptions about these spectra can be made in many cases 59 and the threshold effects can then be well estimated. Historically these were of importance in proton decay estimates in minimal SU(5). Here it was found (in the case of threshold calculations in a momentum subtraction scheme) that grand unified threshold effects spread out the gauge couplings at the naive value of MaGUT so that they actually meet asymptotically beyond this scale [20]. This has the effect of reducing the value of MaGuT required for unification by a factor of 2 from the naive value. Interestingly the electroweak threshold effects are also important, reducing MaGUT by an additional factor of 3 relative to the value obtained using the naive step function approximation. This factor of 6 reduction has dramatic effects on proton decay since from Eq. (3.12) it is proportional to M4. It was through the threshold analysis that it was first x realized that minimal SU(5) predictions for proton decay were dangerously close to the experimental limit. Threshold efects are also potentially important in supersymmetric boundary conditions and in the calculation of renormalization group improved mass relations. They have been used recently to attempt global analyses of the particle spectrum of the MSSM [98,99,100] when contraints coming from the nature of the soft supersymmetry breaking parameters arising from supergravity or string-inspired models are taken into account. One finds that incorporation of supersymmetric particle thresholds tends to increase Msusy from what is naively expected when an average threshold is used. Previous analyses usually treated thresholds beyond the Standard Model by integrating out all heavy particles at a single "average" scale in the step approximation. The inadequacy of this treatment of thresholds has been re-emphasized by Barbieri and Hall [101]. To simplify the argument they consider a general GUT 60 model with a gauge group G containing SU(5) with a naive "average" supersymmetric threshold and a GUT threshold with matching functions incorporating a simple superheavy spectrum. One postulates a superheavy spectrum as follows: superheavy gauge supermultiplets V with common mass My, heavy components of the SU(5) chiral supermultiplets H in which the Higgs doublets lie of mass MH and superheavy Higgs multiplets E involved in breaking G to SU(3)C x SU(2)L x U(1)y of mass ME. For simplicity we consider the analytic solutions to the renormalization group equations of Appendix II. They give [101] M_____ (3 - 15s + 7-a log( ssv 4 5) = I = -2.1 ï¿½ 2.6 + 1.0 , (6.38) MZ 19a where the errors come from the error in as and sin OW = s respectively. The gauge couplings are evaluated at Mz. The 1-loop GUT matching function modifies this result to give log( MgU ) = I + 6 log MV - log MV (6.39) M log HH"(6.39) There is no reason why these logarithmic terms cannot be significant compared to the one loop effects. This implies that one cannot accurately extract Msusy from experiment no matter how small the errors in as and sin2 OW because of uncertainties in the mass splittings in the superheavy sector. New data on the Standard Model parameters must be combined with other experimental or theoretical constraints to make it possible the better estimate Msusy CHAPTER 7 NUMERICAL TECHNIQUES We use the Runge-Kutta method to numerically integrate the /3 functions. There are 18 coupled first order differential equations involved in running the Standard Model couplings. At one loop some equations decouple from the rest. For example, the gauge couplings are decoupled at one loop, although the Yukawa /3 functions depend on both gauge couplings and Yukawas even at one loop as in Eqs. (I.8-9). The two-loop /3 functions we are using are all coupled. To solve these equations we require initial data at some specific scale which we take to be Mg. Unfortunately, as we have discussed in Chapter 5 the Standard Model parameters are experimentally measured at different scales. The method used to obtain all the parametersat a single scale involves the simultaneous solution of N nonlinear equations in N unknowns where N < 18. This can only be done numerically and computer routines to do this are readily available. After all data is obtained at MZ, the Runge-Kutta routines are used to evolve the parameters to any mass scale p. We run the quark masses and CKM angles by diagonalizing the Yukawa matrices at every step in the Runge-Kutta method used in solving the /3 functions. However, in the literature, some authors write down analytic expressions for the running of these masses and angles by making some approximations. Typically, it is assumed that the contribution of the Yukawa couplings matrix is given essentially by the top quark Yukawa since it is much larger than the others. Sometimes a better approximation is made by keeping only the diagonal entries. Our numerical technique represents a minor improvement over these methods. 62 As discussed in Section 5.6, given the evolution of the running parameters mt and A, the physical top quark and Higgs boson masses may be found by solving Eqs. (5.38) and (5.39), respectively. As the top quark and Higgs boson masses are unknown in the Standard Model at present, in the process of our analyses we are free to choose values for these masses at MZ and then proceed to study the consequences. In other models in which there may be certain constraints among couplings, one may incorporate these constraints into the numerical initialization routines. The freedom to choose a value for an unknown parameter may be replaced by such a constraint, and this may result in a definite prediction for that parameter. Constraints from grand unification and supersymmetry were used (14] to arrive at possible values for the top quark and Higgs boson masses. It should be kept in mind that the results of this work were only possible through the painstaking collaborative development of large computer routines which perform the function of evaluating the P functions for each parameter of the model studied, imposing threshold matching conditions at physical particle masses, diagonalizing at each step the mass matrices, extracting the CKM angles and finally plotting the results for investigation of renormalization group patterns. The ultimate aim is to make these routines accessible to other investigators and make it simple to append information concerning other theories. At present these routines are menu-driven and incorporate the full two loop P3 functions of the Standard Model, the MSSM and their low energy counterparts. However it is still in the prototype stage as far as ease of use and improvements are underway. CHAPTER 8 RG ANALYSIS OF THE STANDARD MODEL For completeness we depict the results of numerically integrating the 0 functions for the Standard Model parameters from 1 GeV to Planck mass in Figures 2-3,10-15. We arbitrarily use Mt = MH = 100 GeV and in some plots we superimpose one- and two-loop evolution. Differences between one- and two-loop evolution appear in the high energy regime and are also manifest for the strong coupling at low energies where it becomes large. 0.020 0.015 0.010 0.005 0.000 -0.005 0 5 10 15 1glo(A) 20 Figure 10. Light quark and lepton masses for Mt = 100 GeV and MH = 100 GeV. 64 As previously discussed, we see in Figure 2 the "GUT triangle" signifying the absence of grand unification, assuming the Standard Model as an effective theory in the desert up to the Planck scale. Figures 10, 11, and 3 display the evolution of the light mass fermions (me, mu, and md), the intermediate mass fermions (mp and ms), and the heavy mass fermions (mr, me, and mb), respectively. 0.25 i . i ,. , ..*' - 1 Loop 2 Loop 0.20 Mt=100 GeV ME=100 GeV 0.15 0.10 0.05 -0.00 I i I I I 0 5 10 15 20 logo(P) Figure 11. Intermediate quark and lepton masses for Mt = 100 GeV and MH = 100 GeV. We conclude that the largest differences between one-loop vs. two-loop evolution occur in the bottom, charm, and strange quark masses in these cases. In no case is the minimal SU(5) prediction of quark and lepton unification borne out. 65 In Figure 12, we plot the quartic coupling A and the top Yukawa coupling yt for (Mr = 100 GeV, MH = 100 GeV) and for (Mt = 200 GeV, MH = 195 GeV). 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 10 logo10() Figure 12. Top Yukawa and scalar quartic couplings. These two couplings are the only unknown parameters of the Standard Model. We have studied the effects of changing the values of Mt and MH in our analyses of the running of the other parameters. We observed that, for any Mt between 100 GeV and 200 GeV, varying MH, while maintaining perturbativity and vacuum stability, did not affect appreciably the evolution of any of the other parameters. However, changing Mt itself showed a significant difference in the running of the heavier quarks. To illustrate this, in Figures 13 and 14 we _- rt - - - M.-loo GeV. -1oo GeYV )Lt=2OQ GeV. Mim=195 GeV Yt - N. l- t1111 I ~i~1i 66 display plots similar to Figure 11 and 3 but for Mt = 200 GeV and MH = 195 GeV. 0.25 0.20 M=2oo00 GeV MR=195 GeV S 0.15 m. 0.10 0.05 0.00 0 5 10 15 20 logs0(o) Figure 13. Intermediate quark and lepton masses for Mt = 200 GeV and MH = 195 GeV. In particular, in Figure 14 we note that the intersection point between the bottom quark and the r lepton moves down to a lower scale for this case of a higher top quark mass. This is expected since from Eq. (I.9) one can see that the bottom type Yukawas are driven down by an increased top Yukawa. This is to be contrasted with the SUSY GUT case in which the bottom Yukawa 0 function is such that this crossing point is shifted toward a higher scale with an increased top mass. In an SU(5) SUSY GUT model, the equality of the bottom and 7- Yukawas at the scale of unification was used to get bounds on 67 the top and Higgs masses [14] and we shall discuss this in the next chapter. Lastly, we display the running of the CKM angles in Figure 15. We have taken b = 900, which corresponds to the case of maximal CP violation. As mentioned in Section 5.3, the evolution curves for these angles are effectively flat. 6 m 2 0 0 5 10 15 logt0( ) Figure 14. Heavy quark and MH = 195 GeV. lepton masses for Mt = 200 GeV and In the present case of the Standard Model, we find that two-loop running of the parameters does at times improve on the one-loop running. Indeed, we have tabulated the differences of several parameters in their one- versus two-loop values at various scales, for the cases (Mr = 100 GeV, MH = 100 GeV) and (Mr = 200 GeV, MH = 195 GeV). Table 3 illustrates the difference 68 one-loop vs. two-loop running makes in the ratio mb/mr, for the three scales 102 GeV, 104 GeV, and 1016 GeV. 0.3 0.2 0.1 0.0 -0.1 10 log10o() Figure 15. CKM angles. Table 3. mb/mr Mt = 100 GeV Mt = 200 GeV 102 GeV 104 GeV 1016 GeV 102 GeV 104 GeV 1016 GeV i-loop 1.879 1.455 0.8081 1.868 1.392 0.6647 2-loop 1.782 1.348 0.7336 1.769 1.285 0.6047 Clearly, the difference between one- and two-loop results is more pronounced at higher scales, as expected. Over all these scales the difference is never less than 5.5%. We note that the ratio becomes equal to one well below the scale I I I I I I I I I I 1 I ' I sine, M-100 GeV M=10O0 GeV line s sine* I I I" . 1 1 1 1 " R R 69 of grand unification as noted above in the discussion of Figures 3 and 12. The next table presents a similar comparison for the top Yukawa. Here, two loops represent a smaller correction with the difference at all scales always being less than 5%. Table 4. yt M = 100 GeV Mt = 200 GeV 102 GeV 10 GeV 1016 GeV 102 GeV 104 GeV 1016 GeV 1-loop 0.5405 0.4160 0.1928 1.133 0.9780 0.7145 2-loop 0.5405 0.4071 0.1842 1.143 0.9700 0.6816 Finally, Table 5 displays the same analysis for as for the case Mt = MH = 100 GeV. We observe no appreciable deviation from the tabulated values for any Mt A 200 GeV (except in the low energy regime where the difference is at most - 4%). Table 5. as 1 GeV 102 GeV 104 GeV 1016 GeV 1-loop 0.3128 0.1118 0.07103 0.02229 2/3-loop 0.3788 0.1117 0.07039 0.02208 At scales < MZ, the inclusion of two loops is important in the evolution of the strong coupling (and of the quark masses). Indeed, we find that the pure QCD three-loop contribution is also significant and therefore include it in the running of the strong coupling and of the quark masses in the low energy region. As seen in this table, the combined two and three loops in the low energy regime account for a 17% difference at 1 GeV in as. Although in the cases considered in these last two tables there does not appear to be a significant difference in two-loop over one-loop evolution at scales above Mz, the first table does show a 10% difference at the scale 1016 70 GeV. We expect two-loop effects to be more important when the theory is extended, e.g., to include supersymmetry and/or grand unification. The effects of using a naive step approximation vs. a proper treatment of thresholds are numerically unimportant for the cases discussed above. Indeed they are less important than the two-loop effects. We note however, that the inclusion of non-naive thresholds effects is significant in the numerical analysis of extensions of the Standard Model. CHAPTER 9 RG ANALYSIS OF THE MSSM AND SUSY-SU(5) We present bounds on the mass of the top quark in the MSSM with minimal Higgs structure in the context of a grand unified theory by numerically evolving the couplings using their renormalization group equations. This analysis improves on previous endeavors by taking full account of the Yukawa sector. The MS renormalization group equations for the Standard Model and the MSSM [78] found in Appendix I and II are numerically integrated and used to evolve the parameters of the model to Planck scale. Although it is not possible to analytically express certain parameters (e.g. CKM angles) in terms of the Yukawa couplings, equations for the running of the quantities themselves can be arrived at by making some approximations. Since our approach is numerical we opt to run the quantities by diagonalizing the Yukawa matrices at every step of the Runge-Kutta method as described in Chapter 7. In the expectation that the Standard Model is only the low energy manifestation of some yet unknown GUT or of a possible supersymmetric extension thereof, the three couplings g3, 92, and gl corresponding to the Standard Model gauge groups, SU(3)C x SU(2)L x U(1)y, should meet at some large grand unification scale. Using the accepted values and associated errors of these couplings we observe unification in the SUSY-GUT case but not in the pure GUT case, as noted by several groups [9,11,13] (see Figures 2 and 4). However this should not be viewed as proof of supersymmetry since given the values of al, a2, a3 at some scale, and three unknowns (the value of a at the unification scale, the unification scale, and an extra scale such as the SUSY scale) there is always 72 a solution. The exciting aspect of the analysis of ref. 11 is the numerical output, namely a low SUSY scale, Msusy, and a perturbative solution below the Planck scale which does not violate proton decay bounds1. Furthermore, in the context of a minimal GUT [4] there are constraints on the Yukawa couplings at the scale of unification. We first restrict ourselves to an SU(5) SUSY-GUT [57] where Yb and Yr, the bottom and r Yukawa couplings, are equal at unification. The crossing of these renormalization group flow lines is sensitive to the physical top quark mass, Mr. This can be seen in the downtype Yukawa renormalization group equation (above Msusy, for example), from which we extract the evolution of Yb, since the top contribution is large and appears already at one loop through the up-type Yukawa dependence: dYd 1 dt = 162Yd[ 3dtYd + YutYu + Tr{3YdtYd + Ye Ye 7 7 2 2 16 2) (9.1) - ( 21 + 3g2 + 93). where Yu,d,e are matrices of Yukawa couplings. Demanding that their crossing point be within the unification region determined by the gauge couplings allows one to constrain Mt. This yields an upper and lower bound for Mt which nevertheless is fairly restrictive. The threshold analysis used is as follows. For the electroweak threshold we use one loop matching functions [95] with the two loop # functions valid in the Standard Model regime below the SUSY scale. At the electroweak threshold, near MW, we integrate out the heavy gauge fields and the top quark. Below this threshold there is an effective SU(3)C x U(1)em theory. Thresholds in this region are obtained by integrating out each quark to one loop at a scale equal 1 At the one loop level, due to a cancellation between large numbers, the value of the SUSY scale is extremely sensitive to the value of a3, which means that the proper treatment of thresholds and of two loop effects will determine the actual value of the SUSY scale. (L. Clavelli, private communication). 73 to its physical mass. At these scales the one loop matching functions in the gauge couplings vanish and the threshold dependence appears through steps in the number of quark flavors [97] as the renormalization group scale passes each physical quark mass. There is also a threshold at Msusv. Here the matching condition is the naive one of simple continuity due to the lack of knowledge about the superparticle spectrum. We take this scale to be variable to account for this ignorance. We consider the simplest implementation of supersymmetry and run the couplings above Msusy to one loop. We assume the MSSM above Msusy, and a model with a single light Higgs scalar below it. This is done by integrating out one linear combination of the two doublets at Msusy, thereby leaving the orthogonal combination in the Standard Model regime as the "Higgs doublet": @4(sM) = Idcoss# + $usinf3, (9.2) where = ir2#*, and where tan/3 is also the ratio of the two vacuum expectation values (vu/vd) in the limit under consideration. This sets boundary conditions on the Yukawa couplings at Msusy. Furthermore, in this approximation the quartic self coupling of the surviving Higgs at the SUSY scale is given by AW(Ms usy) = (g + g2)cos2(2#) (9.3) This correlates the mixing angle with the quartic coupling and thereby gives a value for the physical Higgs mass, MH. Using the experimental limits on the MH further constrains some of the results. By using the renormalization group we take into account radiative corrections to the light Higgs mass [102] and hence relax the tree level upper bound, MH . MZ [103]. 74 We determine the bounds on Mt and MH by probing their dependence on 3. In SUSY-SU(5), tan /# is constrained to be larger than one in the one light Higgs limit. It seems natural to us to require that yt - Yb up to the unification scale [104], thereby yielding an upper bound on tan f. The initial values at Mg for the gauge couplings are given in Sections 5.1 and 5.2. For this analysis we use an earlier value the strong coupling, a3 = 0.10900 [1051. We utilize the quark mass values given in Section 5.4. For the bottom mass we use the Gasser and Leutwyler bottom mass value of 5.3 GeV at 1 GeV which translates into a physical mass of Mb = 4.6 GeV [79]. To probe the dependence of our results on Mb we also study the case of Mb = 5 GeV, the typical value obtained from potential model fits for bottom quark bound states [87]. We also investigate the effect of varying Msusv. Given the values of the gauge couplings, we find unification up to a SUSY scale of 8.9 TeV, and as low as MW. For empirical reasons we did not investigate solutions below that scale. From Figure 4 we determine that the lower end scale, MGUT, of the unification region corresponds to an a3 value of 0.104 at MZ, while the higher end scale, MHUT, corresponds to a value of 0.108 at MZ for a3. We find that the unification region is insensitive to the range of top, bottom, and Higgs masses considered. In our analysis of the bounds for Mt, the values for al and a2 are chosen to be the central values since their associated experimental uncertainties are less significant than for a3. Demanding that Yb and Yr cross at MLT and taking a3 = 0.104 then sets a lower bound on Mt. Correspondingly, demanding that Yb and Yr cross at MI-HT and taking a3 = 0.108 yields an upper bound on Mr. These bounds are found for each possible value of f. Figure 16 shows the upper and lower bound curves for both M and MH as a function of #/ and for Msusy = 1 TeV and Mb = 4.6 GeV. When applicable 75 we use the current experimental limit of 38 GeV on the light supersymmetric neutral Higgs mass [106], to determine the lowest possible Mt value consistent with the model. We find 139 < Mt 5 194 GeV and 44 < MH < 120 GeV. We investigated the sensitivity of these results on Msusy in the range, 1.0 + 0.5 TeV. We find that the bounds on Mt are not modified, but the upper bound on the Higgs is changed to 125 GeV, and the lower bound drops below the experimental lower bound. MaMn = 1 TeV, Mb = 4.6 GeV 200 150 a) 0 T 100 50 o 0 " 40 50 60 70 80 90 P (deg.) Figure 16. Plot of Mt, MH, as a function of the mixing angle 0 for the highest value of a3 (high curves) and the lowest value of a3 (low curves) consistent with unification. For Mb = 5.0 GeV, we see an overall decrease in the top and Higgs mass bounds: 116 < Mt 5 181 GeV, MH < 111 GeV. Varying Msusy as above 76 modifies the respective bounds. The top mass lower and upper bounds become 113 and 119 GeV, respectively. The upper bound on MH changes to 115 GeV. We display the results of our analysis for the extreme case, Msusy = 8.9 TeV, in Figure 17, with Mb = 4.6 GeV. This only significantly changes the upper bound on MH to 144 GeV compared to the Msusy = 1 TeV case. Msusr = 8.9 TeV, Mb = 4.6 GeV 200 150 100 50 0 50 60 70 80 6 (deg.) Figure 17. Same as Figure 16 for Msusy = 8.9 TeV and Mb = 4.6 GeV. We have also run yt up to the unification region and compared it with Yb and yr to see what the angle 3 must be for these three couplings to meet [107], as in an SO(10) or E6 model [5 - 6] with a minimal Higgs structure. It is clear that this angle is precisely our upper bound on 3 as described earlier. In Figure 18 we display yt /yb at the GUT scale as a function of tan 3 for Msusy = 1 TeV 77 and for the two bottom masses we have considered. If we demand that the ratio be one we can determine the mixing angles for the low and high ends of the unification region. Then going back to Figure 16, we find as expected a much tighter bound on the masses of the top and of the Higgs. Indeed, for Mb = 4.6 GeV, we have 49.40 < tan / <54.98, which yields 162 < Mt < 176 GeV and 106 < MH < 111 GeV. When Mb = 5.0 GeV, we obtain 31.23 < tan/# 5 41.18, which gives 116 < Mt <5 147 GeV and 93 < MH < 101 GeV. MsMy = 1 TeV 102 101 100 0 10 20 30 40 50 tanfi Figure 18. Plot of the ratio of the top to bottom Yukawas, Yt/Yb, for two different bottom masses (solid and dashed curves) as a function of tan # for the highest value of a3 (high curves) and the lowest value of a 3 (low curves) consistent with unification. Several issues have been left untouched. The effects of soft SUSY breaking terms were not investigated nor was the possible role of a large top mass on 60 78 this breaking. Also, we have integrated out all the supersymmetric particles at the same scale. It would be interesting to study the effect of lifting this restriction. We should also note that our bounds on the top mass are very similar to those of ref. 104, although the physics is very different. Given the relative crudeness of our approximations in this analysis, it is remarkable that the experimental bounds on the p-parameter were satisfied which in our mind gives credence to our program. CHAPTER 10 CONCLUSIONS We have have established an efficient procedure for performing renormalization group analyses of the Standard Model and its extensions. With two loop renormalization group equations at our disposal and numerical routines developed to solve them we simply follow a series of steps. First pick your favorite high energy model with some hierarchy of symmetry breaking. Decide what particles and models are to be placed in the "desert" between MW and Mpl. Implement effective gauge theories between thresholds that break gauge symmetries through the use of matching functions. Run renormalization group equations for each effective theory according to your scenario's physical particle spectrum. Finally analyze output in graphical form to see what patterns emerge and perform careful tests of promising patterns arising from relations among parameters in the high energy theory. We have performed these analyses in the case of the Standard Model and its minimal supersymmetric extension. In the former case we have reproduced the result that gauge and Yukawa coupling relations arising in GUTs with the Standard Model as the low energy theory below MGUT are not valid. No reasonable spectrum of superheavy fields can produce unification in this case, even when thresholds are included. This merely adds to the failure of minimal GUT predictions for proton decay. In the latter case we have found remarkable agreement with the GUT predictions of gauge and Yukawa unification although the latter is in agreement only for the heaviest fermion family. By using the best data available we find constraints on the top quark from the relation mb = mr due to the sensitivity of the renormalization group improvement of this relation to mt. The upper bound 80 on Mt lies below the upper bound coming from radiative corrections to the p parameter. Although the Standard Model can have a 100 GeV top quark, the lower bound from our renormalization group analysis indicates that minimal SUSY-SU(5) favors a somewhat larger value. If Fermilab finds Mt between 100 and 120 GeV this should disfavor minimal SUSY-SU(5) models with one light Higgs doublet below the supersymmetry scale. APPENDIX I THE STANDARD MODEL f FUNCTIONS In this appendix we compile the renormalization group fP functions of the Standard Model. These have appeared in one form or another in various sources. We have endeavored to confirm their validity through a comparative analysis of the literature. Our main source is ref. 78. Following their conventions, L = -QLYutuR + QL,4YdtdR + ?LYeteR + h.c. - 1A(4t4)2, (I.1) 2 where flavor indices have been suppressed, and where QL and eL are the quark and lepton SU(2) doublets, respectively: QL = (UL) , = . (1.2) QL= dL ' eL # and 4 are the Higgs scalar doublet and its SU(2) conjugate: = +0 + ' (1.3) UR, dR, and eR are the quark and lepton SU(2) singlets, and Yu,de are the matrices of the up-type, down-type, and lepton-type Yukawa couplings. The # functions for the gauge couplings are - -bl g'bki dt 167 (1672)2 (1.4) 3 (162) Tr{CluYut Yu + cidYtd + CleYet Ye (161r2)2 where t = lny and 1 = 1, 2, 3, corresponding to the gauge group SU(3)C x SU(2)L x U(1)y of the Standard Model. The various coefficients are defined 1 10, 1 11) (9 3 5 5a 54f 1h 0) - 760 49 3 II9 13, 4 7 0 0 0 , withf = u , d, e , 1 with ng = nfl. In the Yukawa sector the 0 functions are dYu,d,e " (de (1) dt t- ude62 ar 'ude + where the one-loop contributions are given by 3(1) = (YUtY - YdtYd) + Y2(S)2 P(1)= ydtYd - YutYu) + Y2(S)2 1 #(2 ) (16ir2)2p u,d,e 17 2 g91 + (l4 4 9 2 g + 8g), g 8g3) g2 2) Ye~e+ 2(S) 1 (g+g2) with Y2(S) = Tr{3YutYu + 3YdtYd + YetYe}, to be 4 b = -ng - 22 4 1 3 3 g-~ , 4 b3 = 11 - 3ng (0 (bkl) = 0 0 and 0 01 102) /19 I" - ng 3 144 (1.5) (1.6) (1.7) (17 TU (Clf )3= 2 (1.8) (1.9) (I.10) and the two-loop contributions are given by #()u) (y ty~) tt l +y 11(y2 = (2 Y) YUYU Ydd - ~dtYYu + dd2 + Y2(S)( YdtYd - 9YutYu) - X4(S) + 3A2 - 2A(3YutYu + YdtYd) 2232 1352 1 2)43 92 S(-- + 16g + 16g t)Yu _Yu- (-g 9 g + 16g2)YdtYd 806 1 - 2+1693Y~Y 59 29 ,4922 1922 (35gï¿½ 22 +5 Y4(S) + (9 + -ng)gl - +g2 2 --g - ( - ng)g 9g2g 2200 45 1To12-13- g2+23 404 80 4 - ( 3 9 ng)g , (2) (Ydty)2 - YdYdYu tYU - YutYuYdtYd + (Yu Yu)2 Pd 2 4 4u2 + Y2(S)( Yu Yu - 9YdtYd) - X4(S) + 3A2 - 2A(3YdtYd + YutYu) 1872 1352 79 2 92 + (--g1 + j-692 + 16g3)YdtYd - (91 - 2 + 16g )Yu Yu 5 29 1 4 27 2 3122 35 4 + Y4(S)- (20 J-ng)g g - ng)g 2 20 5 20 To4 2 2 404 80 4 +923-( 3 9 ng)g3 fl()3(y ty)2--' 2) 2yet )2 _ _Y2(S)Yetye - X4(S) + A2 - 6AYe Ye 387 135 2 5 5 1 11 27 +( 801 + - g )Ye e + 2Y4(S) - (200 -ng 1 + g2 35 - (_ ng)g4, (I.11) with 172 )rYt} Y4(S) = (-g2 + g + 8g ) Tr{YutY S 49 (I.12) + (1 + + 8g 2)Tr{YdtYd} + ( + g2)Tr{Ye tYe, and x4(S) = Tr {3(YutYu)2 + 3(Ydtyd)2 e)2 4 2(I.13) - 2yutYuYdtYd} In the Higgs sector we present # functions for the quartic coupling and the vacuum expectation of the scalar field. Here we correct a discrepancy in the one-loop contribution to the quartic coupling of ref. 78 dA 1 (1) 1 (2) -= - >+ (1.14) where the one-loop contribution is given by S=12A2- ( g + 9g2)A +9(34 2 9 2 4 25 + +(1.15) + 4Y2(S)A - 4H(S), with H(S) = Trf{3(y ty )2 + 3(dtYd)2 + (Y ty)2} (1.16) and the two-loop contribution is given by p2)= -78A3 + 18( g2 + 3g)A2 [ 9 229 497 + 5 (4 + 2ng)gl ] + (8 9 239 25 2-4 40 42 27 59 -ng)g1 g2 ( " 9 125 24 313 -11722 313 _ 1Ong)g4 + 117g2 2 - 1 3 97 - 8ng)g -5 24 40 + -ng)gl 9 - 64g2 Tr{(YutYu)2 + (YdtYd)2} - g2 Tr{2(YufYu)2 - (ydtyd)2+3(YetYe)2 -5 91 +3(YetYe)2} - gY4(S) + 10A[ (g + 9 + 9 2+ 8 )Tr{Y Yu} +1( g2 + + ( 4g1 + 9 2 32(g2 g2)Tr{y -g2 + 8g') Tr{Ydtyd} + + rYte} ] + g1[ (- g + 21g) Tr{YuftYu} + (3g + 9g2)Tr{YdYd 5 20 + (-~~g1 +11g2 )Tr{Ye ye} ]- 24A22(S) - AH(S) + 6A Tr{YutYuYdtYd} + 20 Tr{3(YutYu)3 + 3(YdtYd)3 + (Ye tYe)3} - 12 Tr {Yu Yu(Yu tYu + YdtYd)YdYd } The #3 function for the vacuum expectation value of the scalar field is dlnv 1 (1) dt 1672r 1 + I (2) (16r2)2 , where the one-loop contribution is given by 7(1) = (g + g2) - Y2(S) , and the two-loop contribution is given by (1.18) (1.19) 7(2) = _ 2 - Y4(S) + x4(S) 1 4 511 2ng)g' + ( 32 8 24 + 8ng)g g2 (1.17) (I.20) 93 - 800 5 4 2 ng)g2 - 2722 - 9 86 These expressions were arrived at using the general formulas provided in ref. 78 for the anomalous dimension of the scalar field, choosing the Landau gauge. In the low energy regime the effective theory is SU(3)C x U(1)em. We employ the general formula of ref. 108 to arrive at the # functions for the respective gauge couplings: dg3 9 [23 +[383 = [(nu + nd) - 11] + (nu + nd) - 102]4 di (47r)2 3 (41r)4 8 2 3 e2 + [-nu + ndld) 43)4 (I.21) 5033 325 2857 g + [ 5033 (nu + nd) - -25(nu + Rd)2_ 857, g 18 54 2 (4r)6 and de 16 4 4 e3 64 4 e5 d-t = [-nu + -nd + -nl] 47 + [-nu + -nd + 4n-] 64 16 (47r)4 (I.22) + [-nu + -nd e3g3 9 9 (47r)4 where nu, nd, and nj are the number of up-type quarks, down-type quarks, and leptons, respectively. In Eq. (I.21) we have also included the three-loop pure QCD contribution to the /3 function of g3 [109]. For the evolution of the fermion masses we used ref. 110. It is known that there is an error in their printed formula [111]. Using the corrected expression, we compute the following mass anomalous dimension. The fermion masses in the low energy theory then evolve as follows: dm dt- 7(l,q)m , (I.23) where the I and q refer to a particular lepton or quark, and where 2 2 1 e 3 93 7Y(,q) =- 7(l,q) (4 + ^(1,q) (4)2 + [11 4 33 4 213 e2g21 I.4 +[(1,q)e + ^1(1,q)93 + 2,(l q)e 3 477r(.24 333 93 (q) (4,)6 87 The superscripts 1 and 3 refer to the U(1)em and SU(3)C contributions, respectively. Explicitly, the above coefficients are given by 1 -- - Q2/q 7(1,q) - 21q) 73 o 7(q) = 0 13 33= ) (1) (1.25) 11 4 80 20 20 2 (1,q) (3Q,q) + 8 + 0d 3 n](1,q) 13= 4Q2q) 7(q) - 4q 33 404 40 ( + d) 7(q) - 3 9(nu + d) 333 2 140 2216 (q) = 14_7(nu + d)2 + (160((3) + 9)(nu + rd) - 3747], where Q(l,q) is the electric charge of a given lepton or quark, and ((3) = 1.2020... is the Riemann zeta function evaluated at three. In the mass anomalous dimension for the quarks above, we have also included the three-loop pure QCD contribution y333 [109]. "(q) APPENDIX II THE MSSM f FUNCTIONS Using some of the notation of Falck [112], the superpotential and soft symmetry breaking potential are as follows: P = ficYusQ + dYd'dQ + CyedL + d~d~u, Voft = m, 2tu + m oto + m2LtL + m2li + m~td+ met + Bp(4u% d + h.c.) + E( Aya Yu'jii4i + AeYdijji'dQj + A"'Yji4dL j + h.c. ), i,3 1 3 ï¿½gaugino = -2 MAX~it + h.c.. 1=1 (II.1) Various a2s have been omitted and a sum over the number of generations is implied. Also, hats imply superfields and tildes the superpartners of the given fields. First the gauge couplings are dgl 1 3 dit 16r2 3 91 g 2 (167r? 2 bkgk - Tr{CluYuYu + Cldydt + CleYetYe} k (11.2) where t = lnp and I = 1, 2, 3, corresponding to gauge group SU(3)c x SU(2)L x U(1)y of the Standard Model. The various coefficients are defined to be 3 bl = -- - 2ng , 5 b2 = 5 - 2ng , b3 = 9 - 2ng , 88 (11.3) /38 6 88) S175 (blk)= 2 14 8 ag + T53 T and /26 T (Clf) = 6 4 9 3 0 9 \ 50 -17 0 0 -54 , withf = u , d, e, with 1 with n9 = Pnfl. In the following we list the # functions for the parameters of the superpotential. dln = 16r2[ Tr{3YutYu + 3YdtYd + YetYe} - 3(g 92 +g2) ] (11.6) In the Yukawa sector the # functions are d,e1 (2) d ud,e( 62 l(1d,e (16r2)2 u,d,e (11.7) where the one-loop contributions are given by (1) = 3YutYu + Ydtyd + 3Tr{YutYu} - ( g + 16g 2) 15 -2 i(1) = 3YdtYd + YutYu + Tr{3YdtYd + YetYe} - (_2 + 39g2 + g3) /3)= 3Y eYe + Tr{3YdtYd + YetYe} - (9g2 + 3g) (II.8) 14 18 ) TT 6 2 4 0 (II.4) (11.5) 90 and the two-loop contributions are given by (2) - _4(YutYu)2 - 2(YdYd)2 - 2YdtYdYutYu - 9 Tr{YutYulYutYu - Tr{3YdtYd + YetYe}YdtYd - 3 Tr{3(YutYu)2 + YdtYdYutYu} +(-g +6g )Yu Yu + + (g+16g3)TrYuyu 26 403 21. 32n 304 4 +(26ng + 40)g + (6ng - -)g + (ng - )g3 22 136 33 + g192 + g193 + 8g2g3 (2) = _-4(Yd Yd)2 - 2(YulYu)2 - 2YutYuYdtYd - 3Tr{YutYu}YutYu - 3 Tr{3YdtYd + YetYelYdtYd - 3Tr{3(YdYd)2 + (YeYe)2 + YdtYdYutYul + (5g)YtYul + (g2 + 6g2)YdtYd + (- 2 + 16g2)Tr{Yd d 2)ydyd +(- V1 + 1693)Tr{ Ydtyd} 6 14 7 4 21 + (g ) Trf{YetYe} + (j-ng + )g + (6ng - 2 32 304 4 22 + 8 g 2 2 2 + (ng - --)g3 + g1g2 + gg + 8g g3 #(2) =-4(YetYe)2 - 3Tr{3YdtYd + YetYe lYetYe - 3 Tr{3(YdtYd)2 + (YetYe)2 + YdtYdYutYu} (6g2 6 2) ryteI+( 2 2 269) kytd +(6g )YetYe + (g)Tr{YetYe} +(- 1 +169)Tr{YdtYd} 5 5 18 27 4 214 922 + (-5ng + -)gj + (6ng - -T)g2 + gg2 (11.9) The evolution of the vacuum expectation values of the Higgs's is given by dlnvoud 1 (1) 1 (2) (.10) dt 167r2ud + (16r2)274u,d ' where the one-loop contribution is given by 7) = ( 9 +g 2) -3Tr{YutYu}, (II.11) 7 =31 + g92) - 3Tr{YdtYd} - Tr{YetYe} and the two-loop contribution is given by 7(2) 3 Tr{3(Yutyu)2 + 3YtyuydtYd} 19 2 9 20g)Tr{YutYu} - T91 + g9 + y~ytu 20 279 1803 207 357 27 92 (80+ 1600ng)g - 2 + --ng)g g -)g (.12) )(11.12) 7 = 4 Tr{3(YdtYd)2 + 3YdtYdYuYu + (Ye tYe)2 2g2 9 2 22) 9 2 3 2)Trlyty - (91 + g2 + 20g)Tr{YdtYd} -(-9l +g2)Tr{YeYe 279 1803 4 207 357 4 27 9 22 - ( -6 + 1ngg1 - (-- + - -ng)g2 - (- + -ng)glg2 800 160 32 6480 8 The soft supersymmetry breaking terms are dAï¿½ 1 Ykyjik " Aie de - [4(YeYet)ikAJ" + 5Aik (YetYe)k3 - 3 e(YeYe Ye)ij dt 167r2[4 Yij Y + 2(Akmlykml2 + 3Akdm ydm2) + 6(3-g2M + g2M2) dA 'j 'r 4kj y ik A"" dt= 4(Y YdikA + 5Ak (YdtYd)kj - 3 d i"dyd) 2 [ 4( I d d dYd .. .ik 2Y ut)ikA kJ Yk + (Ak - A1)(YutYu)kj Yd + 2(dk d dju d S k 14 2 32 2(AemYm2 + 3AdmIdm2) + 9 1Mi + 6g2M2 + 93M3 ], ykj y ik Aui dtA 1 yyyJ yik A, dt - 6[ A + 5A k~(YutYu)k3 .LY' ~ Yy *yklU + (Aik - Aj)(YdtYd)kj i + 2(YuYdt)ikA kji dj UY! yij +6AlmYkml2 + ,1 + 6g'M2 + 3 3 , (15 3.13) (11.13) dm2. 112 0= [ 3 i2(mu + m2. + m(u + A ) +3lTrlYm2} _392 M2 -3 g2 - - a 2M2 dm2 O_ _ 1 ..2 2 + ji12 dt= 8lr21 I2(md + m2i + mi + IAi2 1,3 3+Yl (m + 2(m2di + m + A2) - g Tr{Ym2} - - 12 3g2M dm~1 d = 2IYe12(m 2 + mi + m2l + IA' 12) dt 87r 3i L +3 g2Tr{ym2} 322 1 +2 _ 2 122 - g 91{Y gM -3gM 10 dm2 d [ 21Y 2(m2d + md, + 2 + Aj2) dt d d- 221 m--= 1[ E2Yl2(m2 + m2 + m2q + IA' 2) 2g2 Tr{Ym2} - 62M- 12 2 _ 3 =p [Y(1 12(mY,,, + mi + 2 + IA 2) dt -91 3 11 2,3 + jY I2(m + jim + mi + A2)) +02Tr{Ym2} -2 2 3g 2 + Tr n{Y m 2l _ 1 2 _ 1 g 2 M 32 + g92- 3 g2M2-1- - 32M dB 13 2 3 dB 1[ 3A|Yii2 + 3A iyii2 + AiY 12 + + 3 2 2 21u de(e +3g2M2.14) 87r (11.14) 93 where as in Falck [112], sums are implied over all indices not appearing on the left hand side and where ng Tr{Ym2} = Z(m2 - 2m2 + m - m + m). 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PAGE 1 RENORMALIZATION GROUP ANALYSES OF THE STANDARD MODEL AND ITS MINIMAL SUPERSYMMETRIC EXTENSION By BRIAN D. WRIGHT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1992 UMvaisiTy OF Florida iimm PAGE 2 ACKNOWLEDGEMENTS I would like to acknowledge those who have helped me to pursue the study of particle physics. I have encountered many setbacks over the years and I am indebted to my advisor Pierre Ramond for his patience and guidance. He has taught me most of what I know of particle physics. This work would not have been possible without the contributions of my other collaborators Haukur Arason, Diego Castano, Bettina Keszthelyi, Sam Mikaelian and Eric Piard. I would like to thank Diego and Eric for the use of the /3 function appendices, Sam for his thoughts on quark masses, and Haukur for discussions on the extraction of the strong coupling constant. I am especially indebted to Bettina for her collaboration in the threshold calculations. I would not have been possible to get through these years without Haukur's ice cream and cookie sessions. Dr. Gary Kleppe's good evenings in the morning and burning the midnight oil with Sam and Bettina. I am especially grateful to Bettina for her patience during the writing of this work and helping to make me more human. I must thank my family for their support during my absence from the real world. ii PAGE 3 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii ABSTRACT v CHAPTERS 1 INTRODUCTION 1 2 SUPERSYMMETRY AND THE STANDARD MODEL 6 2.1 Overview and Motivation 6 2.2 Supersymmetry Generalities-Superfields 8 2.3 The Minimal Supersymmetric Standard Model 10 2.4 Supersymmetry Breaking 13 3 SUPERSYMMETRY AND GRAND UNIFIED THEORIES .... 17 3.1 Minimal SU(5) 18 3.2 Supersymmetric SU(5) 24 4 THE RENORMALIZATION GROUP . . . 29 5 STANDARD MODEL PARAMETERS AT 32 5.1 ai(Mz) and a2(-^z) 32 5.2 as{Mz) 34 5.3 Yukawa Couplings 39 5.4 Known Quark Masses 41 5.5 Lepton Masses 42 5.6 Higgs Boson and Top Quark Masses 43 5.7 Vacuum Expectation Value of the Scalar Field 44 6 THRESHOLD EFFECTS 47 6.1 Effective Gauge Theories 47 6.2 Gauge CoupHng Thresholds in the SM 51 6.3 Threshold Effects in Fermion Masses (Yukawas) 55 6.4 Thresholds Beyond the Standard Model 58 iii PAGE 4 7 NUMERICAL TECHNIQUES 61 8 RG ANALYSIS OF THE STANDARD MODEL 63 9 RG ANALYSIS OF THE MSSM and SUSY-SU(5) 71 10 CONCLUSIONS 79 APPENDIX I THE STANDARD MODEL /? FUNCTIONS 81 APPENDIX II THE MSSM /3 FUNCTIONS 88 REFERENCES 94 BIOGRAPHICAL SKETCH 101 IV PAGE 5 A^bstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillmsnt of the Requirements for the Degree of Doctor of Pinlosophy RENORMALIZATION GROUP ANALYSES " OF THE STANDARD MODEL AND ITS MIND^L\L SUPERSYMMETRIC EXTENSION By BRL\N D. WRIGHT December 1992 Chairman: Pierre Ramond Major Department: Physics This -vvork presents some aspects of renomiaUzation group studies m the Standard Model and its minimal supersymmetric extension with an eye toward their application to grand unined predictions. We give an overview of the Minimal Supersymmetric Standard Model (MSSM) and grand unified models to highlight relevant issues. We next present a comprehensive analysis of the running of all the couplings of the Standard Model to two loops, including thresholds effects. The purpose is twofold-to determine what the running of these parameters may indicate for the physics of the Standard Model and to provide a template for the stud^-' of its extensions up to Planck mass. We discuss in detail the subject of e.xtracti-g initial data on Standard Model parameters and numerical methods applied in the analysis. This material is meant to be a complete toolkit for any renormalization group study including the Standard Model in the low energy regime. We then apply these tools to run all the couplings of the MSSM, taking full account of the Yukawa sector. We note the successful uniScation of gauge couplings in this model given the best data available. After identifying the scale V PAGE 6 CHAPTER 1 INTRODUCTION The Standard Model of the strong and electroweak interactions has so far successfully met all experimental challenges. The situation may soon change if the proper resolution of the solar neutrino problem [1] calls for the existence of massive neutrinos [2]. Other unresolved problems are theoretical in nature. There is no explanation of the mass spectrum, the replication of fermion families or the chiral structure of the Standard Model. Nor is there an adequate explanation of the smallness of strong CP violation or the hierarchy of scales between the electroweak scale and Planck mass. Some of these aesthetic problems find partial resolution, when the gauge group of the Standard Model is imbedded in some larger symmetry group. Certain relations between parameters of the theory arise from the higher symmetry and can be extrapolated to lower energies by means of the renormalization group (RG). Thus, one approach to seeking structure beyond the Standard Model is to build such models and attempt to deduce low energy measurable predictions which can test their viabiHty. This has led to the early grand unified theories (GUTs) [3,4,5,6]. A second approach [7] is to use the renormalization group to extrapolate the Standard Model parameters to the unexplored scales. The purpose is to find if those parameters satisfy interesting relations at shorter distances. When used in conjunction with the former approach, this can give powerful hints of the physics expected at shorter scales. Of course, it all depends on having accurate data to input as initial conditions for the renormalization group equations, as well as a strong theoretical basis for the evolution equations themselves. The minimal SU(5) GUT is the prototype for such analyses [8]. There, properties PAGE 7 2 of the model at energies of order 10^^ GeV are translated with the aid of the renormalization group to a prediction for proton decay that is not consistent with experiment. Data on the coupling constants is now sufficiently precise to rule out most simple GUTs, including SU(5), because of the absence of unification of the running coupHngs (GUT triangle) [9,10,11]. The same analysis has recently improved the feasibility of the supersymmetric (SUSY) extensions of these GUTs [9,10,11,12,13]. More recently, constraints coming from Yukawa coupling unification in supersymmetric SU(5) and SO(IO) models have led to bounds on the mass of the top quark [14,15,16]. Renormalization group methods axe of enduring practical importance in the attempts of high energy physicists to glean indications of more fundamental theories from radiative corrections. The purpose of this work is twofold. We first attempt to give a comprehensive guide to the renormalization group techniques necessary to study the Standard Model and its extensions (see also ref. 17). In particular we emphasize the importance of threshold corrections in renormalization group analyses. Such corrections, incorporating information about as yet unseen heavy mass spectra, are becoming increasingly important as parameters of the Standard Model become more precisely known. The possibiHty of probing for such spectra in the deep ultraviolet through the use of the renormalization group is an exciting propect. We discuss some threshold effects not previously described in the literature. Second, we apply these techniques to the minimal supersymmetric extension of the Standard Model (MSSM) in the context of SU(5) grand unified predictions. We find significant theoretical bounds on the mass of the top quark in order to maintain gauge coupling and Yukawa unification in this model. The PAGE 8 3 . bounds are such that the model considered can be ruled out if a top quark neax its present experimental lower bound is discovered. We view these results as a success of our approach. An outline of this work follows. We present an overview of supersymmetry and the MSSM in Chapter 2. The hierarchy problem is discussed as a primary motivation for supersymmetry. A brief introduction to the superfield formalism is given to establish our notational conventions. We then discuss the particle content of the MSSM and consider mechanisms for supersymmetry breaking. Finally we present the form of the scalar potential when soft symmetry breaking effects from a supergravity theory are included. Chapter 3 gives a short review of grand unified theories and their supersymmetric extensions. Similarities and differences in their predictions are pointed out, especially with regard to proton decay. The difference in renormalization behaviors due to the presence of superpartner fields in the supersymmetric case is discussed, translating to very different comparisons with experiment. We find that a minimal SU(5) GUT is ruled out experimentally while its supersymmetric extension has dramatic agreement with available data. An introduction to renormalization and scheme dependence is presented in Chapter 4. Chapters 5, 6 and 7 provide a toolkit for renormalization group studies involving the Standard Model as a low energy theory. A review of initial data extraction from experiments is presented in Chapter 5. Many excellent reviews may be found in the literature, e.g., Marciano's [18] or Peccei's [19]. We identify the values of the various parameters at the different scales where they are most accurately known. The determination of the gauge couplings is discussed in Sections 5.1 and 5.2. Initial data extraction of the Yukawas and the CKM angles is discussed in Section 5.3. PAGE 9 4 The extraction of the quark masses from data is briefly discussed in Section 5.4. In the low energy regime, we consider it necessary to include the pure QCD three-loop contribution to our analysis of the running of the quark masses. Initial data for lepton masses follows in Section 5.5. In Section 5.6 we consider the extraction of and constraints on the physical top and Higgs masses. We address the scale dependence of the renormalized scalar vacuum expectation value in Section 5.7. In Chapter 6, we discuss how threshold effects are incorporated into our analysis. It is well known that in the context of GUTs the effects of particle thresholds are of great importance in analyzing their low energy predictions [20-21], such as decreasing the naive estimate of the proton hfetime [20]. For completeness we present a detailed analysis of threshold effects in the Standard Model where these effects are numerically less important; the two-loop effects dominate the effects of the electroweak threshold. This work is of theoretical interest because the same methods are applicable to other models. We include threshold effects in the running fermion masses, an analysis absent from the present literature. Chapter 7 briefly describes the numerical methods used to solve the complicated coupled two loop ^ functions with threshold matching functions apphed at the one loop level. A quantitative analysis of our results for the case of the Standard Model makes up Chapter 8. We contrast the effects of using one-loop versus two-loop ^ functions and of including a proper versus a naive treatment of thresholds. We include plots of all the running parameters over the entire range of mass scales and also use these plots to display the effects discussed. Furthermore we present some tables with actual numerical differences associated with these effects. PAGE 10 In Chapter 9 we describe the appHcation of our methods to the MSSM. We restrict our attention to testing supersymmetric SU(5) (SUSY-SU(5)) predictions for gauge and Yukawa coupHng unification and what they indicate for Standard Model parameters. We find bounds on the mass of the top quark and the Higgs scalar in a minimal supersymmetric extension of the Standard Model (MSSM) with minimal Higgs structure imbedded in SU(5). These bounds will be subject to direct experimental tests in the next few years. PAGE 11 CHAPTER 2 SUPERSYMMETRY AND THE STANDARD MODEL 2.1 Overview and Motivation Supersymmetry was introduced in the early 70s [22] as a generalization of the Poincaxe group which connects bosonic to fermionic degrees of freedom. At the same time such a symmetry was being studied in the context of dual resonance models [23] and was later applied in a four dimensional field theory [24]. The generators of the Lie superalgebra include those of the usual Poincare group as well as spinorial generators obeying a mixture of commutation and anticommutation relations. The Lie superalgebra for iV = 1 supersymmetry (A^ is essentially the number of sets of spinorial generators) involves the additional relations: _ {Qa,Q^} =2 PAGE 12 enables one to avoid the gauge hierarchy problem [25]. This problem arises in gauge theories in which there is a large separation of energy scales which is unstable in the presence of radiative corrections. In GUTs [3,4,5,6,7], gauge couplings of order g ~ 10~^ create radiative corrections to ratios of the order of Mew /Mgut ~ 10~^^. In general, without extreme fine-tuning of parameters, this separation of scales will not be maintained when radiative corrections are included [25]. The philosophy of this problem is that all extremely small parameters of a theory, e.g. small ratios of mass scales, must obtain radiative corrections of the same order or smaller to avoid such fine-tuning. This is the concept of naturalness and it ensures that the properties of the theory axe stable against small variations in the fundamental parameters [26]. Ordinary GUTs are not natural in this sense as Higgs scalar masses for example receive corrections of the form = mÂ§ g^A^ , (2.2) where mg is bare mass and A is some physical cutoff of order Mqut coming from quadratic divergences in the one loop corrections to the scalar mass. The scalar mass is thus additively renormalized by a term of order 10^^ GeV which requires incredible adjustment of the bare mass at every order of perturbation theory to keep m ~ 100 GeV. Supersymmetric theories, through cancellations between graphs containing boson and fermion loops, eliminate such quadratic divergences and cure the GUT hierarchy problem. This statement must be qualified by the requirement that the effective supersymmetry breaking scale Msusy not be to much larger than the electroweak scale, since this would introduce yet another hierarchy problem. PAGE 13 8 2.2 Supersymmetry Generalities-Superfields Supersymmetric field theories are constructed in terms of irreducible representations of the supersymmetry algebra (2.1). In iV = 1 supersymmetry these representations consist of essentially a pair of boson and fermion fields known as superpartners with the same mass and internal quantum numbers. Supersymmetric Yang-Mills gauge theories are constructed with chiral or scalar supermultiplets of heHcity 0 and 1/2 and vector or gauge supermultiplets of helicity 1/2 and 1. These multiplets can be represented using the well-known superfield formalism. We present below a few aspects of this formalism intended to establish some conventions. Readers seeking more information should consult the literature [27]. Superfields are functions of superspace coordinates z = (x, 9, 9), where 9 and 9 are anticommuting Grassmann variables. The transformations properties of the component fields of a superfield are obtained by applying elements of (2.1). Infinitesimal supersymmetry transformations parametrized by a Grassmann parameters ^, ( are given by where $ is a generic superfield and the generators Q can be represented by S^^x,9,9) = {^Q + ^Q)^ , (2.3) (2.4) It is also useful to define the supersymmetric covariant derivatives Da = d 09Â°' (2.5) 39 which anticommute with the Qs. PAGE 14 9 For each scalar supermultiplet containing a complex scalar field A{x) and a Weyl spinor rp{x), we associate a chiral superfield (j> characterized by Da PAGE 15 10 and \(^F,D) projects out F and D terms. Eliminating the F fields from (2.9) gives / cP'P \ I f)p 2 ij ' i where P, the superpotential, is given by (2.9) before projection over the F terms and where the derivatives in (2.11) are evaluated with the scalar component of <^j. Eliminating D in (2.8) and collecting terms depending only on the scalar components one obtains the scalar potential 2 , ^1 i2 ^(^) = E|ii: +^E|^Â«| ' (2-12) where i and are the generators of the gauge group with couplings gr" for each semisimple or Abelian factor of the group. The Yukawa couplings from the gauge kinetic term can be combined with those in (2.11) to give the Yukawa paxt of the Lagrangian Cy = \/2^5Â„(^r(r)jAÂ„v.j' +h.c.) We now turn to the apphcation of this formalism in the construction of reaHstic supersymmetric models. 2.3 The Minimal Supersymmetric Standard Model The construction of the minimal supersymmetric standard model (MSSM) is straightforward. Each field of the Standard Model is assigned a superpartner and these fields are combined into supermultiplets with definite SU(3)c x PAGE 16 11 SU(2)l X U(1)y quantum numbers. Due to the chiraJ nature of fermion families in the Standard Model, these multiplets are irreducible representations of N = 1 supersymmetry. Models with N > 1 have matter supermultiplets in vectorlike representations, so that ordinary fermion famihes have unobserved mirror counterparts. Attempts to give large masses to these mirror fermions have not been successful. The model must be free of gauge anomalies. One possibility is that the superpartner of the Higgs field is the lepton doublet but more fields must be introduced to cancel the hypercharge anomaly and to give up-type quarks masses and one is beset with phenomenological problems arising from lepton number violation. Another possibility is that the Higgs that gives rise to uptype quark masses is part of a mirror family. This chiral completion must be modified to radiatively generate down-type quark and lepton masses and has difficulty suppressing lepton and baryon number violation. The minimal choice is to introduce a second Higgs doublet and its superpartners of opposite hypercharge from the usual Higgs field to cancel the anomaly. The supermvdtiplets and their component fields are listed in Table 1. The kinetic and interaction terms of the theory are obtained in terms of the superfields of Table 1 by generalizing (2.8) and (2.9) to the semisimple gauge group SU(3)c x SU(2)l x U(1)y, projecting out the D and F terms and eUminating the auxihary fields using their equations of motion. To do this one requires the superpotential which is given by P = KQYuu' + ^dQYjd' + $rfÂ£Yee^ + , (2.15) where the hat denotes a chiral supermultiplet. This is the most general superpotential consistent with conservation of R-parity under which usual standard PAGE 17 12 model fields axe assigned a value of +1 and their superpartners a value Â— 1. This discrete symmetry is a special form of a continuous R-symmetry under which vector and chiral superfields transform as V{x,$,$) ^ V{x,9e-'Â°',ee'Â°') , _ . (2.16) M^.e.B) exp(maa)<^a(a:,^e-'",^e'") . The D terms of a superfield are R-invariant while the F terms of a product of chiral superfields axe invariant only if Yl,a Â— 2. This continuous R-symmetry forbids Majorana gaugino masses and must be broken to account for them, leaving the discrete R-parity. The most important consequence of R-parity conservation is conservation of baryon and lepton numbers. It also implies that superparticles must always be produced in pairs, leading to the existence of a stable lightest supersymmetric particle {e.g. the photino). Table 1. Particle Content of the MSSM Superfield Component Superpartner Quantum Numbers Fields Names SU(3) SU(2) Y Vg 9mi9m gluino 8 1 0 Vw " im'' m wino 1 3 0 Vb BrriiBm bino 1 1 0 HuiHu higgsino 1 2 1 1 2 -1 Q squark 3 2 1 3 3 1 4 3 1 2 3 i mi slepton 1 2 -1 1 1 2 PAGE 18 13 2.4 Supersymmetrv Breaking One major difficulty of supersymmetric theories is the lack of experimental evidence for superparticles. Superparticles must have the same masses as their ordinary partners since is a Casimir of supersymmetry transformations. Thus supersymmetry must be broken to be consistent with phenomenology. It can be broken either explicitly, spontaneously or dynamically. 2.4.1 Spontaneous Breaking In gauge theories with an abelian component in the gauge group it is possible to have a term in Lagrangian proportional to the vector superfield V (since it is gauge invariant in this case), Crr = -2filjV\^ , (2.17) known as the Fayet-Iliopoulos term [28]. This term induces an extra piece in the D part of the scalar potential (2.12): V{Ai)^j\Y,QiA\Ai + fil,\ , (2.18) t where Qi are the charges of the scalars A{. As we shall see this mechanism is not sufficient to break the MSSM and is inappropriate for theories imbedded in a simple gauge group in which case the Fayet-Iliopoulos term is absent. It is possible to breaJc supersymmetry without gauge interactions by picking special forms for the superpotential. This mechanism, invented by O'Raifeartaigh [29], requires at least three chiral superfields A, B, C with associated scalarcomponents. The cleverly chosen superpotential P = UlAB + XiA^ nl)C , (2.19) has no solution which gives V{A, B, C) = 0. One of the auxiliary fields F^ b c has nonvanishing VEV and supersymmetry is broken. This mechanism is also PAGE 19 14 not sufficient to break the MSSM and has the unaesthetic property of requiring many extra supermultiplets. The MSSM scalar potential derived from (2.12) and (2.15) is given by ^\g\{HiHu H\Hd + iti4ttti + ml,f (2.20) ^IgliHirHu + H\rHd + Itf^^f , where the squaxk contribution has been omitted since we forbid SU(3) breaking. Here rupj comes from the Abelian Fayet-Iliopoulos term. A solution to V = 0 exists for a nonzero slepton VEV although this minimum breaks lepton number. By adding more fields to the minimal model one may be able to induce supersymmetry breaking spontaneously; however, for any anomaly free theory broken via D ox F type breaking the following constraint on the mass spectrum holds [30]: J](-1)2-^(2J + l)Tr{M2^)} = 0 , (2.21) J where the trace is over squared particle masses M'^'j^ of spin J. Such a mass relation is phenomenologically unacceptable as it requires the existence of an imobserved massive squark which is lighter than the lightest quark [31]. Thus we shall consider soft explicit breaking of supersymmetry in the low energy form of the model. The most promising mechanisms for inducing such terms are through local supersymmetry or dynamical breaking via gaugino condensation. We shall limit our discussion to the former case. The low energy theory derived from the latter can be parametrized in a similar way. 2.4.2 Induced Breaking from Supergravity An alternative to the troublesome scenarios for breaking supersymmetry spontaneously in the low-energy theory is simply to introduce soft supersymmetry breaking parameters into the effective theory which do not spoil the nice PAGE 20 15 renormalization properties of the theory. In particular, one introduces terms which do not introduce quadratic divergences into the theory so as to maintain an attractive solution to the hierarchy problem. Not all mass dimension 2 and 3 terms are allowed by this criterion. Girardello and Grisaru [32] showed that exphcit masses for the spin 1/2 components of chiral superfields are not allowed, nor axe simple cubic terms in a single scalar field. Such soft supersymmetry breaking parameters are regarded as originating from some unknown higher theory. It is attractive to think that if supersymmetry occurs in nature that it appears as a local symmetry. Such a theory must contain the local extension of the Poincare group, namely general coordinate invariance, and therefore include Einstein gravity. One obvious motivation for such a supergravity theory is that it can solve the gauge hierarchy problem when extended to the Planck scale. It also arises naturally as a low energy theory from superstrings. The attractiveness of supergravity is that supersymmetry breaking can occur in some massive hidden sector of a theory which couples to the usual sector only through gravity [33]. Due to the weakness of the gravitational interactions the symmetry can be broken at a high scale but appear in the low energy effective theory at a much lower scale. The supergravity theory must be broken spontaneously to preserve the Lorentz invariance of the theory. We consider the low energy effective theory arising from an iV = 1 supergravity model interacting with a hidden sector in which supersymmetry is spontaneously broken and a lower energy sector that contains the fields of the MSSM and possibly heavier grand-unified fields. The general scalar potential of globally symmetric YangMills theory in (2.12) is modified in the presence of supergravity. Because supergravity theories, like their nonsupersymmetric PAGE 21 16 counterparts, are not renormalizable, the superpotential can contain arbitrarily high powers of chiral superfields. This leads to certain arbitrary functions of the complex scalar components Ai of the chiral superfields in the expression for the supergravity-induced scalar potential. Remarkably, in the case of supergravity interacting with a single chiral superfield, only one arbitrary function appears [34] while in the case including YangMills and an arbitrary number of chiral superfields, two such functions are present [35]. Soft supersymmetry breaking effects in the MSSM are given by additional terms in the Lagrangian: Vsoft = "^X^i^i + Bfii^u^d + h.c.) i + ^( AilYj^ui^uQj + A^JyJ^di^dQj + Ai^Ye'hi^jLj +h.c. ) , 3 ^gaugino = ~2 X/ ^l^l^l + ' /=1 (2.22) where Vg^fi must be added to Eq. (2.20) and we have omitted a possible Fayetniopoulos term. Here the sum over i includes all scalar fields and A^ d,e-: ^ and m^. are of order rug = 7713/2 where irig is the gravitino mass. We generically denote the scalar mass terms by the scale mg. The gaugino masses are also approximately equal at Mp/ to a common mass Since the breaking of supersymmetry is flavor blind in this scenario, one also has Au = A^ = Ag at the Planck scale which suppresses scalar induced flavor-changing processes in these models. Of course these relations no longer hold at the electroweak scale when the potential is renormalization group improved through the use of the /3 functions of these parameters given in Appendix II. For further analysis and recent reviews see refs. 36,37,38 and 39. PAGE 22 CHAPTER 3 SUPERSYMMETRY AND GRAND UNIFIED THEORIES The subject of grand unified theories and their supersymmetric counterparts is vast and here we give only the essential details. The curious reader should consult one of the many excellent reviews in the subject (see refs. 40 and 41 for reviews of GUTs and refs. 41 and 42 for reviews of SUSY-GUTs). Grand unified theories originated as another step in theorists' search for higher symmetries in particle physics. Such theories axe describable by a single simple gauge group which must contain as a subgroup the SU(3)c x SU(2)l x U(1)y symmetry group of the Standard Model. The minimal group is SU(5), the most predictive of all grand unified models [4] and we will base most of the discussion on this model, although models based on SO(IO), and Eg [5,6] have similar features and predictions. Besides the aesthestics of the larger symmetry manifest in grand unified theories, there are other motivations to study them. One is that GUTs provide simple relations among the parameters of the theory. The penultimate prediction of GUTs is coupling constant unification at the grand-unified scale as is required by imbedding the Standard Model group in a simple group. This decreases the number of gauge parameters in the theory although many more parameters must be introduced in the scalar sector. Typically GUTs also predict definite mass relations between quarks and leptons since they appear together in grand unified multiplets. Grand unified models generically violate C P and baryon number, giving a possible explanation of the cosmological baryon asymmetry [43]. Finally, especially in light of recent observations of 17 PAGE 23 18 solax neutrinos [44,45,46], certain GUTs such as SO(IO) give a simple explanation of the solar neutrino problem [1] by predicting neutrino masses via the so-called seesaw mechanism [47] which give rise to matter induced neutrino oscillations [2] with numerically consistent values for the oscillation parameters [48]. 3.1 Minimal SV(5) The minimal SU(5) GUT of Georgi and Glashow [4] contains only the fermions of the Standard Model. These fall into the 5 and 10 representations of SU(5), which decompose under SU(3)c x SU(2)l as 5 = (3,l)-h(l,2) (3.1) 10 = (3,2) + (3,1)-H(1,1) . The quarks and leptons appear in these multiplets as follows. For the 5 representation we have V^=(e-) , (3.2) V and for the 10 = (5 x 5)^ we have V2 / 0 "3 -U2 ui dl\ -U3 0 ui U2 d2 U2 -ui 0 1*3 dz -ui -U2 -U3 0 e+ v -di -d2 -^3 -e+ 0 ) L (3.3) where the indices on the quark fields are color indices. The gauge bosons lie in the adjoint 24 representation which decomposes as 24 = (3, 2) -f (3, 2) -h (8, 1) + (1, 3) -H (1, 1) , (3.4) and contains the Standard Model gauge bosons and two leptoquark gauge bosons X and Y of charge | and \ respectively which mediate baryon and lepton number violating processes although B L is conserved. PAGE 24 19 Already we can use the simplicity of the gauge group to make a prediction for the Weinberg angle, 9y[r at the GUT scale [7]. Let us write the Standard Model SU(2)l x U(1)y couplings as 512 and 9' = fl'l/C' where C appears in the formula for the U(l)g^ charge generator Q in terms of the third generator of weak isospin and the hypercharge generator: Q = Ts CTo . (3.5) Normalizing the generators of the group in a representation R by Tt(TjiTq) = ^R^AB have Tt(Q^) = (1 + C2) Tr(T|) , (3.6) for any representation of the gauge group. Performing the traces for the 5 representation gives = 5/3 and In fact this prediction also holds for SO(IO) and most Eg models. Note that this prediction holds only at the GUT scale and one must use the renormalization group to find its value at the electroweak scale. The SU(5) symmetry must be broken at a high energy scale, Mgut, to account for the lack of experimental evidence of GUT particles and to make its predictions viable. Masses for the quarks and leptons arise via spontaneous symmetry breaking of SU(2)l x U(1)y to U(l)e^. The SU(5) symmetry breaking is achieved spontaneously by including an adjoint 24 of scalar fields, S^. The fermion mass terms transform as 5 X 10 = 5 + 45 , (3.8) 10 X 10 = 5 + 45 + 50 , PAGE 25 20 so we minimally require a 5 Higgs multiplet H which contains the usual Higgs doublet. With this minimal set of scalar fields both symmetries can be broken and masses given to the fermions leading immediately to the mass relations = ^e, rris = m^, mi, = ttit , (3.9) at the GUT scale. We shall see that only the last relation is pheomenologically feasible. One can replace these extra mass relations by improved ones if more scalar multiplets including 45s are added and if various global symmetries are imposed on the scalar potential. For examples of such models see ref. 49 and in the context of SO(IO) see ref. 50. For recent re-examinations of the renormalization group predictions of these models see refs. 51,52,53 and 54. Perhaps the most exciting prediction of GUTs is the instability of the proton. X and Y leptoquarks mediate baryon number violating processes such as those depicted in Figure 1. "e U d d' d' Figure 1. Some baryon number violating diagrams in minimal SU(5). At low energies these processes give rise to effective dimension 6 operators in the low energy Lagrangian where, omitting quark mixing, (3.11) O2 ={eijk^hf^){uie+ + d^LKRh^'d'j, , PAGE 26 21 and c stands for charge conjugate spinor. Using Mx Â— My Â— Mqut we can naively estimate the proton Ufetime to be of order Tp oc ^ 3.12 on dimensional grounds [7]. However various effects modify the coefficient of the expression. Using renormalization group techniques one can compute the anomalous dimensions of 0\ and Oi coming from virtual gluon, W and Z boson and photon exchange. Large logarithms of order log"(Mx/mp) can be summed, giving rise to enhancement factor for these operators. The fax:tor from gluonic corrections is [55] where // ~ 1 GeV is a typical scale appropriate for the proton decay process and Ug is the number of fermion generations. The actual decay rate requires knowledge of the hadronic matrix elements of Eqs. (3.11), which can be estimated by current algebra or bag model techniques. These estimates indicate a favored decay mode of p Â— > e^'K^ . The best theoretical estimate of this rate, using data from the early 80s is [41] rp/5(p ^ e+7rÂ°) = 4.5 x lo29Â±l-7 yr , (3.14) where the range is due to uncertainty in the QCD scale, ^-jjfgIn the determination of Mx in this rate one must also take care to properly treat mass thresholds effects. We discuss this issue in Chapter 6. The operators of Eqs. (3.11) are not the only dimension 6 operators that can mediate proton decay. One can also have decays mediated by color triplet Higgs scalars. Since these appear in the same SU(5) representation as the ordinary Hght Higgs doublet parameters of the potential must be fine tuned to PAGE 27 22 incredible accuracy to insure that they obtain large masses of order MqutThis is another manifestation of the gauge hierarchy problem. The scalar contribution is largest for processes involving heavy quarks; it is usually suppressed by the ratio of quark or lepton masses to the W boson mass. Huge underground water tanks have been set up around the world to search for this process and have found no signal.^ The ciurent limit from the Kamiokande detector in Japan [56] is ruling out the minimal SU(5) model. The situation is made worse for SU(5) when one uses the renormalization group to obtain low energy predictions from GUT scale coupling constant and mass relations. In Figure 2 we see that using the two loop functions for the gauge couplings (see Appendix I) and inserting their known values at Mz with errors, minimal SU(5) is inconsistent with coupling unification [11,17] This result holds for any GUT model with the Standard Model as the effective theory below MgutThe situation for the only viable mass relation, = is not improved. By using the renormalization group equations of Appendix I, we can determine the prediction for this ratio at low energies [55]. In Figure 3 we see that they meet at a scale far below the required GUT scale, although it might indicate some ununified intermediate scale theory with this relation which breaks around 10^ GeV. These issues will be discussed further in Chapter 8. ^ However these same experiments may have discovered massive neutrinos from SN1987A an SO(IO) grand unified prediction! PAGE 28 23 0 5 10 15 20 logio(M) Figure 2. Running of the Standard Model inverse gauge couplings to two loops using their propagated experimental errors. The picture for minimal grand unified models-those whose low energy theory below the GUT scale is the Standard Model-is bleaJc. This does not, however, rule out models with intermediate scales of symmetry breaking and those with additional intermediate mass fields beyond the Standard Model. One example includes nonminimal SU(5) models with 45s of scalars [49] where asymptotic freedom of the gauge coupling is lost due to the enormous Higgs structure. Another involves analogous SO(IO) models with intermediate breaking scales (for example one can have an intermediate theory with the color group being PAGE 29 24 SU(4) of Pati and Salam [3]) [50]. These theories also predict many interesting mass relations (see for example ref. 51) although they arise from special forms of yukawa and scalar interactions and a plethora of scalar multiplets which make it difficult to take seriously such models as fundamental theories. 0 5 10 15 20 logio(M) Figure 3. Intermediate quark and lepton masses in the Standard Model for M< = 100 GeV and Mh = 100 GeV. 3.2 Supersvmmetric SU(5) The basic predictions of minimal SU(5) also hold for its supersymmetric extension [31,57,58]. However due to the presence of superpartners for all standard particles, the renormalization group equations are modified [59,60] in a manner that gives much better agreement with experimental data. One PAGE 30 25 of the most exciting recent results of improved data on the strong coupling, as and sin^^y, was the striking gauge unification of Figure 4 in the context of supersymmetric theories [9,10,11,12,13]. So far this is the best indirect evidence for the existence of such theories. logio(M) Figure 4. Running of the MSSM inverse gauge couplings to two loops using their propagated experimental errors. The window depicts a blow-up of the area around the unification point. Note the small region where all three couplings intersect. As seen in Figure 5 the mass relation m^, = rrir also is not in perfect agreement for Mt = 100 GeV and is sensitive to this mass. Agreement can only be obtained for a certain range of top quark masses. We shall discuss these renormalization group results and exploit this mass relation in Chapter 9. PAGE 31 26 0 5 10 15 20 log 10 (m) Figure 5. Intermediate quark and lepton masses in the MSSM for Mt = 100 GeV and Mh = 100 GeV. One major distinction between SUSY-GUTs and ordinary GUTs is the mechanism for proton decay. In SUSY-GUTs, in addition to the usual dimension 6 operators, one has dimension 5 operators produced by exchange of color triplet Higgs (Hx) and its superpartner sHiggs field (Hx) [61]. These arise from the diagrams of Figure 6. Proton decay follows from a one loop effect when wino, bino or gluino fields are exchanged between the sfermions as in Figure 7. PAGE 32 27 -1 7 ' ~f y f Figure 6. Diagrams inducing dimension 5 baryon number violating operators in SUSY-SU(5). Here Hx is a colored Higgs triplet and / is fermion. A tilde is used to denote the superpartners of these fields. These contributions axe on the order of [61] Â—M)^^ or A^^l , (3.16) where yjj is is a typical Yukawa coupling and the latter expression holds for squark masses, nisq ^ rrig. Figure 7. Diagrams for colored Higgs and sHiggs mediated proton decay. Anomalous dimensions for the relevant dimension 5 operators give rise to supression factors rather than enhancement factors found in the minimal SU(5) case. This increases the proton lifetime associated with these processes from what is naively expected. The supersymmetric case is clearly less predictive than the minimal one. For sufficiently heavy wino, squark and Higgs and sHiggs color triplet masses, the proton decay rates obtained cannot be ruled out by experiment. However this assumes that the Higgs triplet fields are superheavy. PAGE 33 28 This is not the case in models in which supersymmetry is spontaneously broken in the GUT model as they will always produce light scalar color triplet fields as a consequence of the mass relation (2.21) [31]. We shall assume that supersymmetry is broken by a hidden sector coupled to ordinary and GUT matter via supergravity. We therefore avoid this mass relation as discussed in the previous chapter. Because of the dependence on the Yukawa couplings the preferred decay modes will go into heavier quarks than in gauge boson mediated decay. After taking into account suppression from CKM angles, one finds the dominant decays are strangeness changing: p Â— > fi'^K^ and p Â— > UfxK'^ [61]. The current Umits on the latter decay are Tp/B{p Â— + UfiK'^) ^ 10^^ years [56]. In order to perform detailed analyses of GUT and SUSY-GUT predictions, it is essential to properly evaluate the corrections to tree level relations through the use of the renormalization group. This requires knowledge of the /3 functions of the parameters of the theory and their behavior when mass thresholds are crossed. We have developed essentially a toolkit for such analyses that includes extraction of the measureable parameters of the low energy theory, incorporation of threshold effects that insure decoupling of heavy fields from the low energy effective theory and numerical evaluation of the renormalization group equations according to the hierarchy of theories under study. We therefore devote the next three chapters to these procedures. We first turn to a review of the renormalization group. PAGE 34 CHAPTER 4 THE RENORMALIZATION GROUP Renormalization is a procedure by which finite physical observables are extracted from typically divergent Feynman ampHtudes in quantum field theory. These divergences arise from large quantum fluctuations of the theory when probed at short distances. As the infinities appear in each order in perturbation theory, they are systematically subtracted out via infinite counterterms added to the Lagrangian of the theory. The renormalized Lagrangian yields finite results for physical quantities at every order. This procedure only works when the counterterms are finite in number and typically of the same form as terms present in the bare treelevel Lagrangian CoIn this case the theory is considered renormalizable. The renormalized Lagrangian is the same as the bare one but with its parameters redefined by infinite amounts from the counterterms: 9 = 9o + Sg, (4.2) where go is the bare parameter, g is the renormalized parameter, and 6g is the counterterm. There is of course arbitrariness in this prescription as the finite parts of the counterterms are undetermined. The renormalization group equations for a renormalizable theory state that the theory is invariant under reparametrizations of the theory obtained by choosing different finite parts of Sg^ ^ However, quantities computed at finite order in perturbation theory exhibit a dependence on the scheme by which the renormalized parameters are defined. In QCD where the strong coupling as is large, significant uncertainties arise from this scheme dependence. 29 PAGE 35 30 We shall work in the modified minimal subtraction scheme (MS) commonly used in the literature. The minimal subtraction (MS) prescription [62] is defined by fixing the counterterms by requiring them to consist only of the infinite terms needed to render the theory finite. In this scheme divergent integrals are regularized by continuing them to lower dimensions and an arbitrary mass scale // is introduced in order to keep couplings dimensionless. A family of MS schemes can be defined via where ^ is a constant parametrizing the arbitrariness in the finite parts of divergent integrals in dimensional regularization, e = (4 Â— ci)/2 and d is the spacetime dimension. The MS scheme is given by ^ = 1, while the so-called modified minimal subtraction (MS) prescription [63] is given where 7^ is the Euler-Mascheroni constant. The extra finite piece is typically present in dimensionally regulated integrals and it is mere convenience to subtract it out. The renormalization scheme independence of a physical quantity, P, when expressed in terms of /i and the running parameters of the theory, {5'i(^)}, is given by the renormalization group equation, where the are the /3 functions which determine the evolution of the renormalized parameters with scale. These running parameters are not in general equal to their corresponding physical values (consequently, for the masses, we adopt a convention wherein upper case M refers to physical values and lower case m denotes MS values). This is to be contrasted with the on-shell renormalization 9o(Qfi) ^ = g + 6g , (4.3) .|;P(M.)U) = (.| + a|:)p = o, (4.4) PAGE 36 31 scheme in which, for example, the renormalized masses equal their physical values and the renormalized electromagnetic coupling equals the fine structure constant. Also n cannot be identified with any specific momentum scale such as are encountered in experiments. It is difficult to attach an approximate physical meaning to fi and the uncertainty in the appropriate scale to use to describe a certain process is a significant one in extracting MS quantities from data (see e.g. Section 5.2). Another problem is that the decoupling of massive states in lower energy processes in such mass independent prescriptions is not manifest. This apparent contradiction of the decoupling theorem [64] is solved though the effective gauge theory formalism applied at the thresholds of such massive particles (see Chapter 6). Despite the above drawbacks, the MS schemes have the attractive characteristic that the ^ functions are /i independent and therefore particularly simple to integrate. The two-loop /3 functions of the Standard Model and the MSSM in the MS scheme have been collected in Appendices I and II. To solve these first order differential equation, we require the values of the parameter at one scale, which we take to be PAGE 37 CHAPTER 5 STANDARD MODEL PARAMETERS AT Mz Below we give a brief description of the determination of the parameters of the Standard Model. Details can be found in the cited literature. 5.1 a^{Mz) and aoiMz) The determination of the SU(2)l x U(1)y coupHngs proceeds from the Standard Model relations: 47r cos"^ d^{fi) ' ^^^^ 47r sin2%(^) 2, ^ . . (5-1) where = e^(/x)/47r and C^, defined in (3.5), equals 1 for the Standard Model and | when the Standard Model is incorporated in grand unified theories of the SU(N) and SO(N) type [7]. These couplings can be specified through the MS values of oi{n) and sin"^ dyi/{^). The electromagnetic fine structure constant (<^em ~ 137.036) is extrapolated from zero momentum scale to a scale ^ equal to Mz in our case. The renormalized coupling a(//) is related to the fine structvu-e constant Oem as follows: Q:(//) ^ iTnM ' ^^-^^ where 11(0) is the photon vacuum polarization function at zero momentum and includes radiative effects from charged gauge bosons and fermions as well as hadronic contributions. Threshold effects are treated by including the contributions of charged particles only at scales /z above their masses. This procedure gives [65] a-l(M^) = 127.9 Â±0.3 . (5.3) 32 PAGE 38 33 Many definitions of the renormalized Weinberg angle exist in the literature and many methods are used to extract it from data. We give a brief overview of definitions in current use. The process independent, renormalized weak mixing angle sin^ 9]y of the on-shell scheme is defined to be sin2 0^y = l--^, (5.4) Z where M\y and Mz are the physical masses of the W and Z gauge bosons. One can compute sin by accurately measuring M^r and or one can use the bare relation involving the low energy Fermi constant measured in muon decay and the W boson mass y/2 Ssin^d^VoMwl This may be corrected to order a and rewritten [66,67] (5.5) M^. = M^cos% = (^^^)2 (5.6) V^(^^l sin^^y(l Ar)2 with (iroiem/ V^Gfi)'^ = 37.281 GeV and Ar is a parameter containing order a radiative corrections and which depends on the mass of the top and Higgs. We can absorb the radiative effects using the renormalization group by replacing and aem with corresponding running parameters at MgFor laxge values of Mt and Mjj (Mt, Mh > Mz) one finds [66,68] Ar^l--^^+ ^^"7 In^. (5.7) a{Mz) 167rsin4%M| 487rsin2% M| ^ ' An estimation of the first two terms using (5.3) gives Ar w 0.07 . (5.8) A third way of extracting sin^ 9^ is from neutral current experiments, among which deep inelastic neutrino scattering appears to provide the best determination. PAGE 39 34 A running sin"^ ^wif^) defined in MS and differs from the above sin^ B]^ by order a corrections. The MS running W boson mass m\Y(fi) and the corresponding physical mass M^, identified as the simple pole at = of the W propagator, are related as follows = mlr(fi) + Ajy^viM^, pi) , (5.9) where Aj^yy is the transverse part of the W self-energy. A similar relation holds for the Z boson. In MS renormalization, the following relation defines the running sin^ B]y{fi) sin2MM) = l-47T(5-10) Equation (5.9) and its Z analog may be combined with Eq. (5.10) to give sin^%(//) ^^ cos2% A|^(M|,//) yt^^(M^,/i) sin2 sin2 9w ^ ' ^ ' ^ An explicit expression relating sin^ and sin^^[y(M^) is given in ref. 69. Other relations for sin^ ^jy(A') be arrived at directly linking it to Mz [70] or [71] . A fit to all neutral current data gives sin^ SwiMz) = 0.2324 Â± 0.0011 , (5.12) for arbitrary M< [72]. Using these values of a(M^) and sin^^p^(M^) yields ai{Mz) = 0.01698 Â± 0.00009 , (5.13) a2{Mz) = 0.03364 Â± 0.0002 . 5.2 a,(M^) The value of the strong coupling is known with less precision than most of the parameters of the Standard Model. This is due to large theoretical uncertainties arising from the nonperturbative nature of low energy QCD and the PAGE 40 35 slow convergence of perturbation series in high energy QCD. The determination of the strong coupling constant Â«Â«, is most critical to attempts to probe grand unified structure. The strong coupling constant is measured in a wide variety of processes at various energy scales. Early measurements of as came from deep inelastic electron and muon scattering and W + jet production at pp colliders. This data has been combined in recent years with results from LEP including jet rates, energy-energy correlations and global event shapes. One must also include new results in deep inelastic neutrino scattering, J/^ and T decays, the hadronic decay width of the Z boson, b quark production in pp collisions and the r lepton hadronic width. A recent summary of these results can be found in ref. 73 with details found in the references therein. In Table 2 we give a list of the results for each type of as measurement. Note that the higher energy LEP results tend to give larger values of as than the extrapolated lower energy results. Table 2. Values of as at M^Process a{e^e~ Â— >Â• hadrons) 0.135 Â±0.015 e^e~ hadrons (shapes) 0.119 Â±0.014 JI^^T decay 0 ..0-1-0.007 "Â•Â•^Â•^''-0.005 Deep inelastic scattering (i/, //) 0.112 Â±0.004 pp Â— > hbX pp -fjets 0.121 Â±0.026 Rt (world avg.) 1 1 0-I-0.004 T{Z^ had.) 0.133 Â±0.012 Z^ jets and ev. shapes 0.121 Â±0.004 PAGE 41 36 As an example of the uncertainties encountered in measurements of as consider the determination of as from jet rates (see the reviews of refs. 74,75,76 and 77 for details). One of the dominant uncertainties is theoretical the renormalization scale dependence of the rate. All the processes from which the strong coupling is extracted have such theoretical errors. Depending on the structure of the basic partonic process, the cascades of hadrons produced in an e'^e~ collision may form clusters strongly peaked in energy and constrained in solid angle known as jets. The basic quarkantiquark-gluon (qqg) process can give rise to a 3-jet event if the final state gluon radiation is hard and acollinear with the quark pair. The analysis of jets gives the best indication of the underlying partonic process. The differential cross-section for this process can be written [77] are the center-of-mass energy fractions of the final quarks. In fragmentation models parton structure and hadronic structure can be related by a scale factor (local paxton-hadron duality) with excellent experimental agreement. Once this relation is established one can compute the dependence of the process on asA further difficulty is the problem of deciding experimentally what constitutes a jet. One must work backwards to recombine hadrons into fundamental partons. One algorithm for recombination of two final state particles into one is to take all pairs of particles and compute their weighted invariant mass squared 1 (fia _ 2as + x\ a dxidx2 Btt (1 Â— a;i)(l Â— X2) ' (5.14) where 2Ei (5.15) (1 cos Oij), (5.16) PAGE 42 37 where the second equality holds for massless particles (m,< Ecm)Then find the pair with smallest yij. If < ycut where ycut is a chosen jet resolution parameter then combine this pair into a single pseudoparticle. Continue until all yij > ycut and one is left with a fundamental partonic process. The number of partons remaining corresponds well with the number of jets. The ratio for the n-jet to the total cross-section depends on the jet resolution parameter y and is given by i2Â„(y,^) = ^^, (5.17) where for example J22(y,M) =1 + C2,l{y)Mf^) + C2,2(y,/)Â«?(M) + Â• Â• Â• , (5.18a) Rz{y, f^) =C3,i(y)a.(/x) + C3,2(y, fWsi^i) + ... . (5.18b) Here / is an unknown scale factor presumably of order 1 relating the renormaliation scale fx to the center of mass energy: = fEcm. (5.19) The appearance of the renormalization scale in Rn is disturbing as physical quantities should be independent of this arbitrary scale. However, one works in practice only to finite order in perturbation theory and a dependence on // is introduced. A physical quantity C{a{fi),fi) expressed in terms of /i and the running parameters of the theory, a{fi), must be /i independent: ^Â±C{aM,,) = (^f^+D^)C = 0, (5.20) where the the ^ function is given by M = = /^0Â«^ + + . . . (5.21) PAGE 43 Â• 38 Now write C as a perturbative expansion in a C = Co(//) + Ci(/i)a(/i) + C2(//)a2(//) + . . . + CÂ„(//)a"(//) + . . . (5.22) The independence of C from fi implies that C2(^') C2(m) = -^oClln(^) (5.23) We denote by C^^\fi) and C^^^'{^') the physical quantity C computed to nth order in perturbation theory. The difference between these quantities is of order a""^^. The effect of changing renormalization scale in the estimation of a physical quantity is to introduce higher-order corrections and modify the coefficients in the perturbative expansion. If we write /i' = //i then (5.23) gives = C2 Â— /^oC'ilnf. Thus at finite order in perturbation theory a dependence on renormalization scale is introduced starting at second order in the coupling, justifying Eqs. (5.18). By varying the jet resolution parameter y, a given event can appear as a two or three jet event. Too avoid a given event contributing to different classes of jets, one often defines the differential 2-jet rate for the distribution of y values for a transition from a three to two jet event: Ay Another method used is to measure 7^3 at a fixed value of y. The LEP average for as from jet rates is included in Table 2. The errors reflect a weighted sum of the errors due to statistics, experimental systematics, errors from hadronization and parton shower models, recombination scheme dependence, next order QCD uncertainties and renormalization scale dependence. The largest sources of error are typically those coming from the renormalization scale and the energy PAGE 44 39 scale for transition from parton shower to hadronization. The former error comes from fitting experimental results with / free axid with / = 1 and taking the difference of these extreme values. Values of / from such fits range from 0.008 0.25. A global average of the results of Table 2 which we use is as = 0.117 Â±0.004 , (5.25) where the error can be as large as Â±0.011 depending upon the determination of theoretical uncertainties. 5.3 Yukawa Couplings To take full account of the Yvikawa sector in running all the couplings, initial values for the Yukawa couplings are necessary. They must be extracted from physical data such as quark masses and CKM mixing angles. Furthermore, the interesting parameters to be plotted must be determined step by step in the process of running to Planck mass. These two procedures are not unrelated and require the diagonalization of the up-type, down-type, and leptonic Yukawa matrices. We use Machacek and Vaughn's [78] convention where the interaction Lagrangian for the Yukawa sector is Â£ = Q^^YJuj, + Q^^Y^Un + Ii^YeUn + h.c. . (5.26) PAGE 45 40 The Yukawa couplings are given in terms of 3x3 complex matrices. After electroweak symmetry breaking, these translate into the quaxk and lepton masses Ye = V V me 0 0 0 0 0 0 rrir mj 0 0 0 rris 0 0 0 rrib rriu 0 0 ' 0 rric 0 0 0 (5.27) Yu = Â— I 0 mc OIF, where V is the CKM matrix which appears in the charged current ~ UL^^^Vdi . (5.28) It is a unitary 3x3 matrix often parametrized as follows: (ci 51C3 51S3 \ -s\C2 C1C2C3 52536'* C1C253 + 52036** | , (5.29) -5152 C152C3 + 02536** C15253 C2C36'* / where 5,= sin^j and c,= cos^,-, i = 1,2, 3. The entries of the parametrized CKM matrix can be related simply to the experimentally known CKM entries. Using limits on the magnitudes of these entries from the particle data book [72] (assuming unitarity) we find the following bounds for 5j, i = 1,2,3. 0.2188 < sin^i < 0.2235 , 0.0216 < sin ^2 < 0.0543 , (5.30) 0.0045 < sin ^3 < 0.0290 . These Hmits do not constrain sin^. A set of angles {^1, ^2) ^3) was chosen that falls within the ranges quoted above. The initial data needed to run the Yukawa elements is extracted from the CKM matrix and the quark masses. A problem arises though for the mixing angles, which was solved for the quark masses (see Section 5.4), in that it is not clear at what scale the chosen initial PAGE 46 41 values for these angles should be considered known. However, we have observed that for the whole range of initial values, the running of the mixing angles is quite flat, with a perceptible increase in 62 between Mjy and the Planck scale for higher top masses. This is in accordance with the angles being related to ratios of quark masses, and therefore, the exact knowledge of that scale is not critical. 5.4 Known Quark Masses As QCD is assumed to imply quark confinement, extraction of quark masses from experiment is problematic. On-shell renormalization will not have the same physical significance as it would if quarks comprised the observed particle spectrum. The quark masses then may simply be considered as additional couplings of the model. The light quark masses are the ones least accurately known. They axe determined by a combination of chiral perturbation techniques [79] and QCD spectral sum rules (QSSR) [80,81,82,83]. These techniques yield [17] mÂ«(l GeV) = 5.2 Â± 0.5 MeV , rudil GeV) = 9.2 Â± 0.5 MeV , (5.31) ms(l GeV) = 194 Â±4 MeV . For the heavier quarks, charm and bottom, one can maJce a more precise prediction. Here, the nonrelativistic bound state approximation may be apphed. The physical mass M{q'^ = M^) appearing in the Balmer series may be identified with the gauge and renormalization scheme invariant pole of the quark propagator Siq) = z{q)[j . q M(q'^)]-^ . PAGE 47 .42 ' r . The running mass is determined from the pole mass to three loops via = M^) = ,^1;^=^'' , (5.32) where K = 13.3 for the charm and K = 12.4 for the bottom quarks [84]. The charm and bottom pole masses have been determined from J/rp and T sum rules [85] and recently from CUSB and CLEO II by analysis of the heavy-light B and B* and D and D* meson masses and the semileptonic B and D decays [86]. A weighted average of these results yields Mciq"^ = M|) = 1.53 Â± 0.04 GeV , (5.33) Mbiq"^ = Ml) = 4.89 Â± 0.04 GeV . The running masses at the corresponding pole masses then follow from Eq. (5.32) mc{Mc) = 1.22 Â± 0.06 GeV , (5.34) mi,(Mb) = 4.32 Â± 0.06 GeV . It should be stressed that at the low scales under consideration the three-loop as corrections included in the mass and strong coupling /3 functions are often comparable to the two-loop ones and hence affect the accuracy of the final values noticeably [17]. These heavy quark masses can also be determined by studying nonperturbative potential models (see ref. 87) which predict appreciably higher values of the heavy quark masses than the ones quoted here. These models, however, are not as fundamental as the approach considered here, and their connection to field theory is rather problematic. 5.5 Lepton Masses The physical (pole) masses of the leptons are very well known [72] Me = 0.51099906 Â± 0.00000015 MeV , Mf^ = 105.658387 Â± 0.000034 MeV , (5.35) Mr = l.78Al+l',lll GeV . PAGE 48 43 , > We use these values to determine initial data for the running masses. Some authors neglect QED corrections and use the physical values for the running values at ~ M^, which introduces only a small error. By calculating the oneloop self-energy corrections, one arrives at a QED relation between the running MS masses and the corresponding physical masses mK/i) = Mill ^(ln4 + h . (5.36) 47r mj o Choosing ^ = 1 GeV as in the quark mass case and using Eqs. (5.36) yields the running lepton masses (taking m/ Â— Mi in the log term above is an appropriate approximation to order a) me(l GeV) = 0.4960 MeV , m^(l GeV) = 104.57 MeV , (5.37) mr(l GeV) = 1.7835 GeV . 5.6 Higgs Boson and Top Quark Masses The Higgs boson and top quark masses have not been measured directly at present, however their values affect radiative corrections such as Ar. Consistency with experimental data on sin"^ requires Mf < 197 GeV for Mjj = 1 TeV at 99% CL assuming no physics beyond the Standard Model [9]. Precision measurements of the Z mass and its decay properties combined with low energy neutral current data have been used to set stringent bounds on the top quark mass within the minimal Standard Model. A global analysis of this data yields Mt = I22+32 GeV, for all allowed values of My/ [88]. Recent direct search results set the experimental lower bound Mt ^ 91 GeV. As for the Higgs, the analysis of ref. 88 gives the restrictive bound, Mjj ^ 600 GeV, if Mt < 120 GeV, and < 6 TeV, for all allowed Mt. Since perturbation PAGE 49 44 theory breaks down for Mjj ^ 1 TeV, the latter bound on the Higgs boson mass is not necessarily meaningful. LEP data set a lower bound on the Higgs boson mass of 48 GeV [89]. In our analysis, initial values of the MS running top quark mass m< and of the scalar quartic coupling A at Mz are chosen arbitrarily (consistent with the bounds quoted above). As noted eaxlier in Chapter 1, these running parameters are not equal to their physical counterparts. However, any reasonable prediction for the masses of the top quark and of the Higgs boson that may come from our analysis should be that of experimentally relevant, physical masses. Therefore, formulas similar to Eq. (5.36) relating MS running parameters to physical masses are needed. To calculate the physical or pole mass of the top quark, we use Eq. (5.32) in its general form where M,-, i = 1, ... ,5, represent the masses of the five lighter quarks. Likewise the physical mass of the Higgs boson can be extracted from the following relation: \if.) = ^M]jil + 6ifi)), (5.39) where 6(iJ,) contains the radiative corrections and is given in ref. 90. Equations (5.38) and (5.39) are highly nonlinear functions of Mt and Mjj, respectively. Their solution requires numerical routines that are described in Chapter 7. 5.7 Vacuum Expectation Value of the Scalar Field The vacuum expectation value (vev) of the scalar field may be extracted from the well known lowest order relation V = {V2G^)-'2 = 246.22 GeV . (5.40) PAGE 50 45 From the very well measured value of the muon lifetime, = 2.197035 Â± .000040 X 10-^ s [72], the Fermi constant can be extracted using the following formula [91] 1 _ Glml ml 3 ml a(m^) 25 o.^ -T92^^W^^^ + 5;x^f^ + ^^^T-^^^' (^-^^^ where f{x) = l-8x + 8x^ -x"^ 12x^\nx , (5.42) givmg Gf, = 1.16637 Â± 0.00002 x 10"^ GeV^ . (5.43) This parameter may be viewed as the coefficient of the effective four-fermion operator for muon decay in an effective low energy theory ^l^eAl 75)e][/77^(l Â• (5.44) A direct calculation {e.g., in the Landau gauge) of the electromagnetic corrections yields that the operator is finitely renormalized (i.e., Gfi does not run) [67,92]. Another way to see this is by using a Fierz transformation to rewrite the above expression ^[i7e7/?(l 75)^/.][F7/?(l 75)e] Â• (5.45) The neutrino current does not couple to the photon field, and the e Â— /x current is conserved and is hence not multiplicatively renormalized. We need an initial value for the running vacuum expectation value at some scale fi. Wheater and Llewellyn Smith [93] consider muon decay to order a in the context of the full electroweak theory and derive an equation relating an MS running Gf^ to the experimentally measured value. From this formula we PAGE 51 46 can extract a value for v{Mz). However, the formula is derived in the 't HooftFeynmaxi gauge, and the evolution equation Eq. (1.18) of Appendix I for the vev is valid only in the Landau gauge. Nevertheless, motivated by the discussion of the previous paragraph, we choose the initial condition for the vev to be: t;(Mpy) = 246.22 GeV. Using the initialization algorithm (see Chapter 7), we arrive at v{Mz)We find that this procedure leads to no significant correction, and we therefore take, ab initio, v(Mz) = 246.22 GeV. PAGE 52 CHAPTER 6 THRESHOLD EFFECTS 6.1 Effective Gauge Theories We axe using the MS renormalization scheme to determine the running of the Standard Model coupHngs. From the Appelquist-Carrazzone decouphng theorem [64] we expect the physics at energies below a given mass scale to be independent of the particles with masses higher than this threshold. However, such minimal subtraction schemes are not physical in the sense that they are scale dependent and mass independent so that the decoupling theorem is not manifest. As described in refs. 94,95 and 96, we have to formulate a low energy effective theory by integrating out the heavy fields to one loop and find matching functions in order to take care of the threshold effects. We give a brief review of threshold effects in the running of the gauge couplings for a spontaneously broken GUT and for the Standard Model. Details can be found in the appendices of ref. 95. We next discuss thresholds in running fermion masses, a subject which to our knowledge has not been adequately treated in the literature. In chapter 7 we give a quantitative description of the effect of thresholds in the running parameters. The starting point for treating thresholds in MS and MS schemes is the construction of low energy effective gauge theories [94-96]. The basic idea is to integrate out the heavy fields in such a way the the remaining effective action is gauge invariant under the residual gauge group. Let the simple gauge group G be broken to G, and let $ and PAGE 53 48 Then the action of the effective field theory is obtained from the action l[(f>, $] of the full theory by functional integration over the heavy fields: = J [d^yn'PM . (6.1) Because there are no superheavy fields in the effective theory the decoupling theorem is not needed. However, there is a difficulty having to do with gauge invaxiaixce. Namely, in order to integrate out the heavy fields one has to add a gauge fixing term, and such a term usually spoils the gauge invariance of the low energy theory. The usual gauge fixing action for the full gauge group is V/. = -^E /^^/a(^; PAGE 54 49 presented in previous works [94-96], for completeness we include some of the details here. We begin with the case of a general gauge theory in which some simple group G is broken to a product of simple or Abehan groups, Gi. Let = -lFrFa^u+^iP-MF)
PAGE 55 where in the case ^ = 1, 50 + STritiFtiFlJ^)), (6.8) " i^(-21Tr(V^v) + Tr(<,-5%A) + 8Tr(V PAGE 56 51 This formula can also be used to determine the effect of integrating out a heavy quark in lower energy QCD. Here one need only include the heavy fermion part of the integration to determine the low energy gauge coupling in terms of the coupling of the full theory above the heavy quark threshold. Note that equation (6.12) only holds in the neighborhood of ~ M, M being the heavy scale, and as such provides an initial condition for the running of the effective couplings for /i M. In application to the Standard Model we will integrate out the heavy fields using matching functions as described above. As we are running couplings to two loops, it is sufficient to integrate out these fields to one loop. The heavy gauge bosons, their ghosts, and the top quark are integrated out near M^, the other fermions at their physical masses My. One may integrate out different mass particles at one scale as long as the two-loop contribution to coupling constant renormalization between two threshold scales is negligible. The errors arising from not integrating out fields at a scale // exactly equal to their physical mass M is of order Q!i_2,3ln(M/^), which is negligible within the perturbative regime. 6.2 Gauge CoupHng Thresholds in the SM At the electroweak threshold, the point at which W and Z bosons, their associated Nambu-Goldstone bosons, ghosts, and the top quark are integrated out, one imposes matching conditions similar to (6.12). Above the threshold the theory has the SU(3)c x SU(2)l x U(1)y gauge symmetry of the Standard Model and a SU(3)c x U(l)em effective symmetry below. Following refs. 94 and 95, we gauge fix the Standard Model in such a way that the low energy PAGE 57 52 theory is SU(3)c x U(l)em invariant after the SU(2) gauge fields are integrated out. The gauge fixing part of the Lagrangian is ^9f. = -^((^/^^f + i92ivi{n)ij PAGE 58 53 where ^^^^ (6.18) To get the running relation between e, s and g2 we use the high and low energy /? functions which can be derived from the following: = (6.19) Inserting these expressions into (6.16) yields ^ = (lit ^ ' (6-20) "(/^) "2(/^)^n/^) with the running Weinberg angle s(fj,) defined in (5.1). Using the GUT normalization for ai, one can also write a2 a{fi) where the first term on the right side of each equation gives the usual tree level relation. One can always impose these tree level relations by fixing fx so that the matching function vanishes. Here this occurs for JI = 0.95 Mj^. When the heavy top quark is included in the analysis one may integrate it out separately in which case one has an effective Standard Model without a top quaxk between M]y and Mf. Alternatively one can integrate it out at the same scale as the massive gauge bosons in which case the top queirk loop in the photon propagator contributes to the matching function above O(^) ^ Qifi) + 26g)^^ln(Mt ) , (6.22) PAGE 59 54 where h^Q^jy = l/97r^ is the contribution of the top quark to the coefficient of in the QED /? function. The perturbativity of QED at My^ and the small mass difference between the top quark and gauge bosons (so that aln(Mt/MYv) is small) means that this is a reliable approximation. We incorporate this matching condition which includes the top quark into our numerical integration of the electroweak ^ functions. The strong QCD coupling has thresholds near quark masses. The effect of integrating out a quark is to produce a matching condition of the form 1 1 where -47rQ3(//), (6.23) fi3(/i) = Ih^HÂ—) , (6.24) and 63'^ = l/247r^ appears as the contribution of each quark to the oneloop QCD /3 function: = 63^3. The top quark is integrated out at // = 7* so that the strong coupling does not match continuously across the threshold. For the other heavy quark thresholds however, we choose the renormalization scale for matching to be the physical quark masses so that the matching functions vanish there. The only effect of these thresholds is a step in the number of flavors ny/ as each quark threshold is passed [97]. Light quarks require greater care as the strong coupling becomes nonperturbative at these mass scales and the evaluation of hadronic contributions to vacuum polarization must be evaluated. In this analysis however we remain in the perturbative region. Likewise the QED coupling in the low energy theory has thresholds at charged particle masses, the matching condition having the form PAGE 60 55 Here aem is evaluated in the zero frequency limit and nOÂ£Z,(/.) = T^(8EÂ«M^) + EQiM^)) . (6.26) The F and S subscripts refer to charged fermions and scalars, respectively. Again the matching function vanishes at each particle mass so the effect of the threshold is simply to produce a step in the number of fermions or scalars of a given charge. 6.3 Threshold Effects in Fermion Masses (Yukawas) It is important to realize, that the running fermion masses also experience threshold effects near physical particle masses. To our knowledge there is no mention of these effects in the literature although they axe potentieilly important in analyses of mass relations predicted in many grand unified models and in the full twoloop running of gauge couplings. Since the Higgs field is integrated out at the electroweak scale the Yukawa couplings appear in the low energy theory through particle masses and various nonrenormalizable (nr.) interactions: _ _ ^SM = yfjVa^i^j y'ij PAGE 61 I \ Â— Â» Â— ' > > Figure 9. One-loop corrections to fermion two-point functions. As in (6.13), we work in a specific gauge required for gauge invaxiance of the low energy effective Lagrangian under SU(3)c x U(l)emAs for the gauge bosons the finite piece of the contributions of these diagrams gives the matching function and the divergent piece gives the one-loop ^ function (see Appendix I). In terms of bare parameters and fields the relevant parts of the Standard Model Lagrangian for this calculation are (6.28) Vij^a^PiL^jR vtjMiL^jR + h.c. -f . . . , where 4>i can be a quark or lepton and i is a family index. When the heavy fields are integrated out we generate the following low energy effective Lagrangian, Ciou> = (1 + KiiYliiLi ^tP-^ +{1+ KiR)^iRi ^Vii? (6.29) {rriij + Smij){xpiitpjj^ + h.c.) + ... , where = yfjVaThe ji contain wavefunction renormahzation contributions of the left and right handed fermion fields along with finite parts, and Srriij is the fermion self-energy contribution. PAGE 62 57 We must rescaJe the bare fermion fields in the low energy theory so they have the canonically normalized kinetic term, The relation between the Standard Model and low energy bare masses becomes Note that the left and right handed fermion fields axe differently renormalized due to their different gauge couplings, so Ki ^ Kji . Also note that in the case of quarks the self-energy corrections <5m,j introduce additional nondiagonal contributions to the mass matrix. However, in the limit Mq PAGE 63 give the one-loop /? functions. Those of the finite parts, 4 = (^i' ^i)'(ln^ i) , (6.35) A* give the matching functions. Here gy ^^ = 2(gr^ Â± g^j^), where 9i ji = T^l Â— s^Q^ and T^j^ ^ and Q' are the third component of weaJc isospin and the electric charge, respectively, for a given handedness of the i^^ fermion. For the different quark and lepton charge sectors one has 9\-l = 1 8 ,,2 3 (6.36) s d 1 d 1,42 9a = -1 gv = -l + p Â• Inserting the /3 functions into (6.31) we obtain the relation between the diagonalized, renormalized masses in the Standard Model and the low energy effective theory, 2 mf'^^hf^) = "^P''^(/^)(l + ^^{^Uf^) + li'^W + 4(/^)))) Â• (6.37) We next use the results of the preceding sections to analyze the importance of threshold conditions in the study of theories beyond the Standard Model. 6.4 Thresholds Beyond the Standard Model The difficulty with applying the general formula in (6.8) and (6.12) to higher energy theories such as GUTs is that httle is known of the particle spectrum of such theories to which the thresholds are sensitive. Fortunately some reasonable assumptions about these spectra can be made in many cases PAGE 64 59 and the threshold effects can then be well estimated. Historically these were of importance in proton decay estimates in minimal SU(5). Here it was found (in the case of threshold calculations in a momentum subtraction scheme) that grand unified threshold effects spread out the gauge couplings at the naive value of Mgut so that they actually meet asymptotically beyond this scale [20]. This has the effect of reducing the value of Mqut required for unification by a factor of 2 from the naive value. Interestingly the electroweak threshold effects are also important, reducing Mqut by an additional factor of 3 relative to the value obtained using the naive step function approximation. This factor of 6 reduction has dramatic effects on proton decay since from Eq. (3.12) it is proportional to Mj^. It was through the threshold analysis that it was first realized that minimal SU(5) predictions for proton decay were dangerously close to the experimental limit. Threshold efects are also potentially important in supersymmetric boundary conditions and in the calculation of renormalization group improved mass relations. They have been used recently to attempt global analyses of the particle spectrum of the MSSM [98,99,100] when contraints coming from the nature of the soft supersymmetry breaking parameters arising from supergravity or string-inspired models are taken into account. One finds that incorporation of supersymmetric particle thresholds tends to increase Msusy from what is naively expected when an average threshold is used. Previous analyses usually treated thresholds beyond the Standard Model by integrating out all heavy particles at a single "average" scale in the step approximation. The inadequacy of this treatment of thresholds has been re-emphasized by Barbieri and Hall [101]. To simplify the argument they consider a general GUT PAGE 65 model with a gauge group G containing SU(5) with a naive "average" supersymmetric threshold and a GUT threshold with matching functions incorporating a simple superheavy spectrum. One postulates a superheavy spectrum as follows: superheavy gauge supermultiplets V with common mass My, heavy components of the SU(5) chiral supermultiplets H in which the Higgs doublets lie of mass Mj{ and superheavy Higgs multiplets E involved in breaking G to SU(3)c X SU(2)l X U(1)y of mass M^. For simplicity we consider the analytic solutions to the renormalization group equations of Appendix II. They give [101] MsusY (3 15^^ + 7^) '""^^-Sf-^ = = ' = -2-^ Â± 2.6 Â± 1.0 , (6.38) where the errors come from the error in and sin = s respectively. The gauge couplings are evaluated at M^The 1-loop GUT matching function modifies this result to give MsusY. r, 6i 18 1 oo^ There is no reason why these logarithmic terms cannot be significant compared to the one loop effects. This implies that one cannot accurately extract Msusy from experiment no matter how small the errors in as and sin^ 6y\^ because of uncertainties in the mass spHttings in the superheavy sector. New data on the Standard Model parameters must be combined with other experimental or theoretical constraints to make it possible the better estimate Msusy PAGE 66 CHAPTER 7 NUMERICAL TECHNIQUES We use the Runge-Kutta method to numerically integrate the ^ functions. There are 18 coupled first order differential equations involved in running the Standard Model couplings. At one loop some equations decouple from the rest. For example, the gauge couplings axe decoupled at one loop, although the Yukawa ^ functions depend on both gauge couplings and Yukawas even at one loop as in Eqs. (1.8-9). The two-loop /? functions we axe using axe all coupled. To solve these equations we require initial data at some specific scale which we take to be MzUnfortunately, as we have discussed in Chapter 5 the Standard Model paxameters are experimentally measured at different scales. The method used to obtain all the parametersat a single scale involves the simultaneous solution of N nonlinear equations in N unknowns where N < 18. This can only be done numerically and computer routines to do this axe readily available. After all data is obtained at M^, the Runge-Kutta routines axe used to evolve the parameters to any mass scale /i. We run the quark masses and CKM angles by diagonaUzing the Yukawa matrices at every step in the Runge-Kutta method used in solving the 0 functions. However, in the literature, some authors write down analytic expressions for the running of these masses and angles by making some approximations. Typically, it is assumed that the contribution of the Yukawa coupHngs matrix is given essentially by the top quark Yukawa since it is much larger than the others. Sometimes a better approximation is made by keeping only the diagonal entries. Our numerical technique represents a minor improvement over these methods. 61 PAGE 67 62 As discussed in Section 5.6, given the evolution of the running parameters mt and A, the physical top quark and Higgs boson masses may be found by solving Eqs. (5.38) and (5.39), respectively. As the top quark and Higgs boson masses are unknown in the Standard Model at present, in the process of our analyses we are free to choose values for these masses at Mz and then proceed to study the consequences. In other models in which there may be certain constraints among couplings, one may incorporate these constraints into the numerical initialization routines. The freedom to choose a value for an unknown parameter may be replaced by such a constraint, and this may result in a definite prediction for that parameter. Constraints from grand unification and supersymmetry were used [14] to arrive at possible values for the top quark and Higgs boson masses. It should be kept in mind that the results of this work were only possible through the painstaking collaborative development of large computer routines which perform the function of evaluating the 13 functions for each parameter of the model studied, imposing threshold matching conditions at physical particle masses, diagonalizing at each step the mass matrices, extracting the CKM angles and finally plotting the results for investigation of renormalization group patterns. The ultimate aim is to make these routines accessible to other investigators and make it simple to append information concerning other theories. At present these routines are menu-driven and incorporate the full two loop /3 functions of the Standard Model, the MSSM and their low energy counterparts. However it is still in the prototype stage as far as ease of use and improvements are underway. PAGE 68 CHAPTER 8 RG ANALYSIS OF THE STANDARD MODEL For completeness we depict the results of numerically integrating the ^ functions for the Standard Model parameters from 1 GeV to Planck mass in Figures 2-3,10-15. We arbitrarily use Mt Â— ~ GeV and in some plots we superimpose oneand two-loop evolution. Differences between oneand two-loop evolution appear in the high energy regime and are also manifest for the strong coupling at low energies where it becomes large. 0.020 I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I 1 Loop 2 Loop 0.015 ICt^lOO GeV Mh=100 GeV 0.010 0.005 0.000 m. -0.005 J I 1 L J I I L J I I L I I I I 0 5 10 logio(M) 15 20 Figure 10. Light quark and lepton masses for M< = 100 GeV and Mji = 100 GeV. 63 PAGE 69 64 As previously discussed, we see in Figure 2 the "GUT triangle" signifying the absence of grand unification, assuming the Standard Model as an effective theory in the desert up to the Planck scale. Figures 10, 11, and 3 display the evolution of the light mass fermions (mg, mu, and m^/), the intermediate mass fermions (m^ and rris), and the heavy mass fermions (mr, rric, and mj), respectively. 0.25 0.20 0.15 6 0.10 0.05 0.00 0 5 10 15 20 log 10 (m) Figure 11. Intermediate quark and lepton masses for Mt = 100 GeV and Mjj = 100 GeV. We conclude that the largest differences between one-loop vs. two-loop evolution occur in the bottom, charm, and strange quark masses in these cases. In no case is the minimal SU(5) prediction of quark and lepton unification borne out. PAGE 70 65 In Figure 12, we plot the quartic coupling A and the top Yukawa coupling yt for (Mt = 100 GeV, Mjj = 100 GeV) and for (Mt = 200 GeV, Mh = 195 GeV). Figure 12. Top Yukawa and scalar quartic couplings. These two couplings are the only unknown parameters of the Standard Model. We have studied the effects of changing the values of M< and Mff in our analyses of the running of the other parameters. We observed that, for any M< between 100 GeV and 200 GeV, varying Mjj, while maintaining perturbativity and vacuum stability, did not affect appreciably the evolution of any of the other parameters. However, changing Mt itself showed a significant difference in the running of the heavier quarks. To illustrate this, in Figures 13 and 14 we PAGE 71 66 display plots similar to Figure 1 1 and 3 but for Mt GeV. = 200 GeV and Mh = 195 a' 0.25 0.20 0.15 0.10 0.05 0.00 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 i 1 1 1 1 Â— Mt=200 GeV Â— )fH=195 GeV \ m. -1 , , lilt 1 1 1 1 1 1 1 1 1 1 1 1 0 10 logio(M) 15 20 Figure 13. Intermediate quark and lepton masses for M< = 200 GeV and M;/ = 195 GeV. Â•. v In particular, in Figure 14 we note that the intersection point between the bottom quark and the r lepton moves down to a lower scale for this case of a higher top quark mass. This is expected since from Eq. (1.9) one can see that the bottom type Yukawas are driven down by an increased top Yukawa. This is to be contrasted with the SUSY GUT case in which the bottom Yukawa (3 function is such that this crossing point is shifted toward a higher scale with an increased top mass. In an SU(5) SUSY GUT model, the equality of the bottom and r Yukawas at the scale of unification was used to get bounds on PAGE 72 67 the top and Higgs masses [14] and we shall discuss this in the next chapter. Lastly, we display the running of the CKM angles in Figure 15. We have taken 6 = 90Â°, which corresponds to the case of maximal CP violation. As mentioned in Section 5.3, the evolution curves for these angles are effectively flat. e 4 a o a 2 -_\ 1 1 I 1 1 1 1 1 1 1 1 1 1 I 1 T 1 1 I _ 1 Mt=200 GeV Mb=195 GeV \ nib l mÂ„ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 5 10 15 20 Figure 14. Heavy quark and lepton masses for Mt = 200 GeV and Mff = 195 GeV. In the present case of the Standard Model, we find that two-loop running of the parameters does at times improve on the one-loop running. Indeed, we have tabulated the differences of several parameters in their oneversus two-loop values at various scales, for the cases (M< = 100 GeV, Mjj = 100 GeV) and (Mt = 200 GeV, Mff = 195 GeV). Table 3 illustrates the difference PAGE 73 68 one-loop vs. two-loop running makes in the ratio mi,/mr, for the three scales 102 GeV, 10'' GeV, and 10^^ GeV. T Â— I Â— I Â— j Â— I Â— I Â— 1 Â— I Â— j Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— I Â— rÂ— MiÂ»100 GeV " Mh=100 GeV Bindg Bind, J Â— \ I I I I I I I I I ' I I I I ' I 0 5 10 15 20 logio(M) Figtire 15. CKM angles. Table 3. m^/mr M< = 100 GeV Mt = 200 GeV 102 GeV 10"* GeV 10^6 GeV 10^ GeV 10^ GeV 10^6 GeV 1-loop 1.879 1.455 0.8081 1.868 1.392 0.6647 2loop 1.782 1.348 0.7336 1.769 1.285 0.6047 Clearly, the difference between oneand two-loop results is more pronounced at higher scales, as expected. Over all these scales the difference is never less than 5.5%. We note that the ratio becomes equal to one well below the scale u.o n a Â•iH n 0.2 N CI Â•i-i m 0.1 OS B 0.0 n 1 PAGE 74 69 of grand unification as noted above in the discussion of Figures 3 and 12. The next table presents a similar comparison for the top Yukawa. Here, two loops represent a smaller correction with the difference at all scales always being less than 5%. Table 4. yt Mt = 100 GeV Mt = 200 GeV 102 GeV 10^ GeV 10^^ GeV 102 GeV lO"* GeV 10^^ GeV 1-loop 0.5405 0.4160 0.1928 1.133 0.9780 0.7145 2-loop 0.5405 0.4071 0.1842 1.143 0.9700 0.6816 Finally, Table 5 displays the same analysis for ag for the case Mt = Mjj = 100 GeV. We observe no appreciable deviation from the tabulated values for any Mt ^ 200 GeV (except in the low energy regime where the difference is at most ~ 4%). Table 5. 1 GeV 102 GeV 10^ GeV 10^^ GeV 1-loop 0.3128 0.1118 0.07103 0.02229 2/3-loop 0.3788 0.1117 0.07039 0.02208 At scales ^ M^ , the inclusion of two loops is important in the evolution of the strong coupling (and of the quark masses). Indeed, we find that the pure QCD three-loop contribution is also significant and therefore include it in the running of the strong coupling and of the quark masses in the low energy region. As seen in this table, the combined two and three loops in the low energy regime account for a 17% difference at 1 GeV in as. Although in the cases considered in these last two tables there does not appear to be a significant difference in two-loop over one-loop evolution at I scales above M^, the first table does show a 10% difference at the scale 10^^ PAGE 75 70 GeV. We expect two-loop effects to be more important when the theory is extended, e.^., to include supersymmetry and/or grand unification. The effects of using a naive step approximation vs. a proper treatment of thresholds are numerically unimportant for the cases discussed above. Indeed they are less important than the two-loop effects. We note however, that the inclusion of non-naive thresholds effects is significant in the numerical analysis of extensions of the Standard Model. PAGE 76 CHAPTER 9 RG ANALYSIS OF THE MSSM and SUSY-SU(5) We present bounds on the mass of the top quaxk in the MSSM with minimal Higgs structure in the context of a grand unified theory by numerically evolving the couplings using their renormalization group equations. This analysis improves on previous endeavors by taking full account of the Yukawa sector. The MS renormalization group equations for the Standard Model and the MSSM [78] found in Appendix I and II are numerically integrated and used to evolve the parameters of the model to Planck scale. Although it is not possible to analytically express certain parameters (e.g. CKM angles) in terms of the Yukawa couplings, equations for the running of the quantities themselves can be arrived at by making some approximations. Since our approach is numerical we opt to run the quantities by diagonalizing the Yukawa matrices at every step of the Runge-Kutta method as described in Chapter 7. In the expectation that the Standard Model is only the low energy manifestation of some yet unknown GUT or of a possible supersymmetric extension thereof, the three couplings gz, 92, and g\ corresponding to the Standard Model gauge groups, SU(3)c x SU(2)l x U(1)y, should meet at some large grand unification scale. Using the accepted values and associated errors of these couplings we observe unification in the SUSY-GUT case but not in the pure GUT case, as noted by several groups [9,11,13] (see Figures 2 and 4). However this should not be viewed as proof of supersymmetry since given the values of ai, 02 > at some scale, and three unknowns (the value of a at the unification scale, the unification scale, and an extra scale such as the SUSY scale) there is always 71 PAGE 77 72 a solution. The exciting aspect of the analysis of ref. 11 is the numerical output, namely a low SUSY scale, Msusy, and a perturbative solution below the Planck scale which does not violate proton decay bounds^. Furthermore, in the context of a minimal GUT [4] there are constraints on the Yukawa couplings at the scale of unification. We first restrict ourselves to an SU(5) SUSY-GUT [57] where and yr, the bottom and r Yukawa couplings, are equal at unification. The crossing of these renormalization group flow lines is sensitive to the physical top quark mass, M<. This can be seen in the downtype Yukawa renormalization group equation (above Msusy ? for example), from which we extract the evolution of yj, since the top contribution is large and appears already at one loop through the up-type Yukawa dependence: (9.1) -(y^^i+Sffl + y^i)]. where Yy j g are matrices of Yukawa couplings. Demanding that their crossing point be within the unification region determined by the gauge couplings allows one to constrain Mf. This yields an upper and lower bound for Mt which nevertheless is fairly restrictive. The threshold analysis used is as follows. For the electroweak threshold we use one loop matching functions [95] with the two loop /3 functions valid in the Standard Model regime below the SUSY scale. At the electroweak threshold, near Mpy, we integrate out the heavy gauge fields and the top quark. Below this threshold there is an effective SU(3)c x U(l)em theory. Thresholds in this region are obtained by integrating out each quark to one loop at a scale equal ^ At the one loop level, due to a cancellation between large numbers, the value of the SUSY scale is extremely sensitive to the value of 03, which means that the proper treatment of thresholds and of two loop effects will determine the actual value of the SUSY scale. (L. Clavelli, private communication). PAGE 78 73 to its physical mass. At these scales the one loop matching functions in the gauge couplings vanish and the threshold dependence appears through steps in the number of quark flavors [97] as the renormalization group scale passes each physical quark mass. There is also a threshold at MsusyHere the matching condition is the naive one of simple continuity due to the lax;k of knowledge about the superparticle spectrum. We take this scale to be variable to account for this ignorance. We consider the simplest implementation of supersymmetry and run the couplings above Msusy to one loop. We assume the MSSM above Msusy-, and a model with a single light Higgs scalar below it. This is done by integrating out one linear combination of the two doublets at Msusy , thereby leaving the orthogonal combination in the Standard Model regime as the "Higgs doublet": where $ = 2x2$*, and where tan^^ is also the ratio of the two vacuum expectation values (vu/vj) in the limit under consideration. This sets boundary conditions on the Yukawa couplings at MsusyFurthermore, in this approximation the quartic self coupling of the surviving Higgs at the SUSY scale is given by This correlates the mixing angle with the quartic coupling and thereby gives a value for the physical Higgs mass, Mfj. Using the experimental limits on the Mff further constrains some of the results. By using the renormalization group we take into account radiative corrections to the light Higgs mass [102] and hence relax the tree level upper bound, Mjj ~ Mz [103]. (9.2) X{MsusY) = \{gl + 9l)cos\2/3) . (9.3) PAGE 79 74 We determine the bounds on M< and Mf{ by probing their dependence on /9. In SUSY-SU(5), tan/3 is constrained to be larger than one in the one Hght Higgs Hmit. It seems natural to us to require that yt > yi up to the unification scale [104], thereby yielding an upper bound on tan The initial values at Mz for the gauge coupUngs are given in Sections 5.1 and 5.2. For this analysis we use an earlier value the strong coupling, 03 = O.IOQIq qq^ [105]. We utiUze the quark mass values given in Section 5.4. For the bottom mass we use the Gasser and Leutwyler bottom mass value of 5.3 GeV at 1 GeV which translates into a physical mass of = 4.6 GeV [79]. To probe the dependence of our results on we also study the case of M^, = 5 GeV, the typical value obtained from potential model fits for bottom quark bound states [87]. We also investigate the effect of varying MsusyGiven the values of the gauge couplings, we find unification up to a SUSY scale of 8.9 TeV, and as low as M^y. For empirical reasons we did not investigate solutions below that scale. From Figure 4 we determine that the lower end scale, M^yj,, of the unification region corresponds to an 03 value of 0.104 at Mz, while the higher end scale, Mquj,, corresponds to a value of 0.108 at Mz for a3. We find that the unification region is insensitive to the range of top, bottom, and Higgs masses considered. In our analysis of the bounds for M<, the values for ai and 02 are chosen to be the central values since their associated experimental uncertainties are less significant than for 03. Demanding that yj and j/r cross at M^y^, and taking 03 = 0.104 then sets a lower bound on M<. Correspondingly, demanding that ?/5 and t/r cross at M^j, and taking 03 = 0.108 yields an upper bound on Mt. These bounds are found for each possible value of 0. Figure 16 shows the upper and lower bound curves for both Mt and Mjj as a function of /3 and for Msusy = 1 TeV and Mf, = 4.6 GeV. When apphcable PAGE 80 75 we use the current experimental limit of 38 GeV on the light supersymmetric neutral Higgs mass [106], to determine the lowest possible M< value consistent with the model. We find 139 < Mt < 194 GeV and 44 < M// < 120 GeV. We investigated the sensitivity of these results on Msusy in the range, 1.0 Â± 0.5 TeV. We find that the bounds on Mt are not modified, but the upper bound on the Higgs is changed to 125 GeV, and the lower boimd drops below the experimental lower bound. MgusY = 1 TeV. = 4.6 GeV 0 I ' ' I ' I ' ' I I ' ' I ' I I I I ' I I I I I 40 50 60 70 80 90 iS (deg.) Figure 16. Plot of M<, Mjy, as a function of the mixing angle 13 for the highest value of (high curves) and the lowest value of a3 (low curves) consistent with unification. For Mj = 5.0 GeV, we see an overall decrease in the top and Higgs mass bounds: 116 < M< < 181 GeV, Mh < 111 GeV. Varying Msusy as above PAGE 81 76 modifies the respective bounds. The top mass lower and upper bounds become 113 and 119 GeV, respectively. The upper bound on Mjj changes to 115 GeV. We display the results of our analysis for the extreme case, Msusy Â— 8.9 TeV, in Figure 17, with = 4.6 GeV. This only significantly changes the upper bound on Mjj to 144 GeV compared to the Msusy = 1 TeV case. Msusy = 8.9 TeV. = 4.6 GeV Q I I I I I I I I I I I I I I I I I I I I I I I I I 40 50 60 70 80 90 /S (deg.) Figure 17. Same as Figure 16 for Msusy = 8.9 TeV and Mfc = 4.6 GeV. We have also run yt up to the unification region and compared it with and yr to see what the angle 0 must be for these three coupHngs to meet [107], as in an SO(IO) or Eg model [5 6] with a minimal Higgs structure. It is clear that this angle is precisely our upper bound on /3 as described earher. In Figure 18 we display yt/yi, at the GUT scale as a function of tan/? for Msusy = 1 TeV PAGE 82 77 and for the two bottom masses we have considered. If we demand that the ratio be one we can determine the mixing angles for the low and high ends of the unification region. Then going back to Figure 16, we find as expected a much tighter bound on the masses of the top and of the Higgs. Indeed, for Mb = 4.6 GeV, we have 49.40 < tan/3 < 54.98, which yields 162 < Mt < 176 GeV and 106 < Mh < 111 GeV. When = 5.0 GeV, we obtain 31.23 < tan^ < 41.18, which gives 116 < M< < 147 GeV and 93 < M;/ < 101 GeV. MgusY = 1 TeV I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I ' I I' 'I I I I I I' I ' I I I I I I I I I I I I I I I I 0 10 20 30 40 50 60 tanjg Figure 18. Plot of the ratio of the top to bottom Yukawas, y^/y^,, for two different bottom masses (solid and dashed curves) as a function of tan /? for the highest value of 03 (high curves) and the lowest value of ^3 (low curves) consistent with unification. Several issues have been left untouched. The effects of soft SUSY breaking terms were not investigated nor was the possible role of a large top mass on PAGE 83 78 this breaking. Also, we have integrated out all the supersymmetric particles at the same scale. It would be interesting to study the effect of lifting this restriction. We should also note that our bounds on the top mass axe very similar to those of ref. 104, although the physics is very different. Given the relative crudeness of our approximations in this analysis, it is remarkable that the experimental bounds on the /)Â— parameter were satisfied which in our mind gives credence to our program. PAGE 84 CHAPTER 10 CONCLUSIONS We have have established an efficient procedure for performing renormalization group analyses of the Standard Model and its extensions. With two loop renormalization group equations at our disposal and numerical routines developed to solve them we simply follow a series of steps. First pick your favorite high energy model with some hierarchy of symmetry breaking. Decide what particles and models are to be placed in the "desert" between M\y and Mp/. Implement effective gauge theories between thresholds that brealc gauge symmetries through the use of matching functions. Run renormalization group equations for each effective theory according to your scenario's physical particle spectrum. Finally analyze output in graphical form to see what patterns emerge and perform careful tests of promising patterns arising from relations among parameters in the high energy theory. We have performed these analyses in the case of the Standard Model and its minimal supersymmetric extension. In the former case we have reproduced the result that gauge and Yukawa couphng relations arising in GUTs with the Standard Model as the low energy theory below Mqut are not valid. No reasonable spectrum of superheavy fields can produce unification in this case, even when thresholds are included. This merely adds to the failure of minimal GUT predictions for proton decay. In the latter case we have found remarkable agreement with the GUT predictions of gauge and Yukawa unification although the latter is in agreement only for the heaviest fermion family. By using the best data available we find constraints on the top quark from the relation = mr due to the sensitivity of the renormalization group improvement of this relation to m<. The upper bound 79 PAGE 85 80 on Mt lies below the upper bound coming from radiative corrections to the p parameter. Although the Standard Model can have a 100 GeV top quark, the lower bound from our renormalization group analysis indicates that minimal SUSY-SU(5) favors a somewhat larger value. If Fermilab finds Mt between 100 and 120 GeV this should disfavor minimal SUSY-SU(5) models with one light Higgs doublet below the supersymmetry scale. PAGE 86 APPENDIX I THE STANDARD MODEL /3 FUNCTIONS In this appendix we compile the renormalization group /? functions of the Standard Model. These have appeared in one form or another in various sources. We have endeavored to confirm their validity through a comparative analysis of the literature. Our main source is ref. 78. Following their conventions, C = Q.^Yjun + Q.^YjUj, + Ij^^YeUn + h.c. ^A($t$)2 , (J.i) where flavor indices have been suppressed, and where Ql and Â£i are the quark and lepton SU(2) doublets, respectively: $ and $ are the Higgs scalar doublet and its SU(2) conjugate: . ^ = ^>2^(1-3) Ur, dji, and Cr are the quark and lepton SU(2) singlets, and j g are the matrices of the up-type, down-type, and lepton-type Yukawa couplings. The /? functions for the gauge couplings axe ^ ^ ' (1-4) dt "'I67r2 Z-.^*/ (16^2)2 i (167r2) TtA^ Tr{C/Â„YjYÂ„ + QdY^tY^ + Ci^Y^Ye} , where t Â— In// and / = 1, 2, 3, corresponding to the gauge group SU(3)c x SU(2)l X U(1)y of the Standard Model. The various coefficients are defined 81 PAGE 87 to be h h 82 4 1 "3"'' " To ' 22 4 1 4 ^3 = 11 -^rig , (0 0 0 ^ /19 1 UN / 9 5U 3 TO {hkl) = 0 136 "3" 0 -ng 3 49 T 3 5 9 13 T 0 0 102^ 44 \T5 4 76 0 and (Cif) = /17 1 3\ ' TO 5 5 ^ 3 3 1 5 5 5 2 2 0 , with f =^ u , d , e , with Ug = ^ny/. In the Yukawa sector the ^ functions are where the one-loop contributions are given by = YÂ„tYÂ„) + F2(5) (^^? + + Sgl) , 2' with Y2{S) = Tr{3YÂ„tYÂ„ + SY^^Yj + yJYc} , (1.5) (1.6) (1.7) (1.8) (1.9) (I.IO) PAGE 88 83 and the two-loop contributions axe given by /??^ = ^(YÂ„tYÂ„)2 _ YÂ„tYÂ„Y,tY, jY,tY,YÂ„t YÂ„ + ^(Y,t Yrf)^ + r2(5)(^YdtYd ^YÂ„tYÂ„) X4(S) + 2A(3YÂ„tYÂ„ + Y^tY^) + ( + + 16.|)YÂ„tYÂ„ -(f^gj+ i6,|)Y,tY, with ^T^/cN , / 9 29 ,4 9 2 2 19 2 2 .35 ,4 ^22 2^^^*^^ + ^200 ^ 45"^^^' " 20^1^2 + ng)gi + 9^|fl'3 ,404 80 , 4 ^ = liV^d? YdtY,YÂ„tYÂ„ iYÂ„tYÂ„YrftY, + ^(Y.tYÂ„)2 + r2(5)(^YÂ„tYÂ„ ^YrftYrf) X4{S) + 2A(3YrftYd + YÂ„tYÂ„) + (^.f + '-Â§91 + 16.|)Y,tY, -i'lgl1,1 + 16,i)YÂ„tYÂ„ , 5 , , , 29 1 ,4 27 2 2 , 31 2 2 /35 ,4 + 2^4^^^ ^200 + 45"^^^1 ~ 20^1^2 + Y^9l9z " " n,)52 Â„ 2 2 .404 80 , 4 + 9^253 -(-^--g-" PAGE 89 84 and X4(S) = I Tr{3(YÂ„tYÂ„)2 + 3{Yd^Yjf + (YetYe)^ 2 ^^-''^ -YÂ„tYÂ„YrftYrf} . In the Higgs sector we present ^ functions for the quartic coupUng and the vacuum expectation of the scalar field. Here we correct a discrepancy in the one-loop contribution to the quartic coupling of ref. 78 (1.15) where the one-loop contribution is given by + 4Y2{S)\-iH{S) , with H{S) = 'L:{3(YutYÂ«)2 + 3(Yd^Ydf + (Ye^Yef} , (1.16) PAGE 90 85 and the two-loop contribution is given by ) = -78A3 + 18(^,2 + 3,|)A2 [ 10n,),| + '-^ghl -645|Tr{(YÂ„tYÂ„)2 + (YrftYd)2} ^fir?Tr{2(YÂ„tYÂ„)2 {Yd^Yjf + 3(YetYe)2} IdtYAiS) + 10A[ i^^gl + + S^f) Tr{YÂ„tY,} + i\9l + + 8^i)Tr{Y/Yrf} + ^igf + ^|)Tr{YetYe} ] + Idli i-^gl + 21g|)Tr{YÂ„tYÂ„} + i^gl + 9^1) TriY^^ty^} + (-f 51 + llyi)Tr{YetYe} ] 24A2r2(5) Aff(5) + 6ATr{YÂ«tYÂ„YrftY^} + 20Tr{3(YÂ„tYÂ„)3 + ZiYd^Y^f + (YetYe)3} 12Tr{YÂ„tYÂ„(YÂ„tYÂ„ + Y/Yd)Y/Yd} . The /3 function for the vacuum expectation value of the scalar field is dt 167r2^ ^(167r2)2^ ' where the one-loop contribution is given by 9 ,239 40 , 4 2 27 ,59 40 , Â« 25^ 2^ + Y^(24 + J^9)9l 7 (1.19) and the two-loop contribution is given by 7 -^A2-^F4(5)+X4(5) (1.20) 93 1.4 .511 ^ \ 4 27 2 2 ^ng)g2 ^9192 Â• PAGE 91 86 These expressions were arrived at using the general formulas provided in ref. 78 for the anomalous dimension of the scalar field, choosing the Landau gauge. In the low energy regime the effective theory is SU(3)c x U(l)emWe employ the generzd formula of ref. 108 to arrive at the /5 functions for the respective gauge couplings: ^ = iknu + n,) 11]-^ + [^K + n,) 102]^ ^[^Â«.^n,]g (1.21) .5033, , 325, ,2 2857, gl and de .16 4 4 , M 4 , l-x-nu + x^d + o^iItt^ + [-^^u + 7;=nj + Anil dt ^9"" ' 9"'^ ' 3'''^(47r)2^^27"""27"^"^"'^(47r)4 .64 16 , e^g^ (1.22) 9 " 9 ^^(47r)4 ' where n^, nj, and n/ are the number of up-type quarks, down-type quarks, and leptons, respectively. In Eq. (1.21) we have also included the three-loop pure QCD contribution to the ^ function of ^3 [109]. For the evolution of the fermion masses we used ref. 110. It is known that there is an error in their printed formula [111]. Using the corrected expression, we compute the following mass anomalous dimension. The fermion masses in the low energy theory then evolve as follows: dm dt 7(/,,)m, (1.23) where the / and q refer to a particular lepton or quark, and where ^('. PAGE 92 87 The superscripts 1 and 3 refer to the U(l)em and SU(3)(;; contributions, respectively. ExpHcitly, the above coefficients axe given by 7f,)=0 ^13 _ ^33 _ n 7(0 7(,) 0 (1.25) 11 o^4 rSO 20 20 ,^2 oo 404 40, if^f = + nd? + (160C(3) + + n^) 3747] , where Q^i is the electric charge of a given lepton or quark, and (^(3) = 1.2020 ... is the Riemaiin zeta function evaluated at three. In the mass anomalous dimension for the quarks above, we have also included the three-loop pure QCD contribution -y^^^ [109]. PAGE 93 APPENDIX II THE MSSM 13 FUNCTIONS Using some of the notation of Falck [112], the superpotential and soft symmetry breaking potential axe as follows: + m\u^u + m\d^d-\mleU + Bfi{^u^d + h.c.) + ^( A^Yj^Ui^uQj + A^JrJ^di^dQj + 4^Yjhi^dLj + h.c. ) , hj 1 ^ ^gaugino = ~o X] ^l^l^l + Â• ^ l=\ (11.1) Various PAGE 94 89 6 5 88 \ T5 / 9 9 u (%) = 2 14 8 3 5 -17 0 11 Vt5 3 68 [" 0 -54 j (11.4) and / 26 14 18 \ ' T T T ^ (Cif) = \ 6 6 2 4 4 0 , with / Â— u , d , e , (11.5) with Ug = ^n^/. In the following we list the 0 functions for the parameters of the superpotential. (fln// dt = ^[ Tr{3YÂ„tYÂ„ + 3Yd^Yd + Ye^Ye] 3(^fir? + g|) ] . (II.6) In the Yukawa sector the ^ functions axe dY. dt (11.7) where the one-loop contributions are given by ^ = 3Y,tYÂ„ + Y/Yrf + 3Tr{Y,tYÂ„} {^gl + 3^1 + y^f) , = 3Y/Yrf + YÂ„tYÂ„ + Tr{3Y,tYrf + YetYe} {^gl + Zg^ + ^^|) , ^i^^ = 3YetYe + Tr{3YrftY^ + Ye^Ye} (^y? + Sgj) , (11.8) PAGE 95 90 and the two-loop contributions are given by Z?!^^ = -4(YÂ„tYÂ„)2 2(YdtYd)2 2YrftYdYÂ«tYÂ« 9Tr{YÂ„tYÂ„}YÂ„tYÂ„ Tr{3YdtYj + YetYe}YdtYd 3Tr{3(Y,,tYÂ„)2 + YrftYrfYÂ„tYÂ„} + iloi + 6^|)YÂ„tYÂ« + (^^?)YdtY^ + (1^2 + 16^32) Tr{YÂ„tYÂ„} 26 ^ 403 4 21 4 32 304,4 + ^15"^ + 450^^1 + ^^"^ T^^2 + Â—)gt 9 9 136 9 9 o o = -4(YdtYrf)2 2(YÂ„tYÂ«)2 2YÂ„tYÂ„Y/Yrf 3Tr{YÂ„tYÂ„}YÂ„tYÂ„ -3Tr{3YrftYrf + YetYe}YrftYrf 3Tr{3(YdtYd)2 + (YetYe)^ + YrftYrfYÂ„tYÂ„} + (^^?)YÂ„tYÂ„ + (^^1 + 6^i)YrftYd + (-^^2 ^ 16^2) Tr{YdtYrf} + (^^2)Tr{YetYe} + (^n, + l^)g\ + (6n, |)^| ,32 304 4 99 899 Â„99 + i^^g -^)^3 + 9192 + gfi'i^a + 8Â« /?(^) = -4(YetYe)2 3Tr{3YdtYrf + Ye^YejYe^Ye 3Tr{3(Y/Yrf)2 + (YetYe)^ + Y/YrfYÂ„tYÂ„} + (65i)YetYe + (^5r2)Tr{YetYe} + {-\g\ + 16^1) Tr{ Y Ad} .18 27, 4 ,^ 21, 4 999 (11.9) The evolution of the vacuum expectation values of the Higgs's is given by
PAGE 96 91 where the one-loop contribution is given by PAGE 97 92 ^ = 8^51 E 3|Â«'I'(">1 + "'I + < + + Tr{Fm2} ^g^M^ Zg^Ml ] , + 3|ldiVL + m^,+m| + |A^;|2)) -l9?Tr{rm2}-?9?Mf-3j|M|l, i + ?j?Tr{ymn-fsfMfl, i i hi [ 34|fJ^|2 + 34Vj^|2 + Ai^'lFe'^p + l^fMi + ] . (11.14) dml. dB _ 1 PAGE 98 93 where as in Falck [112], sums are implied over all indices not appearing on the left hand side and where i=l The gaugino masses evolve as follows ^ = -^W. urn PAGE 99 REFERENCES 1) J. N. Bahcall, Neutrino Astrophysics (Cambridge University Press, Cambridge, England, 1989). 2) L. Wolfenstein, Phys. Rev. D 17, 2369 (1978); L. Wolfenstein, ibid. 20, 2634 (1979); S. P. Mikheyev and A. Yu. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985); S. P. Mikheyev and A. Yu. Smirnov, Nuovo Cim. 90, 17 (1986). 3) J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974). 4) H. Georgi and S. Glashow, Phys. Rev. Lett. 32, 438 (1974). 5) H. Georgi, in Particles and Fields-1974, edited by C. E. 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Ross, Nucl. Phys. B157, 273 (1979). 111) R. Tarrach, Nucl. Phys. B183, 384 (1981). 112) N. Falck, Z. Phys. C 30, 247 (1986). PAGE 106 BIOGRAPHICAL SKETCH The author was bom in Baltimore, Maryland, on July 28, 1963. He received a B.A. degree in physics from Cornell University in 1985. During his time at Cornell the author decided to undertalce a serious study of physics, particularly particle physics. He received an M.A. degree from Columbia University in 1987 and left there to continue work in particle theory at the University of Florida. His current interests, besides the subject of this work, also include electroweak baryogenesis and the flavor hierarchy problem. He is currently working on renormalization group aspects of the MSSM at the Phenomenology Institute of the University of Wisconsin. 101 PAGE 107 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierre M. Ramond, Chairman Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for tlie degree of Doctor of Philosophy. Pierre Sikivie Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Charles B. Thorn Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ^ allies N. Fry Associate Professor of J'hysics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. I'aul L. Robinson ~ Assistant Professor of Mathematics PAGE 108 This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. 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