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RENORMALIZATION GROUP STUDY OF THE MINIMAL SUPERSYMMETRIC EXTENSION OF THE STANDARD MODEL WITH SOFTLY BROKEN SUPERSYMMETRY By DIEGO J. CASTANO 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 1993 A Lela Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/renormalizationg00cast ACKNOWLEDGEMENTS I would like to acknowledge all of those who have influenced and helped me in my physics career. I thank my fellow students, particularly Haukur Arason, Sam Mikaelian, and Eric Piard, with whom I have shared illuminating conversations, both in physics and outside. I have learned a lot of physics from my talks with them. I thank all of my teachers and professors for their excellent instruction through the years. I especially thank Pierre Ramond. I have gained much from his tutelage. In being a great particle physicist, he has given me a goal for which to strive. And last but not least, I thank my wife and my family for their continual support, understanding, and encouragement. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ..................... iii ABSTRACT . . . vi CHAPTERS 1 INTRODUCTION ...................... .. 1 2 THE RENORMALIZATION GROUP . 5 3 THE STANDARD MODEL ................. 8 3.1 aI(Mz), a2(Mz), and a3(Mz) . 8 3.2 Yukawas . . . 13 3.3 Quark and Lepton Masses . .. 15 3.4 Top and Higgs Masses ... .............. 17 3.5 Vacuum Expectation Value . .... 19 3.6 Thresholds . . 21 3.7 Analysis and Results . . 22 4 THE MSSM ...... ........... ...... .37 4.1 The Supersymmetric Standard Model . .. 37 4.2 Procedure .. . .. .. ....38 4.3 One Light Higgs Limit ... .. ....... .... 39 4.4 Initial Data . .. .. 40 4.5 Analysis and Results . . .. 41 4.6 Comments ....................... 43 5 SOFT SYMMETRY BREAKING . .... 49 5.1 Minimal Low Energy Supergravity Model . 49 5.2 Radiative ElectroWeak Breaking. . ... 51 5.3 Sfermion Masses .......... .. .. ...... .54 5.4 Higgs Masses .. ... ........ .... .... .54 5.5 Chargino Masses .. . 55 5.6 Neutralino Masses ............... 5.7 Boundary Conditions at M . . 5.8 Numerical Procedure . . 5.9 Thresholds . . . 5.9 Analysis and Results . . 6 CONCLUSIONS ................. APPENDICES A THE STANDARD MODEL 0 FUNCTIONS . B CALCULATING THE MSSM P FUNCTIONS . B.1 Gauge Couplings ............... B.2 Anomalous Dimension of the Scalar Field . B.3 Yukawa Couplings .... ............ B.4 Thresholds . . . C THE MSSM 0 FUNCTIONS ............ D THE P FUNCTION OF THE VEV . . E NUMERICAL TECHNIQUES . . F EXPLICIT FROM OF () . . G 'METAPLECTONS' ............... G.1 Introduction . . . G.2 Anyons . . . G.3 The Metaplectic Representations of OSp(r/2m, R) G.4 New Representations of Anyons . . G.5 Conclusion . . . REFERENCES . . . BIOGRAPHICAL SKETCH ........... .... . 56 . 56 . 58 . 61 . 62 . 69 . 71 . 77 . 78 . 82 . 85 . 88 . 91 . 97 . .. 104 . .. 107 . 108 . 108 . 110 . 115 . 120 .. 123 . 125 . 131 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RENORMALIZATION GROUP STUDY OF THE MINIMAL SUPERSYMMETRIC EXTENSION OF THE STANDARD MODEL WITH SOFTLY BROKEN SUPERSYMMETRY By Diego J. Castafio May 1993 Chairman: Pierre Ramond Major Department: Physics Using the renormalization group all the couplings of the Standard Model and its minimal supersymmetric extension are run to two loops, taking full ac count of the Yukawa sector. It is found that in the standard model the gauge couplings fail to unify, whereas extending the model through supersymmetry achieves unification. Bounds are placed on the top quark mass by requiring equality of the bottom and 7 Yukawa couplings at the scale of unification. In its simplest form, the supersymmetric model has a degenerate superparticle spec trum, and for ISUSY = 1 TeV, and Mb = 4.6 GeV, one finds 139 < Mt < 194 GeV, which remarkably satisfy the pparameter bound. The corresponding bounds on the Higgs mass are found to be 44 < MH < 120 GeV. The model is then coupled to supergravity yielding a richer superparticle spectrum and mak ing it more accountable to experiment. This model has the attractive feature that the electroweak symmetry is radiatively broken. For the special case in which global supersymmetry breaking arises solely from soft gaugino masses, Mt is expected to be less than ~ 130 GeV. A higher upper bound is predicted, if A/h is at the lower end of its experimental uncertainty. CHAPTER 1 INTRODUCTION In the last few years, it has become apparent, using the ever increasing ac curacy in the measurement of the strong coupling, that supersymmetry (SUSY) affords an elegant means to achieve gauge coupling unification [1] at scales con sistent with grand unified theories [2] (GUTs). Whereas in the standard model (SM) the three gauge couplings unify "two by two" forming the "GUT trian gle," in the simplest minimal supersymmetric extension of the standard model (MSSM), these gauge couplings spectacularly unify at a point (within the ex perimental errors in their values). Given that the scale of unification in these models is generally above the lower bound set by proton decay, the socalled SUSYGUTs have gained increasing interest. Constraints coming from Yukawa coupling unification in supersymmetric SU(5) and SO(10) models can be used to yield interesting predictions for various low energy parameters including the top quark mass [3,4]. This analysis employs the renormalization group (RG) [5] to extrapo late the parameters of the standard model and of its minimal supersymmetric extension to unexplored scales [6]. With the ever increasing precision of ex periment, the inclusion of two loop effects is crucial. The complete Yukawa sector contribution is also included. In the standard model case, thresholds effects are implemented in both a simple and a more involved method. The results of these two methods are discussed. Numerical methods are used to evolve the parameters to different scales using the 3 functions found in the 2 literature [7], and the results are plotted for representative values of the Higgs boson and top quark masses. The running of the quark and lepton masses and of the CabibboKobayashiMaskawa (CKM) angles is generally given through the running of the Yukawa matrices (even to one loop). In this work, the quark masses and CKM angles are evolved by diagonalizing the Yukawa matrices at every step in the RungeKutta method used in solving the 03 functions. Often 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 en tries. The present numerical technique represents a small improvement over these methods. Chapter 2 consists of a basic introduction to renormalization and renor malization scheme dependence. Chapter 3 addresses the standard model. A review of initial data extrac tion from experiments is presented. Many excellent reviews may be found in the literature, e.g., Marciano's [8] or Peccei's [9]. The determination of the standard model gauge couplings is discussed as is the initial data extraction of the Yukawas and the CKM angles. The extraction of the quark masses from data is also discussed. This is a complex issue well known to be marred by the nonperturbative nature of QCD. Hence, in the low energy regime, the pure QCD threeloop contribution is included in the analysis of the running of the quark masses. Initial data for lepton masses follow the quark discussion. Then the extraction of and constraints on the physical top and Higgs masses are considered. The scale dependence of the renormalized scalar vacuum expecta tion value is addressed. The method used to obtain the values of all running 3 parameters at the same initial scale is described in Appendix E. In the fol lowing section, threshold effects are discussed. Finally, a quantitative analysis of the results is presented. The effects of using one loop versus two loop 0 functions are contrasted and of including a proper versus a naive treatment of thresholds. Plots of all the running parameters over the entire range of mass scales are included and also used to display the effects discussed. Furthermore some tables are presented with actual numerical differences associated with these effects. Chapter 4 addresses the minimal supersymmetric extension of the stan dard model. A brief discussion of supersymmetric models is presented. The procedure by which top quark mass bounds are determined using the equality of the bottom and tau masses at the scale of unification is discussed as well as the one light Higgs limit employed. The initial data for gauge couplings and quark masses are presented. This is followed by results on the top quark and Higgs boson masses. Finally, a discussion on how to improve the results is presented. Chapter 5 implements some of the improvements discussed at the end of the last chapter. Namely, the supersymmetric two loop 3 functions are included. Also, soft symmetry breaking terms are added. These lead to a nondegenerate superparticle spectrum as well as to the radiative breaking of the electroweak symmetry. Similar analyses have appeared in the literature but use one loop 3 functions and the tree level Higgs potential [10]. In Chapter 5, a brief discussion of the effective one loop potential is presented. The mass formulas for the sfermions, Higgses, charginos, and neutralinos are given. The boundary conditions at the unification scale in these minimal low energy supergravity models are discussed. Next the numerical procedure employed is described. 4 The treatment of thresholds and the "special" form of the 0 functions needed is discussed next. Finally, some preliminary results are presented. Chapter 6 contains the conclusions of this work. Appendix A contains all the needed two loop 3 functions for renormaliza tion group studies of the standard model. Appendix C contains those of the minimal supersymmetric extension of the standard model. Many of these 3 functions have yet to appear in the literature in as general a form. Appendix B presents some examples in calculation of the3 functions of Appendix C. Appendix D deals with issue of the vacuum expectation value's 3 function. A toy model is used to gather insight into the problem. The numerical solution routines used extensively throughout this work are discussed in Appendix E. Appendix F contains a cumbersome formula needed in the extraction of the Higgs boson physical mass. Finally Appendix G presents a novel use of the spinor representations of the orthosymplectic Lie supergroup OSp(r/2m, R) [11]. CHAPTER 2 THE RENORMALIZATION GROUP Renormalization is a reparametrization of a theory which renders Green functions and physical quantities finite order by order in perturbation theory. A specific choice of renormalized parameters defines a renormalization scheme. The physics is, of course, independent of how the theory is renormalized. A common way of relating bare and renormalized parameters is go = g g (2.1) where go is the bare parameter, g is the renormalized parameter, and 6g is the counterterm. Fixing the counterterms by requiring them to consist only of the infinite terms needed to render the theory finite defines the minimal subtraction (MS) prescription [12]. A feature of the MS scheme is a mass scale [ which enters in the process of regularizing divergent integrals using dimensional regularization. Furthermore the unit of mass [i is used to keep couplings dimensionless when continuing to d dimensions in the dimensional regularization procedure. For example, if Eq. (2.1) represents any of the three gauge couplings of the Standard Model, then p is introduced as follows to keep them dimensionless go(e )~ = g 6g (2.2) where g is a constant parametrizing the arbitrariness in the finite parts of diver gent integrals in dimensional regularization and e = (4 d)/2. Equation (2.2) defines a family of MS schemes. Choosing e = 1 is the simplest MS scheme 5 6 which was described above. Choosing e2 = eYET/4r, where yE = 0.5722... is the EulerMascheroni constant, defines the socalled modified minimal sub traction (MS) prescription [13]. This scheme is the most commonly employed in QCD calculations, and it is the one adopted here. The free parameters of the Standard Model in the MS schemes are 1 dependent. Their p evolution is governed by the / functions of the renormalization group. Moreover, these running parameters are not in general equal to their corresponding physical values (consequently, for the masses, a convention is adopted wherein upper case M refers to physical values and lower case m denotes MS values). This is to be contrasted with the onshell renormalization scheme in which, for example, the renormalized masses equal their physical values and the renor malized electromagnetic coupling equals the fine structure constant. However, the MS schemes have the attractive characteristic that the / functions are p independent and therefore particularly simple to integrate. Physical quanti ties P({gi(pi)}, p) expressed in terms of p and the running parameters of the theory, {gi(7)}, must be 1 independent d 0 8 Y P({gi(t)}), ) = (i + 1it3 )P = 0, (2.3) where the 3i are the 3 functions. The twoloop / functions of the Standard Model have been collected in Appendix A. As mentioned above physical quantities are renormalization scheme inde pendent. However, this assumes that calculations can be done without ap proximation. In reality, calculations are only perturbative approximations and these do depend on the renormalization scheme. Consider massless QCD with the one dimensionless coupling, as. Suppose a physical quantity, P. is calculated to nth order in perturbation theory in two 7 renormalization schemes, then the nth order approximation is given by 2 n1 Pn = as(Po + Plas + P2a2 + + Pnlas) (2.4) in one scheme and by p t I + 2 in(2.5) Pn = s(Po + Plas + P'2as +. 2 + Pnlas ) (2.5) in the other scheme. The two couplings, defined in their respective renormal ization schemes, can be related to each other = as(1 + blas + b2a2 + + bnan1) (2.6) Substituting Eq. (2.6) into Eq. (2.5) yields Pn= s(p" + p'as + p2a + + Pnla ) (2.7) +Tas(pas +... +p2nla"1 )( As this is an approximation to the same quantity, P, the first n terms can be identified with Pn. Therefore the two approximations differ by terms of higher order pn 2n 1). (2.8) P n P= as(pns +' ... + P2n (2.8) In QCD where as is large, this difference may be large and thereby lead to renormalization scheme dependence problems. In the Standard Model as in QED where the couplings are small, this is not so great a problem. In QCD where the strong coupling as is large, there will be renormalization scheme dependence problems. In the electroweak model as in QED where the couplings are small, this is not so great a problem. CHAPTER 3 THE STANDARD MODEL 3.1 ai(MZ), ag(Mz), and ca(MZ) The determination of the SU(2)L x U(1)y couplings proceeds from the Standard Model relations a ) g ) _C2 Wa(P) 4 cs2 W() (3.1.1) ( = g2(j) g a(y) a2 47 sin2 W() ' where a(y/) = e2(p)/47 and C2 is a normalization constant which equals 1 for the Standard Model and equals i when the Standard Model is incorpo rated in grand unified theories of the SU(N) and SO(N) type [5]. What is required to specify these couplings are the values of a(/y) and sin2 Ow(i) in the renormalization scheme employed (i.e., MS). The electromagnetic fine structure constant (ar1 137.036) is extrapolated from zero momentum scale to a scale p equal to Mg in the present case. In pure QED with one species of fermion with mass m the MS renormalized vacuum polarization function is given by 1(m) A2 q2 In(q2) = 3 (In 6 dx x(1 x)ln[l x( x) ]) (3.1.2) 37 m m 0 The renormalized coupling a(y) is related to the fine structure constant aem as follows aem () (3.1.3) 8 9 In the Standard Model where there are many species of charged fermions and charged gauge bosons, Eq. (3.1.3) generalizes to [14] a(p) = ae 2 Q2 0( mf)ln + (3.1.4) 37f mf 6Xr The effects of the strong interaction, which enter as a hadronic contribution to the vacuum polarization function, must be included also. The nonperturbative nature of the strong interaction at low momentum is handled by rewriting the hadronic contribution to the vacuum polarization at zero momentum as nI(0) = (IIh(0) II(q2)) + h(q2) (3.1.5) If q2 is chosen large enough, IIh(q2) can be calculated perturbatively. The terms (IIh(O)IIh(q2)) can then be related to the total cross section for e+e * hadrons [14]. Using the optical theorem, one may write Im{Ih(s)} = s (e+e hadrons) (3.1.6) 47trem where s is the square of the center of mass energy. For the process e+e + +fi~, the cross section is calculated to be (taking m, = 0) 47rQa2 (ee + ) = (3.1.7) 3s In terms of the ratio of these two cross sections, S(e+e + hadrons) R(s) = (e+ + ), (3.1.8) o(e+e + M+y) one may write Eq. (3.1.6) Im h)} (3.1.9)_ Im{h (s)} = R(s). (3.1.9) 10 Using an unsubtracted dispersion relation for IIh (q2) the combination (IIh(0) IIh(q2)) can be expressed as q2 aem0 R(s s) nho) _h(q2) = ()(3.1.10) 37r s(q2 s) 4mi This can be evaluated using experimentally known data. This procedure yields a value a(Mz) = 127.9 0.3 (3.1.11) The process independent, renormalized weak mixing angle sin2 OW of the onshell scheme is defined to be sin2 O = 1 (3.1.12) Z where MW and MZ are the physical masses of the W and Z gauge bosons. Knowing the precise values of the W and Z boson masses and using the equa tion above provide one way of extracting the value of sin2 OW. Alternatively, the bare relation involving the low energy Fermi constant measured in muon decay and the W boson mass Go e2 o e8sin(3.1.13) v/2 8 sin2 Wo2 y! 0 may be corrected to order a and rewritten [15,16] 7raem 1 1 Mw = MzcosOw = ( ) (3.1.14) VGu sinyw(1 Ar) with (raem/V'GM) = 37.281 GeV and Ar is a parameter containing order a radiative corrections and which depends on the mass of the top and Higgs. The radiative corrections represented by Ar can be viewed as accounting for the mismatch in the scales associated with the parameters of the relation. G" and aem are low energy parameters whereas .M1 and sin2 9y are associated 11 with the electroweak scale. One can absorb the radiative effects using the renormalization group by replacing G, and aem with corresponding running parameters at MZ (MZ1 (3.1.15) G'G(MZ)M2 v sin2 OW Combining Eqs. (3.1.14) and (3.1.15) gives em G,(MZ) Ar. 1 e G(M) (3.1.16) a(Mz) Ga Using Eq. (3.1.4) and the fact that G,(Mz) ; Gp (see Section 2.8) gives an estimate of the size of the radiative corrections Ar ; 0.07 (3.1.17) For large values of Mt and MH (Mt, MH > MZ) [15,17] aem 3em Mt2 11aem M2 Ar 1 ae 3aem M + 1In (3.1.18) a(Mz) 167rsin4 w M 487rsin2 w M( A third way of extracting sin2 OW is from neutral current experiments, among which deep inelastic neutrino scattering appears to provide the best determi nation. A running sin2 Ow(P) may be defined in MS and differs from the above sin2 OW by order a corrections. The MS running W boson mass mw(Ji) and the corresponding physical mass MW, identified as the simple pole at q2 = M of the W propagator, are related as follows M2 = m () + Aw(Mw, p) (3.1.19) where A y is the transverse part of the W selfenergy. A similar relation holds for the Z boson. In MS renormalization, the following relation defines the running sin2 OW(t) 2sin w ( sin2 0W(z)  1P (3.1.20) ZZr 12 Equation (physmw) and its Z analog may be combined with Eq.(3.1.20) to give sin2 OW( ) 1 cos2 Aw (MA, j(t) AT (M s=w() = 1 s (W ) (3.1.21) sin2 0W sin2 W M M 2 An explicit expression relating sin2 BW and sin2 0y(My) is given in Ref. 18. Another relation for sin2 OW(t) may be arrived at directly linking it to MZ [19] or MW [20]. In particular, if one chooses MW as the input mass, then one introduces a radiative correction parameter AW such that sin2 Ow(Mz)(1 Ary) = sin2 Ow(1 Ar) (3.1.22) from which it follows that (37.271)2 sin2 Ow(Mz) = .21 (3.1.23) My(1 AiW) Similarly one can introduce a radiative correction Aig if one chooses MZ as the input mass sin2 8w(MZ)cos2 O(MZ)(1 Aiz) = sin2W Bcos2 OW(1 Ar) (3.1.24) A fit to all neutral current data gives sin2 OW(MZ) = 0.2324 0.0011 (3.1.25) for arbitrary Mt [21]. Using these values of a(Mz) and sin2 G8(Mz) yields ai(Mz) = 0.01698 0.00009, (3.1.26) a2(MZ) = 0.03364 0.0002. The value of the strong coupling has not been determined experimentally as well as a1 and a2. The nonperturbative nature of low energy QCD leads to rel atively large uncertainties in its determination. Different processes for the ex traction of a3 yield significantly different results. Unless otherwise stated, the 13 value of a3 used in the runs will be that obtained from a Gaussian weighted av erage of the results of several processes, including e+e scattering into hadrons [22], heavy quarkonium decay [23], deep inelastic scattering '24], and e+e scattering into jets [25]. The value found is [6] a3(Mz) = 0.113 0.004 (3.1.27) 3.2 Yukawas 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 uptype, downtype, and leptonic Yukawa matrices. Machacek and Vaughn's [7] convention are used where the interaction Lagrangian for the Yukawa sector is C = i~YutQ + dYdtQ + eYeDtL + h.c. (3.2.1) The Yukawa couplings are given in terms of 3 x 3 complex matrices. After elec troweak symmetry breaking, these translate into the quark and lepton masses V2 me 0 0) Ye = 0 my 0 , v 0 mr V2 mdd 0 0 Yd 0 m 0 (3.2.2) v 0 0 mb 5 m 0 0 V, Yu 0 me 0 V , v 0 0 mt 14 where V is the CKM matrix which appears in the charged current j ~ ,U Ly VdL (3.2.3) It is a unitary 3 x 3 matrix often parametrized as follows ( cl 31c3 1i33 V = lC2 clC2C3 s2s3ei cC2S3 + s2c3e6 (3.2.4) S1S2 clS2c3 + c2s3e6 clS2S3 c2c3ei where si = sin0i and ci = cos i, i = 1,2,3. The entries of the parametrized CKM matrix can be related simply to the experimentally known CKM entries. The particle data book [21] gives the following ranges of values (assuming unitarity) for the magnitudes of the elements of the CKM matrix / 0.97470.9759 0.2180.224 0.0010.007 IVI= 0.2180.224 0.97340.9752 0.0300.058 (3.2.5) \0.0030.019 0.0290.058 0.99830.9996 These ranges of values can be converted to bounds for si, i = 1, 2, 3, and sin 6. These bounds are arrived at by finding values for the four angles such that the entries of the CKM matrix obtained from these satisfy the conditions imposed by Eq.(3.2.5). One finds 0.2188 < sin 0 < 0.2235 , 0.0216 < sin 2 < 0.0543 (3.2.6) 0.0045 < sin 03 < 0.0290 . However, the accuracy with which IVI is known does not constrain sin 6. A set of angles {01,02, 3,6} was chosen that falls within the ranges quoted above. The initial data needed to run the Yukawa elements are 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 2.5), in that it is not clear at what scale the chosen initial values for these angles should 15 be considered known. However, it is 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 (or lack thereof) is not as critical as might be feared a prior. 3.3 Quark and Lepton Masses There are large theoretical uncertainties in the extraction of the masses of the three lightest quarks from experiment. They are determined by chiral perturbation techniques and QCD spectral sum rules [26,27]. In the following, the MS running masses at 1 GeV of the three lightest quark will be taken to be [6] mu(1 GeV) = 5.2 0.5 MeV , md(l GeV) = 9.2 0.5 MeV (3.3.1) ms(1 GeV) = 194 4 MeV , For the charm and bottom, the nonrelativistic bound state approximation may be applied. One speaks of physical masses and associates these with the pole of the quark propagator. A weighted average, based on results from J/1 and T sum rules [28] and from heavylight, B and B*, D and D* meson masses and semileptonic B and D decays [29], yields the following physical and running masses at 1 GeV for the charm and bottom [6] Me = 1.60 0.05 GeV, mc(1 GeV) = 1.41 0.06 GeV , (3.3.2) Mb = 4.89 0.04 GeV, mb(l GeV) = 6.33 0.06 GeV , Although the associated errors on these averages are restrictive, there is larger uncertainty in the actual central values. For example, in the bottom quark, values as small as mb(l GeV) = 5.7 GeV and as large as mb(l GeV) = 6.5 16 GeV are acceptable. In the case of the lighter quarks, their mass ratios are known more accurately than the actual value of their respective masses. The ratios md/mu = 1.8 and ms/md = 21 are widely accepted. The physical (pole) masses of the leptons are very well known [21] Me = 0.51099906 0.00000015 MeV, My = 105.658387 0.000034 MeV (3.3.3) M = 1.7841+00027 GeV . M 0.0036 These masses represent an example of a physical quantity as discussed in Chap ter 2. Indeed expressed in terms of the renormalized parameters (i.e., masses and couplings) of the theory, the physical mass is just the simple pole of the rel evant field's propagator. Suppose S(f; g, m, p) is the renormalized propagator of some fermion with a simple pole at #j = M(g, m, pi) such that z(2/;g,m,p) S(O; g, m, P) = ,(g,m, ) (3.3.4) # M(g, m, P) with finite residue, z(f/ = M; g, m, p). The relevant renormalization group equation is G( + 0 + mym + 27)S(Y; g, m, p) = 0 (3.3.5) where 7 is the anomalous dimension and 7m is the mass anomalous dimension. Inserting Eq. (3.3.4) in Eq.(3.3.5), then multiplying the resulting expression by (/ M)2, and lastly setting M = M gives the renormalization group equation for M d 9 a a M(g, m, I) = ( + + mym )M(g, m, ) = 0 (3.3.6) dy apy og am These values are used to determine initial data for the running masses. Some authors neglect QED corrections and use the physical values for the running 17 values at ~ MiZ, which introduces only a small error. By calculating the one loop selfenergy corrections, one arrives at a QED relation between the running MS masses and the corresponding physical masses mt(p) = M[1 3(In)1n2 + 4 (3.3.7) 47r ml 3 Choosing p = 1 GeV as in the quark mass case and using Eqs. (3.3.7) and (3.1.4) yields the running lepton masses (taking mi = MI in the log term above is an appropriate approximation to order a) me(1 GeV) = 0.4960 MeV , mp(l GeV) = 104.57 MeV (3.3.8) mr(1 GeV) = 1.7835 GeV . 3.4 Top and Higgs Masses The Higgs boson and top quark masses have not been measured directly at present; however, their values affect radiative corrections such as Ar. Consis tency with experimental data on sin2 OW requires Mt < 197 GeV for MH = 1 TeV at 99% CL assuming no physics beyond the Standard Model [30]. Pre cision 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 [31]. Recent direct search results set the experimental lower bound Mt Z 91 GeV. As for the Higgs, the analysis of Ref. 31 gives the restrictive bound, MH $i 600 GeV, if Mt < 120 GeV, and MH < 6 TeV, for all allowed Mh. Since perturbation theory breaks down for. MH ; 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 [32]. 18 In the present analysis, initial values of the MS running top quark mass mt and of the scalar quartic selfcoupling A at MZ are chosen arbitrarily (con sistent with the bounds quoted above). As noted earlier in Chapter 2, 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 this analysis should be that of experimentally relevant, physical masses. Therefore, formulas similar to Eq. (3.3.7) relating MS running parameters to physical masses are needed. To calculate the physical or pole mass of the top quark, the following equation is used [33] 1 4 as(Mt) MA as(Mt)2 (3.4.1) Mt = 1+4 (M [16.11 1.04 (1 ()2 (3.4.1) mt(Mt) 3 r i=1 where Mi, i = 1,..., 5, represent the masses of the five lighter quarks. Like wise the physical mass of the Higgs boson can be extracted from the following relation [34] A(Mp) = G MH(1 + 6)) (3.4.2) where S(/p) contains the radiative corrections. Its form is rather elaborate and it is relegated to Appendix B. Equations (3.4.1) and (3.4.2) are highly nonlin ear functions of Mt and MH, respectively. Their solution requires numerical routines that are described in Appendix C. At the one loop level, the gauge couplings are unaffected by the other couplings in the theory. On the other hand, the Yukawa couplings are affected at one loop by both the gauge and Yukawa couplings. Since the top Yukawa coupling is at least as big as the gauge couplings at low energy, that means the running of the Yukawas is sensitive to mostly the top Yukawa and the QCD gauge couplings. Thus one expects the mass and mixing relations just described to be sensitive to the value of the top quark mass. The Higgs quartic 19 selfcoupling enters in the running of the other couplings only at the two loop level, so that its effect on the quark and lepton parameters is small. However, its own running is very sensitive to the top quark mass; it can become negative as easily as it can blow up, corresponding to vacuum instability or to strong self interaction of the Higgs (triviality bound), respectively [35]. The discovery of the Higgs with mass outside these bounds would be a signal for physics beyond the Standard Model. The graphs in Figs. 14 summarize these bounds for representative values of the top quark mass. For example, if Mt = 150 GeV, one can see from the corresponding plot that a Higgs mass between 95 and 150 GeV need not imply any new physics up to Planck scale. However, if the Higgs were observed outside of this range, then some new physics must appear at the scale indicated by the curve, either because of vacuum instability if MH < 95 GeV or because the Higgs interaction becomes too strong if MH > 150 GeV. It is amusing to note that it is for comparable values of the top and Higgs masses that these bounds are least restrictive, but it is important to emphasize that a high value of the top with a relatively low value of the Higgs necessarily indicates the presence of new physics within reach of the SSC. 3.5 Vacuum Expectation Value The vacuum expectation value (vev) of the scalar field may be extracted from the well known lowest order relation v = ( G)5 = 246.22 GeV (3.5.1) From the very well measured value of the muon lifetime, r = 2.197035 .000040 x 106 s [21], the Fermi constant can be extracted using the following formula [36] 1 G 3 m2 a(m) 25 2 123 )( + 3mW12 2 + a~(m) 7r), (3.5.2) 1927r3 M 5 m 2x 4 J" W where f(x) = 1 8x + 8x3 4 12x21nx, (3.5.3) giving Gp = 1.16637 0.00002 x 105 GeV2 (3.5.4) This parameter may be viewed as the coefficient of the effective fourfermion operator for muon decay in an effective low energy theory 2[e (1 75)e] [W (1 75)v] (3.5.5) A direct calculation (e.g., in the Landau gauge) of the electromagnetic correc tions yields that the operator is finitely renormalized (i.e., G1 does not run) [16,37]. Another way to see this is by using a Fierz transformation to rewrite the above expression S[ve77 (1 t5)v][YY(1 75)e] (3.5.6) The neutrino current does not couple to the photon field, and the e p current is conserved and is hence not multiplicatively renormalized. An initial value is needed for the running vacuum expectation value at some scale p. Wheater and Llewellyn Smith [38] 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 one can extract a value for v(MZ). However, the formula is derived in the 't HooftFeynman gauge, and the evolution equation, Eq. (1.18) of Appendix A for the vev, is valid only in the Landau gauge (see Appendix D). Nevertheless, motivated by the discussion of the previous paragraph, the initial condition for the vev is chosen to be v(MW) = 246.22 GeV. Using the initialization algorithm (see Appendix C), one arrives at v(Mz). It is found that this procedure leads 21 to no significant correction, and one therefore takes, ab initio, v(Mz) = 246.22 GeV. 3.6 Thresholds In mass independent renormalization schemes, the running couplings are unphysical. From the decoupling theorem [39] one expects the physics at en ergies below a given mass scale to be independent of the particles with masses higher than this threshold. Therefore, for a correct interpretation of these run ning couplings, one must take into account the thresholds [40,41,42]. For the electroweak threshold, one loop matching functions [42] are used with the two loop beta functions valid in the Standard Model regime below the SUSY scale. These matching functions are obtained in MS renormalization by integrating out the heavy gauge fields in such a way that the remaining effective action is invariant under the residual gauge group [41]. At the electroweak threshold, near My, the heavy gauge fields and the top quark are integrated out. 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 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 [43] as the renormalization group scale passes each physical quark mass. In the SM runs (see next section), in which an analysis of the relative importance of including a proper treatment of thresholds effects instead of using a simple step function technique, it is found that the former method does not improve significantly over the latter [6]. 22 3.7 Analysis and Results The results of numerically integrating the 0 functions of the Standard Model parameters from 1 GeV to Planck mass are depicted in the following figures. For most of these plots, the arbitrary choice, Mt = MH = 100 GeV, is made. Figure 5 displays the evolution of the inverse of each of the three gauge couplings including the associated uncertainties in their values. In it, one sees 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. Here, the differences between one and twoloop evolution appear in the high energy regime. Differences are also present for the strong coupling at low en ergies where it becomes large. Note that the uncertainties do not fill in the "GUT triangle." Figures 6, 7, and 8 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. The largest differ ences between oneloop and twoloop evolution occur in the bottom, charm, and strange quark masses in these cases. In Fig. 9, the quartic selfcoupling A and the top Yukawa coupling yt for (Mt = 100 GeV, MH = 100 GeV) and for (Mr = 200 GeV, MH = 195 GeV) are plotted. These two couplings are the only unknown parameters of the Standard Model. The effects of changing the values of Mt and MH in the analyses of the running of the other parameters have been studied. It is 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 heav ier quarks. In particular, Fig. 10 shows that the intersection point between the bottom quark and the r lepton moves down to a lower scale for higher 23 top quark masses. 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 / function is such that this crossing point is shifted toward a higher scale with an increased top mass. The relation mb = mr (I) is the most natural one in the SU(5) theory [44], and it could be expected to be valid at scales where the Standard Model gauge couplings are the closest to one another. Its validity is examined for three different physical values of the top and Higgs masses in the Standard Model. The noteworthy feature of Fig. 10 is that this simplest of the SU(5) relations is valid at an energy scale many orders of magnitude removed from that at which the gauge couplings tend to converge. This re sult is vastly different from that of the original investigations [45]. This work improves upon that work by including two loop effects in the running of the quark Yukawas, by taking into account the full Yukawa sector, and most im portantly by incorporating QCD corrections in the extraction of the bottom quark mass. Other mass relations were studied in the SM context [46]: (II) md = 3me, 3ms = mp [47], (III) tan Cabbibo = (md/ms)1/2 [48], and (IV) Vcb = (mc/mt)1/2 [49]. There is no scale at which all of these can be satisfied. The scale at which relation (I) tends to be satisfied does not coincide with that at which the others are valid. Still the disagreement is never too large. In an SU(5) SUSY GUT model, the equality of the bottom and r Yukawas at the scale of unification will be used to get bounds on the top and Higgs masses [3]. Lastly, the running of the CKM angles is displayed in Fig. 11. The initial data used are sin01 = 0.2206, sin 2 = 0.0298, and sin 3 = 0.0106. Also 6 has been taken to be 900, which corresponds to the case of maximal CP violation. As 24 mentioned in Section 2.4, the evolution curves for these angles are effectively flat. In the present case of the Standard Model, it is found that twoloop running of the parameters does at times improve on the oneloop running. Indeed, the differences of several parameters in their one versus twoloop values at various scales have been tabulated, for the cases (Mt = 100 GeV, MH = 100 GeV) and (Mt = 200 GeV, MH = 195 GeV). Table 1 illustrates the difference between oneloop and twoloop running in the ratio mb/mr, for the three scales 102 GeV, 104 GeV, and 1016 GeV. 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%. Note that the ratio becomes equal to one well below the scale of grand unification as noted above in the discussion of Fig. 10. Table 2 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%. Finally, Table 3 displays the same analysis for as for the case Mt = MH = 100 GeV. No appreciable deviation from the tabulated values is observed for any Mt ~ 200 GeV (except in the low energy regime where the difference is at most ~ 4%). At scales MZ, the inclusion of two loops is important in the evolution of the strong coupling (and of the quark masses). Indeed, it is found that the pure QCD threeloop 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 twoloop over oneloop evolution at scales above MZ, the first table does show a 10% difference at the scale 1016 GeV. 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 twoloop effects. Table 1: mb/mr Mt = 100 GeV Mt = 200 GeV 102 GeV 104 GeV 1016 GeV 102 GeV 104 GeV 1016 GeV One loop 1.879 1.455 0.8081 1.868 1.392 0.6647 Two loop 1.782 1.348 0.7336 1.769 1.285 0.6047 Table 2: yt Mt = 100 GeV Mt = 200 GeV 102 GeV 104 GeV 1016 GeV 102 GeV 104 GeV 1016 GeV One loop 0.5405 0.4160 0.1928 1.133 0.9780 0.7145 Two loop 0.5405 0.4071 0.1842 1.143 0.9700 0.6816 Table 3: as 1 GeV 102 GeV 104 GeV 1016 GeV One loop 0.3128 0.1118 0.07103 0.02229 Two and three loop 0.3788 0.1117 0.07039 0.02208 STANDARD MODEL; Mt=100 GeV 20 15 10 5 Q > 0 50 100 150 200 MH (GeV) Figure 1. Vacuum stability and triviality bounds on the Higgs mass for Mt = 100 GeV giving scales of expected new physics beyond the Standard Model. STANDARD MODEL; Mt=125 GeV 50 100 150 200 MH (GeV) Figure 2. Vacuum stability and triviality bounds on the Higgs mass for Mt = 125 GeV giving scales of expected new physics beyond the Standard Model. to 0 P 0 'i4 0 STANDARD MODEL; Mt=150 GeV 20 0 O4 =t &0 0D 50 100 150 200 MH (GeV) Figure 3. Vacuum stability and triviality bounds on the Higgs mass for Mt = 150 GeV giving scales of expected new physics beyond the Standard Model. STANDARD MODEL; M,=200 GeV 20 15 10 5 50 100 150 200 250 MH (GeV) Figure 4. Vacuum stability and triviality bounds on the Higgs mass for Mt = 200 GeV giving scales of expected new physics beyond the Standard Model. (D 0 r*i hi) 4 60 50 a  40 I( 30 1 20 20 10 0 1 0 5 10 15 20 log 0(o(/1GeV) Figure 5. Running of the inverse gauge couplings using their prop agated experimental errors for the twoloop case only. 0.020 i i i 0.015 Mt=100 GeV MH= 100 GeV 0.010 md 0.005 0.000  me 0.005 I I I I 0 5 10 15 20 logo(p//l1GeV) Figure 6. Light quark and lepton masses for Mt = 100 GeV and MH = 100 GeV. 0.25 i i i' i 0.20 Mt= 100 GeV MH=100 GeV Q 0.15  0.10 Ins 0.05 0.00 I I I I 0 5 10 15 20 logo10 (/1GeV) Figure 7. Intermediate quark and lepton masses for Mt = 100 GeV and MH = 100 GeV. 6 M= 100 GeV MH=100 GeV mb  0 m 0 5 10 15 20 logio([/lGeV) Figure 8. Heavy quark and lepton masses for Mt = 100 GeV and MH = 100 GeV. .2 i l l i l l I I I I I I I I I I : yI Mt=100 GeV, MZ=100 GeV 1.0 Mt=200 GeV, Mg=195 GeV 0.8 0.6 Yt\ 0.4  0.2  0.0 I I 0 5 10 15 20 log0 o(b//1GeV) Figure 9. Top Yukawa and scalar quartic couplings. STANDARD MODEL 1.75 1.50 1.25 1.00 0.75 0.50 Figure 10. Plot of mb/mr as a function of scale in the Standard Model for various top and Higgs masses. H \ / I I / 'T  Mt=100 GeV, M= 0) GeV Mt=150 GeV, MH=150 GeV    Mt=190 GeV, MH=180 GeV  .\  ........ 0 5 10 15 20 log10 (,/1 GeV) 0.3 S sin01 "' 0.2 SMt=100 GeV . o.1 MH=100 GeV  SlnUp o sin__ S 0.0 sin  0.1 0 5 10 15 20 loglo(/,/1 GeV) Figure 11. CKM mixing angles for Mt = 100 GeV and MH = 100 GeV. CHAPTER 4 THE MSSM 4.1 The Supersymmetric Standard Model In the minimal supersymmetric extension of the standard model (MSSM), every particle has a supersymmetric partner, their spins differing by a half [50]. Also required is a second Higgs field with opposite hypercharge to the first as the superpotential cannot contain both a field and its complex conjugate. The second Higgs is also needed for anomaly cancellation and to give this sector a mass. For renormalizable theories, the superpotential can have at most degree three interactions. The superpotential for the MSSM is (suppressing the SU(2) and Weyl metrics) W = UYuDuQ + dYd dQ + Ye dL + APud + h.c., (4.1.1) where the hat indicates a chiral superfield and the overline denotes a left handed CP conjugate of a righthanded field, 7 = ia2,*. The usual Yukawa interactions are accompanied by new Yukawa interactions among the scalar quarks and leptons and the Higgsinos in the supersymmetric Lagrangian. There are also new gauge Yukawa interactions involving the gauginos. The new purely scalar interactions form the scalar potential which is positive definite in super symmetric theories. The scalar potential will be discussed in a subsequent section. A remarkable aspect of supersymmetry is that all these new interactions require no new couplings. The p term is only present to avoid a PecceiQuinn 37 38 (PQ) symmetry. Omitting it would lead to exact PQ symmetry and to a visible axion which is experimentally ruled out. An alternative way to break the PQ symmetry is to omit the f term and add an explicit soft symmetry breaking term, m3 uDd. The f can be interpreted dynamically as essentially the vacuum expectation value of a singlet chiral superfield, N, through the following additional interactions ANu d + KNN (4.1.2) This approach also provides a natural explanation for p ~ O(Mw). The cubic term now explicitly breaks the PQ symmetry. Table 4 displays the SU(3) x SU(2) x U(1) quantum numbers of the chiral (all lefthanded) and vector superfields of the MSSM. 4.2 Procedure Bounds are presented for the mass of the top quark in a minimal supersym metric extension of the Standard Model (MSSM) with minimal Higgs structure in the context of a grand unified theory (GUT) by numerically evolving the couplings using their renormalization group equations. This analysis improves on previous endeavors by taking full account of the Yukawa sector. In the expectation that the Standard Model is only the low energy mani festation of some yet unknown GUT or of a possible supersymmetric (SUSY) extension thereof, the three couplings g3, g2, and gl corresponding to the Stan dard 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 unification is observed in the SUSYGUT case but not in the pure GUT case, as noted by several groups [1,51] (see Fig. 12). However, this should not be viewed as proof of supersymmetry since given the values of 39 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 a solution. The exciting aspect of the analysis of Ref. 1 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 [52]. Furthermore, in the context of a minimal GUT [44] there are constraints on the Yukawa couplings at the scale of unification. One first restrict oneself to an SU(5) SUSYGUT [53] 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, Mt. This can be seen in the down type Yukawa renormalization group equation (above MSUSY, for example), from which the evolution of yb is extracted, since the top contribution is large and appears already at one loop through the uptype Yukawa dependence dYd 1 dt 162Yd[ 3YdtYd + YuYu + Tr{3YdtYd + YtYe) 7 2 16 (4.2.1) (1gl + 392 +93) * where Yu,d,e are the matrices of Yukawa couplings. Demanding that their crossing point be within the unification region determined by the gauge cou plings allows one to constrain Mt. This yields an upper and lower bound for fI( which nevertheless is fairly restrictive. There is a threshold at MSUSyy. Here the matching condition is the naive one of simple continuity due to the lack of knowledge about the superparticle spectrum. The scale is taken to be variable to account for this ignorance. 4.3 One Light Higgs Limit The simplest implementation of supersymmetry is considered and the cou plings are run above AMUSY to one loop. The MSSM is assumed above 40 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" #(SM) = 'dcos3 + iusin/ (4.3.1) where $ = irT2*, and where tan3 is also the ratio of the two vacuum ex pectation values (vu/vd) in the limit under consideration. This sets boundary conditions on the Yukawa couplings at MSUSY. Furthermore, in this approx imation the quartic selfcoupling of the surviving Higgs at the SUSY scale is given by A(MSy) = g 2cos2(2) (4.3.2) 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 one takes into account radiative corrections to the light Higgs mass [54] and hence relax the tree level upper bound, MH MZ [55]. 4.4 Initial Data The bounds on Mt and MH are determined by probing these masses de pendence on 0. In SUSYSU(5), tan f is constrained to be larger than one in the one light Higgs limit. It seems natural to require that yj > yb up to the unification scale [56], thereby yielding an upper bound on tan #. The initial values at MZ for the gauge couplings are taken to be [1,57] al = 0.016887 0.000040, a2 = 0.03322 0.00025 (4.4.1) 3 = 0.109+04 , a3 0.005 ' 41 where GUT normalization for al is used. The following set of four quark running masses defined at 1 GeV by the Particle Data book [21] are used mu = 5.6 MeV, md = 9.9 MeV, ms = 199 MeV, and me = 1.35 GeV. For the bottom mass, the Gasser and Leutwyler bottom mass value of 5.3 GeV at 1 GeV is used which translates into a physical mass of Mb = 4.6 GeV [26]. To probe the dependence of the results on Mb, the case Mb = 5 GeV is also studied, this is the typical value obtained from potential model fits for bottom quark bound states [58]. The effect of varying MSUSY is also investigated. Given the values of the gauge couplings, unification holds for SUSY scales up to 8.9 TeV and as low as MW. For empirical reasons solutions below MW were not investigated. 4.5 Analysis and Results The inclusion of supersymmetry collapses the GUT triangle. This is il lustrated in Fig. 12 taking MASUSY = 1 TeV. As mentioned above, a range MyW MSUSY 5 9 TeV will achieve unification within one sigma error. From Fig. 13 (the magnified unification region of Fig. 12), one determines that the lower end scale, MGIjUT, of the unification region corresponds to an a3 value of 0.104 at MZ, while the higher end scale, MHGT, corresponds to a value of 0.108 at MZ for a3. It is found that the unification region is insensitive to the range of top, bottom, and Higgs masses considered. In the analysis of the bounds for Mt, the values for a1 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 M UT and taking 03 = 0.104 then sets a lower bound on Mt. Correspondingly, demanding that Yb and yr cross 42 at MgUT and taking a3 = 0.108 yields an upper bound on Mt. These bounds are found for each possible value of /. Figure 14 shows the upper and lower bound curves for both Mt and MH as a function of f and for MgSSY = 1 TeV and Mb = 4.6 GeV. When applicable the current experimental limit of 38 GeV on the light supersymmetric neutral Higgs mass [59] is used to determine the lowest possible Mt value consistent with the model. It is found that 139 < Mt < 194 GeV and 44 < MH < 120 GeV. The sensitivity of these results on MSUSY is investigated in the range, 1.0 0.5 TeV. It is found 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. For Mb = 5.0 GeV, an overall decrease in the top and Higgs mass bounds is observed 116 < Mt < 181 GeV, MH < 111 GeV. Varying MSUSY as above 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. The results of the analysis are displayed for the extreme case, MSUSy = 8.9 TeV, in Fig. 15, with Mb = 4.6 GeV. This only significantly changes the upper bound on MH to 144 GeV compared to the MSUSY = 1 TeV case. yt has also been run up to the unification region and compared with yb and yr to see what the angle must be for these three couplings to meet [60], as in an SO(10) or E6 model [61,62] with a minimal Higgs structure. It is clear that this angle is precisely the upper bound on / as described earlier. In Fig. 16 yt/yb is displayed at the GUT scale as a function of tan 3 for MsuSY = 1 TeV and for the two bottom masses considered. If one demands that the ratio be one, one can determine the mixing angles for the low and high ends of the unification region. Then going back to Fig. 14. one finds as expected a much 43 tighter bound on the masses of the top and of the Higgs. Indeed, for Mb = 4.6 GeV, one has 49.40 < tan3 < 54.98, which yields 162 < Mt < 176 GeV and 106 < MH < 111 GeV. When Mb = 5.0 GeV, one obtains 31.23 < tan/3 < 41.18, which gives 116 < Mt < 147 GeV and 93 < MH < 101 GeV. The four mass relations of Section 3.7 were also studied in the context of the MSSM [46]. In this case, these relations can all be satisfied at the scale of gauge coupling unification. However, for this to be true, several things must happen: first Vcb must be larger than its presently measured central value of 0.043; second the top quark mass must be around 190 GeV (if it is lighter, then agreement dictates that V4b should be larger still); third the Higgs boson mass should be around 120 GeV. An analysis which recently appeared in the literature has reached similar conclusions [63]. 4.6 Comments To improve on this analysis, one should implement the supersymmetric two loop beta functions and the corresponding thresholds. The effects of soft SUSY breaking terms should be investigated. Also, all the supersymmetric particles have been integrated out at the same scale. It would be interesting to study the effect of lifting this restriction. It should be noted that the bounds on the top mass are very similar to those of Ref. 56, although the physics is very different. These issues will be addressed in a subsequent chapter. However, given the relative crudeness of the approximations made here, it is remarkable that the experimental bounds on the pparameter were satisfied. A sign which gives credence to the program. 60 50 40 30 20 10 0 0 5 10 15 20 logo 10(//1GeV) Figure 12. Plot of the running of the inverse couplings. The dot ted lines above and below the solid lines represent the experimental error for each coupling. Table 4: Particle Quantum Numbers in the MSSM Q u d L T u 'i dA ra fB U(1) + 1 +_ + +1 + o o o SU(2) 2 1 1 2 1 2 2 1 3 1 SU(3) 3 3 3 1 1 1 1 8 1 1 ( I 00 8 *^ 1 I CM Q * 11 I l a 2 8 I li I 1  1 1 I I Ii i I 27 .... ..... d. .. ............** . ....................... . 26 25 15.6 15.8 16 16.2 16.4 logio(Ol/1GeV) Figure 13. The plot depicts a blowup of the area around the unifi cation point. (Note the small region where all three cou plings intersect. This region reduced to a point when MSUSY = 8.9 TeV and was nonexistent above that scale.) MsuY = 1 TeV, Mb = 4.6 GeV 200 150 100 50 0 40 50 60 70 80 90 13 (deg.) Figure 14. Plot of the top quark mass, Mt, and of the Higgs mass, MH, as a function of the mixing angle # for the highest value of a3 (high curves) and the lowest value of a3 (low curves) consistent with unification as per Fig. 12. 0 0 TeV, Mb = 4.6 GeV 50 60 70 80 P (deg.) 90 Figure 15. Same as Fig. 14 for GeV. MSUSY = 8.9 TeV and Mb = 4.6 MsusY = 8.9 200 150 100 50 > Q) 0 rn a) ao 0i 0 1 TeV 0 10 20 30 40 50 60 tanfl 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 a3 (low curves) consis tent with unification as per Fig. 12. 0 E,0 0 101 Figure 16. 102 CHAPTER 5 SOFT SYMMETRY BREAKING 5.1 Minimal Low Energy Supergravity Model Since no super particles have been observed experimentally, supersymme try, if truly present in nature, must be broken. One way to accomplish this breaking is to couple the standard model to N = 1 supergravity (SUGRA). In the minimal low energy supergravity model considered, supersymmetry is explicitly broken by the addition of supergravity induced soft terms (including gaugino mass terms) 2 (bt ( 2 + m l 1t + B A( ,u~ d h c . Vsoft = "M.uu + md d + d + h.c.) + 2 t t 2 t t 2 t +e 2m t + mQi iQi mL ii +mui uii +md idi + meii ) + Z( Aiy iiuQj + AYJYd'di;4dQj + AYeiei'tdLj + h.c.) , ij 1 3 Vgaugino = M1AAl + h.c. , 1=1 (5.1.1) where Vgaugino is the Majorana mass terms for the gaugino fields, Al (suppress ing the group index), corresponding to U(1), SU(2), and SU(3), respectively. From the supersymmetry algebra, one deduces that spontaneous symmetry breaking occurs if and only if the vacuum energy is not zero. In global su persymmetric theories, the scalar potential is a sum of F and D terms. Su persymmetry is spontaneously broken if either the vacuum values of the F term ( [64]) or D term ( [65]) are nonzero. A consequence of the spontaneous 49 50 symmetry breaking is a massless fermion in analogy with the breaking of an ordinary global symmetry. We will assume that the spontaneous breaking of the local N=1 supersymmetry is communicated to the "visible" sector by weak gravitational interactions from some "hidden" sector. Spontaneous symmetry breaking in supergravity occurs via the superHiggs mechanism. The goldstone fermion, or goldstino, associated with the breaking of global supersymmetry is eaten by the gravitino thereby providing it with a mass. This spontaneous symmetry breaking of supergravity manifests itself at low energy as explicit soft breaking terms of global supersymmetry. This leads to a common (grav itino) mass, m0, for the scalars of the model and masses, MI, for the gauginos at the GUT scale, MX. By assuming gauge coupling unification, we can take the three gaugino masses equal. Furthermore, the trilinear soft couplings A?, A and A" are all equal to a common value A0. The bilinear soft coupling BO may be related to A0 (BO = A0 mo), if the SUGRA model has only canonical kinetic terms for the chiral superfields. This scenario is to be contrasted with one in which general soft breaking terms are added ad hoc to the Lagrangian. In the most general case, there are sixtythree soft symmetry breaking param eters. Their number can be reduced by invoking certain symmetries such as flavor and family blindness. In either case, all of these couplings will evolve to different values under the renormalization group. The complete scalar poten tial appears as V = VF + VD + Vsoft (5.1.2) where VF contains the potential contributions from the Fterms VF = ISYUQ + I'd + IdYdQ + ,eL + ,,2 + IYQiul2 + IYd Qd + YeLd l2 (5.1.3) + iYuu + dYdd2 + I Ye d2, 51 and VD contains the potential contributions from the Dterms 9 1 tr 1 VD = ( d Q d+ 4 26 3 3 2 2 U 2 2)2 + ( t + Lt ++ + F+ d + ( QtQ *tu ~xd )2 (5.1.4) where ? = (ri,r2,r3) are the SU(2) Pauli matrices and A = (A ,..., A) are the GellMann matrices. In general, one must impose constraints on the pa rameters to avoid charge and color breaking minima in the scalar potential. Some necessary constraints have been formulated, such as A2 < 3(m +m +m2 ) , A < 3(m + m + m ), (5.1.5) A2 < 3(m + m + md). However, these are in general neither sufficient nor indeed always necessary [66]. They involve very specific assumptions about the spontaneous symmetry breaking. 5.2 Radiative ElectroWeak Breaking An appealing feature of the models being considering is that they can lead to the breaking of the electroweak symmetry radiatively. The one loop potential responsible for the breaking is Vltoop(Y) = Vtree(p) + Vi(p) (5.2.1) where 1 m 3 S64r2 (_)2S(2sp + 1)m4(ln 2 P 52 where M2 is the field dependent squared mass matrix of the model and mp is the eigenvalue mass of the pth particle of spin sp. The tree level potential is Vtree(L) = m2 d d + m2 u + m3( u0d + h.c. ) '2 g2 (5.2.3) + ( tL ~d )+ 2 + ( + d )2 where m1 = md + It m2 = m2 + 2 (5.2.4) m2 = Bf . Minimization gives 1 2 m2 _ 2 tan2 p m = m (5.2.5) 2Z tan2 # 1 where m2 = (g2 + g2)v2/2, v2 = u + v2 and Bf = 2 M) sin 23 (5.2.6) where tan/3 = vu/vd, also 2 2 AV1 m = m + (5.2.7) where v = vd, v2 = vu and 9AV 1 Z( )a2 m2 am2 A = (1)22s (2sp + 1)m2(ln_ 1) (5.2.8) av2 327r2 P 2 1v,528 p The parameters of the potential are taken as running ones, that is, they vary with scale as dictated by the renormalization group. Because the one loop correction to the potential is not negligible at all scales, an appropriate scale must be chosen if the tree level formulas are to be valid [67]. Indeed, the tree level analysis may lead to incorrect conclusions about the regions of parameter space that yield electroweak breaking and consistent scenarios. When AV1 is included, the value of I is not critical as long as p is in the neighborhood of ArI . 53 Although, as stated, the tree level results cannot always be trusted, one can get some idea under what conditions electroweak breaking occurs. The renormalization group evolution of mn (see Appendix C) can be such that it turns negative at low energies if the top quark mass is large enough, whereas 2d runs positive. From Eq. (5.2.3), the scale at which this occurs is set by the condition m (b) m2(pb) m (pb) = 0. (5.2.9) If the free parameters are adjusted properly, then the correct value of the ZO mass (MZ = 91.17 GeV) can be achieved. At tree level there is another critical scale that must be considered. Again from Eq. (5.2.3), it is evident that the potential becomes unbounded from below along the equal field (neutral components) direction, if m 2(s) + mi(ps) < 2m (I) (5.2.10) Since mm2 m4 > 0 implies m2 + m2 > 2m, condition ((5.2.10)) can only occur at scales lower than condition ((5.2.9)), so p, < Pb. As mentioned above, when working with the tree level potential an appropriate scale to minimize it must be used at which one loop corrections may be safely neglected. Gamberini et al. [67] give a prescription for the choice of this scale. In the present work, the one loop corrections are incorporated. Contribu tions from the third generation are included, that is, the topstop, bottom sbottom, and taustau. The one loop effective potential should be constant against the renormalization group to this order. The Zo mass is chosen as the scale at which to evaluate the minimization conditions Eqs. (5.2.9)(5.2.10). Equation (5.2.5) can be written m2 m tan2 9 1 i2 (MZ) d . 2 (5.2.11) tan2 3 1 ( 54 where m,d = mudA+a A1/v2ud and used to solve for p(Mz) (see Numerical Procedure Section) given the value of all the relevant parameters at MZ. A choice for the sign of p must be made (A is multiplicatively renormalized; see Appendix C). It is evident that no adequate minimum exists if the parameters are such that p2 < 0. 5.3 Sfermion Masses The mass matrices for scalar matter are constructed from Eq. (5.1.2). For example, in the up squark sector the relevant mass matrix appears as / ML2 M2l m1 MLi Lj La Rj\ M2 L M2 (5.3.1) \ RLj LiRj / where i,j = 1, 2,3 are flavor indices and M2 2 +v2(v2 g)g2 LL,L = mQij + u2(YuYu)ij J )(Y(uL)g'2 T3(uL) ij, M R, = i + v (YuY)ij J(vd vu)(Y(uR)g'2)j , M Lj = AvdY + vuAYu' , M2Ri = M2. Li Rj Rj Li (5.3.2) Note in Table 4 that Y = Q T3 in the adopted notation. Similar matrices follow for the other sfermions. These mass formulas as well as the ones to follow are given in terms of running parameters. The domain of validity of these formulas is at low energies ( MZ) with the parameters taking on their renormalization group evolved values at this scale. 5.4 Higgs Masses If the following notation is employed '1= \2 2= 55 then the physical masses of the Higgs at tree level are calculated from the following three matrices 1 2 Vtree 12 tan M sn( tanM 1 2 =9_9)~(si 2 = 1 cot 3 1 2 tree 1 2 (tan/ 1 2 ( )( ) M sin 2 2 1 cot 0 +m1 M2 cot3 15.4.1) 2 z + 1 tan 2Vtree 1 2 tan f 1 2a(7a(+si) 2 n2 1 cot ) where M 2 = m + m, M2 = M2 + m2 and m2 g2v2/2. The eigenval ues for the first matrix are 0 corresponding to the Goldstone boson and M2 corresponding to the CP odd scalar. The second matrix gives the masses of the light and heavy Higgs bosons MHi,h + 2mA) Z / + )2 4MA2m cos2 20] (5.4.2) This tree level result predicts Mh < MZ. One loop calculations show that this need not be the case [68]. The third matrix has eigenvalues 0 and M2 corresponding to a massless, charged Goldstone boson and a charged scalar. In cluding Eq. (5.2.2) in the calculations leads to corresponding one loop versions of these masses [69,70]. 5.5 Chargino Masses The following four terms contribute to the chargino masses i29g2'u tr i i 2ti I 4d 2+ M2WiTWi + h.c.. (5.5.1) 2 d 2 56 The first two terms are the supersymmetric Yukawagauge terms. Letting A = (Wt2 ilW)/V\2, the mass matrix follows 0 0 M2 92vd A+ A(+ A 0 0 2vu (5.5.2) {A d ) ^ 2 M2 gu 0 0 A 92vd A 0 0 7 d Diagonalization yields two charged Dirac fermions with masses M2Hh 1 2 2+2 2= [(M22 +2 + 2m ) (5.5.3) (5.5.3) y/(M2 + A2 + 2m2 )2 4(M2A m2 sin 20)2] . 5.6 Neutralino Masses Contributing to the neutralino masses are the terms in Eq. (5.5.1) and ig'v/2u(+1 )~u ig'd/20i(2 df + MI BB (5.6.1) 2 2 2 The neutralino mass matrix follows SMI 0 0 _aV. 0 M2 3 2V 0. 1 0 iW3 d2 2 0 0d (5.6.2) 5.7 Boundary Conditions at My In this work, the modified minimal subtraction scheme (MS) of renormal ization is employed. The parameters of the Lagrangian are not in general equal to any corresponding physical constant. For example, in the case of masses, except for those of the bottom and top quark (see Eq. (3.4.1)), all other physi cal masses will be determined from their corresponding running masses by the simpler relation S= m() =(5.7.1) 57 The renormalization group P functions of the gauge and Yukawa couplings have been calculated to two loops without making any approximations in the Yukawa sector of the model. These have been included in Appendix C. Because SUGRA models make certain simplifying predictions about the soft parameters at the unification (Planck) scale, the evolution of the renor malization group equations is initiated at this scale. It has been demonstrated that the introduction of supersymmetry leads to gauge coupling unification at approximately 1016 GeV. Therefore one takes AMX = 1016 GeV. At the unification scale, MX, all scalars have a common mass mQi(MAX) = mui(MY) = md (My) = mL (My) (5.7.2) = me,(MX) = r (M ) = mM (MX) = m0 , as do the gauginos M1(Mx) = M2(Mx) = M3(Mx) = m (5.7.3) 2 The trilinear soft scalar couplings will be taken diagonal and equal at MX Az'(Mx) = A" (Mx) = A?(Mx) = AO (5.7.4) Also the bilinear soft scalar coupling and the mixing mass at MX are given by B(Mx) = BO, p(Mx) = po (5.7.5) Furthermore, to constrain the parameter space, the bottom and tau masses will be assumed equal at My mb(MX) = mr(MX) (5.7.6) Also for the sake of CPU time, we will take the masses of the two lightest families to be zero in all the runs. 58 5.8 Numerical Procedure There are seven free parameters in the model considered. These are A0, B0, mO, ml/2, O0, tan/3, and mt. The two minimization constraints (5.2.5) and (5.2.6) reduce this set to five, which are taken to be A0, mi, m1/2, tan 3, and mt. In the present framework, B0 and 40 will be determined using the numerical solutions routines described in Appendix E in conjunction with the minimization of the one loop effective potential at MZ. Minimization at MZ will give B(MZ) and p(Mz). To arrive at B0 and to (their corresponding values at MX), the solution routine are employed as follows. A guess for B0 and [0 is made at VMX and then the parameters of the model are run to MZ at which scale the evolved value of B is compared to the minimization out put value for B at IZ. The same is done for Ap. If the compared values agree to some set accuracy, then B0 and Po are the required values. Other analyses that also extract B(MZ) and p(Mz) simply evolve these two param eters via their renormalization group equations back to MX to find B0 and Po relying on their near decoupling from the full set of renormalization group equations. Notice from Eq. (5.2.11) that a choice must be made for the sign of p. To constrain the parameter space further, the bottom quark and tau lepton masses will be taken equal at Mx. This equality is a characteristic of many SUSYGUTs. This constrains the model to four free parameters, A0, mo, m1/2, and tan #. Demanding that mb(MX) = mr(MX) and achieving the correct physical masses for the bottom quark and tau lepton fixes the mass of the top quark which affects the evolution of the bottom Yukawa significantly. Gauge coupling unification shall be assumed, an assumption which appears reasonable when one considers SUSY models with SUSY breaking scales 6 10 59 TeV. The solution routines could be used to find the precise (and similar) val ues of al, a2, and a3 at Mx that will evolve to the experimentally known values at MZ, however this increases the CPU time considerably. Therefore some precision shall be sacrificed in their MZ values by taking them exactly equal at MX. This is already a theoretical oversimplification since one does not expect the gauge couplings to be exactly equal due to threshold effects at the GUT scale. It is found that for all cases studied, the common value a 1(Mx) = a2(Mx) = a3(My) = 25.31 leads to errors no bigger than 1%, 5%, and 10% in al(M.z), a2(MZ), and a3(Mz), respectively. This is not so bad considering that the (combined experimental and theoretical) errors on a3(MZ) from some processes can be as large as this. It is well known that there is a fine tuning problem inherent in the ra diatively induced electroweak models. For certain values of the parameters, the top quark mass must be tuned to an "unnaturally" high degree of accu racy to achieve the correct value of MZ. This problem is generally handled by rejecting models that require "too much" tuning. The amount of tuning is usually defined quite arbitrarily. The usual procedure is to define fine tuning parameters x2 8M2 ci = M2 a (5.8.1) Z i where xi are parameters of the theory such as mg, ml/2, p, or mt. One then demands that the ci be less than some chosen value that is typically taken to be 10. The differences in using the tree level vs. one loop effective potential were analyzed to some extent. The results agree generally with those of Ref. 67. Moreover it is found that the fine tuning problem is exacerbated in the tree level analysis. The basis for the "theoretical" fine tuning problem can be seen, 60 if one makes some simplifying assumptions, in the dependence of MW on the top quark Yukawa coupling yt [71] MW ~ MXe1Y (5.8.2) In the tree level analysis, one encounters another fine tuning problem. The vacuum expectation value coming from the minimization conditions changes rapidly from 0 to infinity over the interval (ps, pb). Using the prescription of Ref. 67 for the scale A at which to adequately minimize the tree level potential, find v(A), and thereby arrive at a value for MZ, one finds that although a small variation in yt(MX) may lead to a small variation in 11b, the steepness in the tree level vacuum expectation value will lead to a large variation in the value of v(A) and therefore in MZ. Hence, in the tree level analysis, solutions which may be within the bounds of the "theoretical" fine tuning may nevertheless display a fine tuning aspect because of this "tree level" fine tuning problem. The use of the one loop effective potential levels off the vev around the MZ scale. The vevs depend on scale through wave function renormalization effects which are never large as can be seen from the form of the renormalization group equations for the vevs in Appendix C. In the present framework, solutions shall also be rejected based on fine tuning considerations, however the present method differs somewhat from the usual one in that it is incorporated in the solution routine described above. The routine is an iterative one which determines the convergence properties of the solution which reflect an inherent fine tuning. If the convergence is too slow, the solution will be rejected. Effectively any solution which the computer cannot pinpoint within an allotted number of iterations is rejected. Given values for AO, mn, ml/2, tan and sign(p), the solution routines search for 61 the values of v(Mx), mb,r(MX), mt(Mx), B0, and A0. The process by which B0 and 0 are found was described above. The remaining three parameters are determined similarly. The routine makes a guess for v(Mx), mb,r(MX), and mt(MX), then the full renormalization group equations are evolved to 1 GeV calculating superparticle threshold masses in the process and minimizing the one loop effective potential at MZ. The merits of the guess for v(MX), mb,r(MX), and mt(Mx) is assessed by comparing the resulting values of MZ, mr(1 GeV), and mb(l GeV) with the expected ones. The process is iterated until the correct values are achieved to within some tolerance. 5.9 Thresholds In the minimal low energy supergravity model being considered, the super particle spectrum is no longer degenerate as in the simple global supersymme try model in which all the super particles are given a common mass, MSUSY In the simple case, one makes one course correction in the renormalization group evolution at MSUSY. In the model with soft symmetry breaking, the nondegenerate spectrum should lead to various course corrections at the super particle mass thresholds. To this end, the renormalization group / functions must be cast in a new form which makes the implementation of the thresholds effects (albeit naive) evident (see Appendix B). Since the MS renormalization group equations are mass independent, particle thresholds must be handled us ing the decoupling theorem [39], and each super particle mass has associated with it a boundary between two effective theories. Above a particular mass threshold the associated particle is present in the effective theory, below the threshold the particle is absent. 62 The simplest way to incorporate this is to (naively) treat the thresholds as steps in the particle content of the renormalization group fl functions. This method is not always entirely adequate. For example, in the case of the SU(2) gauge coupling there will be scales in the integration process at which there are effectively a half integer number of doublets using this method. Nevertheless, this method should yield the correct, general behavior of the evolution. It is a simple means of implementing the smearing effects of the nondegenerate super particle spectrum. The determination of the spectrum of masses is done without iteration as is common in other analyses. The method employed in this work deduces the physical masses by solving the equation m(p) = P for each superparticle in the process of evolving from MX to 1 GeV. The usual iterative method requires several runs to find a consistent solution. 5.9 Analysis and Results The tremendous computing task involved in analyzing the full parameter space of the soft symmetry breaking models, using the methods described as designed, would be far too time consuming given the available computing facilities. Therefore, in the following analysis, some simplifications will be made in the procedural method. First, only the heaviest family of quarks and leptons will have nonzero mass. Second, as stated previously, the value of the strong coupling at MZ will be allowed to vary from its central value of .113 by at most 10%. This translates into a similar error in the bottom quark mass. Third, the allotted number of RungeKutta steps, involved in numerically integrating the renormalization group equations, will be cut down to ~ 100. The present analysis will restrict itself to a subclass of soft symmetry break ing models that have two or three soft parameters equal to zero at MX. One 63 such class follows from the noscale model [72] and has AO = m0 = 0. The strict noscale model has A0 = m0 = B0 = 0. Another class coming from string derived models has A0 = B0 = 0. Since Bo is not a free parameter in the procedure adopted in this work (see Section 5.8), B0 = 0 results must be inferred. Because the GUT inspired constraint, mb(AX) = mr(MX), is enforced in this analysis, the results will depend on the mass of the bottom quark. Most results will be reported for the case mb(l GeV) = 6.00 GeV, but lower mass (5.70 GeV) and higher mass (6.33 GeV) cases were also studied. The running value of 6.00 GeV for mb(l GeV) corresponds to a physical bottom mass Mb = 4.85 .15 GeV, with the uncertainty coming from the error in the strong coupling, as discussed above. The results of Chapter 4, indicate that the top quark mass predictions increase with decreasing bottom mass, and the present analysis corroborates this. Given the noscale or stringinspired cases, the phase space is explored by setting AO = m0 = 0 or A0 = 0, respectively, and coarse graining the remaining hyperslice. tan j was most commonly coarse grained as 2, 5, and 10. The mass of the lowest supersymmetric particle (LSP), when it is a neutralino, is observed to be correlated with the value of ml/2. Cosmological considerations indicate that the LSP must be neutral and colorless and have a mass less than ~ 200 GeV. Points in parameter space that lead to LSPs other than neutralinos with masses less than 200 GeV (or sneutrinos) are rejected. Figure 17 shows a slice of the available phase space, in the noscale case, plotted against the top quark mass. The present unofficial bound on the top quark mass is 120 GeV. The figure indicates that the top quark cannot have a mass greater than 132 GeV in this model, if mb(l GeV) = 6.00 GeV 64 (This upper bound is raised to ~ 160 GeV, if mb(l GeV) = 5.70 GeV, and the model is ruled out, if mb(l GeV) = 6.33 GeV). The figure also indicates that 190 GeV r ml/2 < 265 GeV; however, nothing significant can be said about the value of tan 3. The points to the right of the allowed region were found to lead to charged LSPs (iR) and are not displayed. Points to the left of the region do not lead to electroweak breaking. Furthermore, as Figs. 14 and 15 indicated, the top quark mass does not grow monotonically with tan /, rather it reaches a maximum for some high tan # value then decreases. For the choice sign(I) = , the top quark mass upper bound falls below the experimental limit, so this case is ruled out. Analysis of the available data indicates that the strict noscale case lowers the top quark mass upper bound slightly to ~ 128. However, there is an allowed range of 3 9 for tan /. Figure 18 is a plot of the evolution of the lefthanded and righthanded stop, bottom, stau, and gaugino soft mass parameters for the particular strict noscale case ml/2 = 240 GeV, tan # = 8.3, sign(i) = +. The resulting spectrum of super particle masses is presented in Table 5. The LSP is a neutralino in this case and has a mass of 92 GeV. The top quark mass is just above the experimental limit at 126 GeV, and the Higgs boson mass is 77 GeV. The strict noscale case is a special case of the string inspired one, therefore the 128 GeV top quark mass upper bound is not expected to decrease but rather to increase in the "stringy" case. The data indicate that top quark masses as high as 150 GeV are possible. Finally, a representative stringy scenario is presented. Figure 19 displays the variation of Mt and BO with tan 3 for the slice of parameter space with A0 = 0, mo = 100 GeV, and ml/2 = 250 GeV. The scenario inferred from the figure has Mt = 130 and tan / = 7.8. 65 Table 5 A0 0 m0 0 mr/l2 240 pto 171 tan 3 8.3 sign(p) + Mt 126 d 473 506 ii 390 532 e 94 176 iL 180 7 92 Z, TV 153, 147 557 Ho 213, 271 fH* 270 h, H 77, 158 H, A 235, 222 66 Ao=O, mo=0, s(u,)=+ 150 100 50 0 I I I I I I  I I I I I * . , I 100 I I I I I I 200 300 ml/2 (GeV) Figure 17. Slice of parameter space in noscale case displaying Mt against the common gaugino mass ml/2. The points represent radiative electroweak breaking solutions with mb = mr at MX and a neutralino LSP with mass ; 200 GeV. a, 4~J I I I I I I I, 67 600 i i ' 500 400  300 200 0o I 0 5 10 15 20 logoo/ 1 GeV) Figure 18. Evolution of squark (dashes), slepton (dots), and gaug ino (solid) soft mass terms for AO = 0, mo = 0, ml/2 240 GeV, and tan 3 = 8.3. 50 68 Ao=O, mo=100, ml/2=250 ,I I   \ \ I I I I I I I I I I I  tan# Figure 19. Mt (solid) and Bo (dashes) plotted against tan 3 for A0 = 0, mo = 100 GeV, and ml/ = 250 GeV to infer the BO = 0 case. 150 V> 0/ 0 100 50 0 I I L I I I I I I I 1 CHAPTER 6 CONCLUSIONS This work has studied, using one and two loop renormalization group 3 functions, both the standard model and its minimal supersymmetric extension. The parameters of these models, i.e., the gauge couplings, the quark, squark, lepton, and slepton masses, the Yukawa sector mixing angles and phase, the scalar quartic coupling, and soft symmetry breaking parameters, were run over scales ranging from 1 GeV to Planck mass. The aspects of the standard model and its minimal supersymmetric extension were reviewed. In the standard model case, thresholds effects were served well by naive step functions. The more sophisticated implementation using one loop matching functions did not significantly improve on the step function method. The difference between two loops and one loop represented a more significant effect, albeit sometimes small. Plots exhibiting these different features for all the parameters of the Standard Model were included. Gauge coupling unification was shown to fail, and some interesting mass and mixing angle relations were considered, but failed to hold simultaneously at a common scale. In the supersymmetric case, in which gauge coupling unification is achieved, bounds on the top quark and Higgs boson masses were determined using the SU(5) inspired constraint that the bottom and tau masses be equal at the scale of unification. Remarkably, the top quark bounds were consistent with the p parameter bounds. In supersymmetric context, the mass relations could 70 all be accommodated at the scale of unification provided the top quark mass was high ~ 190. Minimal low energy supergravity models were considered. They have the appealing feature that the electroweak symmetry is radiatively broken for cer tain ranges of the soft breaking parameters. The study of specific models, with some soft parameter fixed, resulted in upper bounds for the top quark mass. Noscale models in which only gaugino masses provide global supersymmetry breaking yield top quarks with masses less than ~ 130 GeV. The results are sensitive to the value of the bottom quark mass. Lower bottom quark masses, within the experimental uncertainty, lead to higher top quark upper bounds. In these models, the ratio of vacuum expectation values of the two Higgs fields is expected to be larger than ~ 720. APPENDIX A THE STANDARD MODEL 0 FUNCTIONS In this appendix, the renormalization group P functions of the Standard Model are compiled [6]. These have appeared in one form or another in various sources. Effort has gone into confirming their validity through a comparative analysis of the literature. The main source is Ref. 7. Following their conven tions, S= +Yu6Q + Yd+tQ + YetL + h.c. IA(Itl)2, (A.1) where flavor indices have been suppressed, and where Q and L are the left handed quark and lepton SU(2) doublets, respectively, Q( UL) L (vL (A.2) 1 and i are the Higgs scalar doublet and its SU(2) conjugate S= ( ) i2s. (A.3) UR, dR, and eR are the quark and lepton SU(2) singlets, and Yu,d,e are the matrices of the uptype, downtype, and leptontype Yukawa couplings. The 3 functions for the gauge couplings are dgl 3 2 3 1b bkl 2 dt 167r2 (12)2 k (A.4) S(l Tr{CluYutY + CldYdtYd + CieYeYe} , (1672)2 72 where t = Iny and I = 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 to be and (Cf) = ( 4 1 b= 23ng 1, 22 4 1 b2 3 3 ng 6 , 4 b3 = 11 f ng, 3 /19 1 11 n 3 49 3 14 T 7 ( 17 1 3\ TU 1 2 3 3 1 with =u d e , 3 with ng = . In the Yukawa sector the f functions are (A.5) 0)\ (A.6) (A.7) dYu_,d,e 1 + (1) 1 (2) dt Yu,d,e( 1Gi2 ude (162)2 u,d,e where the oneloop contributions are given by 1) = 3(YuYu YdYd) + Y2(S) (1g + g2 + 8g2) = (Ydd YYu) + Y2(S) ( 1 g2 + 9 + 8) 3Y1 + + 8g+) S+YeYe + Y2(S) (g + with (A.8) (A.9) Y2(S) = Tr{3YutYu + 3YdtYd + YeYe} , (A.10) (bkl) 0 196 00 T 1002) 73 and the twoloop contributions are given by 2232 135 43 9 (2) = 3(YtY)2 YtYYr+r tydtyd + dty(YYd21 + Y2(S)( Ydt Y Yutyu) X4(S) + 3 A2 2A(3YdtY, + YdtYd) 223 2 135 g7_ 9 2 9 2 2)ydtYd + ( 2 + + 16gY ( g + 163)Y 5 29 4 9 2 2 1922 35 4 2 +Y4(S) + (200+ 4ng)g1 Ng2 + 3 ( ng)g2 + 9g2gg 404 80 4 9( 3 9 ng)3 'g (d2) = (Ydtyd)2 YdtYdyutyu 1YutYuYdtYd + 11(YtY)2 SY2(S)( 5YutYu YdtY X4(S) + A2 2A(3YdtYd + YutYu) 4 4 92 1872 135 5 7 1 92 + 279 + ( 9g + 9 22 + 16g3)YetY4 (_1 2gg + 16g3Y Y 5 29 1 4 2722 3122 35 _ S2 0 + n) 2 + 4 ) 22 404 80 4 (2) (YetYe)2 y2(S)YetYe X4(S)+ 3X2 6AYetYe 3872 135 5 51 11 4 2722 +( 1 + g2)YetYe y4(S) + ( + ng)g4 + 2 2 35 ( ng)4 , (A.11) with Y4(S) (1g2 + 9 + 8g1)Tr{YutYu} 10 4 (A.12) +( g2+ g + 8g )Tr{YttYd} + (g + g2)Tr{Ye Ye and X4(S)= Tr{3(Y ~Yu)2 + 3(YdtYd)2 + (YetYe)2 94 (A.13) YutYYdtYd} 3 In the Higgs sector, the 0 functions for the quartic coupling and the vacuum expectation of the scalar field are presented. Here a discrepancy in the oneloop contribution to the quartic coupling of Ref. 7 is corrected dA 1 (1) 1 (2) (A.14) dt 16_2 + (16r2)2 A 74 where the oneloop contribution is given by 12A2 (99 21 + 92+ 9 3 4 2g2g4 1 2) g1 1 2 2g4) = 22 (5 + (A.15) + 4Y2(S)A 4H(S) , with H(S) = Tr{3(YutYu)2 + 3(YdtYd)2 + (YYet)2} (A.16) and the twoloop contribution is given by (discrepancies found by Ford et al. [73] in Machacek and Vaughn [7] are corrected), (2) 182 22 (313 11722 ( 78A3 + 18(2 + 3g )A2 [( 10ng)g 92 9 229 50 4 497 6 3 97 8 4 + + 4 ng)g1 ] (y 8ng)g2 54 + ng)glg2 9 239 40 4 2 27 59 40 g ( ng)g lg2 "( ng)g1 25 24 +9 ng)g 125 24 9 g 64gTTr{(YtYu)2 + (YdtYd)2} SgTr{2(YutY)2 (YdtYd)2 + 3(YetYe)2} 4Y4(S) + 10A[ (11 9 2 + g2 +8g2)Tr{YutYu} (A.17) +( + + 8g +8 )Tr{YdtY} + 3( + )Tr{Ye+Ye} i + g ?[( g + 2192)Tr{YUtY,} +( 2 + 9)Tr{YtYd} 5 10 2 + ( 9g2 + llg2)Tr{YYeYe} 24A2y2(S) AH(S) + 6ATr{YutYuYdtYd} + 20Tr{3(YutYu)3 + 3(YdtYd)3 + (YetYe)3} 12Tr{YutYu(YutYu + YdtYd)YdtYd} The / function for the vacuum expectation value of the scalar field is nv 1 (1) 1 (2) (A.18) dt 16ir2 (16r2)2 where the oneloop contribution is given by (1) = ( + g2 ) Y2(S) (A.19) 7 9 9 75 and the twoloop contribution is given by (2) _3A2 4(S)+ X4(S) 93 1 4 511 5 4 27 2 2(A.20 ( + ng) + ( n 2 ng)g2 80192 " These expressions were arrived at using the general formulas provided in Ref. 7 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. The general formula of Ref. 74 is used to arrive at the / functions for the respective gauge couplings dg3 2 38 S[(nu + nd) 11] + [3(nu + d) 102] dt 3 (4X)2 3 (47r)4 8 2 3e2 + [nu + nd] (A.21) 9 9 (4r)4 5033 325 2 2857 g + [ (nu + nd) (nu + nd)2 7] , 18 54 2 (47r)6 and de 16 4 4 e3 64 4 e5 t [nu + nd + + [nu + nd + 4nl 64 16 e3g2 9 9 (4r)4 where nu, nd, and nj are the number of uptype quarks, downtype quarks, and leptons, respectively. In Eq. (A.21), the threeloop pure QCD contribution to the / function of g3 have also been included [75]. For the evolution of the fermion masses, Ref. 76 is used. It is known that there is an error in their printed formula [77]. This typographical error is found and corrected. The calculation of the twoloop contribution to the mass anomalous dimension in QCD using the corrected formula agrees with the result obtained in Ref. 77. Equation (2.26) of Ref. 76 should read i17 148 7m = 6ij(1 (C' + C C12 (C4C + C12C + 3 3 F FR (C' C + C'RC ) + T(Cm 2Cj 2C )bi . (A.23) 76 Using the corrected expression, the following mass anomalous dimension is computed. The fermion masses in the low energy theory then evolve as follows dm t = 7(l,q)m (A.24) where the I and q refer to a particular lepton or quark, and where 2 2 1 e2 3 93 (1,q) = (,q) T2 + (l,q) ()2 + [11 4 33 4 13 2 1 + [7(1 q)e4 (1,3q)g3 + 2^g()e2 4 (A.25) 6 333 93 +^(1) (4r)6 The superscripts 1 and 3 refer to the U(1)EM and SU(3)C contributions, respectively. Explicitly, the above coefficients are given by 7(l,q) 16Q,q) 3 7(3) = 0 3) 8 13 33 7() = 7(1) = 1(1 ) 3Q4 80 20 20 n 2 (A.26) 7(l,q) (,q) +9 9 3 + 1(,q) 13 _4/Q2 7q)= 4q) 33 404 40 7( 3 + 9 (nu + nd) 333 2 140 2 2216 7) 3[7 (nu + id) + (160((3) + )(nu + nd) 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 anoma lous dimension for the quarks above, the threeloop pure QCD contribution 333 [75] have also been included. _(q) APPENDIX B CALCULATING THE MSSM 3 FUNCTIONS In this appendix, some useful results in calculating the / function of the MISSM are included. Also some sample calculations are presented. Although, to one loop the MSSM P functions have appeared in the literature, one is not aware of the two loop ones' appearance except in some approximate form, such as keeping only the contribution of the heaviest family. There are at least two ways to proceed. References [78,79] give general formulas valid to two loops to compute the 3 functions of gauge and Yukawa couplings in a supersymmetric gauge theory. These were used to calculate the two loop gauge and Yukawa coupling / functions appearing in the next appendix. Reference [7] gives formulas valid to two loops to compute the 3 functions of gauge, Yukawa, and scalar quartic couplings in a general gauge theory. For the purpose of including thresholds, this approach is more useful, and it is this manner that the following examples are done. The results of this second approach were checked against the first one, and they agreed (as they must). Not all calculations are included, as this would be useless and not enlight ening. However, it is hoped that the examples will give some flavor of the endeavor and that the results will prove useful to anyone wishing to pursue such calculations further. 78 B.1 Gauge Couplings B.1.1 Group Invariants Tables 6 and 7 will be useful in the calculations of the/3 functions of the MSSM. Also useful are the following group invariants C2(R) = TA(R)TA(R) = C2(R) I(R) (B.1.1.1) Using this definition and Tr(TA(R)TB(R)} = T2(R)6AB (B.1.1.2) the following identity is derived Tr{C2(R)} = d(G)T2(R) C2(R)Tr{I(R)} = rT2(R) (B.1.1.3) C2(R)d(R) = T2(R)r , where r = d(G) is the rank of the group, and SAA = d(G). From this, it follows that C2(G) = T2(G). (B.1.1.4) The Yukawa interactions arising in the SUSY Lagrangian can be written S wa = iNg[f* T jAA Aj Ti ,i (B.1.1.5) where TA are the group generators, and overlined fermi (Grassmann) fields are righthanded. Now make the correspondence Y SUSY , Ya Y = ivgT , (B.1.1.6) jo(i) = +i gT Written this way, the i (i.e., 0(i)) subscript represents the ith group component of the scalar, and the j represents the group component of the fermion. The i 79 encodes the scalar information in SUSY whereas the a does so in the notation of Ref. 7. When taking a Yukawa term from Ref. 7 and "supersymmetrizing" it, the trace need not represent a trace unless all scalar indices (i.e., the superscript on the Y's) are contracted. For example, in the oneloop contribution to the scalar anomalous dimension Tr{YaYbt} =[i/gTA[iVgT = 2g2(TATA)ij (B.1.1.7) = 2g2C2ij = 2g2C2(S)6ij Now to compute Y4(F). Tr{C2(F)YaYat} = 2g2C2(F)Tr{ I TBTB} = 2g2C2(F)Tr{TBTC}6CB (B.l.1.8) = 2g2C2(F)T2(R)6BC CB = 2g2d(G)C2(F)T2(R) Y4(F) Tr( C2(F) 3aat} d(G) (B.1.1.9) = 2g2C2(F)T2(R) These results may be used to derive the SUSY gauge beta function. In the general (but single simple group G) case, at one loop (47r22(1)/3= 2 1 11 (47r)2p()/ = T 2(F) + T2(S) C2(G) (B.1.1.10) where F, S, and G stand for the fermion, scalar, and adjoint representations, respectively. To go to the SUSY from, the following correspondences are used SUSY FR +G . SUSY (B.1.l.11) S >R. Whence (47)21SY/3 = [T(R) + C2(G) + 1 2() C2(G) ( 3 3 32 (B.1.1.12) = T2(R) 3C2(G) . At two loops, the only extra complication is a Yukawa contribution in the general case that must be supersymmetrized as discussed above (4x)43(2)/g5 = [10 C2(G) + 2C2(F)IT2(F) + [ C2(G) + 4C2(S)]T(S) C2(G)2 Y4(F). 3 (B.1.1.13) Applying the stated correspondences yields (47r2(2) 10 (4w)2 S g5 = C2([2() + T2(G)] + 2C2(G)T2(R) + 2[C2(R)T2(R) + C2(2(G)T(G)] + 4C2(R)T2(R) C2(G)2 [Y4(R) + Y4(G)] 3 = 4C2(G)T2(R) + 6C2(R)T2(R) 6C2(G)2 [2C2(R)T2(R) + 2C2(G)T2(R)] = 2C2(G)T2(R) + 4C2(R)T2(R) 6C2(G)2 (B.1.1.14) B.1.2 Fierzing A matrix M can be expressed in terms of the group G's generators M fATA + f0 I. (B.1.2.1) Taking the trace of both sides yields Tr{M} = f0d(R) (B.1.2.2) Multiplying by TB and taking the trace gives Tr{MTB} = fAT2(R)bAB (B.1.2.3) Hence 1 1 M = TrMTA TA + Tr{M} I. (B.1.2.4) T2(R) d(R) If M = 4(t, it follows that t = 1 (4tTA 4)TA + 1 ( ) (B.1.2.5) T2(R) d(R) where use has been made of Tr{ TTA} = Tr{ftTAq} = tTA[. Now multiplying on the left by ,t and on the right by gives ($t$)2 = T (tTA)(tTA) (t2 (B.1.2.6) T2(R) d(R) Combining like terms yields (tTAX)(tTA ) = T2(R)[1 ](t)2 (B.1.2.7) In the fundamental representation of SU(n), T2(R) = 1/2 and d(R) = n. Taking SU(2) as an example, with 4 a doublet, it follows that (=tTA Xt)()TA) ( 2 (B.1.2.8) It is also useful to have the analogous expression for the case when there are two different SU(2) fields involved _1 Dt T t,(a> I)(,t Ta 2) + 1 l 2) , (~1T2)( T2(R) d(R)= () (B.1.2.9) which follows immediately from Eq. (B.1.2.5). 82 B.2 Anomalous Dimension of the Scalar Field Using the MachacekVaughn convention, the Yukawa sector of the SUSY Standard Model is written C = ULU~uQL + ULUulQL + UU~,uQL+ dLDddQL + dLDddQL + dLDdQL+ TLEDdLL + LEidLL + eLEidLL + c.c. = ULiaUij uA(iU2)ABQLj3B oa3 + *uA(i'2)ABOQLjB Uj ULi +'' (B.2.1) The ir2's involved in the Weyl spinor product have been suppressed and so have the ia2's involved in the SU(2) product, although the ia2's are displayed in the second step. The SU(3) product follows since the barred fields are anti quarks. All fermions are represented by Grassmann fields. The c.c. is explicitly calculated in the example below [XL(ir2)7L] = *(i72) * = XL(i2)7L (B.2.2) = [XR(ir2)](ir2)[(ir2)R] = YR(i2)7R = j(ir2)R, where the identity, 4R = ir2~, has been used. The following identifications can be made from Eq.(B.2.1) Y+u((A) )ABiu(A) a 6(ia 2)BAUi , [(uia).](Qj} +g a*(2)Asjij (O (Qjm3B)[,(uif B] y+u(uia) u( ia) [.( A)](QjI3B +ba 3(i2)ABUij Y(Qj/3B)[(4A) 3a(i'2)BA ji y+u(Qji3B) ,((Qji3B) (iT2)BAi [(uia)(OA)]. =6ai2)ABUij Y[(A)(uia)] = (2BA (B.2.3) y+d(fA) .rd(fA) _(io2)BAD i Y[(di)(Qjp3) +6aO(ia)ABDij Y (QB)[*(dia] )BAD y+d(dia) Vd(dia) a(i2)BADi [.*(A)](Qj +#a)?(i2)ABDij (Qj3B)[(4A)i 3aQ~2BA , y+d(Qj#B) d(Qjl3B) (io) [(dia)(OA)]. = + 3(iT2)ABDij Y((A)(dia)] = 0a(2)BAD (B.2.4) +d(A) d(OA) (i2)BAEi [(ei).(LB) = +(i2)ABEi (LB)(ei)] +d(ei) ]L d(kei) (io2)BAEi , ](LjB)= +(iO2)ABEij Y(L[(A) ( y+d(LjB) + 2)A j d(LjB) (i.)BEi [(ei)(A)]. = +(i2)ABEi [( A)(ei)] BA (B.2.5) The + and on the Y's denote whether or not the 4 is complex conju gated, respectively. The upper index always denotes the scalar, and a bullet is placed in its unoccupied lower position. Since two of the three fields are either transposed or not (in the spinor, SU(2) doublet, SU(3) triplet, or generation sense), they appear grouped in square brackets. Dots over Y's indicate that the Yukawa coupling matrix it represents is daggerred. Consider the following example from Ref. 7 X = Tr{YbytaycYtc + Tr{YbYtcyaytc} (B.2.6) a quantity that enters the scalar 2loop anomalous dimension. In the above notation (suppressing some indices at first), the first term reads (in the fu case) X(+A)(B) = Tr{y+u(OA)yu(B)YY} (B.2.7) but to this must be added another term (+A)(B) = Tr{Y+u(A)yu(B)YY} (B.2.8) since there must be alternating daggerred Y's in these expressions, and both equations (B.2.7) and (B.2.8) above satisfy this (the underlines on two Y's indicate that their scalar indices are to be contracted). Furthermore, the result must be symmetric, in this case, in A and B, so the next two terms must also be included X(A)(+B) = Tryu(OA)y+u(OB)yy} (B.2.9) X(A)(+B) = Tr {Yu(4A)y+u(B)yy} . 2 84 Putting it all together, the first term in Eq. (B.2.6) is given by 1 X = ((X X+) + (X+ + X2)]. 2 1 (B.2.10) Similarly the second term in Eq. (B.2.6) is (B.2.11) Now to compute these contributions (the flavor and color indices are suppressed since these contract in a straightforward way) X+ Tr{+u(A) u(B) [+u(E),u(E) y+u(QE)Vu(QE) I 1 =" u(QC) (QC).u u.(QD) (QD)*u u(4D). *(4D)u = 3Tr{(UtU)2 (iT2)AC(io2)CB[ (ia2)ED(io2)DE + (i"2)DE(iO2)ED = 12Tr{(UtU)26AB , X+ =0, xi = 0 , X+ Tr u(A) v+u(B) u(E) y+u(E) + d(4E) +d(OE) 2 (QC).u u.(QD) (QD)*u u.(QC) (QD).d d.(QC) + U(U) v+U(U) + d(d) v+d(d) (QD)(OE)* *( E)(QC) (QD)(OE)* (OE)(QC) = 2.3Tr{(UtU)2UtUDtD}(ia2)CA(ia2)BD(i2)DE(i2)EC = 6Tr{(UtU)2 + UtUDtD} AB (B.2.12) Some of the 3's come from the trace over the suppressed color delta functions. Finally, X = 3Tr{3(UtU)2 + UtUDtD}AB . y+ = Tr{Y+u(OA)_yu(OB)Y} = 0, Y2+ = Tr{Y +u(A)Yu(4B)Y} = 0, Y+ = Tr{YU(OA)YY+U(B)Y} = 0, y+ = Tr{fYu(OA)YY+u(B)Y} = 0. Similarly, (B.2.13) (B.2.14) Y = [(Y1 + Y2)+ + (Y + Y2)+] 2 The full answer is then Xu = 3X + Y = Tr{3(UtU)2 + UtUDtD} (B.2.15) 2 2 B.3 Yukawa Couplings The Yukawa interactions arising from the Dterms in the SUSY Lagrangian can be written YU = ivzg[ Ti jR i T j i R], (B.3.1) where the overlined fermi (Grassmann) fields are righthanded. Now make the correspondence SUSY +(AR) = iVgT ij (D,)(oj) (B.3.2) Y(AR) = +i RgT (o0 )(3D) The o's a's, o's, etc. represent fieldtypes (i.e., Q, u, d, etc.). Note that the Y's which represent TR's are diagonal in type (i.e., both indices are o's as exemplified above). The + and indicate that the scalar index is complex conjugated (equivalently, is the left subscript on Y's) or not complex conju gated (equivalently, is the right subscript on Y's), respectively. Written this way, the i above represents the ith group component of the scalar, and the j represents the group component of the fermion. Contracting two Y's (i.e., using underscores) implies in their case contracting the superscript, group gen erator indices (the R's). 86 In order to construct the one loop SUSY beta function formula for the Yukawa matrices the following must be supersymmetrized yytya SUSY (.r(AR)r:t(AR)y+(7))[3.]& YY Ya SUSY Y Y Y (AR) +(AR)y+(O) + 1r+(AR),r(AR)y+(O) = 0Oo 0OI []o.& 10 [o.],a =(C2(0)+ C2))Y ) 2g2C2(R)Y(O) (B.3.3) yayty SUS (R)) y +(0) ,(AR)m^+(AR) =Y+() C2(a) 2g2Y() C2(R) Furthermore, in obtaining the formulas for the beta functions of the different Yukawas the "external subscripts" are generally two fermions (not scalarinos) which are represent by o's, a's, o's, etc., and the "external superscript" is a scalar (a Higgs not an sfermion) which is represented by a (0). Hence, it should be clear from inspection that the following terms cannot be constructed for any choice of "internal subscripts" YTr{Ytya SUSY (Y(AR)Tr{F(R)y()})o 0 , (B.3.4) yyaty SUSY (( AR)y()Y:F(AR)) 0 . Application of Eq. (B.3.3) to the upYukawa (Y ) ; see section where the SUSY Y's are defined) yields C2(R)Y) C Y)u + =B) + C[ 2PoYO C(R .](QB) = C2u [o ) [u](QB) (B = (C2() + +0 U (( B(B.3.5)  (62() + C2(BU,][uo](QB) , 87 where use was made of the diagonal nature of C20S ~ 6&AC2(0). Note that C2 on the left can act on both left subscripts. Likewise (YC2(R) Y+(u)o) +U(C) [u](QB) [u C2(QB) (B.3.6) = WC2 *[u](QB) ' The standard one loop beta function for the Yukawas is (4r)2/a 1 [yytya + yayty] + 2YYatY + YTr{Ytya} 2L (B.3.7) 3g2 C2(F), ya Supersymmetrizing according to the results derived above gives (47)2a = 1 [yytya + yayty] + 2yyatY + YTr{YtYa} Sr (B.3.8) (3 g)g2 C2(R), Ya where 6g equals 1 or 0 depending on whether one is above or below, respectively, the mass threshold of the gaugino in question. The labels Ti, i = 1,..., 5, will be used to denote the five terms appearing in Eq. (B.3.8) when computing the supersymmetric beta function for the upYukawa T = (yty+u(.))[u](QB) [Y+u(OD) Vu(OD) v+u(QD) u(QD)] +u(OA) [ [u.](QC) (QC)[.u] + [u(C)]* [(OC)u] Y[u.](QB) (B.3.9) = [(ia2)DC(i2)CD + (iu2)CD(i2)Dc ](ia2)ABUUtU = (1i2)ABU[4UtU] , T = (y+u()YY)Y [u.](QB) = y+u(OA) [ru(D) v+u(OD) +Y(u) v+U(u) [u.](QC) [ (QC)[.u] [u.](QB) (QC)[(4D).] I[.(D)](QB) + d()d(D) Y+d(D) + (d(d) v+d(d) (QC)[.dl [d.](QB) (QC)[(OD).] [.(OD)](QB)( (B.3.10) = (io2)ACU [(iD2)CD(i 2)DBUtU + (i2)CD(i02)DBUtU + (ia2)CD(ia2)DBDtD + (io2)CD(ia2)DBDD] = (ia2)ABU[2UtU + 2DtD , 88 T3 = (YY^+uY_) ) SY [nu.](QB) (B.3.11) =0, T4 = (YTr{YtYu()}) [u.](B) = v+u((C) Tv u( Y C) v+u(A) [u.](QB) (QD)[u] [u.](QD) (B.3.12) = (ia2)CBUTr{3UtU}(iO2)DC(iO2)AD = (i2)ABU[Tr{3UtU}] , T5 = (C2), +u(}) ) ( 3 = (ia2)ABU Z [C+'i) + C2(Q) + C2k(+)] k=1 .(4 1 1 2 3 3 2 (B.3.13) = (ia2)ABU[( + + + (0 + + )g +( + + +0) [13 3 2 8 2] = (i2)ABU [ + 2 + 3 In T4 the 3 comes from the trace over the suppressed color indices. Also, in T5 use was made of the fact that if the gauge group is not simple but a direct product G1 x ... x Gn with couplings gl,... ,g9n, then g2C2  k =1 gC2k B.4 Thresholds To implement the super particle thresholds in the minimal low energy super gravity model, the renormalization group P function must be calculated in a form that has not appeared in the literature. In the following example, the one loop p function of g3 is considered. Starting from Eq. (B.1.1.12) (47r)2/31) /3 = 2T2(F3) + T2(S3) + C2(G3)g C2(G3) (B.4.1) 3 3 3 3 where F3, S3 refer to the fermion and scalar representations, and G3 = SU(3). Also 01= .< Mg9 (B.4.2) 89 where Mg is the mass of the gluino. When dealing with a direct product group, like SU(3) x SU(2) x U(1), T2(R3) 2(R3)d(R)d(R2) (B.4.3) From the definition of T2(R) in Eq. (B.1.1.2), one obtains the following result in the SU(3) case T2(R3) = 2( )NQ + ( )N + ( )N (B.4.4) where Np equals the number of generations of particle p. This result is valid for both fermion (R = F) and scalar (R = S) representations. The notation will be such that N = n for the particle and N = n for the SUSY partner. Equations (B.4.1) and (B.4.4) lead to (420(1)/3 2 12 1_ (43)2 1)/g3 = (nu + nd) + nQ + 1 + 6 + 26g 11 (B.4.5) It has been assumed that nQ = (nu + nd)/2. Also the fact that lefthanded and righthanded quarks of a given flavor have the same mass implies nu = nU and nd = n. Note that this reduces to the right standard model result when h = Sg = 0, and to the right supersymmetric result (J. > MSUSY) when h = 3, g = 1. Similar formulas are calculated for gl and g2 (4.)2(l)/g 2.17 5 5 1 1 4 1 (47r)2 3 (nu + " d + ne + nv) + iiQ + +  + nd 512 12 4 4 3015 15 1 1 1 1 + 10 L 5h + 1 ni + 1 "nu + ) + 10 (no + n d) ' (47r2l)/g3 22 1 1 1 1 (4  + ("u + nd) + (ne + nv) + 2nQ + niL 1 1 42 + 3(" + i)+ ) + (n + n ) + W w. (B.4.6) For the gauge couplings, the two loop contributions were also calculated in this manner. For the Yukawa couplings, this form of the j function was calculated to one loop. Table 7 G C2(G) d(G) U1 0 1 SU2 2 3 SU3 3 8 Table 6: Group Theory Factors T2(R) C2(R) d(R) U1 SU2 SU3 Ui SU2 SU3 UI SU2 SU3 Q_ (+1)2 +)2 1 4 1 2 3 yT (2)2 0 1 (2)2 0 1 1 3 d (+3)2 0 1 ()2 02 1 1 3 L (2)2 0 ()2 T 0 1 2 1 e (+1)2 0 0 (+1)2 0 0 1 1 1 _, (+0)2 1 0 (+ )2 3 0 1 2 1 d (_1)2 1 0 (4)2 3 0 1 2 1 l~  1 :  1  APPENDIX C THE MSSM 0 FUNCTIONS Using some of the notation of Falck [80], the superpotential and soft sym metry breaking potential are as follows W = YvQU + dYd dQ + eYe dL + Apiud + h.c., V'oft = M2 I + m d + B(ud d+ h.c.) + m tQ + m2 Lu t + m + 2f 2 + ( AY~Y~i Qj + A d didQj + AYeidLj + h.c. ), i,j 1 3 Vgaugino = 2 MiAA + h.c.. l=1 (C.1) Various a2's have been omitted, and a sum over the number of generations is implied in the squark and slepton mass terms. Also, hats imply superfields and tildes the superpartners of the given fields. First the gauge couplings dg, 1 3 dt 16b,2 I (16r2)2 [Y 1 Tr{CluYutYu + CldYdYd + CleYeYe}] k (C.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 = e 2ng , b2 = 5 2ng (C.3) b3 = 9 2ng , 38 6 88 9 9 17 5 IT M 5 0 (blk)= 14 8 ng 3 17 (C.4) 11 3 68 0 54 and 26 1r 14 8 (Cl) = with f = nu, d, e, (C.5) 4 4 0 with ng= =nfl. In the following, the beta functions for the parameters of the superpotential are listed. = 162[ Tr{3YutYu + 3YdYd + YeYe} 3( 2 +2) g (C.6) In the Yukawa sector the 3 functions are dYue 1 (1) 1 (2) (C.7) dt Yu,d,e( 162 (ud,e + 16ir2)2 u,d,e) where the oneloop contributions are given by ,31) = 3YutYu + YdtYd + 3Tr{YutYu} (13 2 + 16 2 (1) = 3YdtYd + Y tYu + Tr{3YdtYd + YetYe} (792 3g2 9 ) 31) = 3YetYe + Tr{3YdtYd + YetYe} (9g2 + 3922) (C.8) 93 and the twoloop contributions are given by 32) = 4(YutYu)2 2(YdtYd)2 YdtYdYutYu 9Tr{YtYu}YYu Tr{3YdtYd + YetYelYdtYd 3Tr{3(Yu Yu)2 + YdtYdYutYu 2 2 2 + d 4 2 2)Trlyutyu, + ( 9 + 6g )YutY + (J2)Y d + ( + Y16 Yu 26 403 21 32 3044 + (ng + 45)g + (6ng 32 92+ 9)g3 22 136 22 + 8 33 + 9192 + 3 + g2g3 2) = 4(YdtYd)2 2(YutYu)2 2YutYuYdtYd 3Tr{YutYu}YutYu 3Tr{3YdtYd + YeYe}YdtYd 3Tr{3(YdtYd)2 + (Yete)2 + YdYdYU Y} + (4gu)yytYu +( (g + 6g2)YdtYd + ( g + 16g )Tr{YdfY} 6 14 7 21 +(g )Tr{YeY} + ( 15n + +(6n 2 +32 304, 4 22 8 22 22 + ( g )g3 + 9192 + 9g3 + 23 e2) = 4(YetYe)2 3Tr{3YdtYd + YetY }etYe 3Tr{3(YdtYd)2 + (Yetye)2 + YdtYdYutu} + (6g2)YetYe + ( 2)Tr{Yet} + + 2 + 16g )Tr{YtYd} 18 27 214 9 2 2 + ( 1ng + 7)g + (6ng 2) + 9192 (C.9) The evolution of the vacuum expectation values of the Higgs's is given by dlnvbuOd 1 (1) 1 (2) dt 1672 .,d + (167r2)2 ,,d ( where the oneloop contribution is given by 3 1 2 2) = ( +g) 3Tr{YutYu} (C.11) 7 = ( +2 2 3Tr{YdtYd) Tr{YeYe) , Od =(591+92 and the twoloop contribution is given by (2) = Tr{3(YutYu)2 + 3YutYuYdtYd} 19 2 9 2+ 20rg3)Trjyutyu (g+ g92 + 2093T 279 1803 4 207 357 4 27 9 22 (0+ n1g)g1 ( + ng)g2 ( + 9)g192 2 800 1600 32 64 80 80 ( = 3Tr{3(Ydtd)2 + 3YdtYdYutY + (Yetye)2} (g2 + g + 20g)Tr{YtYd} (92 + 2)Tr{YeYe} 279 1803 4 207 357 4 27 9 2 2 (8 + 1600ng)gl (2 + ng)g2 + Ong)g2 800 1600 32 64 80 80 The renormalization group equations for soft symmetry breaking terms are dA= 1[ 4(YeYet)ik Ae y dt 16x2 Y, + 5Ak y (Y )kj 3 Ae(YeYetYe) Ye Ye + 2(Akm Yekm2 + 3AdmIY~m) 6( 21M1 + gM2)] , kYtk yik [ 4(YdYdt Ad + 5A~ (YYd)kj 3 d (YYdYd)i 16 d d d i .. yikk__ 2)ikkuj + (Ak A')(YutYu)kj + 2(YdYut)ikAy + + dd d 3d dk 2 + 2(Amlykm2 + 3Akm iym2) 1Mi 6g2M2 g32 23 15 3 dA" 1 4y" Sdt6 4(Y uYt)^kAkJ + dt 1672 4YIU Yu 5Aik u(yty)kj 3 A(YuYtYu) Y2 Yu.j kyik + (A' Aj )(YdtYdj + 2(YuYdt)ikA d + 6Amiyrm2 2621M 692M2 M ]3 , 15 3 (C.13) (C.12) dAz dt 