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FERMION MASSES AND MIXINGS, FLAVOR VIOLATION, AND THE HIGGS BOSON MASS IN SUPERSYMMETRIC UNIFIED FRAMEWORK By ABDELHAMID ALBAID Bachelor of Arts/Science Applied Physics Jordan University of Science and Technology Irbid, Amman, Jordan 2001 Master of Arts/Science in Physics University of Jordan Amman, Amman, Jordan 2004 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial ful¯llment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2011 COPYRIGHT °c By ABDELHAMID ALBAID July, 2011 FERMION MASSES AND MIXINGS, FLAVOR VIOLATION, AND THE HIGGS BOSON MASS IN SUPERSYMMETRIC UNIFIED FRAMEWORK Dissertation Approved: Kaladi S. Babu Dissertation Advisor Flera Rizatdinova Jacques H. H. Perk Birne Binegar Mark Payton Dean of the Graduate College iii ACKNOWLEDGMENTS All praises and gratitude to Allah Almighty who guides me with His mercy and bounty to ¯nish this project. I would like to thank my advisor Professor Kaladi S. Babu for his guidance and teaching. He has been the important person in my learning process. Without his advising and collaboration, this project would have been impossible. I would like also to thank Prof. S. Nandi from whom I have learnt a lot. I would like to thank my thesis committee: Prof. Jacques H. H. Perk, Prof. Flera Rizatdinova and Prof. Birne Binegar for their advice. I would like to express my gratitude to Prof. Paul Westhaus for being my advisor in the beginning of my Ph.D program. He was very helpful and dedicated to the students. I want to thank my o±ce mates for useful discussions and conversations. Special thanks for Benjamin Grossmann and Julio for being very helpful. I want to especially thank my parents for their support and love and to thank my wife, Rema Alradwan, who was supportive, cooperative and full of love. iv TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 A Brief Review of the Standard Model (SM) . . . . . . . . . . . . . . 1 1.2 Seesaw Mechanism and Leptonic Mixing Matrix . . . . . . . . . . . . 7 1.3 Shortcomings of the SM and the Need for New Physics. . . . . . . . . 10 1.3.1 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.2 Discrete Flavor Symmetry A4 . . . . . . . . . . . . . . . . . . 16 1.4 Minimal SUSYSU(5) . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.1 SU(5) Matter Fields . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.2 Higgs Sectors and Yukawa Couplings in the minimal SUSYSU(5) 18 1.4.3 Gauge Sector of Minimal SU(5) . . . . . . . . . . . . . . . . . 20 1.5 Minimal SUSYSO(10) . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5.1 Matter Fields in SO(10) GUTs . . . . . . . . . . . . . . . . . 22 1.5.2 The Higgs Fields and Yukawa Couplings in SO(10) GUTs . . 22 1.5.3 Neutrino Masses . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Fermion Masses and Mixings in a Minimal SO(10)£A4 SUSY GUT 29 2.1 Fermion Mass Structure in SO(10) £ A4 Symmetry . . . . . . . . . . 31 2.2 Extension to the First Generation and Doubly Lopsided Structure . 35 2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4 Right Handed Neutrino Mass Structure . . . . . . . . . . . . . . . . . 43 3 Flavor Violation in a Minimal v SO(10) £ A4 SUSY GUT 48 3.1 A Brief Review of Minimal SO(10) £ A4 SUSY GUT . . . . . . . . . 51 3.2 Sources of Flavor Violation in SO(10) £ A4 Model . . . . . . . . . . . 56 3.2.1 The Scalar Mass Insertion Parameters . . . . . . . . . . . . . 57 3.2.2 The Chirality Flipping Mass Insertion (Aterms) . . . . . . . . 59 3.2.3 Mass Insertion Parameters Induced Below MGUT . . . . . . . 60 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Higgs Boson Mass in GaugeMediating Supersymmetry Breaking with MessengerMatter Mixing 68 4.1 Higgs Mass Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.1 Higgs Mass Bounds in the 5 + 5 Model . . . . . . . . . . . . . 73 4.1.2 Higgs Mass Bounds in the 10 + 10 Model . . . . . . . . . . . . 77 4.2 Flavor Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.1 Flavour Violation in 5 + 5 Model . . . . . . . . . . . . . . . . 82 4.2.2 Flavour Violation in 10 + 10 Model . . . . . . . . . . . . . . . 85 5 CONCLUSION 88 BIBLIOGRAPHY 92 A Diagonalization of Fermion Mass Matrix 100 A.1 Derivation of the Light Fermion Mass Matrix . . . . . . . . . . . . . 100 A.2 Light Neutrino Mass Matrix . . . . . . . . . . . . . . . . . . . . . . . 106 B RGE from the Scale M¤ to the GUT Scale in the SO(10) £ A4 Model 113 C Yukawa Couplings RGEs 115 C.1 MSSM with 5 + 5 Messenger Fields . . . . . . . . . . . . . . . . . . . 115 vi C.2 MSSM with 10 + 10 Messenger Fields . . . . . . . . . . . . . . . . . . 116 D Generated Scalar Masses due to MessengerMatter Mixing 117 D.1 5 + 5 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 D.2 10 + 10 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 vii LIST OF TABLES Table Page 1.1 The transformation of the lepton (Li,ec), quark (Q,uc,dc), and Higgs (H) ¯elds under SM gauge group SU(3)c £ SU(2)L £ U(1)Y . . . . . . 2 2.1 The transformation of the matter ¯elds under SO(10)£A4 and Z2 £ Z4 £ Z2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 The transformation of the Higgs ¯elds under SO(10)£A4 and Z2£Z4£Z2. 35 2.3 This Table shows the comparison of the model predictions at low scale and the experimental data. . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 The transformation of the matter ¯elds under SO(10) £ A4 and Z2 £ Z4 £ Z2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 The transformation of the Higgs ¯elds under SO(10)£A4 and Z2£Z4£Z2. 52 3.3 The fermion masses and mixings and their experimental values. The fermion masses, except the neutrino masses, are in GeV. . . . . . . . 64 3.4 The mass insertion parameters predicted by SO(10) £ A4 model and their experimental upper bounds obtained from [55]. . . . . . . . . . . 66 3.5 Branching ratio of ¹ ! e° for di®erent choices of input parameters at the GUT scale. Cases I and II correspond to ln M¤ MGUT = 1 and cases III and IV correspond to ln M¤ MGUT = 4:6. ~mÃi and M1=2 are given in GeV 67 4.1 We show the values of the minimal GMSB input parameters, ¤, ¸ex and Mmess that lead to the highest mh values at tan ¯ = 10. . . . . . 73 viii 4.2 The spectra corresponding to 10 + 10 model and 5 + 5 model. All the masses are in GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3 We show the values of the GMSB input parameters, ¤, ¸0 0 and Mmess that lead to the highest mh values. These values correspond to ¸0 m0 = 0 and tan ¯ = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 We show the values of the GMSB input parameters, ¤, ¸0 0 and Mmess that lead to the highest mh values. These values correspond to ¸0 m0 = 1:6 and tan ¯ = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5 The U(1) charge assignments to the messenger, MSSM, Z and S ¯elds. 83 4.6 The U(1) charge assignments to the 10 + 10 messenger, MSSM, Z and S super¯elds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.7 The calculated mass insertion parameters for the 5 + 5 and 10 + 10 models and their experimental upper bounds. The numerical values of ·'s are ·d5 = 0:0066, ·l 5 = 0:032, ·d 10 = 0:0028 and ·l 10 = 0:0025. . . . . 87 ix LIST OF FIGURES Figure Page 1.1 Feynman diagrams of K0 $ K0 induced by higher order corrections in the SM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 The evolution of the inverse gauge couplings ®¡1 i in the standard model (dashed lines) and in the MSSM (solid lines). . . . . . . . . . . . . . 12 1.3 These three diagrams contribute to K0 $ K0 mixing in supersymmet ric models. They put constraints on the o®diagonal elements of the soft breaking scalar down mass matrix that is indicated by £. . . . . 15 1.4 The most common breaking chains of SO(10) gauge group to the SM gauge group (GSM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1 This ¯gure shows a diagrammatic representation of the couplings in the superpotential W1. . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 This ¯gure leads to the °avor symmetric contribution to the down quarks and charged leptons. . . . . . . . . . . . . . . . . . . . . . . . 36 2.3 This ¯gure leads to the °avorantisymmetric contribution to the down quarks and charged leptons. . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 This ¯gure leads to the righthanded neutrino mass matrix. . . . . . . 45 3.1 The above graphs show the plot of Log of Br(¹ ! e°) divided by experimental bound (1:2£10¡11) versus mÃ for two cases I and II with M1=2=787 GeV, 437 GeV and 175 GeV. . . . . . . . . . . . . . . . . . 65 x 4.1 The evolutions of the gauge couplings with Mmess = 108 GeV and tan ¯ = 10. Solid lines correspond to MSSM. Dashed lines are for MSSM+10 + 10 and dotted lines are for MSSM+5 + 5. . . . . . . . . 70 4.2 The left graph is ~m2¿ c versus ¸0 0 at the scale Mmess for two di®erent messenger scales. The right graph is ~m2t c versus ¸0 0 at the low energy scale for two di®erent messenger scales. . . . . . . . . . . . . . . . . . 74 4.3 The left (right) graph shows the running of two exotic Yukawa cou plings from the GUT scale MGUT = 2 £ 1016 GeV to the messenger scale Mmess = 108 GeV for the 5+5 (10+10) model where the uni¯ed Yukawa coupling is taken to be ¸0 0 = 1:6. . . . . . . . . . . . . . . . . 77 4.4 The left graph is a plot of mh versus ¤ for ¸0 0 = 0 and ¸0 0 = 1:2. The right graph is mh versus ¸0 0 for di®erent messenger scales at ¤ = 105 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 xi CHAPTER 1 INTRODUCTION 1.1 A Brief Review of the Standard Model (SM) The standard model (SM) of particle physics is based on the nonabelian gauge sym metry SU(3)c£SU(2)L£U(1)Y . Here the SU(3)c gauge group describes the theory of strong interaction called quantum chromodynamics (QCD). This type of interactions holds the quarks and the gluons together to form hadrons. Each quark type is called °avor. For example, up, charm and top denoted respectively by u, c and t are three °avors of the uptype quark. Each °avor of quark transforms as the fundamental color triplet of SU(3)c while the gauge bosons, the gluons, are assigned to the adjoint octet representation of SU(3)c. In this case, we have eight gluons associated with the eight SU(3)c generators. The SU(2)L £ U(1)Y is the gauge group of the Glashow, Wein berg, and Salam model [1] which successfully combines the electromagnetic and weak interactions in one theory called electroweak theory. The total number of generators of SU(2)L £ U(1)Y is four. Accordingly, this theory contains four electroweak gauge bosons (three of them conventionally are denoted Wi and the forth one is denoted B). This SU(2)L£U(1)Y symmetry is respected above roughly 100 GeV (the electroweak scale). The electromagnetic interaction arises below the electroweak scale where the electroweak symmetry is broken spontaneously by the Higgs mechanism. In order to understand how the electroweak symmetry breaking is implemented in the SM, let us ¯rst point out that the invariance of the Lagrangian for both quantum electrodynamics (QED) and QCD under local gauge transformations leads respectively to massless photons and gluons. However, this idea can not be applied 1 SU(3)c £ SU(2)L £ U(1)Y Li= 0 BB@ ºi ei 1 CCA , (1,2,¡1) eci , (1,1,2) Qi= 0 BB@ ui di 1 CCA , (3,2, 1 3 ) uci , (3,1,¡4 3 ) dci , (3,1, 2 3 ) H= 0 BB@ H+ H0 1 CCA , (1,2,1) Table 1.1: The transformation of the lepton (Li,ec), quark (Q,uc,dc), and Higgs (H) ¯elds under SM gauge group SU(3)c £ SU(2)L £ U(1)Y . to the weak interaction since the gauge bosons of the weak interaction are massive (of order 90 GeV). One way out of this problem is to consider the situation of a hidden symmetry; the Lagrangian still respects the local gauge symmetry, but picks one of all possible ground states that result from minimizing the potential for a Higgs ¯eld as the physical vacuum which breaks the symmetry. The spontaneous symmetry breaking is implemented by including a doublet of scalar Higgs boson to the SM. The transformations of the quark, lepton and Higgs ¯elds under the SM gauge group are shown Table 1.1. In this Table, all fermion ¯elds are left handed and the generation index i runs from 1 to 3. Let us study the spontaneous symmetry breaking of the gauge group SU(2)L £ U(1)Y to U(1)em by 2 writing down the Higgs potential for the Higgs ¯eld H: V (H) = ¡¹2HyH + ¸(HyH)2; ¹2 > 0: (1.1) The above potential is invariant under the SM gauge group. Minimizing the potential V (H), one obtains hHi = h0jHj0i = v p 2 0 BB@ 0 1 1 CCA ; (1.2) where v = ¹= p ¸. The generator that remains unbroken is Q = T3 + Y 2 . Y refers to the electroweak hypercharge. Q is identi¯ed as the electric charge. The unbroken charge is easily checked by QhHi = 0: (1.3) The parameter Y needs to be adjusted such that the electric charges of the quarks and the leptons come out right. In general, the broken generators correspond to the gauge bosons that pick up mass, and the unbroken generators correspond to the massless gauge bosons. In this case, there are three broken generators associated with three massive gauge bosons (W+, W¡, Z0), and the unbroken charge Q asso ciated with massless gauge boson ° (the electromagnetic ¯eld A¹). The electroweak symmetry breaking scale is around the masses of the gauge bosons (i.e., 100 GeV). We can calculate the masses of electroweak gauge bosons by substituting the vacuum expectation value (VEV) of the Higgs ¯eld from Eq.(1.2) into the following gauge invariant kinetic term of the Higgs ¯eld: (D¹H)(D¹H)y = j@¹H ¡ ig 2 ¡!¿ : ¡! W¹H ¡ ig0 2 B¹Hj2; (1.4) where the gauge coupling constants g and g0 are associated respectively to the gauge groups SU(2)L and U(1)Y . The masses of the electroweak gauge bosons are then mW = ev 2 sin µW ; (1.5) mZ = ev 2 sin µW cos µW : (1.6) 3 The gauge coupling constants are parameterized in terms of an angle µW (known as the Weinberg angle) de¯ned as follows: tan µW = g0 g ; (1.7) and e = g sin µW. The mass term of fermions cannot be added to the Lagrangian by hand because the lefthanded and the righthanded fermions transform di®erently under SU(2)L £U(1)Y . Therefore, one employs the Higgs mechanism that generates mass to the fermions via Yukawa couplings. The Higgs ¯eld and its charge conjugate are given respectively by H = 0 BB@ H+ H0 1 CCA ~H = i¿2H¤ = 0 BB@ H¤ 0 ¡H¡ 1 CCA : (1.8) The transformation of ~H under SU(3)c£SU(2)L£U(1)Y is (1, 2, ¡1). We can write the gauge invariant Yukawa couplings as follows: LY = Y d ijdcT i HyQj + Y e ijecT i HyLj + Y u ij ucT i ~H yQj + h:c:; (1.9) where a charge conjugation C is understood to be sandwiched between the fermion ¯elds. As a consequence of spontaneous symmetry breaking, LY leads to mass terms for fermions as follows: LY = DcTMdD + UcTMuU + EcTMeE + h:c:; (1.10) where U = 0 BBBBBB@ u c t 1 CCCCCCA ; D = 0 BBBBBB@ d s b 1 CCCCCCA ; E = 0 BBBBBB@ e ¹ ¿ 1 CCCCCCA ; Uc = 0 BBBBBB@ uc cc tc 1 CCCCCCA ; Dc = 0 BBBBBB@ dc sc bc 1 CCCCCCA ; Ec = 0 BBBBBB@ ec ¹c ¿ c 1 CCCCCCA : (1.11) 4 The mass matrix elements for upand downquarks as well as charged leptons are given by MF ij = v p 2 Y F ij ; F = u; d; e: (1.12) Note that in the standard model the right handed neutrino does not exist. Therefore, the neutrinos are massless. The weak eigenstates are not eigenstates of the Hamil tonian. In order to write the Lagrangian in terms of the Hamiltonian eigenstates (i.e mass eigenstates), we need to diagonalize the fermion mass matrices given by Eq.(1.12) by means of biunitary transformation given as: V F R y MFV F L = MF diag:; (1.13) where Mu diag: = diag(mu;mc;mt); Md diag: = diag(md;ms;mb); Me diag: = diag(me;m¹;m¿ ): (1.14) The fermion mass matrices (MF ) are in general neither symmetric nor hermitian. but, MF y MF is hermitian and can be diagonalized as follows: V F L y MF y MFV F L = MFy diag:MF diag:: (1.15) The mass eigenstates (D0, U0, E0, Dc0 , Uc 0 , Ec 0) can be written in terms of the weak eigenstates as follows: D0 = V d L y D; Dc0 = V d R T Dc; U0 = V u L yU; Uc 0 = V u R TUc; E0 = V e L yE; Ec 0 = V e R TDc: (1.16) The charged current weak interactions for quarks are given as Lcc = g p 2 WyU°¹D + h:c: = g p 2 WyU 0 VCKM°¹D0 + h:c: (1.17) 5 It is clear from the above equation that the charged current W§ interactions couple to the physical u0j and d0 k quarks with a couplings matrix represented by VCKM = V u L yV d L = 0 BBBBBB@ Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb 1 CCCCCCA : (1.18) This is called the CabibboKobayashiMaskawa mixing matrix [2, 3]. It is a unitary matrix that can be parameterized by three mixing angles and one CPviolation phase: VCKM = 0 BBBBBB@ c12c13 s12c13 s13e¡i± ¡s12c23 ¡ c12s23s13ei± c12c23 ¡ s12s23s13ei± s23c13 s12s23 ¡ c12c23s13ei± ¡c12s23 ¡ s12c23s13ei± c23c13 1 CCCCCCA ; (1.19) where sij = sin µij , cij = cos µij and ± is the phase factor responsible for the violation of CP symmetry [3]. All other phases can be removed by ¯eld rede¯nition. It is known experimentally that the CKM mixing angles are small (i.e s13 << s23 << s12 << 1). It is convenient to write down an expression for a CPviolation parameter which is phaseconventionindependent: ´ = ¡Im(VudV¤ ub=VcdV¤ cb): (1.20) Unlike the situation in the case of charged current interactions, no °avor mixings exist for neutral current interactions of SU(2)L £ U(1)Y at tree level and this has been con¯rmed to a great accuracy by experiments. However, °avor changing neutral currents (FCNC), which have been measured, but which are strongly suppressed, can be induced by considering higher order corrections. For example, FCNC can be induced in the process K0 $ K0 transition which arises from box diagrams shown in Fig1.1. The calculation on the K0 $ K0 mass di®erence ¢mk has been done [4], and the result is close to the experimental value of ¢mk = 3:5£10¡15 GeV. This can be considered as a successful prediction of the SM. 6 u,c,t W + W u,c,t W u,c,t u,c,t _ _ d s d s d s d s Figure 1.1: Feynman diagrams of K0 $ K0 induced by higher order corrections in the SM. 1.2 Seesaw Mechanism and Leptonic Mixing Matrix In the previous section, we have seen that SM contains left and right chiral projections for all fermions except the neutrinos. This looks unnatural. Besides, the absence of a righthanded neutrino from Eq.(1.9) leads to massless neutrinos. However, neutrino experiments indicate that the neutrinos have tiny masses. The current experimental values for neutrino masses are [7] ¢m2 21 = (7:59 § 0:20) £ 10¡5eV2; ¢m2 32 = (2:43 § 0:13) £ 10¡3eV2; (1.21) where ¢m2 2;1 = m22 ¡ m21 and ¢m23 ;2 = m23 ¡ m22 . To explain this, let us add to the SM righthanded neutrinos (ºc i ) corresponding to each charged lepton. The ºc i ¯elds transform as (1,1,0) under the SM gauge group. Thus, we can write down the Yukawa couplings for the neutrino sector as follows: Lº Y = Y º ij ºcT i ~H yLj + h:c: (1.22) With a VEV of ~H y, this gives the following neutrino Dirac mass term Lº Y = MDºcT º + h:c; (1.23) where (MD)ij = Y º ij v= p 2. Since the ºc ¯elds are singlets under the SM gauge sym metry, they can posses a gauge invariant bare mass term (Majorana mass): Lbare = 1 2 MRºcT ºc + h:c: (1.24) 7 We can write the combination of Majorana and Dirac neutrino masses as a matrix for the (º, ºc) system as: Mº = 0 BB@ 0 MTD MD MR 1 CCA ; (1.25) where MD and MR are 3 £ 3 matrices. The invariance of the righthanded neutrino mass terms under SM gauge symmetry suggests that they can be above the weak interaction scale. So after integrating out these heavy ¯elds (or equivalently by ¯nding the eigenvalues of the matrix in Eq.(1.25)), the light neutrino masses are suppressed by MR via: Mº = ¡MTD M¡1 R MD; (1.26) where MD should not exceed about 100 GeV. This idea, known as the seesaw mech anism [5], is an elegant way to explain the smallness of neutrino masses. The light neutrino mass matrix given by Eq.(1.26) can be diagonalized as: V T º MºVº = 0 BBBBBB@ m1 m2 m3 1 CCCCCCA; (1.27) with m1;2;3 being the tiny masses of the three light neutrinos. Now, we can write the leptonic charge current interaction in terms of the mass eigenstates as follows: Lcc = g p 2 [e0°¹VPMNSº0]W¡¹ + h:c: (1.28) where VPMNS = V y LVº is the leptonic mixing matrix, or the PontecorvoMakiNakagawa Sakata (PMNS) matrix [6]. In general, the PMNS matrix can be written as VPMNS = 0 BBBBBB@ Ve1 Ve2 Ve3 V¹1 V¹2 V¹3 V¿1 V¿2 V¿3 1 CCCCCCA ; (1.29) 8 which can be parameterized in terms of three Euler angles and three phases one \Dirac phase" and two \Majorana phases". The standard parametrization [7] has VPMNS = V:P where V = 0 BBBBBB@ c12c13 s12c13 s13e¡i± ¡s12c23 ¡ c12s23s13ei± c12c23 ¡ s12s23s13ei± s23c13 s12s23 ¡ c12c23s13ei± ¡c12s23 ¡ s12c23s13ei± c23c13 1 CCCCCCA (1.30) P = 0 BBBBBB@ ei® ei¯ 1 1 CCCCCCA : (1.31) Here sij = sin µij , cij = cos µij which should not be confused with the angles in the quark sector, given in 1.19. The parameters ® and ¯ are the Majorana phases, while ± is the Dirac phase. Present constraints on the neutrino mixing angles can be summarized by (2¾ error bars quoted)[8] sin2 µ12 = 0:27 ¡ 0:35; (1.32) sin2 µ23 = 0:39 ¡ 0:63; (1.33) sin2 µ13 · 0:040: (1.34) The above data can be well represented by the tribimaximal mixing of the form [9] V = 0 BBBBBB@ q 2 3 q 1 3 0 ¡ q 1 6 q 1 3 ¡ q 1 2 ¡ q 1 6 q 1 3 q 1 2 1 CCCCCCA P; (1.35) which corresponds to sin2 µ12 = 1=3, sin2 µ23 = 1=2 and sin2 µ13 = 0. No information on the Dirac phase ± and on the Majorana phases (¯, ®) is known at present. There are several thoughts to reproduce the structure in Eq.(1.35). One interesting idea is to employ the discrete °avor symmetry A4 [10] which will be further discussed in chapter 2. 9 1.3 Shortcomings of the SM and the Need for New Physics. The standard model is a trustful theory in the energy range of few 100 GeV. However, things become more obscure beyond the electroweak energy scale. Understanding how nature behaves at higher energy scales might answer many of the standard model's puzzles. For example, the SM has no real explanation of the di®erent strengths of the three gauge couplings associated with the three gauge groups. Also, there is no reason why the fermions transform under the local gauge interactions of the SM in the way shown in Table 1.1, except for the posteriori justi¯cation of ¯tting the data. Grand uni¯cation theory (GUT) provides an understanding of the origin of the three gauge couplings and consequently an understanding of three gauge groups. The GUT idea is described by a uni¯ed gauge group which necessitates a single uni¯ed gauge coupling. This uni¯ed gauge group will be broken at a certain high energy scale (GUT scale) to the SM gauge group. Thus, strong, weak and electromagnetic forces are described in the framework of a single grand uni¯ed theory. Moreover, if the uni¯ed gauge group is simple, quantization of electric charge will follow automatically because the eigenvalues of the nonabelian group generators are discrete as opposed to the eigenvalues of the abelian U(1) group generator which are continuous. The most popular simple nonabelian groups that are chosen as grand uni¯cation groups are SU(5) and SO(10). We will study these GUT groups in details in sections 1.4 and 1.5. Arbitrary Parameters The SM has 19 arbitrary parameters. 3 gauge coupling constants (gs, g, and g0 as sociated respectively with SU(3)c, SU(2)L and U(1)Y ), 9 charged fermion masses, 4 quark mixing parameters, and v, ¸ (or equivalently to Mz, mh) and the QCD µ parameter. Besides, if we consider the neutrino sector, there are at least 9 additional parameters: 3 light neutrino masses, 3 mixing angles, and 3 phases (assuming Majo 10 rana neutrinos). Thus, the SM has too many arbitrary parameters which are chosen in order to ¯t the data. On the other hand, GUTs do not contain that many arbi trary parameters. Another advantage of GUT is that the seesaw mechanism can be implemented naturally within SO(10) GUT, since the gauge structure requires the existence of ºc, as we will see in section 1.5.3. Grand uni¯cation theory describes the three interactions (strong, weak, and elec tromagnetic) by one gauge coupling constant. However, it is known that these inter actions are described by three distinct gauge couplings at low energy (E ¼ 100 GeV). So the question is how does the grand uni¯cation idea reconcile with these three disparate couplings? This question can be answered by the suggestion [11] that the three gauge coupling constants are scale dependent quantities, and if the hypothesis of grand uni¯cation holds, the three gauge coupling constants of the SM will meet to a uni¯ed value at the GUT scale MGUT . Above the scale MGUT we have one gauge coupling described by a simple uni¯ed group. The renormalization group running of the gauge couplings determines the GUT scale. In the SM, however, the gauge couplings come only close to one another forming what is called the GUT triangle as shown in Fig.1.2. This can be ¯xed by introducing new physics around the TeV scale. The most promising new physics scenario is supersymmetry, which will be further discussed below Hierarchy Problem Another problem that needs to be ¯xed is the hierarchy problem of the SM. This problem occurs because the mass of the Higgs boson receives a quadratically divergent loop correction given by: m2 HSM(phys) ' m2 HSM + c 16¼2¤2; (1.36) where m2 HSM is the Higgs mass squared parameter in the Lagrangian and the second term denotes the quadratically divergent loop correction. The cuto® scale ¤ is in 11 a1 1 a2 1 a3 1 5 10 15 0 10 20 30 40 50 60 Log10 Hm GeVL a 1 Figure 1.2: The evolution of the inverse gauge couplings ®¡1 i in the standard model (dashed lines) and in the MSSM (solid lines). terpreted as the scale at which the SM ceases to be valid. Reasonable values of the energy scale ¤ at which the new physics becomes important are chosen such that any extremely ¯netuned cancelation between the two terms on the righthand side of Eq.(1.36) is avoided. The physical Higgs boson mass mHSM(phys) has to be smaller than a few hundred GeV [12]. Therefore, reasonable values of ¤ might be around the TeV scale. A promising scenario that solves the hierarchy problem of the SM and allows the uni¯cation of the three gauge coupling constants is supersymmetry (SUSY). In order to avoid extreme ¯netuning, SUSY should exist above an energy scale of order 1 TeV which is being probed at the Large Hadron Collider. Problems in the Flavor Sector The SM does not provide an explanation for the existence of three families of fermions, and the observed masses and mixings of the fermions, and the smallness of the quark mixing angles compared to the largeness of the neutrino mixing angles. These prob lems can be understood either through GUTs and/or by adding a family symmetry. 12 Some of the features of the fermions such as the three fold replication of fermion gen erations, mixing properties of the lepton sectorthat is two large mixing angles and one small mixing anglecannot be explained successfully by GUT symmetry alone. So in order to meet these challenges, one may consider the possibility of introducing a °avor symmetry (family symmetry) group which is the symmetry between genera tions. In this case, the three known generations can be assigned to a representation of the family group. There are many possible candidates for the family symmetry group. Basically, we can divide them into two categories: continuous and discrete groups. The general feature of the global continuous groups is that they lead to undesired Goldstone bosons. On the other hand, it is suggestive to consider discrete nonabelian symmetry because in this case there is no problem with unwanted Goldstone bosons. Combining grand uni¯cation gauge symmetry and family symmetry (GGUT £ GFAM) in the framework of supersymmetric theory leads certainly to new physics beyond the SM that solves most of the standard model's puzzles. Many grand uni¯cation models with discrete family symmetry have been studied so far [13, 14, 15, 16]. In particular, employing SO(10) £ A4 symmetry may give the tribimaximal mixings structure in Eq.(1.35) [15]. 1.3.1 Supersymmetry Supersymmetry is a symmetry that relates bosons and fermions. It predicts new yet to be discovered superpartner states for each known particle in the SM. The SM particle and its supersymmetric partner belong together to the same supermultiplet which is collectively described in terms of a super¯eld. In this way a spin0 boson and a spin1/2 fermion are described as a chiral super¯eld and a spin1 vector boson and a spin1/2 fermion form a vector super¯eld. The supersymmetric extension of the SM assumes that all quarks and leptons of the SM are accompanied by their scalar su perpartners which are called respectively squarks and sleptons, and the gauge bosons 13 with their fermionic superpartners which are called gauginos. This supersymmetric extension of the SM is called Minimal Supersymmetric Standard Model (MSSM), it is minimal in the sense that it contains the smallest number of new particle states. The SM contains one Higgs doublet ¯eld to achieve electroweak symmetry breaking while the MSSM contains two Higgs doublets Hu and Hd which give mass to the uptype and downtype quarks respectively. Their superpartners are called higgsinos. This setup helps in solving the quadratic divergence correction of the Higgs mass due to the fact that the loops involving particles are canceled by the loops involving their su perpartners. Another feature in favor of the MSSM is that the gauge couplings unify around 2 £ 1016 GeV as shown in Fig 1.2. These features motivate the consideration of supersymmetric GUTs. Unlike the SM where the baryon and lepton numbers are conserved automatically, there are additional superpotential terms in the case of MSSM that are consistent with SU(3)c £ SU(2)L £ U(1)Y symmetry, which break the lepton and baryon num bers. These terms are dangerous since the lepton and baryon violating processes are strongly constrained by experiment, especially from proton stability. These un wanted terms can be prohibited by requiring the superpotential to be invariant under Rparity de¯ned by, R = (¡1)3(B¡L)+2s; (1.37) where s is the spin of the ¯eld, and B and L are the baryon and the lepton number respectively. For example B = 1=3(¡1=3) for quark (antiquark) super¯elds, L = 1(¡1) for lepton (antilepton) super¯elds, and zero for the Higgs and gauge super¯elds. Supersymmetry Breaking The supersymmetry algebra tells us that the particle and its superpartner acquire the same mass. However, this is not consistent with experiment since for instance no spin0 particle has been detected so far with the same mass as the electron. Therefore, 14 d*R ~ g~ d s g~ s _ R s~* d _ d s g~ s _ sR* ~ d _ d s g~ s _ sR* ~ d _ ~ dR s~ R d*R ~ dL* ~ sL d ~ ~ L d ~ L s~ R (a) (b) (c) Figure 1.3: These three diagrams contribute to K0 $ K0 mixing in supersymmetric models. They put constraints on the o®diagonal elements of the soft breaking scalar down mass matrix that is indicated by £. SUSY must be broken somewhere above the energy scale that has been probed so far. SUSY should preferably be broken spontaneously. In other words, the generators of the SUSY does not annihilate the vacuum. Although many models of SUSY breaking have been proposed, there is no complete theory where this is achieved satisfactorily at present. In order to maintain the remarkable cancelation of quadratic divergencies in ¯eld theoretical models, SUSY should be broken softly in the e®ective low energy theory. This can be done by assuming that the outcome of symmetry breaking is extra terms (soft terms), such as additional masses for the scalars. The common philosophy of all the scenarios of SUSY breaking is that SUSY is broken in a \hidden sector" of particles which is decoupled from the visible sector of MSSM particles. The e®ects of SUSY breaking in the hidden sector are communicated to the visible sector by messengers, resulting in the MSSM soft SUSY breaking terms. The soft SUSY breaking terms imply °avor mixing. For example, suppose ~m2 Q is not diagonal in the soft term ~ dy Li(m2 Q)ij ~ dLi. In this case, the e®ective Hamiltonian for K0 $ K0 mixing gets contributions from the box diagrams involving squarks and gluinos, such as the ones shown in Fig.1.3. The experimental value of ¢mK puts constraints on the soft SUSY breaking mixing of the three diagrams in Fig1.3. The most striking limit applies to the diagram in Fig1.3(b) [28]: jRe[ ~m2s ¤ RdR ~m2s ¤ LdL j1=2 ~m2q < ~mq £ 10¡3 500 GeV ; (1.38) 15 where ~mq is the average mass of squarks ~md and ~ms and the gluino mass has been assumed equal to the average squark mass. Thus, in order to suppress the o®diagonal entries of ~m2 Q, we need to assume the masses of the squarks are nearly degenerate. This can be achieved by adding a nonAbelian discrete symmetry group. This can be done either by grouping the ¯rst two families into an irreducible doublet [29] or by grouping all three families into an irreducible triplet of the °avor group. For example, the group could be A4, which is the smallest discrete group that contains a triplet in its irreducible representations. Another natural solution to the °avor violation problem is obtained by adopting gaugemediated supersymmetry breaking (GMSB) scenario [59, 60, 61]. In this sce nario the supersymmetry breaking is transmitted to the visible sector by SM gauge interactions. In this case the soft masses are generated through loops such that the scalar masses with the same gauge quantum number are automatically degenerate. A model based on the GMSB scenario will be discussed in chapter 4. 1.3.2 Discrete Flavor Symmetry A4 The nonabelian ¯nite group A4 is the symmetry group of even permutations of four objects. It has twelve elements and four irreducible representations (irreps): 1, 10, 100, 3s, and 3a with the multiplication rule 3 £ 3 = 1 + 10 + 100 + 3s + 3a: (1.39) For example, let (a1; a2; a3), and (b1; b2; b3) transform as triplets under A4, then the multiplication of 3 £ 3 can be decomposed as a1b1 + a2b2 + a3b3 » 1; (1.40) a1b1 + !2a2b2 + !a3b3 » 10; (1.41) a1b1 + !a2b2 + !2a3b3 » 100; (1.42) (a2b3 + a3b2; a3b1 + a1b3; a1b2 + a2b1) » 3s; (1.43) 16 (a2b3 ¡ a3b2; a3b1 ¡ a1b3; a1b2 ¡ a2b1) » 3a; (1.44) where ! = exp[2¼i=3]. One advantage of the discrete A4 symmetry is that it is the smallest group that contains a 3dimensional irrep so that the three generations of the fermions can be accommodated within this triplet. Another advantage is that the FCNC problem might be solved if one considers the combinations of A4 and SUSY SO(10) GUT. This is due to the fact that the SO(10)£A4 symmetry allows us to write down one universal mass term for the three generations of sfermions. Consequently, the degeneracy of sfermions is satis¯ed. 1.4 Minimal SUSYSU(5) We have pointed out previously that the running behavior of the three gauge couplings with energy scale indicates that they should unify at some point at a high energy scale. This uni¯cation of the gauge couplings does not occur exactly in the SM. However, in the case of the MSSM, the uni¯cation occurs with impressive precision at MGUT ¼ 2 £ 1016 GeV. This strongly suggests that MSSM might be remnant of some sort of supersymmetric grand uni¯cation theory. Therefore, it is logical to propose a larger gauge group associated with one gauge coupling constant. The ¯rst approach of ¯nding a simple gauge group that contains the SM group was the GeorgiGlashow SU(5) model [17]. In this section we will discuss this SU(5) model, its predictions and its experimental implications because it is considered the simplest example of grand uni¯cation models and it is a subgroup of SO(10). 1.4.1 SU(5) Matter Fields The SM gauge group has rank 4. Hence the rank of the grand uni¯cation group should be at least 4. There are many possibilities for a rank 4 simple group with one gauge couplings. Among all possibilities, SU(5) is found to be the only choice that meets all the required features: It has complex representation for fermions and it 17 accommodates both integer and fractionally charged fermions. The 15 lefthanded SM fermions for one family can be embedded into just two irreps, the antifundamental 5F and the twoindex antisymmetric tensor 10F . This can be seen by writing the decomposition of 5F and 10F irreps of SU(5) under SU(3)c £ SU(2)L £ U(1)Y as follows: 5 = (3; 1; +2=3) © (1; 2;¡1); 10 = (3; 1;¡4=3) © (3; 2; +1=3) © (1; 1; +2): (1.45) Also, this embedding can be depicted in matrix representation as 5 = 0 BBBBBBBBBBBBBB@ dc1 dc2 dc3 e¡ º 1 CCCCCCCCCCCCCCA ; 10 = 1 p 2 0 BBBBBBBBBBBBBB@ 0 uc 3 uc 2 u1 d1 ¡uc 3 0 uc 1 u2 d2 ¡uc 2 ¡uc 1 0 u3 d3 ¡u1 ¡u2 ¡u3 0 e+ ¡d1 ¡d2 ¡d3 ¡e+ 0 1 CCCCCCCCCCCCCCA : (1.46) This assignment is free of chiral anomalies. In the SUSY version of SU(5), these multiplets are promoted to super¯elds. 1.4.2 Higgs Sectors and Yukawa Couplings in the minimal SUSYSU(5) In order to test the viability of minimal SUSY SU(5), let us ¯rst construct the invariant Yukawa couplings by writing down the SU(5) decomposition of all possible multiplications of the irreps 5 and 10. 5 £ 5 = 10 + 15; (1.47) 10 £ 10 = 5 + 45 + 50; (1.48) 5 £ 10 = 5 + 45: (1.49) It is easy to check that the MSSM super¯eld Higgs doublet Hu is contained in 5 and 45, and Hd in 5 and 45. Therefore, two quintets 5H and 5H are introduced minimally in 18 the SUSYminimal SU(5). These two quintets are responsible for breaking SU(3)c £ SU(2)L £ U(1)Y to SU(3)c £ U(1)em. Based on the above analysis, the invariant superpotential that contains only the Yukawa couplings is given as follows: ^ f 3 Y u ®¯²ijklm10ij F®10kl F¯5mH + Y d ®¯10ij F®5Fi¯5Hj : (1.50) The mass matrices generated by the VEVs of the the SU(2)L doublets in both 5H and 5H then read Md = ML = Y dh5Hi; Mu = Y uh5Hi: (1.51) Since the ¯rst term in Eq.(1.50) contains two identical 10s, the upquark Yukawa couplings are symmetric in the generation indices, i.e., Mu = M> u . Diagonalization of the down quarks and charged leptons mass matrix leads to me = md m¹ = ms m¿ = mb: (1.52) Note that the above mass relations are only valid at mass scales where the SU(5) is a good symmetry. But the light fermion masses are observed at low energy scale of order (25) GeV. Therefore, the above mass relations should be extrapolated to low energy scale. The results are the following: the ¯rst two mass relations in Eq.(1.52) are violated by experiment, while the third one is considered as a successful prediction of minimal SUSY SU(5). One way to correct the bad mass relations for the ¯rst and second generations is to employ the 45H [18]. In this case, the price that we have to pay is including several Higgs multiplets. It is obvious that the Higgs multiplets 5H and 5H do not break SU(5) to SU(3)c£ SU(2)L £ U(1)Y since they do not contain a SM singlet. The smallest dimensional Higgs representation that contains the SM singlet is the adjoint of SU(5). The adjoint Higgs representation 24H decomposes under SU(3)c £ SU(2)L £ U(1)Y to 24H = (1; 1; 0) © (8; 1; 0) © (1; 3; 0) © (3; 2;¡5=6) © (3; 2; +5=6); (1.53) 19 and the (1,1,0) component can acquire a GUTscale VEV. Equivalently, one can show [19] h24Hi = ¾ 0 BBBBBBBBBBBBBB@ 2 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 ¡3 0 0 0 0 0 ¡3 1 CCCCCCCCCCCCCCA : (1.54) The two Higgs ¯elds 24H and 5H develop hugely di®erent VEVs (i.e., h24Hi of order MGUT ¼ 1016 GeV and h5Hi of order MW ¼ 102 GeV). Consequently, this leads to a huge hierarchy of the gauge symmetry. In nonSUSY model, the param eters at tree level of the Higgs potential should be ¯netuned in order to maintain this huge hierarchy. On the other hand, this ¯netuning gets worse via radiative cor rections. However, in the minimal SUSY SU(5), once the parameters of the Higgs superpotential ^ f 3 m55H5H + m24Tr[24H24H] + ¸1Tr[24H24H24H] + ¸25H24H5H (1.55) are ¯netuned properly at tree level, the SUSY nonrenormalization theorem of Gris aru, Rocek and Siegel [67] ensures that it does not get upset by radiative corrections, since according to this theorem these parameters do not receive either ¯nite or in¯nite corrections. 1.4.3 Gauge Sector of Minimal SU(5) The adjoint representation of SU(5) has the dimension 52 ¡1 = 24. Hence, there are 24 gauge bosons associated with SU(5). They decompose under SU(3)c £ SU(2)L £ U(1)Y as given in Eq.(1.53). The gauge bosons of SM are contained within 24 gauge bosons of SU(5) as follows: (8; 1; 0) are SU(3)c gluons , (1; 3; 0) are the three SU(2)L vector ¯elds W, and (1; 1; 0) is the U(1) B¯eld. The remaining 12 gauge bosons, 20 which transform under the SM gauge group as (3; 2; 5 3 ) and (3¤; 2;¡5 3 ) are called leptoquark gauge bosons denoted respectively by X and Y . These gauge bosons can be collectively described by a 5 £ 5 matrix form, A¹ = Aa¸a=2, where ¸a are the SU(5) generators (a runs from 1 to 24) and the summation over index a is implied. As we have discussed before, the Higgs phenomenon can provide masses to the gauge bosons by developing a VEV to the Higgs ¯eld. This can be seen by writing down the invariant kinetic term of the Higgs ¯elds as follows LKE = Tr[(D¹24H)(D¹24H)¤]: (1.56) Here the covariant derivative of the adjoint representation 24H is de¯ned as follows: D¹24H = @¹24H + ig5[A¹; 24H]; (1.57) where [A¹; 24H] = A¹24H ¡ 24HA¹, and g5 is the SU(5) gauge coupling. The factor g2 5Tr[A¹; h24Hi]2 contains the mass term for the gauge bosons. Since 24H commutes with the generators of the SM gauge group, the gauge bosons of the SM (Wr, B, G®¯ ) do not pick up mass, while the X and Y gauge bosons acquire masses according to MX = MY = 5 p 2g5¾ (1.58) 1.5 Minimal SUSYSO(10) We have seen that the SM fermions can be accommodated within two irreducible representations of the simplest uni¯ed model based on SU(5) gauge symmetry. This leads to the uni¯cation of the Yukawa couplings of the down quarks and charged lep tons. On the other hand, a single 16dimensional chiral spinor of SO(10) is enough to accommodate all the SM model fermions of one generation. This brings the follow ing bene¯ts: First, the righthanded neutrino is automatically accommodated within the same multiplet. Second, the number of independent parameters of the e®ective fermion masses and mixing matrices can be reduced considerably. These observations motivate us to consider the SO(10) gauge symmetry. 21 1.5.1 Matter Fields in SO(10) GUTs The reducible spinorial representation of SO(10) splits into a pair of spinorial repre sentations 16 and 16 under a chiral projection operator, for details see Ref. [21]. All the femions reside in only one chirality of a SO(10) spinorial representation (i.e, 16 dimensional representation of SO(10)). In order to see how the SM fermions can be ¯tted within a 16dimensional irrep of SO(10), let us write down its decompositions under SU(3)c £ SU(2)L £ U(1)Y : 16 = (3; 2; +1=3) © (1; 2;¡1) © (3; 1;¡4=3) © (3; 1; +2=3) © (1; 1; +2) © (1; 1; 0); (1.59) where the quantum numbers on the righthand side (except the last one) are those for the SM fermions (see Table 1), while the last one is the righthanded neutrino. Equivalently, the 16dimensional irrep of SO(10) can be written in terms of the SU(5) basis as follows: 16 = 5 © 10 © 1; (1.60) where the matrix representations of the irreducible representations of SU(5) (5 and 10) are given in Eq.(1.46). The righthanded neutrino (or equivalently ºc) is assigned to the singlet of SU(5). 1.5.2 The Higgs Fields and Yukawa Couplings in SO(10) GUTs The Higgs sector of any realistic SO(10) model should be chosen appropriately in order to satisfy the following requirements. First, the Yukawa couplings should be invariant under SO(10) and compatible with the current data on the quark and the lepton masses and mixings. Second, the Higgs sector should lead to the proper spontaneous symmetry breaking of SO(10) gauge symmetry down to the SU(3)c £ SU(2)L £ U(1)Y of the MSSM. The invariant Yukawa couplings follow 22 from the decomposition of 16 16 = 10 © 126 © 120: (1.61) Thus, there are three types of SO(10) Higgs multiplets that can give masses to the matter fermions: the 10dimensional vector representation 10H, the 126dimensional 5index antisymmetric tensor 126H and the 120dimensional threeindex antisymmet ric tensor 120H. Then, the most general Yukawa couplings are WY = Y ®¯ 10 16F®16F¯10H + Y ®¯ 12016F®16F¯120H + Y ®¯ 12616F®16F¯126H: (1.62) The good feature of the 10dimensional Higgs multiplet of SUSYSO(10) is that 10H contains the SUSYSU(5) Higgs multiplets 5H and 5H that give masses to the uptype and the downtype quarks respectively. The fermion masses are generated by giving VEVs to the Higgs ¯elds in Eq.(1.62). The fermion masses with Higgs ¯eld belonging to the 10dimensional irrep can be calculated by writing the irreps of SO(10) matter and Higgs ¯elds in terms of SU(5) £ U1 basis as [22]: 10 = 5(2) + 5(¡2); 16 = 1(¡5) + 5(3) + 10(¡1): (1.63) where the numbers in the bracket are quantum numbers of U1. Then we construct the invariant combinations of SU(5) £ U1 multiplets as Y ®¯ 10 1F®(¡5)5F¯i(3)5j H(2) + Y ®¯ 10 ²ijklm10ij F®(¡1)10kl F¯(¡1)5mH (2) + Y ®¯ 10 5F®i(3)10ij F¯(¡1)5Hj(¡2): (1.64) We remind the reader that 5F and 10F are the usual SU(5) representations of Georgi and Glashow given in Eq.(1.46). The ¯rst line in Eq.(1.64) shows that the Dirac neutrinos and upquarks couple with the same Higgs multiplets 5H while the second line tell us that the charged leptons and down quarks couple with the other Higgs muliplets 5H. Thus, M®¯ d = M®¯ e = Y ®¯ 10 h5Hi M®¯ u = M®¯ º = Y ®¯ 10 h5Hi: (1.65) 23 The above fermion mass matrices are symmetric. Since the up and down quark mass matrices in Eq.(1.65) can be diagonalized by the same unitary matrix, the quark mixing matrix is an identity matrix. This can be considered as a zeroth order ap proximation for the CKM mixing matrix. The 120dimensional Higgs representation is antisymmetric under the °avor index, however it contributes to mixings between various generations. On the other hand, the 126dimensional is symmetric under the °avor index and by itself would lead to the following mass relations [21]: Me = ¡3Y126v126 d = ¡3Md; Mº = ¡3Yºv126 d = ¡3Mu: (1.66) A realistic Higgs spectrum would include, for example, 10H © 126H. In order to achieve the spontaneous symmetry breaking of SO(10) gauge symmetry down to SU(3)c£SU(2)L£U(1)Y a (GSM) of the MSSM, we need to consider all possible Higgs ¯elds that contain GSM singlet in their decomposition under the SM gauge group such as 45H, 54H, 210H and 126H. Since SO(10) is a rank 5 group, there are many symmetry breaking chains leading to the rank4 GSM. The most common breaking chains and the Higgs representation that has been used to break the intermediate symmetries at each step are represented in Fig1.4. In any SO(10) breaking chain, there must be a Higgs multiplet capable to break the considered symmetry down to the subsequent one by giving a VEV to the com ponent that transforms as a singlet under the lower intermediate symmetry group. Being a rank 5 group, there should be at least two Higgs ¯elds to break SO(10) down to the SM. One is needed to break the rank of SO(10) from 5 to 4 while the other breaks the remnant symmetry down to the SM gauge group. There are two simple choices of the Higgs ¯elds that not only break the rank of SO(10) but also give a superlarge mass to the right handed neutrino as shown in section 1.5.3. The choices are an antisymmetric ¯ve index tensor 126H or a spinor 16H. In either case, there 24 16 + 16 or 126 + 126 16 + 16 or 126 + 126 16 + 16 or 126 + 126 45H or 54H 45H 54H or 210H SU(5) U(1) SU(4)c SU(2) L R SU(2) (1) (2) SO(10) (1) (2) GSM (1) (2) SO(10) (1) B L SU(3)c SU(2) U(1) _ L R SU(2) (2) GSM (1) (2) SO(10) (1) (2) GSM Figure 1.4: The most common breaking chains of SO(10) gauge group to the SM gauge group (GSM) should be a Higgs ¯eld in the conjugate representation, 126H or 16H, to go along with it, in order to obtain Dterm cancelation and consequently maintain the invariance of supersymmetry down to the electroweak scale. Breaking the rank of SO(10) by either 16H or 126H leaves SU(5) unbroken because both 16H and 126H contain a SU(5) singlet in their decomposition under SU(5) as shown below [22]: 126H = 1 © 5 © 10 © 15 © 45 © 50; 16H = 1 © 5 © 10: (1.67) Therefore, a second Higgs ¯eld is needed to break SU(5) down to the SM. The appropriate Higgs multiplets of SO(10), that can break SU(5), should contain a 24dimensional representation with neutral U(1) charge in their SU(5) £ U(1) com ponents (recall that the adjoint of SU(5) (24H) is used to break SU(5) to G321 of SM). For example, the decomposition of the following Higgs multiplets 45H , 54H, 25 and 210H under SU(5) £ U(1) [22] 45H = 1(0) © 10(4) © 10(¡4) © 24(0); 54H = 15(4) © 15(¡4) © 24(0); 210H = 1(0) © 5(¡8) © 5(8) © 10(4) © 10(¡4) ©24(0) © 40(¡4) © 40(¡4) © 75(0) (1.68) makes them capable of breaking SU(5) down to the SM. There are two approaches that have been adopted so far in order to break the SO(10) gauge group to the SM gauge group. One uses large Higgs representations such as 210H, 126H, and 126H [23]. Although this approach has the advantage that R parity is automatic, the uni¯ed gauge coupling diverges in this case just above the GUT scale. On the other hand, the other approach uses only small Higgs representations [24, 25]. This choice of Higgs representations guarantees that the theory is perturbative up to the Planck scale [26] and also has the potential to arise from string theory. Therefore, we shall adopt the simplest breaking scheme; a pair of spinors 16H and 16H is used to break the rank of SO(10) and only one adjoint 45H is used to break SU(5). The general VEV direction of 45H required to break SU(5) gauge symmetry is given by [19] h45Hi = diag(b; b; a; a; a) i¿2: (1.69) The h45Hi is proportional to the generator of B¡L when b = 0 and it is proportional to T3R when a = 0. The former VEV direction is preferred in the DimopoulosWilczek (DW) [27] mechanism in order to solve the doublettriplet splitting problem. 1.5.3 Neutrino Masses The existence of righthanded neutrinos is important to understand the smallness of the neutrino mass as we have seen in the seesaw mechanism in the context of SM. The accommodation of righthanded neutrinos within the 16dimensional irreps of SO(10) 26 indicates that the seesaw mechanism can be implemented in SO(10) models. In order to see this, let us assume that the only source for the quark and lepton masses is the 10dimensional Higgs representation of SO(10), causing Mu = Mº. The following coupling Y12616F 16F 126H; (1.70) can be used to generate a Majorana mass term for righthanded neutrinos by giving VEV to the SU(5) singlet component of 126H, so the combination of the Dirac and Majorana neutrino mass terms are given by L = ºcMDº + 1 2 MRºcº; (1.71) Here MR = Y126h1(126H)i = Y126v126 and the notation p(q) refers to p of SU(5) contained in q of SO(10). This can be written in a 2£2 mass matrix for the (º,ºc) system as given in Eq.(1.25). If we ignore the mixing among generations, the light neutrino masses for the three generations are given by mºe ¼ m2 u MR1 ; mº¹ ¼ m2c MR2 ; mº¿ ¼ m2t MR3 ; (1.72) where we have used MD = Mu. The magnitude of the scale h1(126H)i is model dependent. For example, if the MSSM is a valid symmetry all the way until the GUT scale, then v126 = MU ¼ 2£1016. It is important to point out that the assumption we have made that the fermion masses arise only from 10H is not good, because it leads to the undesirable relation md=ms = me=m¹. Therefore, we need additional ¯elds, in order to have a realistic SO(10) GUT model. Another way to give Majorana masses to righthanded neutrinos is by using a bilinear product of 16H. The relevant interaction is the e®ective nonrenormalizable 27 interaction fij16i16j16H16H=M which may arise from integrating out a heavy state with mass M. Several realistic models were published along these lines [30]. By giving a VEV to the component of 16 in the SU(5) singlet direction, the righthanded neutrino mass matrix is generated as follows: MRij = fij h16Hi2 M : (1.73) If we assume that both 16H and 16H break the rank of SO(10) at the GUT scale, then h16Hi ¼ 2 £ 1016 GeV. In order to obtain the heaviest right handed neutrino mass to be of order 2 £ 1014 GeV, the mass of the heavy state should be around the Planck scale (2 £ 1018 GeV) [31] One advantage of 126H is that it leads to a theory that conserves R parity auto matically. This is because 126H breaks B¡L by two units. Plugging B¡L = 2 back into the R formula in Eq (1.36), one can see that R parity remains invariant even after symmetry breaking. While in the case of 16H, B ¡ L is broken by one unit, then R parity is not conserved after symmetry breaking. However, the superpotential terms that contain 16H and break B ¡L by one unit can be avoided by imposing a discrete symmetry. Besides, as we mentioned in the previous section, the choice of 16H and 16H is inspired by string theory, and the fact that using small Higgs representations leads to make the uni¯ed gauge coupling perturbative up to the Planck scale. 28 CHAPTER 2 Fermion Masses and Mixings in a Minimal SO(10) £ A4 SUSY GUT We have seen that the GUT models unify the strong and electroweak interactions into a simple group. The simplest GUT model is based on SU(5) gauge symmetry. The minimal SU(5) model predicts a good mass relation for the third generation (i.e., m0b = m0¿ at GUT scale). However, it gives bad prediction for the ¯rst and second generation masses (i.e., m0s = m0 ¹, m0 d = m0e at the GUT scale). In addi tion, SU(5) does not naturally accommodate the righthanded neutrino. On the other hand, SO(10) models accommodate all chiral fermions of one generation plus a right handedneutrino within a 16dimensional irreducible representation (irrep). Also, minimal SO(10) with only 10H involved in Yukawa couplings leads to the up quark mass matrix being proportional to the down quark mass matrix, so it is consid ered a good zeroth order approximation for CKM mixings. Models based on SO(10) symmetry, without including any family symmetry, were proposed to explain most of the features of quarks and leptons [32, 33]. However, one is not really fully satis¯ed with only producing the fermion masses and mixing angles without explaining why we have three generations and without understanding the relation among generations, such as the mass hierarchy and features of the mixing angles. For example, the °a vor symmetry A4 [34] can be employed to explain why the observed neutrino mixing matrix is in very good agreement with the so called tribimaximal (TBM) mixing structure given by Eq(1.35). Thus, it may be important to consider the underlying family symmetry. One of the best candidates for °avor symmetry is the nonAbelian discrete symmetry A4, for the following reasons. First, it is the smallest group that 29 has a 3dimensional irrep. Second, SUSYSO(10)£ A4 symmetry solves the FCNC problem since the scalar fermions, which belong to the 16irrep of SO(10) and trans form as a triplet under A4, have degenerate masses. Finally, it was shown that the TBM mixing structure for the neutrinos can be obtained by imposing A4 symmetry [34]. Several models based on the SO(10)£A4 group have been studied [14, 15, 16]. In these models, large Higgs representations are employed. For example, in Ref.[16], the authors employed a (126H,3) representation, where the ¯rst (second) entry indicates the transformation under SO(10) (A4), in order to produce the fermion masses and mixing angles for both normal and inverted neutrino mass spectra. Besides employ ing the large Higgs representation 126H, the models in Refs.[14, 15] contain more than one adjoint 45H representation. It has been shown that only one adjoint Higgs ¯eld is required to break SO(10) while preserving the gauge coupling uni¯cation [35]. Also, using large Higgs representations like 126H leads to the uni¯ed gauge coupling being nonperturbative before the Planck scale, which might be hard to obtain from superstring theory [36]. Therefore, the purpose of this chapter is to construct an SO(10)£A4 model in which SO(10) is broken to the standard model (SM) group in the minimal breaking scheme. This means using only a spinorantispinor (16H,16H) to break the rank of SO(10) from ¯ve to four, and the righthanded neutrino gets a heavy mass from the antispinor Higgs ¯eld (16H). Then one adjoint representation 45H is used to break the group all the way to the SM group. Recently, a numerical analysis for quark and charged lepton masses and mixings based on nonsupersymmet ric SO(10) without °avor symmetry was done [33]. The authors did not include the neutrino sector in the numerical ¯tting. Their result for the atmospheric angle was sin µatm = 0:89. However, as this work shows, when the neutrino sector is included, not only is the result a better ¯t for the atmospheric angle sin µatm = 0:776, but the known light neutrino mass di®erences are also accommodated. 30 This chapter is organized as follows. In section 2.1, a general structure of the fermion mass matrices for the second and third generations is constructed. Then, based on that structure, the fermion mass hierarchy and relations are explained. In section 2.2, it is shown that introducing several 10plets of matter ¯elds to the model leads to the doubly lopsided structure which produces large neutrino mixing angles and small quark mixing angles simultaneously [37]. Then, some analytical expressions for quark masses and mixing angles at the GUT scale are derived in a certain approximation on the model parameters. In Sec 2.3, an exact numerical analysis is done to ¯nd the outputs at the GUT scale. To get predictions of fermion masses and mixings at low scale, the quark masses and mixings at the GUT scale will be run to the low scale by using renormalization group equations. section 2.4 shows how to get a suitable righthanded neutrino mass structure that gives the correct ¯ts for the atmospheric angle after adding the charged lepton contribution. 2.1 Fermion Mass Structure in SO(10) £ A4 Symmetry In this section, the renormalizable Yukawa couplings of the SM fermions with the extra spinorantispinor matter ¯elds are considered as a concrete example of the model. The known matter ¯elds of the SM (quarks and leptons) plus the right handed neutrino are contained in the three spinors (16,3). The ordinary fermions, 16i, do not couple with 45H in the minimal SO(10). As a result, some of the predictions of the minimal SO(10) such as m¹ = ms and mc=mt = ms=mb will follow; these are badly broken in nature. Therefore, extra heavy fermion ¯elds must be introduced in order to allow the 45H to couple directly with the quarks and leptons of the standard model. The transformation of the ordinary fermions and the extra matter ¯elds under A4 and the additional symmetry Z2 £ Z4 £ Z2 are summarized in Table 2.1. Let us consider ¯rst the invariant superpotential W1 under the assigned symmetry that contains the 31 16 161 16 16 16 16 16 16 1 45 1’ H H 10 i 3 3 2 2 j Hi Hj M1 M3 M2 1 Figure 2.1: This ¯gure shows a diagrammatic representation of the couplings in the superpotential W1. coupling of ordinary fermions with the spinorantispinor matter ¯elds. W1 = b116i1611Hi + b216i16210 Hi + 16116345H + a16316210H +M1161161 +M2162162 +M3163163: (2.1) Table 2.2 summarizes the transformation of the Higgs ¯elds that are needed to achieve a minimum breaking scheme as well as the Higgs singlets that are needed to break the A4 symmetry. Although in this model, the structure in Eq.(2.1) does not include the Yukawa term 16i16i10H which is forbidden by the discrete symmetry Z2 £ Z4 £ Z2, the ordinary standard model fermions get their masses through their coupling with heavy extra ¯elds. This is similar to how the light neutrinos get their masses through coupling with the heavy righthanded neutrinos in the known seesaw mechanism. The coupling terms in the superpotential W1 can be represented diagrammatically as shown in Fig.2.1. After integrating out the heavy states, the approximate e®ective operators can be read from the diagram, i.e., Wij ¼ X ij 16i16jh45Hih10Hih1Hiih10 Hji M1M2M3 : (2.2) The VEVs of the Higgs ¯elds can be written down in a general form as h45Hi = Q; (2.3) h1Hii = 0 BBBBBB@ ²1 ²2 ²3 1 CCCCCCA ; (2.4) 32 SO(10) 16i 161,161 162,162 163,163 1ci A4 3 1 1 1 3 Z2 £ Z4 £ Z2 +,+,+ +,,+ ,+,+ +,+, +,+,+ SO(10) 10i 100 i 1000 i 10000 i 1i A4 3 3 3 3 3 Z2 £ Z4 £ Z2 +,i,+ +,¡i,+ +,i, +,¡i, +,¡i,+ Table 2.1: The transformation of the matter ¯elds under SO(10)£A4 and Z2£Z4£Z2. h10 Hii = 0 BBBBBB@ s1 s2 s3 1 CCCCCCA ; (2.5) h5(10)i = vu; h5(10)i = vd: (2.6) Here the notation hp(q)i refers to a p of SU(5) contained in a q of SO(10). The Q from Eq.(2.3) is a linear combination of SO(10) generators. One can rede¯ne, without loss of generality, the light fermion states as 161²1 + 162²2 + 163²3 = ²160 3; 161s1 + 162s2 + 163s3 = S(160 2sµ + 160 3cµ); (2.7) where ² = q ²21 + ²22 + ²23 and S = q s21 + s22 + s23 . In terms of the rede¯ned light fermion states, after dropping the prime notation and plugging in the VEVs, one gets W0 ¼ ²Sh10Hi M1M2M3 (163162Q(163)sµ + 163163Q(163)cµ): (2.8) In general, the above e®ective operator can be written in terms of quark and lepton ¯elds as WF ¼ ²Sh10Hi M1M2M3 (F3Fc 2QF sµ + Fc 3F2QFcsµ + F3Fc 3 (QF + QFc)cµ): (2.9) Here F is a general notation for up quarks (U), neutrinos (N), charged leptons (L), and down quarks (D). The quantity QF (QFc) refers to the assigned charge of the 33 lefthanded fermion (charge conjugate of the righthanded fermions) after breaking the SO(10) group down to the SM group. The unbroken charge Q can be written as a linear combination of two generators that commute with SU(3)c £SU(2)L £U(1)Y as: Q = 2I3R + 6 5 ±( Y 2 ); (2.10) where I3R is the third generator of SU(2)R and Y is the hypercharge of the Abelian U(1) group. The charge Q for di®erent quarks and leptons is given by. Qu = Qd = 1 5 ±; Quc = ¡1 ¡ 4 5 ±; Qdc = 1 + 2 5 ±; Ql = Q¹ = ¡ 3 5 ±; Qlc = 1 + 6 5 ±; Qºc = ¡1: (2.11) Eq.(2.9) can be expressed in the following matrix form: WF ¼ µ Fc 1 Fc 2 Fc 3 ¶ ( ²Sh10Hi M1M2M3 ) 0 BBBBBB@ 0 0 0 0 0 QF sµ 0 QFcsµ (QF + QFc)cµ 1 CCCCCCA 0 BBBBBB@ F1 F2 F3 1 CCCCCCA : (2.12) Some factors that arise from doing the algebra exactly should be included in the above mass matrix as we are going to see later. Finding these factors that we have assumed to be of order one is important in the °avor violation analysis. The ¯rst feature of the general mass matrix of the light fermions in Eq.(2.12) is an explanation for the mass hierarchy between the second and third generations in the limit sµ ! 0. It is remarkable that a relation among generations is related to the vacuum alignment of the A4 Higgs. Another feature of the above light fermion mass matrix m0b = m0¿ is obtained through MD33 = ML33, which follows from the relation Qdc + Qd = Qlc + Ql. This relation occurs because both down quarks and charged leptons get their masses from the same Higgs. A further consequence of the light fermion mass structure is that m0s 6= m0 ¹. This inequality relation follows from m0 ¹=m0s = L32L23=D32D23 = QlcQl=QdcQd, which 34 SO(10) 10H 45H 16H 16H 1Hi 10 Hi 100 Hi 1000 Hi A4 1 1 1 1 3 3 3 3 Z2 £ Z4 £ Z2 ,+, +,, +,¡i,+ +,¡i,+ +,,+ ,+,+ +,+, +,i,+ Table 2.2: The transformation of the Higgs ¯elds under SO(10)£A4 and Z2£Z4£Z2. is not necessarily equal to 1. This leads to the following question: What VEV direction should be given to 45H in order to obtain the GeorgiJarlskog relation jm0 ¹j = 3jm0s j? There are two choices, either ± ! 0 or ± ! ¡1:25. The former choice gives the unwanted relation (m0c =m0t )=(m0s =m0¿ ) ! 1, while the latter leads to (m0c =m0t )=(m0s =m0¿ ) ! 0. Thus, a good ¯t for ± should be around ¡1:25. 2.2 Extension to the First Generation and Doubly Lopsided Structure In this section, vector 10plet fermions are added to the model to generate masses and mixings of the ¯rst generation. These vector multiplets do not contribute to the upquark mass matrix since 10plets do not contain a charge of (§2=3). Therefore, the upquark matrix is still rank 2, and this is consistent with m0 u m0t ¼ 10¡5 being much smaller than m0 d m0b ¼ 10¡3 and m0e m0¿ ¼ 0:3£10¡5. First, I will show how the model leads to the doubly lopsided structure by employing these vector multiplets; then some analytical expressions for masses and mixing angles of fermions at the GUT scale will be derived. Let us ¯rst consider the invariant couplings under the assigned symmetry, which can be read from the Feynman diagram in Fig.2.2. The allowed couplings in the superpotential W2 are W2 = 16i10i16H +M1010i100 i + h0 ijk100 i100 j1Hk + hijk10i10j1Hk: (2.13) The important point is that Fig.2.2 gives a °avorsymmetric contribution to the down quark and charged lepton mass matrices. In order to understand this, recall that the general product of three triplets(a1, a2, a3), (b1, b2, b3), and (c1, c2, c3)that 35 16 16 16 10 10’ 10’ 10 16 1 i i j j H Hk H i j M M 10 10 Figure 2.2: This ¯gure leads to the °avor symmetric contribution to the down quarks and charged leptons. transform as a singlet under A4 is given by h1(a2b3c1 + a3b1c2 + a1b2c3) + h2(a3b2c1 + a1b3c2 + a2b1c3): (2.14) The third term of Eq.(2.13) gives a symmetric contribution since there are two iden tical 10plets. The last term in Eq.(2.13) has been ignored by assuming the Yukawa couplings hijk to be very small. The contribution of Fig.2.2 to the mass matrices of the down quarks and charged leptons, after integrating out the extra vector multiplets is then MsL = MsD / 0 BBBBBB@ 0 c12 c13 c12 0 c23 c13 c23 0 1 CCCCCCA; (2.15) where c12, c13, and c23 are proportional to ²1, ²2, ²3, respectively. To obtain the desired fermion mass structure (the doubly lopsided structure, which is going to be explained later in this section), other couplings need to be included by employing four vector 10plets plus adding another Higgs singlet 100 iH to the model (their transformations under the assigned symmetry are shown in Tables 2.1 and 2.2). The purpose of these couplings is to give a °avorantisymmetric contribution to the downquark and charged lepton mass matrices. Since the adjoint of SO(10) (45H) is an antisymmetric tensor which changes its sign under the interchange 100 i $ 10000 i , one can consider employing the Yukawa coupling 10000 i 100 i45H. Also, due to the fact that when we write 36 16 16 10 10 10’ 10’’ 10’’’ 10’ 1 45 16 M 10 M 10 m i i i j j j j j H Hk H H 16 Figure 2.3: This ¯gure leads to the °avorantisymmetric contribution to the down quarks and charged leptons. the SO(10)vectors in the SU(5) basis such as 10i = 5i + 5i, the charged lepton and down quark contents of 5i or 5i have di®erent chiralities, the structures of matrices ML and MD therefore have opposite signs [look at the mass structures in Eqs.(2.182.19). It is important to emphasize that the minimum Higgs breaking scheme assumption does not allow us to add another adjoint to the model. Therefore, the same adjoint 45H Higgs representation that breaks the SO(10) group to the SM group is going to be used. Additional couplings to the previous superpotential can be read from Fig.2.3, i.e., W3 = 100 i1000 j 100 Hk + m1000 i 10000 i + 10000 i 100 i45H; (2.16) where h45Hi has been de¯ned previously. The VEV of the Higgs singlet 100 H is given below: h100 Hi = 0 BBBBBB@ ±1 ±2 ±3 1 CCCCCCA : (2.17) After integrating out the heavy states, the following contribution to the ML and MD is obtained: MA L / 0 BBBBBB@ 0 ¡±3Ql ±2Ql ±3Ql 0 ¡±1Ql ¡±2Ql ±1Ql 0 1 CCCCCCA ; (2.18) 37 MA D / 0 BBBBBB@ 0 ±3Qdc ¡±2Qdc ¡±3Qdc 0 ±1Qdc ±2Qdc ¡±1Qdc 0 1 CCCCCCA ; (2.19) where the overall constant has been absorbed in the rede¯nition of ±1, ±2, and ±3. Equations (2.182.19) show that the o®diagonal elements of MA D (MA L ) are propor tional to Qdc (Ql). This is because 5i(10) contains, in its representation, the charge conjugation of a color triplet of the lefthanded down quarks dc Li and the lefthanded charged leptons eLi. The full treelevel mass matrices, which are obtained by adding the three superpotentials W1 +W2 +W3, have the following forms: ML = m0 d 0 BBBBBBBBBBBBBBBBBBBBBB@ 0 c12 + 3±3(¡1+® 5 ) ¡±2® + ³ c12 ¡ 3±3(¡1+® 5 ) 0 ±1® + ¯ ¡3s(¡1+® 5 ) ³ ¡ ±2 6¡® 5 ±1( 6¡® 5 ) + ¯ 1 +s(¡1+6® 5 ) 1 CCCCCCCCCCCCCCCCCCCCCCA ; (2.20) MD = m0 d 0 BBBBBBBBBBBBBBBBBB@ 0 c12 + ±3(3+2® 5 ) ¡2±2(3+2® 5 ) + ³ c12 ¡ ±3(3+2® 5 ) 0 2±1(3+2® 5 ) + ¯ +s(¡1+® 5 ) ³ s(3+2® 5 ) + ¯ 1 1 CCCCCCCCCCCCCCCCCCA ; (2.21) MU = m0 u 0 BBBBBB@ 0 0 0 0 0 ( 1¡® 5 )s 0 (1+4® 5 )s 1 1 CCCCCCA ; (2.22) 38 MN = m0 u 0 BBBBBB@ 0 0 0 0 0 (¡3+3® 5 )s 0 s 1 1 CCCCCCA ; (2.23) the convention being used here is the lefthanded fermions multiplied from the right. The parameters of the model have been de¯ned as follows: ³ = c13 + ±2Qdc ; ¯ = c23 + ±1Qdc ; ± = ¡1 + ®; (2.24) s = sµ ( 3 5± + 1)cµ : The above fermion mass structure has eight parameters. If ® goes to zero, the fermion mass matrices in Eqs.(2.202.21) go to the SU(5) limit (m0b = m0¿ , m0s = m0 ¹, m0 d = m0e ). To avoid the bad prediction of SU(5) for lighter generations, a good numerical ¯tting for ® should deviate from zero. On the other hand, to keep the good SU(5) prediction for the third generation, the parameter ® should satisfy ® << 1. If ±1 and ±2 are of order 1 and the other model parameters are very small (¯; ³; ®; ±3; c12; s << ±1; ±2), the model leads to the doubly lopsided structure. To see this clearly, let us go to the limit where the small parameters are zero (except s). So the MD and ML go to the following form: ML = MTD = m0 d 0 BBBBBB@ 0 0 0 0 0 ( 3s 5 ) ¡±2 6 5 (¡s 5 ) + ±1( 6 5 ) 1 1 CCCCCCA : (2.25) In diagonalizing ML of Eq.(2.25), the large o®diagonal elements ±1 and ±2 that appear asymmetrically in MD and ML must be eliminated from the right by a large left handed rotation angle µsol in the 12 plane, where tan(µsol) = ¡±2 ±1 . The next step of diagonalization is to remove the large element ¾ ¼ (±2 1 +±2 2) 1 2 that has been produced 39 after doing the ¯rst diagonalization, where the (3,2) element of the matrix in Eq.(2.25) is replaced by ¾. This can be done by a rotation acting from the right by a large lefthanded angle µ23 in the 23 plane, where tan(µ23) ¼ ¡¾. On the other hand, there are no corresponding large lefthanded rotation angles in diagonalizing MD since ML = MTD . However, the large o®diagonal elements in MD can be eliminated by large righthanded rotation angles acting from the left on the MD in Eq.(2.25), while the lefthanded rotation angles are small. This explains how the doubly lopsided structure leads to small CKM mixing angles and large neutrino mixing angles simultaneously. If the parameters c12, ±3, and ³ are zero, analytical expressions can be written down for the ratios of quark and lepton masses of the second and third generations, Vcb, and neutrino mixing angles (tan µ12 and tan µ23) in terms of ±1, ±2, s , ®, and ¯: m0c m0t = s2(1 ¡ ®)(1 + 4®) 25 ; m0s m0b = ¡2(3 + 2®)(¯ + s(3+2® 5 )) q ±2 1 + ±2 2 5(1 + 4 25(3 + 2®)2(±2 1 + ±2 2)) ; m0 ¹ m0¿ = q (¡3s 5 (¡1 + ®) + ±1® + ¯)2 + ±2 2®2 q (±2 1 + ±2 2)(6 ¡ ®) 5(1 + (6¡®)2 25 (±2 1 + ±2 2)) ; (2.26) V D cb = ¯ + s(3+2®) 5 (1 + 4 25(3 + 2®)2(±2 1 + ±2 2)) ; V U cb = ¡s(1 + 4®) 5 ; tan µ12 = ±2( 6¡® 5 ) ±1( 6¡® 5 ) + s(¡1+6® 5 ) + ¯ ; tan µ23 = ¡( 6 ¡ ® 5 ) q ±2 2 + ±2 1: These expressions are derived by using the approximation ®; s; ¯ << ±1; ±2, and are useful for ¯tting the data. The best ¯t for the data is obtained by setting tan µ23 = ¡2 and tan µ12 = 0:68, which correspond to µ23 = ¡63o and µ12 = 34o. The central value of the atmospheric angle is around 45o. In order to bring 63o close to the central value, the neutrino sector is required to be included as shown in Sec 2.5. Also, it will be shown that the contribution of the neutrino sector to the solar angle is small. 40 2.3 Numerical Results The model can be shown to be concrete by giving numerical values to the parameters of the model, and producing the six mass ratios of quarks and leptons, CKM mixing angles (Vus, Vub, and Vcb), the CP violation parameter ´ = ¡Im(VubVcs=VusVcb), and neutrino mixing angles (sin µ12, and sin µ13). The ten parameters (±1, ±2, ±3, ®, ¯, s, ³, c12, m0 d, and m0 u) appearing in Eqs(2.202.23) are in general complex. Five phases of the complex parameters can be removed by rede¯ning the phases of the quark and lepton ¯elds. Then, we have ten real parameters and ¯ve phases in order to ¯t the 16 quantities appearing in Table 2.3. However, the best numerical ¯t is obtained when two parameters (±3; c12) are complex while the others are real. If ±1 = ¡1:302, ±2 = 1:0142, ±3 = 0:015 £ e4:95i, ® = ¡0:05801, s = 0:29, ³ = 0:0105, c12 = ¡0:00153e1:1126i, and ¯ = ¡0:12303, the following excellent ¯t at the GUT scale is obtained : m0c m0t = 0:002717, m0b m0¿ = 0:958, m0e m0 ¹ = 0:00473, m0 ¹ m0¿ = 0:0585, m0 d m0e = 3:63, m0s m0 ¹ = 0:302, ´ = 0:357, Vus = 0:2264, Vub = 0:0037, Vcb = 0:0362, sin µ12 = 0:569, and sin µ13 = 0:0653. The above numerical ¯ttings lead to sin µL 23 = 0:904, which is not close to the central value sin µatm 23 = 0:707. One can see from the superscript L that the mixing angle µL 23 comes only from the charged lepton contribution. To obtain close to the expected atmospheric angle and the correct neutrino mass di®erences, it is important to include the neutrino sector contribution to the atmospheric angle by ¯nding out a suitable righthanded neutrino structure which respects the assigned symmetry of the model. In order to compare with experiment, the predicted fermion masses and mixing angles at the low energy scale need to be found. The above numerical values of the fermion masses and mixing angles which are obtained at the GUT scale have been evolved to the low scale in two steps. First, the running from the GUT scale to MSUSY = 1 TeV is done by using the twoloop MSSM beta function. The running factors denoted by ´i depend on the value of tan ¯. The known fermion masses and 41 mixing data are best ¯tted with tan ¯ = 10. The running factors for tan ¯ = 10 are (´s=b, ´¹=¿ , ´b=¿ , ´c=t, ´cb= ´ub)=(0.8736, 0.9968, 0.5207, 0.73986, 0.910335), where ´i=j = (m0i =m0j )=(mi(1TeV)=mj(1TeV)) and ´cb;ub = V 0 cb;ub=Vcb;ub(1TeV). The second step is to evolve the fermion masses and mixing angles from MSUSY = 1 TeV to the low scale. The renormalization factors ´i that run fermion masses from their respective masses up to the supersymmetric scale MSUSY = 1 TeV are computed using threeloop QCD and oneloop QED, or the electroweak renormalization group equation with inputs ®s(MZ) = 0:118, ®(MZ) = 1=127:9, and sin µw(MZ) = 0:2315. The relevant renormalization equations can be found in [38][39]. The results are (´c, ´b, ´e, ´¹, ´¿ , ´t, ´ub=´cb)=(0.4456, 0.5309, 0.8188, 0.83606, 0.8454, 0.98833, 1.0151). By using the above renormalization factors, m¿ = 1776 MeV, and mt = 172:5 GeV, the following predictions at the low scale can be obtained: mc(mc) = 1:4 GeV, mb(mb) = 5:2 GeV, me(me) = 0:511 MeV, m¹(m¹) = 105:6 MeV, md(2 GeV) = 7:5 MeV, ms(2 GeV) = 132 MeV, ´ = 0:357, Vus = 0:2264, Vub = 0:004, Vcb = 0:0392, sin µ12 = 0:569, and sin µ13 = 0:0653. Note that the numerical value of mb is not in perfect agreement with the exper imental value mb = 4:20+0:17 ¡0:07 GeV [40]. In order to ¯x this, the ¯nite gluino and chargino loop corrections [41] are required to be included in the downtype quark masses (md, ms, mb). The total contributions are denoted as (1+¢d), (1+¢s), and (1+¢b). These corrections are proportional to the supersymmetric particle spectrum: ¢b ¼ tan ¯ Ã 2®3 3¼ ¹M~g m2 ~b L ¡m2 ~b R h f(m2 ~b L =M2 ~g ) ¡f(m2 ~b R =M2 ~g ) i + ¸2t 16¼2 ¹At m2 ~tL ¡m2 ~tR h f(m2 ~tL =¹2)¡ f(m2 ~tR =¹2) i´ , where f(x) = ln(x)=(1 ¡ x) and the ¯rst (second) term refers to the gluino (chargino) correction. Similar expressions exist for ¢s and ¢d, but without the chargino contribution and ~b ! ~s; ~ d. If the chargino loop corrections are neg ligible and m~ d, m~s, and m~b are degenerate, the equality relation ¢d = ¢s = ¢b is approximately satis¯ed. In order to get a better ¯tting for downtype quark masses, let us take ¢d = ¢s = ¢b = ¡0:17, which gives md(2 GeV) = 6:24 MeV, 42 m0s (2 GeV) = 109:65 MeV, and mb(mb) = 4:31 GeV. The comparison of the model predictions and experimental data at the low scale is summarized in Table 2.3, where the quark and charged lepton masses, the CKM mixing angles (Vub, Vus, Vcb), the neutrino mixing angles (sin µsol, sin µatm, sin µ13), and the CP violation parameter (´) are taken from [40]. The masses are all in GeV. Although the model here predicts mu(GUT) = 0, the quantity mud = (mu + md)=2 is considered in Table 2.3, where it is assumed that the tiny up quark mass at GUT scale may be generated either by including the coupling 16i16i10H into the model or by considering higher dimensional operators. If mu(2 GeV) = 2:4 MeV, the model predictions of the quantities mud and ms mud , which are wellknown from lattice calculations [42], are given in Table 2.3. The asterisks in Table 2.3 indicate that the model predictions of neutrino mixing angles are obtained after including the neutrino sector in section 2.5. 2.4 Right Handed Neutrino Mass Structure So far, the model gives excellent agreement with the known values for the CKM mixings, the quark masses, the charged lepton masses, the CP violation parameter, and the neutrino mixing angles (sin µ12 and sin µ13). However, the whole picture is still not complete and the following question arises. What is the appropriate light neutrino mass matrix (Mº = ¡MTN M¡1 R MN) that gives not only the correct contribution to the atmospheric angle, but also the correct neutrino mass di®erences: ¢m2 21 = (7:59 § 0:2)£10¡5eV2, j¢m2 32j = (2:43§0:13)£10¡3eV2 [37]? In other words, we are looking for a suitable structure of righthanded neutrino mass matrix MR since MN is ¯xed. Recall that the MNS mixing matrix is given by UMNS = Uy LUº; (2.27) 43 where UL and Uº are the unitary matrices needed to diagonalize the Hermitian lepton matrix My LML and the light neutrino matrix Mº, respectively. Mdiagy L Mdiag L = Uy LMy LMLUL; Mdiag º = UT º MºUº; (2.28) where Mº is assumed to be real and symmetric. The Dirac neutrino mass matrix MN in Eq. (2.23) has vanishing ¯rst row and column, and the same is true for Mº. So the matrix required to diagonalize Mº is simply a rotation in the 23 plane by an angle µº, while Uy L is determined numerically from the charged lepton mass matrix. Thus, the mixing matrix of neutrinos is given by UMNS = 0 BBBBB@ ¡0:14 ¡ 0:81i 0:13 + 0:55i 0:065 0:25 + 0:06i 0:34 ¡ 0:04i 0:90 ¡0:51 ¡0:75 0:42 1 CCCCCA 0 BBBBB@ 1 0 0 0 cos µº sin µº 0 ¡sin µº cos µº 1 CCCCCA : (2.29) One can conclude that the correct contribution of the neutrino sector to the atmo spheric angle is around µº=¡20o. For example, if we take µº=¡20o, the neutrino mixing angles (sin µatm, sin µsol, sin µ13) become (0.707, 0.53, 0.21). In order to ¯nd the suitable righthanded neutrino mass structure, one can easily prove the inverse of the seesaw relation, MR = ¡MNUº(Mdiag º )¡1UT º MT º : (2.30) A similar technique was used in Ref [43]. Note that one of the eigenvalues of Mº is zero (i.e. Mdiag º is singular), so the inverse of Mdiag º does not exist. To overcome this problem, one can generally de¯ne Mdiag º =diag( m1, m2, m3 ), and m1 will not appear in MR. By using the numerical result of MN, µº=¡20o, and m2/m3=0.178, the righthanded mass structure can be presented numerically. 0 BBBBBB@ 0 0 0 0 0:0186 ¡0:13 0 ¡0:13 1 1 CCCCCCA : (2.31) 44 16i 1 1 1 1 16j c l k Hm H m1 16H H 16 i j c l 1’’’ 1’’’ Figure 2.4: This ¯gure leads to the righthanded neutrino mass matrix. From the above numerical mass matrix, one concludes (MR)23 £ (MR)23 ¼ (MR)22, so to a good approximation, the above numerical structure can be represented ana lytically as follows: 0 BBBBBB@ 0 0 0 0 r2 ar 0 ar 1 1 CCCCCCA : (2.32) The constant a should not be equal to 1 because then MR would be singular. Now our mission is to ¯nd the Yukawa couplings that respect the symmetry of the model and lead to an analytical structure similar to Eq.(2.32). This can be accomplished by considering the following Yukawa couplings represented by the Feynman diagram in Fig.2.4, i.e., W4 = 16i16H1i + hijk1i1cj 1000 Hk + m11ci 1ci ; (2.33) where two fermion singlets 1i and 1ci , which couple with the singlet Higgs 1000 iH, have been introduced (their transformation under SO(10)£A4 and the additional symme try are shown in Tables 2.12.2). The product of the three triplets of the second term in Eq. (2.33) that transform as a singlet under A4 is given by h1(N1Nc 2®3 + N2Nc 3®1 + N3Nc 1®2) + h2(N1Nc 3®2 + N3Nc 2®1 + N2Nc 1®3), where ®1, ®2, and ®3 are the VEV's components of 1000 iH. By assuming h1=h2, Fig.2.4 leads to the desired right 45 handedneutrino mass structure. MR = ¤ 0 BBBBBB@ ®21 ®23 ®1®2(¡1 ®23 + 2 ®21 +®22 +®23 ) ¡®1(®21 ¡®22 +®23 ) ®3(®21 +®22 +®23 ) ®1®2(¡1 ®23 + 2 ®21 +®22 +®23 ) ®22 ®23 ¡®2(¡®21 +®22 +®23 ) ®3(®21 +®22 +®23 ) ¡®1(®21 ¡®22 +®23 ) ®3(®21 +®22 +®23 ) ¡®2(¡®21 +®22 +®23 ) ®3(®21 +®22 +®23 ) 1 1 CCCCCCA : (2.34) By comparing the 23 block of the above structure with the mass structure in Eq. (2.32), one can see the constant a is equivalent to the quantity ((¡®2 1 + ®2 2 + ®2 3)=(®2 1 + ®2 2 + ®2 3)), which is equal to 1 in the limit ®1 ! 0. So, let us expand the eigenvalues of the right handed neutrino mass structure in Eq(2.34) around ®1. MR1 = 1 + ®2 2 ®2 3 + ®2 1(®4 2 ¡ 6®2 2®2 3 + ®4 3) ®2 3(®2 2 + ®2 3)2 + O(®4 1); MR2 = 4®2 1®2 2 (®2 2 + ®2 3)2 ¡ 8®3 1®3 2®3 (®2 2 + ®2 3)7=2 + O(®4 1); (2.35) MR3 = 4®2 1®2 2 (®2 2 + ®2 3)2 + 8®3 1®3 2®3 (®2 2 + ®2 3)7=2 + O(®4 1): One can see that two of the righthanded neutrino masses are approximately de generate for small values of ®1 (i.e. MR2 ¼ MR3). By setting (®1, ®2, ®3, ¤)=(¡0:05, 0.125, 0.994, 8:42 £ 1015), the numerical ¯t for the neutrino mixing angles, the light neutrino masses, and the right handedneutrino masses are obtained as follows: m1 = 0 eV; sin µsol = 0:551; MR1 = 8:57 £ 1015 GeV; m2 = 0:01 eV; sin µatm = 0:776; MR2 = 1:3 £ 1012 GeV; m3 = 0:056 eV; sin µ13 = 0:154; MR3 = 1:28 £ 1012 GeV: As can be seen from Table 2.3, the masses and mixing angles of the quarks and leptons after including the neutrino sector are predicted in this model to be within 2¾ error bars of their experimental values. 46 Model predictions Experiment Pull me(me) 0.511£10¡3 0.511£10¡3 ... m¹(m¹) 105.6£10¡3 105.6£10¡3 ... m¿ (m¿ ) 1.776 1.776 ... mud 4:32 £ 10¡3 (3:85 § 0:52)£10¡3 0.9 mc(mc) 1.4 1:27+0:07 ¡0:11 1.85 mt(mt) 172.5 171.3§2.3 0.52 ms mud 25.36 27:3 § 1:5 1.29 ms(2Gev) 109.6£10¡3 105+25 ¡35 £ 10¡3 0.184 mb(mb) 4.31 4:2+0:17 ¡0:07 0.58 Vus 0.2264 0.2255§0.0019 0.473 Vcb 39.2£10¡3 (41.2§1.1)£10¡3 1.82 Vub 4.00£10¡3 (3.93§0.36)£10¡3 0.194 ´ 0.3569 0:349+0:015 ¡0:017 0.526 sin µsol 12 0.551 0.566§0.018 0.83 sin µatm 23 0.776 0.707§0.108 0.63 sin µ13 0.154 < 0:22  Table 2.3: This Table shows the comparison of the model predictions at low scale and the experimental data. 47 CHAPTER 3 Flavor Violation in a Minimal SO(10) £ A4 SUSY GUT Flavor changing neutral current (FCNC) processes impose severe constraints on the soft supersymmetric breaking (SSB) sector of the minimal supersymmetric standard model (MSSM). The simplest way to satisfy the FCNC constraints is to adopt univer sality in the scalar masses at a high energy scale where the e®ects of supersymmetry (SUSY) breaking in the hidden sector is communicated to the scalar masses of MSSM via gravitational interactions. For example, in the the minimal supergravity model (mSUGRA) [44] the MSSM is a valid symmetry between the weak scale and grand uni¯cation scale (MGUT) at which the universality conditions are assumed to hold. In this case, the leptonic °avor violation (LFV) is not induced. However, in a di®erent class of models studied in Refs [45, 46, 47, 48, 49, 50] the universality of the scalar masses will be broken by radiative corrections. Consequently, FCNC will be induced in these models as discussed below. If the universality conditions hold at the grand uni¯cation scale MGUT, the LFV is induced below GUT scale by radiative corrections in the MSSM with righthanded neutrino [45, 46, 47] or SUSYSU(5) [48] models. Unfortunately, it is di±cult to predict LFV decay rates in these models because the Dirac neutrino Yukawa couplings are arbitrary within MSSM. However, in an SO(10) GUT model, we can predict the LFV decay rates below the GUT scale because the Dirac neutrino couplings are related to the uptype quark Yukawa couplings and are thus ¯xed. The FCNC could also be induced above the GUT scale by radiative corrections. 48 It was shown that as a consequence of the large top Yukawa coupling at the uni¯ cation scale, SUSY GUTs with universality conditions valid at the scale M¤, where MGUT < M¤ · MPlanck, predict lepton °avor violating processes with observable rates [49, 50]. The experimental search for these processes provides a signi¯cant test for supersymmetric grand uni¯cation theory (SUSY GUT). Both contributions of FCNC that are induced above and below MGUT will be studied in our model. In this chapter, the °avor violation processes for charged lepton and quark sectors are investigated in the framework of a realistic SUSY GUT model based on the gauge group SO(10) and a discrete nonabelian A4 °avor symmetry [51]. This model is realistic because it successfully describes the fermion masses, CKM mixings and neutrino mixing angles. This work di®ers from other studies in several aspects. First, it is di®erent from those based on MSSM with righthanded neutrino masses or SUSY SU(5) in the sense that the Dirac neutrino Yukawa couplings are determined from the fermion masses and mixing ¯t of the SO(10) £ A4 model. Thus, this model predicts the lepton °avor violation arising from the renormalization group (RG) running from MGUT to the righthanded neutrino mass scales. Second, it is di®erent from those based on SUSY SO(10) studied in [52] in the sense that the FCNC processes are closely tied to fermion masses and mixings. Finally, in the SO(10)£A4 model °avor violation is induced at the GUT scale at which A4 symmetry is broken due to large (order one) mixing of the third generation of MSSM ¯elds (Ã3) with the exotic heavy ¯elds (Âi, i runs from 1 to 3). This large mixing arises when the A4 °avor symmetry is broken at the GUT scale. This is di®erent from the case where the °avor violation is induced due to large top Yukawa coupling at the GUT scale [49, 50]. The reason for introducing the exotic heavy fermion ¯elds in our model is to obtain the correct fermion mass relations at the GUT scale as we shall see in section 1. The mass scales of these exotic ¯elds range from 1014 GeV to 1018 GeV depending on the values of the Yukawa couplings and the scale of A4 °avor symmetry breaking. 49 In this chapter we study °avor violation of the hadronic and leptonic processes by calculating the °avor violating scalar fermion mass insertion parameters (±AB)ij = (m2 AB)ij ~m2 , for (A;B) = (L;R), with ~m being the average mass of the relevant scalar partner of standard model fermions (sfermions). All the °avor violation sources are included in our calculations. The sfermion mass insertions, ±LL;RR;LR, arise from the large mixing between the Ã3 and Âi and the mass insertions, (±ij LL)RHN, arise from RG running from MGUT to the righthanded neutrino mass scales. These scalar mass insertion parameters are analyzed in the framework of our model; then they are compared with their experimental upper bounds. We found that the most stringent constraint on °avor violation comes from the ¹ ! e° process. This constraint requires a high degree of degeneracy of the soft masses of MSSM ¯elds and the exotic ¯elds. Therefore, in this model we assume that these soft masses are universal at the scale M¤ with M¤ > MGUT, then we run them down to the GUT scale. The branching ratio Br(¹ ! e°) close to experimental bound (i.e. Br(¹ ! e°)=1:2 £ 10¡11) is obtained when the slepton masses of order 1 TeV , while the Yukawa couplings remain perturbative at the scale M¤. We also found in the framework of our model that once the constraint from Br(¹ ! e°) is satis¯ed, all the FCNC processes will be automatically consistent with experiments. This chapter is organized as follows. In section 1, we show how the fermion mass matrices are constructed in SO(10)£A4 model. In section 2, we discuss the sources of °avor violation by ¯nding the sfermion mass insertion parameters ±ij LL;RR at the GUT scale at which A4 symmetry is assumed to be broken as well as below the GUT scale. The results of the SO(10)£A4 model regarding °avor violation analysis are presented in section 4. Section 5 has our conclusion. The derivation of the light fermion mass matrices and the light neutrino mass matrix after disentangling the exotic fermions is shown in appendix A. In appendix B, we list the renormalization group equations (RGEs) for various SUSY preserving and breaking parameters between MGUT and 50 M¤ relevant for FCNC analysis. 3.1 A Brief Review of Minimal SO(10) £ A4 SUSY GUT In the SO(10) gauge group, all the quarks and leptons of the SM are naturally accommodated within a 16dimensional irreducible representation. However, minimal SO(10) (i.e., with only one 10dimensional Higgs representation) leads to fermion mass relations at the GUT scale, such as m0c m0t = m0s m0b and m0 ¹ = m0s , that are inconsistent with experiment. This can be ¯xed by introducing exotic 16 + 16 fermions and by coupling 16i with these exotic ¯elds via 45H, which is used for SO(10) symmetry breaking. The nonabelian discrete A4 symmetry is chosen in our model because it is the smallest group that has a 3dimensional representation, so the three generations of SM ¯elds transform as triplet under A4. Besides, FCNC is not induced in the SUSY SO(10) £ A4 as long as A4 symmetry is preserved. However, as we will see later, the breaking of A4 symmetry at the GUT scale will reintroduce the FCNC via large mixing between the exotic and light ¯elds. Based on the above reasons, a SO(10)£A4 model is proposed in [51]. In this model, a minimal set of Higgs representations are used to break the SO(10) gauge group to the SM gauge group so the uni¯ed gauge coupling remains perturbative all the way to the Planck scale. Employing this minimal Higgs representation and A4 symmetry, our model successfully accommodates small mixings of the quark sector and large mixings of the neutrino sector in the uni¯ed framework as shown summarized below. The fermion mass matrices of the model proposed in [51] were constructed approx imately. In this section, we construct these matrices by doing the algebra exactly and show that the excellent ¯t for fermion masses and mixings is obtained by slightly modifying the numerical values of the input parameters of Ref.[51]. There are two superpotentials of the model. The ¯rst one (Wspin:) describes the couplings of the standard model ¯elds (Ãi(16i), i runs from 13) with the exotic heavy spinorantispinor 51 SO(10) Ãi Â1,Â1 Â2,Â2 Â3,Â3 Zc i A4 3 1 1 1 3 Z2 £ Z4 £ Z2 +,+,+ +,,+ ,+,+ +,+, +,+,+ SO(10) Ái Á0 i Á00 i Á000 i Zi A4 3 3 3 3 3 Z2 £ Z4 £ Z2 +,i,+ +,¡i,+ +,i, +,¡i, +,¡i,+ Table 3.1: The transformation of the matter ¯elds under SO(10)£A4 and Z2£Z4£Z2. SO(10) 10H 45H 16H 16H 1Hi 10 Hi 100 Hi 1000 Hi A4 1 1 1 1 3 3 3 3 Z2 £ Z4 £ Z2 ,+, +,, +,¡i,+ +,¡i,+ +,,+ ,+,+ +,+, +,i,+ Table 3.2: The transformation of the Higgs ¯elds under SO(10)£A4 and Z2£Z4£Z2. ¯elds (Âi(16i), Âi(16i), i runs from 1 to 3), while the second one (Wvect:) describes the couplings of Ãi with the exotic 10vector ¯elds (Ái, Á0 i, Á00 i , Á000 i , i runs from 1 to 3) as given below: Wspin: = b1ÃiÂ11Hi + b2ÃiÂ210 Hi + k1Â1Â345H + aÂ3Â210H +M®Â®Â®; (3.1) Wvect: = b3ÃiÁi16H +M10ÁiÁ0 i + h0 ijkÁ0 iÁ0 j1Hk + hijkÁiÁj1Hk +AijkÁ0 iÁ00 j 100 Hk + mÁ00 i Á000 i + k2Á000 i Á0 i45H: (3.2) The above superpotentials are invariant under A4 and the additional symmetry Z2 £ Z4 £ Z2. The transformations of the matter ¯elds (i.e., the ordinary and exotic fermion ¯elds) and the Higgs ¯elds under the assigned symmetry are given in Table 3.1 and 3.2. The general fermion mass matrix structure that results from integrating out the 52 exotic heavy spinorantispinor ¯elds in Wspin: is: MF (spin:) = Ã aT1T2T3f2h10Hi rF rFc ! 0 BBBBBB@ 0 0 0 0 0 QF sµ rFc f 0 QFcsµ rF f (QF + QFc)cµ 1 CCCCCCA ; (3.3) where we have made the following transformation: Ã1²1 + Ã2²2 + Ã3²3 = ²Ã03 and Ã1s1 + Ã2s2 + Ã3s3 = S(Ã02 sµ + Ã03 cµ). Here ²i and si are VEVcomponents of h1Hi and h10 Hi respectively and sµ(cµ) is sin µ(cos µ). f = (1 + T2 2 + T2 1 (1 + s2µ T2 2 ))¡1=2 and rF = (1 + Q2 FT2 3 T2 1 (1 + s2µ T2 2 )f2)1=2 are factors that come from doing the algebra exactly (see appendix A). Here T1 = b1² M1 , T2 = b2S M2 , T3 = k1 M3 and Q = 2I3R + 6 5±(Y 2 ) is the unbroken charge that results from breaking SO(10) to the SM gauge group by giving a VEV to 45H, where h45Hi = Q. The charge Q for di®erent quarks and leptons is given as. Qu = Qd = 1 5 ±; Quc = ¡1 ¡ 4 5 ±; Qdc = 1 + 2 5 ±; Ql = Q¹ = ¡ 3 5 ±; Qlc = 1 + 6 5 ±; Qºc = ¡1: (3.4) The above general structure of fermion mass matrix has the following interesting features: (1) The relation m0b = m0¿ automatically follows from Qd +Qdc = Qe +Qec , (2) The hierarchy of the the second and third masses generation is obtained by taking the limit sµ ! 0, and (3) The approximate GeorgiJarlskog relation m0 ¹ = 3m0s leads to two possible values for ±, either ± ! 0 or ± ! ¡1:25, (4) the former possibility is excluded by experiment since it leads to (m0c =m0t )=(m0s =m0b ) ! 1 at the GUT scale, while the latter possibility leads to (m0c =m0t )=(m0s =m0b ) ! 0 which is closer to experiments. Let us de¯ne ± = 1+®. The masses and mixings of the ¯rst generation arise from Wvector. The full mass matrices arising from Wspinor and Wvector have the 53 following form: MD = m0 d 0 BBBBBBBBBBBBBBBBB@ 0 (c12 + ±3( 3+2® 5 ))rdrdc (¡2±2( 3+2® 5 ) + ³)rdc (c12 0 (2±1( 3+2® 5 ) ¡±3( 3+2® 5 ))rdrdc +s(¡1+® 5 ) + ¯)rdc ³rd (s( 3+2® 5 ) + ¯)rd 1 ¡2(¯ + 3+2® 5 ±1)fcµsµT2 2 1 CCCCCCCCCCCCCCCCCA ; MU = m0 u 0 BBBB@ 0 0 0 0 0 ( 1¡® 5 )sruc 0 ( 1+4® 5 )sru 1 1 CCCCA ; (3.5) ML = m0 d 0 BBBBBBBBBBBBBBBBB@ 0 (c12 + 3±3(¡1+® 5 ))rerec (¡±2® + ³)rec (c12 0 (±1® ¡3±3(¡1+® 5 ))rerec ¡3s(¡1+® 5 ) + ¯)rec (³ (s(¡1+6® 5 ) + ±1( 6¡® 5 ) 1 ¡±2 6¡® 5 )re +¯)re ¡2(¯ + 3+2® 5 ±1)fcµsµT2 2 1 CCCCCCCCCCCCCCCCCA ; MN = m0 u 0 BBBB@ 0 0 0 0 0 (¡3+3® 5 )srºc 0 srº 1 1 CCCCA ; where the parameters are de¯ned in terms of the Yukawa couplings of the super potential (Wspin: + Wvect:) and the VEVs of the Higgs ¯elds as shown in appendix A. These matrices are multiplied by lefthanded fermions on the right and right handed fermions on the left. A doubly lopsided structure for the charged lepton and down quark mass matrices of Eq.(3.5) can be obtained by going to the limit ¯; ³; ®; ±3; c12; s ¿ 1 and ±1; ±2 are of order one. This doubly lopsided form leads simultaneously to large neutrino mixing angles and to small quark mixing angles. Based only on the above fermion mass matrices in Eq.(3.5), an excellent ¯t is found for fermion masses (except for the neutrino masses), quark mixing angles and neu 54 trino mixing angles (except the atmospheric angle) by giving the input parame ters, appearing in Eq.(3.5), the following numerical values: ±1 = ¡1:28, ±2 = 1:01, ±3 = 0:015 £ e4:95i, ® = ¡0:0668, s = 0:2897, ³ = 0:0126, c12 = ¡0:0011e1:124i, and ¯ = ¡0:11218. The above numerical values lead to sin µL 23 = 0:92 which is not close to the experimental central value of atmospheric angle sin µatm 23 = 0:707 [7]. This con tribution to the atmospheric angle is only from the charged lepton sector. Therefore, the neutrino sector should be included by considering the following superpotential: WN = b4ÃiZi16H + hijkZiZc j 1000 Hk + m1Zc i Zc i ; (3.6) where two fermion singlets Zi and Zc i that couple with the Higgs singlet 1000 Hk have been introduced. The full neutrino mass matrix is constructed in Appendix B. The Higgs singlet 1000 Hk has the VEVcomponents (®1, ®2, ®3). The light neutrino mass matrix is obtained by employing the seesaw mechanism. The numerical values (®1 = 0:075, ®2 = 0:07, ®3 = 0:9, and ¸ = 0:0465 eV), where ¸ is de¯ned in appendix B, lead to not only the correct contribution to the atmospheric angles (sin µatm 23 = 0:811) but also to the correct light neutrino mass di®erences. The predictions of the fermion masses and mixings are slightly altered by doing the algebra exactly compared to the analysis of Ref.[51]. These predictions and their updated experimental values obtained from [7] are shown in Table 3.3. The right handedneutrino masses arise from integrating out the exotic fermion singlets Zi and Zc i in Eq.(3.6). The right handedneutrino mass matrix is MR = ¤ 0 BBBBBB@ ®21 ®23 ®1®2(¡1 ®23 + 2 ®21 +®22 +®23 ) ¡®1(®21 ¡®22 +®23 ) ®3(®21 +®22 +®23 ) ®1®2(¡1 ®23 + 2 ®21 +®22 +®23 ) ®22 ®23 ¡®2(¡®21 +®22 +®23 ) ®3(®21 +®22 +®23 ) ¡®1(®21 ¡®22 +®23 ) ®3(®21 +®22 +®23 ) ¡®2(¡®21 +®22 +®23 ) ®3(®21 +®22 +®23 ) 1 1 CCCCCCA ; (3.7) where ¤ = 8:45 £ 1015 GeV and the righthanded neutrino masses are given by MR1 ¼ MR2 ¼ 1:4 £ 1012 GeV and MR3 = 8:5 £ 1015 GeV. 55 Another interesting feature of this model is that it contains a minimal set of Higgs ¯elds needed to break SO(10) to the SM gauge group. Consequently, the uni¯ed gauge coupling remains perturbative all the way up to the Planck scale. This can be understood from the running of the uni¯ed gauge coupling with energy scale ¹ > MGUT as 1 ® = 1 ®G ¡ bG 2¼ log( ¹ MGUT ); (3.8) where ® = g2=(4¼) and bG = S(R) ¡ 3C(G). Here C(G) is the quadratic Casimir invariant and S(R) is the Dynkin index summed over all chiral multiplets of the model. The uni¯ed gauge coupling stays perturbative at the Planck scale (i.e g(MP ) < p 2) as long as bG < 26. Employing large Higgs representations might lead to bG ¸ 26. For example, using 126H+126H gives bG = 46. On the other hand, the SO(10)£A4model gives bG = 19 which is consistent with the uni¯ed gauge coupling being perturbative till the Planck scale. We will use the same ¯t for fermion masses and mixings to calculate the mass insertion parameters ±ij LL;RR, and ±ij LR;RL in the quark and lepton sectors and conse quently investigate the FCNC in this model. The charged lepton and down quark mass matrices in Eq.(3.5) are diagonalized at the GUT scale by biunitary transfor mation: Mdiag: d;l = V yd;l R MD;LV d;l L ; (3.9) where V u;d;l R;L are known numerically. Now, we discuss the sources of FCNC in this model. 3.2 Sources of Flavor Violation in SO(10) £ A4 Model We assume in our °avor violation analysis that A4 °avor symmetry is preserved above GUT scale and it is only broken at GUT scale. In this case °avor violation is induced 56 at GUT scale where A4 symmetry is broken. In this section we discuss the °avor violation induced at the GUT scale by studying the sfermion mass insertion parameter ±ij LL;RR and the chirality °ipping mass insertion (Aterms) parameter ±ij LR;RL. We will see that these °avor violation sources arise from large mixing of the light ¯elds with the heavy ¯elds. This large mixing is due to the breaking of A4 symmetry. In addition, we discuss the induced °avor violation arising below GUT scale through the RG running from MGUT to the righthanded neutrino mass scales. 3.2.1 The Scalar Mass Insertion Parameters Let us assume the soft supersymmetry breaking terms originate at the messenger scale M¤, where MGUT < M¤ · MPlanck. The quadratic soft mass terms of the matter super¯elds that appear in the superpotential Wspin: are ¡L = ~m2 ÃÃy iÃi + ~m2 ÂiÂy iÂi + ~m2 Âi Ây iÂi: (3.10) The MSSM scalar fermions that reside in Ãi transform as triplets under the non abelian A4 symmetry. Since the A4 symmetry is intact, they have common mass ( ~m2 Ã) at the scale M¤. On the other hand, the exotic ¯elds each of which transforms as singlet under A4 symmetry have di®erent masses ( ~m2 Âi , ~m2Â i , i runs 13) at the scale M¤. The MSSM scalars remain degenerate above the GUT scale where the A4 symme try is broken. In order to ¯nd the scalar masses in the fermion mass eigenstates, two transformations are required. The ¯rst transformation is needed to blockdiagonalize the fermion mass matrix into a light and a heavy blocks as shown in Appendix A. The upper left corner represents the 3 £ 3 light fermions mass matrix. The second trans formation is the complete diagonalization of the light fermion mass matrix. Applying the ¯rst transformation to the quadratic soft mass terms of Eq.(3.10) by going to the new orthogonal basis (L2, L3, H1, H2, H3) as de¯ned in appendix A, the quadratic 57 soft mass matrix of the light states is transformed as follows: ~m2 ÃI ! ~m2 ÃI + ± ~m2 Ã; (3.11) where, ± ~m2 Ã = 0 BBBBBB@ 0 0 0 0 0 ² 0 ² ± 1 CCCCCCA ; (3.12) ² = f rF T2 2 sµ( ~m2 Â2 ¡ ~m2 Ã), ± = (( f rF )2 ¡ 1) ~m2 Ã + ( f rF )2( ~m2 Â1T2 1 + ~m2 Â2T2 2 + ~m2 Â3Q2T2 1 T2 3 ), and we have safely ignored the terms that contain s2µ ¿ 1. It is obvious that the ¯rst two generations of the light scalars are almost degenerate because the mixing of the second light generation (L2) with the heavy states is proportional to sµ ¿ 1. On the other hand, since the mixing of the third light generation (L3) with the heavy states is of order one, its mass splits from those of the ¯rst two generations. The top Yukawa coupling is given in terms of T1, T2, and T3 as: Yt = af2(Qu + Quc)T1T2T3 rucru : (3.13) The numerical values of T1 = 0:0305, T2 = 2, T3 = 100 and a » 1:2 are consistent with the top Yukawa coupling at the GUT scale to be of order ¸GUT t » 0:5 and ru;uc to be of order one. Plugging these numerical values and sµ = 0:0465 into the expressions for ² and ± gives us: (±d; ±dc ; ±e; ±ec) = (0:81; 0:87; 0:88; 0:82)( ~m2 Â ¡ ~m2 Ã); (²d; ²dc ; ²e; ²ec) = (0:061; 0:05; 0:048; 0:06)( ~m2 Â ¡ ~m2 Ã): (3.14) Here we have dropped ~m2 Â1 terms because their coe±cients are negligible. Also, the RGE expressions of ~m2 Â2 and ~m2 Â3 are the same (see Eq.(B.13)), so we have assumed that ~m2 Â2 = ~m2 Â3 = ~m2 Â. The next step is to apply the second transformation by evaluating V yd;l L ±m2 ÃV d;l L and similarly for L ! R. The unitary matrices V d;l L are numerically known from the 58 ¯tting for fermion masses and mixings. So, the mass insertion parameters for charged leptons and down quarks are given respectively by (±d;e LL;RR)ij = (V yd;l L;R ± ~m2 d;lV d;l L;R)ij= ~m2 d;l: (3.15) The above mass insertion analysis without including the superpotential Wvect: is good enough because we assumed in our analysis that the mixing of the 10 vector multiplets with the ordinary spinor ¯elds is small. 3.2.2 The Chirality Flipping Mass Insertion (Aterms) The FV processes are also induced from the o®diagonal entries of the chirality °ipping mass matrix ~M RL. The chirality °ipping soft terms are divided into two parts Lspin and Lvect: ¡Lspin = ~b 1b1 ~ Ãi~Â 11Hi +~b 2b2 ~ Ãi~Â 210 Hi + ~k1k1 ~Â1~Â 345H +~aa~Â3 ~Â210H + ~G iMi ~Âi~Â i; (3.16) ¡Lvect = ~b 3b3 ~ Ãi ~Ái16H + ~B 10M10 ~Ái ~ Á0 i + ~ h0 ijkh0 ijk ~ Á0 i ~ Á0 j1Hk +~h ijkhijk ~Ái ~Áj1Hk +A~ijkAijk ~ Á0 i Á~00 j100 Hk + g~mÁ~00 i Á~000 i + ~k2k2 Á~000 i ~ Á0 i45H: (3.17) The fourth term of Eq.(3.16) induces the o®diagonal elements of the chirality °ipping mass matrix, if it is written in terms of the new orthogonal basis de¯ned in Eqs.(A.1). This transformation can be represented by ~M 2R L(spin:) ! ~aMF (spin:); (3.18) where MF (spin:) is de¯ned in Eq.(3.3). The entire chirality °ipping mass matrix in the new orthogonal basis is obtained by including ¡Lvect. The biunitary transfor mations that blockdiagonalize the full fermion mass matrix is applied on the entire chirality °ipping mass matrix (see Appendix A). Accordingly, the 3 £ 3 quadratic mass matrix ( ~M 2 LR) associated with the light states is transformed as follows: ~M 2R L ! ~aMF (spin:) +~b 3MF (vector); (3.19) 59 where MF (vect:) = ¡mM¡1M0 (see Eq.(A.6)) and we have assumed for simplicity that the soft parameters appearing in Eq.(3.17) are all of the same order. Then, the M2 LR matrix is written in the fermion mass eigenstate basis as: ~M 2R L ! V y R(~aMF (spin:) +~b 3MF (vect:))VL: (3.20) It is straightforward to show that the chirality mass insertion parameters are given by: (±RL)ij = ~b 3 ~m2f Mdiag: Fi ±ij + (~zV y RMF (spinor)VL)ij ; (3.21) where Mdiag: F = V y RMFVL and ~z = ~a¡~b 3 ~m2f . The induced FV arises only from the second term of Eq.(3.21). 3.2.3 Mass Insertion Parameters Induced Below MGUT The Dirac neutrino Yukawa couplings (YN)ij induce °avor violating o®diagonal ele ments in the lefthanded slepton mass matrix through the RG running from MGUT to the righthanded neutrino mass scales. The RGEs for MSSM with righthanded neutrinos are given in Ref.[46]. The righthanded neutrinos MRi are determined in the SO(10) £ A4 model. In this case, the induced mass insertion parameters for lefthanded sleptons are given by [50], (±l LL)RHN ij = ¡ 3m2 Ã + ~a2 8m2 Ã¼2 X3 k=1 (YN)ik(Y ¤ N)jkln MGUT MRk ; (3.22) where the matrix YN is written in the mass eigenstates of charged leptons and right handed neutrinos. The total LL contribution for the charged leptons is given by (±l LL)Tot ij = (±l LL)RHN ij + (±l LL)ij : (3.23) 3.3 Results In this section, we investigate the °avor violating processes by calculating the mass insertion parameters ±LL, ±RR, and ±LR;RL, then we compare them with their exper 60 imental bounds. These bounds in the quark and lepton sectors were obtained by comparing the hadronic and leptonic °avor changing processes to their experimental values/limits [54, 55]. Eq.(3.12), Eq.(3.14) and Eq.(3.15) are used to calculate ±LL;RR and Eq.(3.21) is used to calculate ±LR;RL for both charged leptons and down quarks. The result of mass insertion calculations and their experimental bounds are presented in Table 3.4. In this table, we have de¯ned ¾ = ~m2 Â2 ¡~m2 Ãi ~m2 Ãi and ~k = ~zmb;¿ The stringent bounds on leptonic ±12, ±13, and ±23 in Table 3.4 come only from the decay rates li ! lj°. The experimental bounds on the mass insertion parameters listed in column 3 were obtained by making a scan of m0 and M1=2 over the ranges m0 < 380 GeV and M1=2 < 160 GeV , where m0 and M1=2 are the scalar universal mass and the gaugino mass respectively [55]. Glancing at Table 3.4, we note that the stringent constraint on leptonic °avor violation arises from ±l 12 which corresponds to the decay rate of ¹ ! e°. On the other hand, there is a weaker constraint that arises from ±d 12 on the quark sector. One can do an arrangement such that ~a ¡~b 3 = 200 GeV and ~mf = 800 GeV (equivalent to ~k = 2:6£10¡4) so that all the chirality °ipping mass insertions will be within their experimental bounds. This arrangement is possible if the trilinear soft terms vanish at the scale M¤. Since the stringent constraint comes from the ¹ ! e° process, let us discuss the branching ratio of this process in more details. In general, the branching ratio of li ! lj° is given by BR(li ! lj°) BR(li ! ljºi ¹ ºj) = 48¼3® G2 F (jAij L j2 + jAij Rj2): (3.24) We have used the general expressions for the amplitudes Aij L;R given by Ref.[57] where the contributions from both chargino and the neutralino loops are included. These expressions are written in terms of mass insertion parameters. The correct suppression of the decay rate ¡(¹ ! e°) requires a high degree of degeneracy of the soft mass terms of MSSM ¯elds and the exotic ¯elds. For example, 61 ¾ ¼ 0:01, as can be seen from Table 3.4. In order to obtain high degree of degeneracy, let us assume that the SSB terms which are generated at the messenger scale M¤ satisfy the universality boundary conditions at the scale M¤ given by ~m2 Ãi = ~m2 Âi = ~m2 Âi = ~m2 10H = ~m21 H = ~m21 0 H = m0; M¸ = M0; ~a = ~b 1 =~b 2 = 0; (3.25) where M¸ is the gaugino mass of SO(10) gauge group. Solving the RGE listed in Appendix C with the boundary conditions given by Eq.(3.25) determines the value of ¾. In Table 3.5 we give the branching ratio of the process ¹ ! e° predicted by the SO(10) £ A4 model for di®erent choices of the input parameters a, b1, b2, ~mÃ and M1=2 at the GUT scale. The experimental searches have put the upper limit on the branching ratio of ¹ ! e° as Br(¹ ! e°) · 1:2 £ 10¡11 [56]. Note that ~mÃ and M1=2 originate respectively from m0 and M0 through RGEs. In this Table we consider ln M¤ MGUT = 1 and ln M¤ MGUT = 4:6 that correspond respectively to M¤ ¼ 3MGUT and M¤ ¼ MPlanck. Let us analyze the four cases in the Table 3.5. In the cases (I, II and III), the chosen values of the parameters a are consistent with the top Yukawa coupling of order 0:5 at the GUT scale and with the ¯tting for fermion masses and mixing. On the other hand, the choice of a = 0:68 in Case IV is not consistent with the ¯t. Although the medium slepton masses of order 550 GeV are obtained in Case I, the choice b1 = b2 = 1:9 corresponds to nonperturbative Yukawa couplings at the scale M¤ (i.e. b1 = b2 = 4 at M¤). In this case, the solutions of the 1loop RGEs are not trusted since the Yukawa couplings b1 and b2 go nonperturbative above the GUT scale. Also, it is important to point out that the °avor violation constraint on ¹ ! e° in Case III requires heavy slepton masses (¸ 3 TeV) while it requires slepton masses of order » 900 GeV in Case II. In other words, Case II is preferred in our model 62 in the sense that the decay rate of ¹ ! e° is close to the experimental limit with a reasonable supersymmetric mass spectrum, so it might be tested in the ongoing MEG experiment[58]. Besides, the Yukawa couplings remain perturbative at the messenger scale M¤. Figure 3.1 shows the allowed values of mÃ that correspond to the graphs below the xaxis for the cases I and II. 63 Predictions Expt. Pull mc(mc) 1.4 1:27+0:07 ¡0:11 1.85 mt(mt) 172.5 171.3§2.3 0.52 ms=md 19.4 19:5 § 2:5 0.04 ms(2Gev) 109.6£10¡3 105+25 ¡35 £ 10¡3 0.184 mb(mb) 4.31 4:2+0:17 ¡0:07 0.58 Vus 0.223 0.2255§0.0019 1.3 Vcb 38.9£10¡3 (41.2§1.1)£10¡3 2 Vub 4.00£10¡3 (3.93§0.36)£10¡3 0.7 ´ 0.319 0:349+0:015 ¡0:017 1.7 me(me) 0.511£10¡3 0.511£10¡3  m¹(m¹) 105.6£10¡3 105.6£10¡3  m¿ (m¿ ) 1.776 1.776  ¢m2 21 7:69 £ 10¡3eV2 (7:59 § 0:2) £ 10¡3eV2 0.5 ¢m2 32 2:36 £ 10¡3eV2 (2:43 § 0:13) £ 10¡3eV2 0.5 sin µsol 12 0.555 0.566§0.018 0.61 sin µl 23 0.811 0.707§0.108 0.96 sin µ13 0.141 < 0:22 Table 3.3: The fermion masses and mixings and their experimental values. The fermion masses, except the neutrino masses, are in GeV. 64 500 1000 1500 myHGeVL 0.5 0.5 1.0 1.5 2.0 2.5 log10@ Br Hm egL Expt limit D 175 GeV 437 GeV 787 GeV Case II 500 1000 1500 myHGeVL 6 4 2 log10@ Br Hm egL Expt limit D 175 GeV 437 GeV 787 GeV Case I Figure 3.1: The above graphs show the plot of Log of Br(¹ ! e°) divided by exper imental bound (1:2 £ 10¡11) versus mÃ for two cases I and II with M1=2=787 GeV, 437 GeV and 175 GeV. 65 Mass Insertion (±) Model Predictions Exp. Upper Bounds (±l 12)LL 0.062 ¾+(±l 12)RHN LL 6 £ 10¡4 (±l 12)RR 6.1 £ 10¡4 ¾ 0.09 (±l 12)RL;LR (0.084, 0.0096) ~k 10¡5 (±l 13)LL 0.022 ¾+(±l 13)RHN LL 0.15 (±l 13)RR 0.028 ¾  (±l 13)RL;LR (0.0335, 0.076) ~k 0.04 (±l 23)LL 0.27 ¾+(±l 13)RHN LL 0.12 (±l 23)RR 0.034 ¾  (±l 23)RL;LR (0.055, 0.899) ~k 0.03 (±d 12)LL 1.9 £ 10¡4 ¾ 0.014 (±d 12)RR 0.15 ¾ 0.009 (±d 12)LR;RL (0.029, 0.035) ~k 9 £ 10¡5 (±d 13)LL 0.014 ¾ 0.09 (±d 13)RR 0.061 ¾ 0.07 (±d 13)LR;RL (0.173, 0.016) ~k 1:7 £ 10¡2 (±d 23)LL 0.054 ¾ 0.16 (±d 23)RR 0.29 ¾ 0.22 (±d 23)LR;RL (0.875, 0.064) ~k (0.006, 0.0045) Table 3.4: The mass insertion parameters predicted by SO(10)£A4 model and their experimental upper bounds obtained from [55]. 66 I II III IV a 1.14 1.07 1.14 0.62 b1 1.9 1.5 1.24 1.24 b2 1.9 1.5 1.24 1.24 ~mÃi 542 886 2932 675 M1=2 350 787 1924 350 BR(¹ ! e°) 1:4 £ 10¡13 1:16 £ 10¡11 1:2 £ 10¡11 2:2 £ 10¡12 Table 3.5: Branching ratio of ¹ ! e° for di®erent choices of input parameters at the GUT scale. Cases I and II correspond to ln M¤ MGUT = 1 and cases III and IV correspond to ln M¤ MGUT = 4:6. ~mÃi and M1=2 are given in GeV 67 CHAPTER 4 Higgs Boson Mass in GaugeMediating Supersymmetry Breaking with MessengerMatter Mixing Supersymmetric (SUSY) grand uni¯cation theories (GUTs) are promising candidates for physics beyond the standard model (SM). However, supersymmetry is not an exact symmetry at the lowenergy scale and it must be broken somehow to be relevant to nature. SUSY can not be broken at tree level since the supertrace theorem leads to nonphenomenological particle spectra. Therefore, it is assumed that SUSY breaking occurs in the hidden sector which has no renormalizable tree level couplings with the observable sector. SUSY breaking is transmitted to the visible sector either via gravitational interactions as inspired by supergravity models (SUGRA)[44], or by SM gauge interactions as in theories with gaugemediated SUSY breaking (GMSB)[59, 60, 61]. In the ¯rst scenario, the soft terms are generated at the Planck scale. In general, these soft terms are not °avorinvariant. The gravitymediated scenario can only give realistic models if the universality or an approximate alignment between particle and sparticle masses is imposed in order to suppress the °avor violation processes. On the other hand, the universality condition is naturally satis¯ed in the GMSB where the soft terms are generated at the messenger scale, below the GUT scale, from radiative corrections. In GMSB theories, messenger ¯elds communicate the SUSY breaking from the hidden sector to the visible sector. In addition to the observable sector, at least one gauge singlet super¯eld (Z) is needed in order to give mass to the messenger ¯elds and break SUSY by giving vacuum expectation values (VEVs) to its scalar 68 component (hZi) and to its auxiliary Fcomponent (hFZi) respectively. The SUSY breaking factor (i.e. hFZi) that appears in the mass splitting between the fermionic and scalar components of the messenger ¯eld is communicated to the MSSM particles through radiative corrections. For example, the gauginos and the scalars of MSSM get their masses at the messenger scale Mmess from oneloop and twoloop Feynman diagrams respectively as fellows: M¸r = gNmess ®r 4¼ ¤; (4.1) ~m2 = 2f X3 r=1 NmessC ~ f r ®2 r (4¼)2¤2; (4.2) where Nmess is called the messenger index. For example, Nmess = 1 (Nmess = 3) for messenger ¯elds belong to 5 + 5 (10 + 10) of SU(5). Here, ¤ = hFZi hZi is the e®ective SUSY breaking scale, C ~ f r are the quadratic Casimir invariants for the scalar ¯elds, and ®r are the gauge coupling constants at the scale Mmess. These gauge couplings are all equal at the GUT scale. In Eqs.(4.1) and (4.2), f and g are the 1loop and 2loop functions whose exact expressions can be found e.g. in Ref.[61]. The universal scalar masses in Eq.(4.2) are obtained when the messenger and matter ¯elds are completely separated. There are additional contributions to universal masses if messengermatter mixing is allowed. Two interesting features of GMSB are concluded from Eqs.(4.1) and (4.2). Firstly, the scalar masses are only functions of gauge quantum number so scalar masses with the same gauge quantum number are degenerate. As a result, the supersymmetric °avor problem is solved. Secondly, GMSB is highly predictive since all soft terms at the messenger scale are determined by only two parameters ¤ and Nmess. In order to preserve the successful gauge coupling uni¯cation of MSSM, the messenger ¯elds should reside in complete SU(5) multiplets. In this chapter, we consider two cases when the messenger ¯elds belong to 5 + 5 and 10 + 10 of SU(5). In both cases the perturbative uni¯cation is still maintained, as shown in Fig. 4.1. 69 a3 1 a2  1 a1  1 2 4 6 8 10 12 14 16 10 20 30 40 50 60 Log10 Hm GeVL a 1 Figure 4.1: The evolutions of the gauge couplings with Mmess = 108 GeV and tan ¯ = 10. Solid lines correspond to MSSM. Dashed lines are for MSSM+10+10 and dotted lines are for MSSM+5 + 5. The complete separation of messenger sector and visible sector is problematic in cosmology because this leads to models possessing stable particles [62]. Besides, messengermatter couplings are allowed by gauge symmetry and they can only be forbidden by imposing discrete °avor symmetry. If one allows these couplings, ad ditional contributions to the universal scalar mass given by Eq.(4.1) and (4.2) are obtained [63, 64, 65]. These new contributions reintroduce °avor violation either in the leptonic or the quark sector depending on the structure of the messenger ¯elds. In this chapter, we have shown that the induced °avor violation from messengermatter mixing that occurs mainly with the third generation is still su±ciently suppressed. Another advantage of the messengermatter mixingthe main result of this chapter is that it might increase the lightest Higgs mass to value as large as 125 GeV, which is di±cult to realize without such mixing. In order to reproduce the known qualitative features of quark and lepton masses and mixings, we consider the FroggattNielsen mechanism [66]. This mechanism leads to the lopsided structure of downquark and charged lepton mass matrix. It 70 was shown that in this kind of structure the ¹ ! e° decay rate is generally large by adopting gravity mediated SUSY breaking and it is consistent with the experimental limit of Br(¹ ! e°) only with a heavy SUSY spectrum [67]. On the other hand, the lopsided structure works well in the GMSB regarding the °avor violation processes even with light SUSY spectra as we show in this chapter. This chapter is organized as fellows: In section 4.1 the Higgs mass bounds are considered in two models. The ¯rst is 5+5 model in which the messenger ¯elds belong to the 5 + 5 representation of SU(5) while the second is 10 + 10 model in which the messenger ¯elds belong to the 10 + 10 representation of SU(5). In both models, the messengermatter couplings (i.e. the exotic couplings) are allowed. We investigate the e®ect of these couplings on the lightest Higgs mass of MSSM. In section 4.2, we construct the general structure of the superpotential of both models by employing the U(1) °avor symmetry of the FroggattNielsen mechanism as discussed in section 4.2.1. We ¯nd that the FCNC processes that are induced by the exotic Yukawa couplings are in agreement with experimental bounds. The Yukawa RGEs between messenger and GUT scales for both models are listed in Appendix C. The soft terms which are induced by the exotic Yukawa couplings are evaluated in Appendix D. 4.1 Higgs Mass Bounds One of the interesting features of MSSM is setting upper bounds on the lightest Higgs mass. The tree level bound on the lightest Higgs mass equal to Mz has been already excluded by the LEP2 lower bound mh > 114:4 [68]. However, radiative corrections push this mass above the LEP2 bound. The leading 1 and 2 loop contributions to the CPeven Higgs boson mass in the MSSM are given by [70, 71] m2 h = M2 z cos2 2¯(1 ¡ 3 8¼2 m2t v2 t) + 3 4¼2 m4t v2 [ 1 2 Ât + t + 1 16¼2 ( 3 2 m2t v2 ¡ 32¼®3)(Âtt + t2)]; (4.3) 71 where v2 = v2 d + v2u , t = log( M2 s M2 t ); Ât = 2 ~ A2t M2 s (1 ¡ ~ A2t 12M2 s ): (4.4) Here the scale Ms has been de¯ned in terms of the stop mass eigenvalues as M2 s = m~ t1m~ t2 ; (4.5) A~t = At¡¹ cot ¯, where At denotes the stop left and stop right soft mixing parameter. The upper bound on the lightest Higgs mass depends crucially on the soft su persymmetry breaking terms. For example, the upper bound of around 125 GeV corresponds to the maximal mixing condition, A~t = p 6Ms. Since there are restric tions on these soft terms from GMSB, it will be interesting to study the e®ect of these restrictions on the lightest CPeven Higgs mass. In the following subsections we will investigate the e®ect of allowing messengermatter couplings on the soft terms of MSSM and consequently on the lightest CPeven Higgs mass. In the ordinary GMSB (i.e. without messengermatter mixing), both Aterms and the soft breaking param eter B vanish at the messenger scale. However, B can be induced in the process of running. By using the following equations that result from minimizing the Higgs potential, M2 z 2 = ¡¹2 ¡ m2 Hu tan2 ¯ ¡ m2 Hd tan2 ¯ ¡ 1 ; (4.6) sin 2¯ = 2B¹ 2¹2 + m2 Hu + m2 Hd ; (4.7) one can solve for the parameters tan ¯ and ¹. Then tan ¯ turns out to be large (around 3545) when the messenger scale is close to the e®ective SUSY breaking scale ¤. On the other hand, by allowing messengermatter couplings B is induced signi¯cantly at low energy scale. This can be understood from the following RGE for the parameter B: dB dt = 1 2¼ (3®tAt + 3®2M2 + 3 5 ®1M1); (4.8) 72 ¸0 0 mh(GeV) ¤(105GeV) M(1013GeV) ~mt1(GeV) ~mt2(GeV) 0 117 2 1.78 1634 2012 0.8 118 2 10 1590 1857 1.2 119 2 10 1065 2788 Table 4.1: We show the values of the minimal GMSB input parameters, ¤, ¸ex and Mmess that lead to the highest mh values at tan ¯ = 10. where ®t = ¸t 4¼ and ¸t is the top Yukawa coupling. Since At does not vanish in the presence of messengermatter mixing as shown in Eqs.(4.13) and (4.21), the ¯rst term of Eq.(4.8) that pushes B to large values becomes more signi¯cant than in the case when At is zero. This leads to small tan ¯. For example in the 10 + 10 model, the range 1:64 · tan ¯ · 7 corresponds to 105 GeV ·Mmess ·1014 GeV. In the subsequent analysis, we will give the scalar mass spectrum that leads to the highest mh for two cases. The ¯rst case is to assume a nonvanishing B is somehow generated at the messenger scale such that tan ¯ = 10 is obtained by using Eqs.(4.6) and( 4.7). The potential solution to the ¹ problem based on °avor symmetries was suggested by Ref. [72]. The authors of Ref. [72] gave an example of B¹ » ¹2 that leads to unconstrained values on tan ¯ by introducing three singlets that are charged under U(1) °avor symmetry. The second case is having a vanishing B at the messenger scale as predicted by both 5 + 5 and 10 + 10 models. In this case tan ¯ is determined by Eqs.(4.6) and (4.7) where B at low energy scales is obtained by solving the RGE with the boundary condition of vanishing B at the messenger scale. 4.1.1 Higgs Mass Bounds in the 5 + 5 Model The messenger ¯elds belonging to 5 + 5 of SU(5) decompose to downquark singlets dc m and dc m, and to lepton doublets Lm and Lm. The additional contributions to the 73 0 2 4 6 8 10 0 0.5 1 1.5 2 ~m2 tc /105 (GeV2) l’0 Mmess=107 GeV Mmess=1014 GeV 0 2 4 6 8 10 0 0.5 1 1.5 2 m~2 tc /105 (GeV2) l’0 Mmess=107 GeV Mmess=1014 GeV Figure 4.2: The left graph is ~m2¿ c versus ¸0 0 at the scale Mmess for two di®erent messenger scales. The right graph is ~m2t c versus ¸0 0 at the low energy scale for two di®erent messenger scales. MSSM superpotential due to messengermatter couplings is W5+5 = fddc mdc mZ + ¸0 bQ3dc mHd + feLmLmZ + ¸0 ¿cLmec 3Hd: (4.9) We assume the messenger ¯elds couple only with the third generation of MSSM. We will show later that the superpotential W5+5 can be obtained by imposing the U(1) °avor symmetry of the FroggattNielsen mechanism. Also, we have assumed that the exotic Yukawa couplings ¸0 b and ¸0 ¿c (fd and fe) are obtained from one uni¯ed coupling ¸0 0(f0) at the GUT scale by solving the RGEs listed in the Appendix C.1 between the messenger scale and the GUT scale. In the universal case (i.e. without including messengermatter couplings), the scalar masses are obtained by employing Eqs.(4.1) and (4.2), while the trilinear soft terms (Aterms) vanish at the scale Mmess. There are new contributions to the u
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Title  Fermion Masses and Mixings, Flavor Violation, and the Higgs Boson Mass in Sypersymmetric Unified Framework 
Date  20110701 
Author  Albaid, Abdelhamid 
Keywords  Flavor, gauge symmetry, Higgs, Standard Model, Supersymmetry, Unification 
Department  Physics 
Document Type  
Full Text Type  Open Access 
Abstract  In spite of the impressive success of the standard model (SM) in explaining a wide variety of low energy experiments, many fundamental questions remain unanswered. Combining grand unification symmetry and flavor symmetry in the supersymmetric framework leads to interesting new physics that might solve many of the puzzles of the SM. I construct a supersymmetric grand unification model based on the gauge symmetry SO(10) and the nonabelian discrete symmetry A 4. In the framework of this model the small mixings of the quark sector and the large mixings of the lepton sector are successfully accommodated by employing exotic heavy fermion fields with superheavy masses. An excellent fit to the quark and lepton masses and mixings as well as to the CP violation parameter is obtained. Flavor violating processes in the quark and lepton sectors within this realistic supersymmetric SO(10)×A 4 model are also investigated. I find that flavor violation is induced at grand unification scale as a consequence of the large mixing of the light fermion fields and the exotic heavy fields. The stringent experimental constraint from &mu&rarr e&gamma decay rate requires a high degree of degeneracy of the supersymmetry breaking soft scalar masses of the exotic heavy fields and supersymmetric scalar partners of the light fermion fields. The choice of slepton masses of order 1 TeV is found to be consistent with the constraints from branching ratio of &mu&rarr e&gamma and with all other flavor changing neutral current (FCNC) processes being sufficiently suppressed. In a related project, we study the effect of allowing messengermatter mixing in a class of models with gaugemediated supersymmetry breaking in the unification framework. We find that the maximal mixing condition that leads to the upper limit of the lightest Higgs mass (~125 GeV ) of MSSM can be obtained in models with the messenger fields belonging to $10+\overline{10}$ representations of $SU(5)$ gauge symmetry. Consistent with the cosmological preference for the messenger scale of < 3 × 10^8 GeV, the lightest Higgs mass of order 121 GeV is obtained, along with all superparticle masses below 1 TeV. Our results are also consistent with the gauge and exotic Yukawa couplings being perturbative and unified at the GUT scale as well as with all FCNC processes being suppressed in agreement with experimental bounds. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  FERMION MASSES AND MIXINGS, FLAVOR VIOLATION, AND THE HIGGS BOSON MASS IN SUPERSYMMETRIC UNIFIED FRAMEWORK By ABDELHAMID ALBAID Bachelor of Arts/Science Applied Physics Jordan University of Science and Technology Irbid, Amman, Jordan 2001 Master of Arts/Science in Physics University of Jordan Amman, Amman, Jordan 2004 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial ful¯llment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2011 COPYRIGHT °c By ABDELHAMID ALBAID July, 2011 FERMION MASSES AND MIXINGS, FLAVOR VIOLATION, AND THE HIGGS BOSON MASS IN SUPERSYMMETRIC UNIFIED FRAMEWORK Dissertation Approved: Kaladi S. Babu Dissertation Advisor Flera Rizatdinova Jacques H. H. Perk Birne Binegar Mark Payton Dean of the Graduate College iii ACKNOWLEDGMENTS All praises and gratitude to Allah Almighty who guides me with His mercy and bounty to ¯nish this project. I would like to thank my advisor Professor Kaladi S. Babu for his guidance and teaching. He has been the important person in my learning process. Without his advising and collaboration, this project would have been impossible. I would like also to thank Prof. S. Nandi from whom I have learnt a lot. I would like to thank my thesis committee: Prof. Jacques H. H. Perk, Prof. Flera Rizatdinova and Prof. Birne Binegar for their advice. I would like to express my gratitude to Prof. Paul Westhaus for being my advisor in the beginning of my Ph.D program. He was very helpful and dedicated to the students. I want to thank my o±ce mates for useful discussions and conversations. Special thanks for Benjamin Grossmann and Julio for being very helpful. I want to especially thank my parents for their support and love and to thank my wife, Rema Alradwan, who was supportive, cooperative and full of love. iv TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 1.1 A Brief Review of the Standard Model (SM) . . . . . . . . . . . . . . 1 1.2 Seesaw Mechanism and Leptonic Mixing Matrix . . . . . . . . . . . . 7 1.3 Shortcomings of the SM and the Need for New Physics. . . . . . . . . 10 1.3.1 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.2 Discrete Flavor Symmetry A4 . . . . . . . . . . . . . . . . . . 16 1.4 Minimal SUSYSU(5) . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.1 SU(5) Matter Fields . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.2 Higgs Sectors and Yukawa Couplings in the minimal SUSYSU(5) 18 1.4.3 Gauge Sector of Minimal SU(5) . . . . . . . . . . . . . . . . . 20 1.5 Minimal SUSYSO(10) . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5.1 Matter Fields in SO(10) GUTs . . . . . . . . . . . . . . . . . 22 1.5.2 The Higgs Fields and Yukawa Couplings in SO(10) GUTs . . 22 1.5.3 Neutrino Masses . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Fermion Masses and Mixings in a Minimal SO(10)£A4 SUSY GUT 29 2.1 Fermion Mass Structure in SO(10) £ A4 Symmetry . . . . . . . . . . 31 2.2 Extension to the First Generation and Doubly Lopsided Structure . 35 2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4 Right Handed Neutrino Mass Structure . . . . . . . . . . . . . . . . . 43 3 Flavor Violation in a Minimal v SO(10) £ A4 SUSY GUT 48 3.1 A Brief Review of Minimal SO(10) £ A4 SUSY GUT . . . . . . . . . 51 3.2 Sources of Flavor Violation in SO(10) £ A4 Model . . . . . . . . . . . 56 3.2.1 The Scalar Mass Insertion Parameters . . . . . . . . . . . . . 57 3.2.2 The Chirality Flipping Mass Insertion (Aterms) . . . . . . . . 59 3.2.3 Mass Insertion Parameters Induced Below MGUT . . . . . . . 60 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Higgs Boson Mass in GaugeMediating Supersymmetry Breaking with MessengerMatter Mixing 68 4.1 Higgs Mass Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.1 Higgs Mass Bounds in the 5 + 5 Model . . . . . . . . . . . . . 73 4.1.2 Higgs Mass Bounds in the 10 + 10 Model . . . . . . . . . . . . 77 4.2 Flavor Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.1 Flavour Violation in 5 + 5 Model . . . . . . . . . . . . . . . . 82 4.2.2 Flavour Violation in 10 + 10 Model . . . . . . . . . . . . . . . 85 5 CONCLUSION 88 BIBLIOGRAPHY 92 A Diagonalization of Fermion Mass Matrix 100 A.1 Derivation of the Light Fermion Mass Matrix . . . . . . . . . . . . . 100 A.2 Light Neutrino Mass Matrix . . . . . . . . . . . . . . . . . . . . . . . 106 B RGE from the Scale M¤ to the GUT Scale in the SO(10) £ A4 Model 113 C Yukawa Couplings RGEs 115 C.1 MSSM with 5 + 5 Messenger Fields . . . . . . . . . . . . . . . . . . . 115 vi C.2 MSSM with 10 + 10 Messenger Fields . . . . . . . . . . . . . . . . . . 116 D Generated Scalar Masses due to MessengerMatter Mixing 117 D.1 5 + 5 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 D.2 10 + 10 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 vii LIST OF TABLES Table Page 1.1 The transformation of the lepton (Li,ec), quark (Q,uc,dc), and Higgs (H) ¯elds under SM gauge group SU(3)c £ SU(2)L £ U(1)Y . . . . . . 2 2.1 The transformation of the matter ¯elds under SO(10)£A4 and Z2 £ Z4 £ Z2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 The transformation of the Higgs ¯elds under SO(10)£A4 and Z2£Z4£Z2. 35 2.3 This Table shows the comparison of the model predictions at low scale and the experimental data. . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 The transformation of the matter ¯elds under SO(10) £ A4 and Z2 £ Z4 £ Z2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 The transformation of the Higgs ¯elds under SO(10)£A4 and Z2£Z4£Z2. 52 3.3 The fermion masses and mixings and their experimental values. The fermion masses, except the neutrino masses, are in GeV. . . . . . . . 64 3.4 The mass insertion parameters predicted by SO(10) £ A4 model and their experimental upper bounds obtained from [55]. . . . . . . . . . . 66 3.5 Branching ratio of ¹ ! e° for di®erent choices of input parameters at the GUT scale. Cases I and II correspond to ln M¤ MGUT = 1 and cases III and IV correspond to ln M¤ MGUT = 4:6. ~mÃi and M1=2 are given in GeV 67 4.1 We show the values of the minimal GMSB input parameters, ¤, ¸ex and Mmess that lead to the highest mh values at tan ¯ = 10. . . . . . 73 viii 4.2 The spectra corresponding to 10 + 10 model and 5 + 5 model. All the masses are in GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3 We show the values of the GMSB input parameters, ¤, ¸0 0 and Mmess that lead to the highest mh values. These values correspond to ¸0 m0 = 0 and tan ¯ = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 We show the values of the GMSB input parameters, ¤, ¸0 0 and Mmess that lead to the highest mh values. These values correspond to ¸0 m0 = 1:6 and tan ¯ = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5 The U(1) charge assignments to the messenger, MSSM, Z and S ¯elds. 83 4.6 The U(1) charge assignments to the 10 + 10 messenger, MSSM, Z and S super¯elds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.7 The calculated mass insertion parameters for the 5 + 5 and 10 + 10 models and their experimental upper bounds. The numerical values of ·'s are ·d5 = 0:0066, ·l 5 = 0:032, ·d 10 = 0:0028 and ·l 10 = 0:0025. . . . . 87 ix LIST OF FIGURES Figure Page 1.1 Feynman diagrams of K0 $ K0 induced by higher order corrections in the SM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 The evolution of the inverse gauge couplings ®¡1 i in the standard model (dashed lines) and in the MSSM (solid lines). . . . . . . . . . . . . . 12 1.3 These three diagrams contribute to K0 $ K0 mixing in supersymmet ric models. They put constraints on the o®diagonal elements of the soft breaking scalar down mass matrix that is indicated by £. . . . . 15 1.4 The most common breaking chains of SO(10) gauge group to the SM gauge group (GSM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1 This ¯gure shows a diagrammatic representation of the couplings in the superpotential W1. . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 This ¯gure leads to the °avor symmetric contribution to the down quarks and charged leptons. . . . . . . . . . . . . . . . . . . . . . . . 36 2.3 This ¯gure leads to the °avorantisymmetric contribution to the down quarks and charged leptons. . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 This ¯gure leads to the righthanded neutrino mass matrix. . . . . . . 45 3.1 The above graphs show the plot of Log of Br(¹ ! e°) divided by experimental bound (1:2£10¡11) versus mÃ for two cases I and II with M1=2=787 GeV, 437 GeV and 175 GeV. . . . . . . . . . . . . . . . . . 65 x 4.1 The evolutions of the gauge couplings with Mmess = 108 GeV and tan ¯ = 10. Solid lines correspond to MSSM. Dashed lines are for MSSM+10 + 10 and dotted lines are for MSSM+5 + 5. . . . . . . . . 70 4.2 The left graph is ~m2¿ c versus ¸0 0 at the scale Mmess for two di®erent messenger scales. The right graph is ~m2t c versus ¸0 0 at the low energy scale for two di®erent messenger scales. . . . . . . . . . . . . . . . . . 74 4.3 The left (right) graph shows the running of two exotic Yukawa cou plings from the GUT scale MGUT = 2 £ 1016 GeV to the messenger scale Mmess = 108 GeV for the 5+5 (10+10) model where the uni¯ed Yukawa coupling is taken to be ¸0 0 = 1:6. . . . . . . . . . . . . . . . . 77 4.4 The left graph is a plot of mh versus ¤ for ¸0 0 = 0 and ¸0 0 = 1:2. The right graph is mh versus ¸0 0 for di®erent messenger scales at ¤ = 105 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 xi CHAPTER 1 INTRODUCTION 1.1 A Brief Review of the Standard Model (SM) The standard model (SM) of particle physics is based on the nonabelian gauge sym metry SU(3)c£SU(2)L£U(1)Y . Here the SU(3)c gauge group describes the theory of strong interaction called quantum chromodynamics (QCD). This type of interactions holds the quarks and the gluons together to form hadrons. Each quark type is called °avor. For example, up, charm and top denoted respectively by u, c and t are three °avors of the uptype quark. Each °avor of quark transforms as the fundamental color triplet of SU(3)c while the gauge bosons, the gluons, are assigned to the adjoint octet representation of SU(3)c. In this case, we have eight gluons associated with the eight SU(3)c generators. The SU(2)L £ U(1)Y is the gauge group of the Glashow, Wein berg, and Salam model [1] which successfully combines the electromagnetic and weak interactions in one theory called electroweak theory. The total number of generators of SU(2)L £ U(1)Y is four. Accordingly, this theory contains four electroweak gauge bosons (three of them conventionally are denoted Wi and the forth one is denoted B). This SU(2)L£U(1)Y symmetry is respected above roughly 100 GeV (the electroweak scale). The electromagnetic interaction arises below the electroweak scale where the electroweak symmetry is broken spontaneously by the Higgs mechanism. In order to understand how the electroweak symmetry breaking is implemented in the SM, let us ¯rst point out that the invariance of the Lagrangian for both quantum electrodynamics (QED) and QCD under local gauge transformations leads respectively to massless photons and gluons. However, this idea can not be applied 1 SU(3)c £ SU(2)L £ U(1)Y Li= 0 BB@ ºi ei 1 CCA , (1,2,¡1) eci , (1,1,2) Qi= 0 BB@ ui di 1 CCA , (3,2, 1 3 ) uci , (3,1,¡4 3 ) dci , (3,1, 2 3 ) H= 0 BB@ H+ H0 1 CCA , (1,2,1) Table 1.1: The transformation of the lepton (Li,ec), quark (Q,uc,dc), and Higgs (H) ¯elds under SM gauge group SU(3)c £ SU(2)L £ U(1)Y . to the weak interaction since the gauge bosons of the weak interaction are massive (of order 90 GeV). One way out of this problem is to consider the situation of a hidden symmetry; the Lagrangian still respects the local gauge symmetry, but picks one of all possible ground states that result from minimizing the potential for a Higgs ¯eld as the physical vacuum which breaks the symmetry. The spontaneous symmetry breaking is implemented by including a doublet of scalar Higgs boson to the SM. The transformations of the quark, lepton and Higgs ¯elds under the SM gauge group are shown Table 1.1. In this Table, all fermion ¯elds are left handed and the generation index i runs from 1 to 3. Let us study the spontaneous symmetry breaking of the gauge group SU(2)L £ U(1)Y to U(1)em by 2 writing down the Higgs potential for the Higgs ¯eld H: V (H) = ¡¹2HyH + ¸(HyH)2; ¹2 > 0: (1.1) The above potential is invariant under the SM gauge group. Minimizing the potential V (H), one obtains hHi = h0jHj0i = v p 2 0 BB@ 0 1 1 CCA ; (1.2) where v = ¹= p ¸. The generator that remains unbroken is Q = T3 + Y 2 . Y refers to the electroweak hypercharge. Q is identi¯ed as the electric charge. The unbroken charge is easily checked by QhHi = 0: (1.3) The parameter Y needs to be adjusted such that the electric charges of the quarks and the leptons come out right. In general, the broken generators correspond to the gauge bosons that pick up mass, and the unbroken generators correspond to the massless gauge bosons. In this case, there are three broken generators associated with three massive gauge bosons (W+, W¡, Z0), and the unbroken charge Q asso ciated with massless gauge boson ° (the electromagnetic ¯eld A¹). The electroweak symmetry breaking scale is around the masses of the gauge bosons (i.e., 100 GeV). We can calculate the masses of electroweak gauge bosons by substituting the vacuum expectation value (VEV) of the Higgs ¯eld from Eq.(1.2) into the following gauge invariant kinetic term of the Higgs ¯eld: (D¹H)(D¹H)y = j@¹H ¡ ig 2 ¡!¿ : ¡! W¹H ¡ ig0 2 B¹Hj2; (1.4) where the gauge coupling constants g and g0 are associated respectively to the gauge groups SU(2)L and U(1)Y . The masses of the electroweak gauge bosons are then mW = ev 2 sin µW ; (1.5) mZ = ev 2 sin µW cos µW : (1.6) 3 The gauge coupling constants are parameterized in terms of an angle µW (known as the Weinberg angle) de¯ned as follows: tan µW = g0 g ; (1.7) and e = g sin µW. The mass term of fermions cannot be added to the Lagrangian by hand because the lefthanded and the righthanded fermions transform di®erently under SU(2)L £U(1)Y . Therefore, one employs the Higgs mechanism that generates mass to the fermions via Yukawa couplings. The Higgs ¯eld and its charge conjugate are given respectively by H = 0 BB@ H+ H0 1 CCA ~H = i¿2H¤ = 0 BB@ H¤ 0 ¡H¡ 1 CCA : (1.8) The transformation of ~H under SU(3)c£SU(2)L£U(1)Y is (1, 2, ¡1). We can write the gauge invariant Yukawa couplings as follows: LY = Y d ijdcT i HyQj + Y e ijecT i HyLj + Y u ij ucT i ~H yQj + h:c:; (1.9) where a charge conjugation C is understood to be sandwiched between the fermion ¯elds. As a consequence of spontaneous symmetry breaking, LY leads to mass terms for fermions as follows: LY = DcTMdD + UcTMuU + EcTMeE + h:c:; (1.10) where U = 0 BBBBBB@ u c t 1 CCCCCCA ; D = 0 BBBBBB@ d s b 1 CCCCCCA ; E = 0 BBBBBB@ e ¹ ¿ 1 CCCCCCA ; Uc = 0 BBBBBB@ uc cc tc 1 CCCCCCA ; Dc = 0 BBBBBB@ dc sc bc 1 CCCCCCA ; Ec = 0 BBBBBB@ ec ¹c ¿ c 1 CCCCCCA : (1.11) 4 The mass matrix elements for upand downquarks as well as charged leptons are given by MF ij = v p 2 Y F ij ; F = u; d; e: (1.12) Note that in the standard model the right handed neutrino does not exist. Therefore, the neutrinos are massless. The weak eigenstates are not eigenstates of the Hamil tonian. In order to write the Lagrangian in terms of the Hamiltonian eigenstates (i.e mass eigenstates), we need to diagonalize the fermion mass matrices given by Eq.(1.12) by means of biunitary transformation given as: V F R y MFV F L = MF diag:; (1.13) where Mu diag: = diag(mu;mc;mt); Md diag: = diag(md;ms;mb); Me diag: = diag(me;m¹;m¿ ): (1.14) The fermion mass matrices (MF ) are in general neither symmetric nor hermitian. but, MF y MF is hermitian and can be diagonalized as follows: V F L y MF y MFV F L = MFy diag:MF diag:: (1.15) The mass eigenstates (D0, U0, E0, Dc0 , Uc 0 , Ec 0) can be written in terms of the weak eigenstates as follows: D0 = V d L y D; Dc0 = V d R T Dc; U0 = V u L yU; Uc 0 = V u R TUc; E0 = V e L yE; Ec 0 = V e R TDc: (1.16) The charged current weak interactions for quarks are given as Lcc = g p 2 WyU°¹D + h:c: = g p 2 WyU 0 VCKM°¹D0 + h:c: (1.17) 5 It is clear from the above equation that the charged current W§ interactions couple to the physical u0j and d0 k quarks with a couplings matrix represented by VCKM = V u L yV d L = 0 BBBBBB@ Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb 1 CCCCCCA : (1.18) This is called the CabibboKobayashiMaskawa mixing matrix [2, 3]. It is a unitary matrix that can be parameterized by three mixing angles and one CPviolation phase: VCKM = 0 BBBBBB@ c12c13 s12c13 s13e¡i± ¡s12c23 ¡ c12s23s13ei± c12c23 ¡ s12s23s13ei± s23c13 s12s23 ¡ c12c23s13ei± ¡c12s23 ¡ s12c23s13ei± c23c13 1 CCCCCCA ; (1.19) where sij = sin µij , cij = cos µij and ± is the phase factor responsible for the violation of CP symmetry [3]. All other phases can be removed by ¯eld rede¯nition. It is known experimentally that the CKM mixing angles are small (i.e s13 << s23 << s12 << 1). It is convenient to write down an expression for a CPviolation parameter which is phaseconventionindependent: ´ = ¡Im(VudV¤ ub=VcdV¤ cb): (1.20) Unlike the situation in the case of charged current interactions, no °avor mixings exist for neutral current interactions of SU(2)L £ U(1)Y at tree level and this has been con¯rmed to a great accuracy by experiments. However, °avor changing neutral currents (FCNC), which have been measured, but which are strongly suppressed, can be induced by considering higher order corrections. For example, FCNC can be induced in the process K0 $ K0 transition which arises from box diagrams shown in Fig1.1. The calculation on the K0 $ K0 mass di®erence ¢mk has been done [4], and the result is close to the experimental value of ¢mk = 3:5£10¡15 GeV. This can be considered as a successful prediction of the SM. 6 u,c,t W + W u,c,t W u,c,t u,c,t _ _ d s d s d s d s Figure 1.1: Feynman diagrams of K0 $ K0 induced by higher order corrections in the SM. 1.2 Seesaw Mechanism and Leptonic Mixing Matrix In the previous section, we have seen that SM contains left and right chiral projections for all fermions except the neutrinos. This looks unnatural. Besides, the absence of a righthanded neutrino from Eq.(1.9) leads to massless neutrinos. However, neutrino experiments indicate that the neutrinos have tiny masses. The current experimental values for neutrino masses are [7] ¢m2 21 = (7:59 § 0:20) £ 10¡5eV2; ¢m2 32 = (2:43 § 0:13) £ 10¡3eV2; (1.21) where ¢m2 2;1 = m22 ¡ m21 and ¢m23 ;2 = m23 ¡ m22 . To explain this, let us add to the SM righthanded neutrinos (ºc i ) corresponding to each charged lepton. The ºc i ¯elds transform as (1,1,0) under the SM gauge group. Thus, we can write down the Yukawa couplings for the neutrino sector as follows: Lº Y = Y º ij ºcT i ~H yLj + h:c: (1.22) With a VEV of ~H y, this gives the following neutrino Dirac mass term Lº Y = MDºcT º + h:c; (1.23) where (MD)ij = Y º ij v= p 2. Since the ºc ¯elds are singlets under the SM gauge sym metry, they can posses a gauge invariant bare mass term (Majorana mass): Lbare = 1 2 MRºcT ºc + h:c: (1.24) 7 We can write the combination of Majorana and Dirac neutrino masses as a matrix for the (º, ºc) system as: Mº = 0 BB@ 0 MTD MD MR 1 CCA ; (1.25) where MD and MR are 3 £ 3 matrices. The invariance of the righthanded neutrino mass terms under SM gauge symmetry suggests that they can be above the weak interaction scale. So after integrating out these heavy ¯elds (or equivalently by ¯nding the eigenvalues of the matrix in Eq.(1.25)), the light neutrino masses are suppressed by MR via: Mº = ¡MTD M¡1 R MD; (1.26) where MD should not exceed about 100 GeV. This idea, known as the seesaw mech anism [5], is an elegant way to explain the smallness of neutrino masses. The light neutrino mass matrix given by Eq.(1.26) can be diagonalized as: V T º MºVº = 0 BBBBBB@ m1 m2 m3 1 CCCCCCA; (1.27) with m1;2;3 being the tiny masses of the three light neutrinos. Now, we can write the leptonic charge current interaction in terms of the mass eigenstates as follows: Lcc = g p 2 [e0°¹VPMNSº0]W¡¹ + h:c: (1.28) where VPMNS = V y LVº is the leptonic mixing matrix, or the PontecorvoMakiNakagawa Sakata (PMNS) matrix [6]. In general, the PMNS matrix can be written as VPMNS = 0 BBBBBB@ Ve1 Ve2 Ve3 V¹1 V¹2 V¹3 V¿1 V¿2 V¿3 1 CCCCCCA ; (1.29) 8 which can be parameterized in terms of three Euler angles and three phases one \Dirac phase" and two \Majorana phases". The standard parametrization [7] has VPMNS = V:P where V = 0 BBBBBB@ c12c13 s12c13 s13e¡i± ¡s12c23 ¡ c12s23s13ei± c12c23 ¡ s12s23s13ei± s23c13 s12s23 ¡ c12c23s13ei± ¡c12s23 ¡ s12c23s13ei± c23c13 1 CCCCCCA (1.30) P = 0 BBBBBB@ ei® ei¯ 1 1 CCCCCCA : (1.31) Here sij = sin µij , cij = cos µij which should not be confused with the angles in the quark sector, given in 1.19. The parameters ® and ¯ are the Majorana phases, while ± is the Dirac phase. Present constraints on the neutrino mixing angles can be summarized by (2¾ error bars quoted)[8] sin2 µ12 = 0:27 ¡ 0:35; (1.32) sin2 µ23 = 0:39 ¡ 0:63; (1.33) sin2 µ13 · 0:040: (1.34) The above data can be well represented by the tribimaximal mixing of the form [9] V = 0 BBBBBB@ q 2 3 q 1 3 0 ¡ q 1 6 q 1 3 ¡ q 1 2 ¡ q 1 6 q 1 3 q 1 2 1 CCCCCCA P; (1.35) which corresponds to sin2 µ12 = 1=3, sin2 µ23 = 1=2 and sin2 µ13 = 0. No information on the Dirac phase ± and on the Majorana phases (¯, ®) is known at present. There are several thoughts to reproduce the structure in Eq.(1.35). One interesting idea is to employ the discrete °avor symmetry A4 [10] which will be further discussed in chapter 2. 9 1.3 Shortcomings of the SM and the Need for New Physics. The standard model is a trustful theory in the energy range of few 100 GeV. However, things become more obscure beyond the electroweak energy scale. Understanding how nature behaves at higher energy scales might answer many of the standard model's puzzles. For example, the SM has no real explanation of the di®erent strengths of the three gauge couplings associated with the three gauge groups. Also, there is no reason why the fermions transform under the local gauge interactions of the SM in the way shown in Table 1.1, except for the posteriori justi¯cation of ¯tting the data. Grand uni¯cation theory (GUT) provides an understanding of the origin of the three gauge couplings and consequently an understanding of three gauge groups. The GUT idea is described by a uni¯ed gauge group which necessitates a single uni¯ed gauge coupling. This uni¯ed gauge group will be broken at a certain high energy scale (GUT scale) to the SM gauge group. Thus, strong, weak and electromagnetic forces are described in the framework of a single grand uni¯ed theory. Moreover, if the uni¯ed gauge group is simple, quantization of electric charge will follow automatically because the eigenvalues of the nonabelian group generators are discrete as opposed to the eigenvalues of the abelian U(1) group generator which are continuous. The most popular simple nonabelian groups that are chosen as grand uni¯cation groups are SU(5) and SO(10). We will study these GUT groups in details in sections 1.4 and 1.5. Arbitrary Parameters The SM has 19 arbitrary parameters. 3 gauge coupling constants (gs, g, and g0 as sociated respectively with SU(3)c, SU(2)L and U(1)Y ), 9 charged fermion masses, 4 quark mixing parameters, and v, ¸ (or equivalently to Mz, mh) and the QCD µ parameter. Besides, if we consider the neutrino sector, there are at least 9 additional parameters: 3 light neutrino masses, 3 mixing angles, and 3 phases (assuming Majo 10 rana neutrinos). Thus, the SM has too many arbitrary parameters which are chosen in order to ¯t the data. On the other hand, GUTs do not contain that many arbi trary parameters. Another advantage of GUT is that the seesaw mechanism can be implemented naturally within SO(10) GUT, since the gauge structure requires the existence of ºc, as we will see in section 1.5.3. Grand uni¯cation theory describes the three interactions (strong, weak, and elec tromagnetic) by one gauge coupling constant. However, it is known that these inter actions are described by three distinct gauge couplings at low energy (E ¼ 100 GeV). So the question is how does the grand uni¯cation idea reconcile with these three disparate couplings? This question can be answered by the suggestion [11] that the three gauge coupling constants are scale dependent quantities, and if the hypothesis of grand uni¯cation holds, the three gauge coupling constants of the SM will meet to a uni¯ed value at the GUT scale MGUT . Above the scale MGUT we have one gauge coupling described by a simple uni¯ed group. The renormalization group running of the gauge couplings determines the GUT scale. In the SM, however, the gauge couplings come only close to one another forming what is called the GUT triangle as shown in Fig.1.2. This can be ¯xed by introducing new physics around the TeV scale. The most promising new physics scenario is supersymmetry, which will be further discussed below Hierarchy Problem Another problem that needs to be ¯xed is the hierarchy problem of the SM. This problem occurs because the mass of the Higgs boson receives a quadratically divergent loop correction given by: m2 HSM(phys) ' m2 HSM + c 16¼2¤2; (1.36) where m2 HSM is the Higgs mass squared parameter in the Lagrangian and the second term denotes the quadratically divergent loop correction. The cuto® scale ¤ is in 11 a1 1 a2 1 a3 1 5 10 15 0 10 20 30 40 50 60 Log10 Hm GeVL a 1 Figure 1.2: The evolution of the inverse gauge couplings ®¡1 i in the standard model (dashed lines) and in the MSSM (solid lines). terpreted as the scale at which the SM ceases to be valid. Reasonable values of the energy scale ¤ at which the new physics becomes important are chosen such that any extremely ¯netuned cancelation between the two terms on the righthand side of Eq.(1.36) is avoided. The physical Higgs boson mass mHSM(phys) has to be smaller than a few hundred GeV [12]. Therefore, reasonable values of ¤ might be around the TeV scale. A promising scenario that solves the hierarchy problem of the SM and allows the uni¯cation of the three gauge coupling constants is supersymmetry (SUSY). In order to avoid extreme ¯netuning, SUSY should exist above an energy scale of order 1 TeV which is being probed at the Large Hadron Collider. Problems in the Flavor Sector The SM does not provide an explanation for the existence of three families of fermions, and the observed masses and mixings of the fermions, and the smallness of the quark mixing angles compared to the largeness of the neutrino mixing angles. These prob lems can be understood either through GUTs and/or by adding a family symmetry. 12 Some of the features of the fermions such as the three fold replication of fermion gen erations, mixing properties of the lepton sectorthat is two large mixing angles and one small mixing anglecannot be explained successfully by GUT symmetry alone. So in order to meet these challenges, one may consider the possibility of introducing a °avor symmetry (family symmetry) group which is the symmetry between genera tions. In this case, the three known generations can be assigned to a representation of the family group. There are many possible candidates for the family symmetry group. Basically, we can divide them into two categories: continuous and discrete groups. The general feature of the global continuous groups is that they lead to undesired Goldstone bosons. On the other hand, it is suggestive to consider discrete nonabelian symmetry because in this case there is no problem with unwanted Goldstone bosons. Combining grand uni¯cation gauge symmetry and family symmetry (GGUT £ GFAM) in the framework of supersymmetric theory leads certainly to new physics beyond the SM that solves most of the standard model's puzzles. Many grand uni¯cation models with discrete family symmetry have been studied so far [13, 14, 15, 16]. In particular, employing SO(10) £ A4 symmetry may give the tribimaximal mixings structure in Eq.(1.35) [15]. 1.3.1 Supersymmetry Supersymmetry is a symmetry that relates bosons and fermions. It predicts new yet to be discovered superpartner states for each known particle in the SM. The SM particle and its supersymmetric partner belong together to the same supermultiplet which is collectively described in terms of a super¯eld. In this way a spin0 boson and a spin1/2 fermion are described as a chiral super¯eld and a spin1 vector boson and a spin1/2 fermion form a vector super¯eld. The supersymmetric extension of the SM assumes that all quarks and leptons of the SM are accompanied by their scalar su perpartners which are called respectively squarks and sleptons, and the gauge bosons 13 with their fermionic superpartners which are called gauginos. This supersymmetric extension of the SM is called Minimal Supersymmetric Standard Model (MSSM), it is minimal in the sense that it contains the smallest number of new particle states. The SM contains one Higgs doublet ¯eld to achieve electroweak symmetry breaking while the MSSM contains two Higgs doublets Hu and Hd which give mass to the uptype and downtype quarks respectively. Their superpartners are called higgsinos. This setup helps in solving the quadratic divergence correction of the Higgs mass due to the fact that the loops involving particles are canceled by the loops involving their su perpartners. Another feature in favor of the MSSM is that the gauge couplings unify around 2 £ 1016 GeV as shown in Fig 1.2. These features motivate the consideration of supersymmetric GUTs. Unlike the SM where the baryon and lepton numbers are conserved automatically, there are additional superpotential terms in the case of MSSM that are consistent with SU(3)c £ SU(2)L £ U(1)Y symmetry, which break the lepton and baryon num bers. These terms are dangerous since the lepton and baryon violating processes are strongly constrained by experiment, especially from proton stability. These un wanted terms can be prohibited by requiring the superpotential to be invariant under Rparity de¯ned by, R = (¡1)3(B¡L)+2s; (1.37) where s is the spin of the ¯eld, and B and L are the baryon and the lepton number respectively. For example B = 1=3(¡1=3) for quark (antiquark) super¯elds, L = 1(¡1) for lepton (antilepton) super¯elds, and zero for the Higgs and gauge super¯elds. Supersymmetry Breaking The supersymmetry algebra tells us that the particle and its superpartner acquire the same mass. However, this is not consistent with experiment since for instance no spin0 particle has been detected so far with the same mass as the electron. Therefore, 14 d*R ~ g~ d s g~ s _ R s~* d _ d s g~ s _ sR* ~ d _ d s g~ s _ sR* ~ d _ ~ dR s~ R d*R ~ dL* ~ sL d ~ ~ L d ~ L s~ R (a) (b) (c) Figure 1.3: These three diagrams contribute to K0 $ K0 mixing in supersymmetric models. They put constraints on the o®diagonal elements of the soft breaking scalar down mass matrix that is indicated by £. SUSY must be broken somewhere above the energy scale that has been probed so far. SUSY should preferably be broken spontaneously. In other words, the generators of the SUSY does not annihilate the vacuum. Although many models of SUSY breaking have been proposed, there is no complete theory where this is achieved satisfactorily at present. In order to maintain the remarkable cancelation of quadratic divergencies in ¯eld theoretical models, SUSY should be broken softly in the e®ective low energy theory. This can be done by assuming that the outcome of symmetry breaking is extra terms (soft terms), such as additional masses for the scalars. The common philosophy of all the scenarios of SUSY breaking is that SUSY is broken in a \hidden sector" of particles which is decoupled from the visible sector of MSSM particles. The e®ects of SUSY breaking in the hidden sector are communicated to the visible sector by messengers, resulting in the MSSM soft SUSY breaking terms. The soft SUSY breaking terms imply °avor mixing. For example, suppose ~m2 Q is not diagonal in the soft term ~ dy Li(m2 Q)ij ~ dLi. In this case, the e®ective Hamiltonian for K0 $ K0 mixing gets contributions from the box diagrams involving squarks and gluinos, such as the ones shown in Fig.1.3. The experimental value of ¢mK puts constraints on the soft SUSY breaking mixing of the three diagrams in Fig1.3. The most striking limit applies to the diagram in Fig1.3(b) [28]: jRe[ ~m2s ¤ RdR ~m2s ¤ LdL j1=2 ~m2q < ~mq £ 10¡3 500 GeV ; (1.38) 15 where ~mq is the average mass of squarks ~md and ~ms and the gluino mass has been assumed equal to the average squark mass. Thus, in order to suppress the o®diagonal entries of ~m2 Q, we need to assume the masses of the squarks are nearly degenerate. This can be achieved by adding a nonAbelian discrete symmetry group. This can be done either by grouping the ¯rst two families into an irreducible doublet [29] or by grouping all three families into an irreducible triplet of the °avor group. For example, the group could be A4, which is the smallest discrete group that contains a triplet in its irreducible representations. Another natural solution to the °avor violation problem is obtained by adopting gaugemediated supersymmetry breaking (GMSB) scenario [59, 60, 61]. In this sce nario the supersymmetry breaking is transmitted to the visible sector by SM gauge interactions. In this case the soft masses are generated through loops such that the scalar masses with the same gauge quantum number are automatically degenerate. A model based on the GMSB scenario will be discussed in chapter 4. 1.3.2 Discrete Flavor Symmetry A4 The nonabelian ¯nite group A4 is the symmetry group of even permutations of four objects. It has twelve elements and four irreducible representations (irreps): 1, 10, 100, 3s, and 3a with the multiplication rule 3 £ 3 = 1 + 10 + 100 + 3s + 3a: (1.39) For example, let (a1; a2; a3), and (b1; b2; b3) transform as triplets under A4, then the multiplication of 3 £ 3 can be decomposed as a1b1 + a2b2 + a3b3 » 1; (1.40) a1b1 + !2a2b2 + !a3b3 » 10; (1.41) a1b1 + !a2b2 + !2a3b3 » 100; (1.42) (a2b3 + a3b2; a3b1 + a1b3; a1b2 + a2b1) » 3s; (1.43) 16 (a2b3 ¡ a3b2; a3b1 ¡ a1b3; a1b2 ¡ a2b1) » 3a; (1.44) where ! = exp[2¼i=3]. One advantage of the discrete A4 symmetry is that it is the smallest group that contains a 3dimensional irrep so that the three generations of the fermions can be accommodated within this triplet. Another advantage is that the FCNC problem might be solved if one considers the combinations of A4 and SUSY SO(10) GUT. This is due to the fact that the SO(10)£A4 symmetry allows us to write down one universal mass term for the three generations of sfermions. Consequently, the degeneracy of sfermions is satis¯ed. 1.4 Minimal SUSYSU(5) We have pointed out previously that the running behavior of the three gauge couplings with energy scale indicates that they should unify at some point at a high energy scale. This uni¯cation of the gauge couplings does not occur exactly in the SM. However, in the case of the MSSM, the uni¯cation occurs with impressive precision at MGUT ¼ 2 £ 1016 GeV. This strongly suggests that MSSM might be remnant of some sort of supersymmetric grand uni¯cation theory. Therefore, it is logical to propose a larger gauge group associated with one gauge coupling constant. The ¯rst approach of ¯nding a simple gauge group that contains the SM group was the GeorgiGlashow SU(5) model [17]. In this section we will discuss this SU(5) model, its predictions and its experimental implications because it is considered the simplest example of grand uni¯cation models and it is a subgroup of SO(10). 1.4.1 SU(5) Matter Fields The SM gauge group has rank 4. Hence the rank of the grand uni¯cation group should be at least 4. There are many possibilities for a rank 4 simple group with one gauge couplings. Among all possibilities, SU(5) is found to be the only choice that meets all the required features: It has complex representation for fermions and it 17 accommodates both integer and fractionally charged fermions. The 15 lefthanded SM fermions for one family can be embedded into just two irreps, the antifundamental 5F and the twoindex antisymmetric tensor 10F . This can be seen by writing the decomposition of 5F and 10F irreps of SU(5) under SU(3)c £ SU(2)L £ U(1)Y as follows: 5 = (3; 1; +2=3) © (1; 2;¡1); 10 = (3; 1;¡4=3) © (3; 2; +1=3) © (1; 1; +2): (1.45) Also, this embedding can be depicted in matrix representation as 5 = 0 BBBBBBBBBBBBBB@ dc1 dc2 dc3 e¡ º 1 CCCCCCCCCCCCCCA ; 10 = 1 p 2 0 BBBBBBBBBBBBBB@ 0 uc 3 uc 2 u1 d1 ¡uc 3 0 uc 1 u2 d2 ¡uc 2 ¡uc 1 0 u3 d3 ¡u1 ¡u2 ¡u3 0 e+ ¡d1 ¡d2 ¡d3 ¡e+ 0 1 CCCCCCCCCCCCCCA : (1.46) This assignment is free of chiral anomalies. In the SUSY version of SU(5), these multiplets are promoted to super¯elds. 1.4.2 Higgs Sectors and Yukawa Couplings in the minimal SUSYSU(5) In order to test the viability of minimal SUSY SU(5), let us ¯rst construct the invariant Yukawa couplings by writing down the SU(5) decomposition of all possible multiplications of the irreps 5 and 10. 5 £ 5 = 10 + 15; (1.47) 10 £ 10 = 5 + 45 + 50; (1.48) 5 £ 10 = 5 + 45: (1.49) It is easy to check that the MSSM super¯eld Higgs doublet Hu is contained in 5 and 45, and Hd in 5 and 45. Therefore, two quintets 5H and 5H are introduced minimally in 18 the SUSYminimal SU(5). These two quintets are responsible for breaking SU(3)c £ SU(2)L £ U(1)Y to SU(3)c £ U(1)em. Based on the above analysis, the invariant superpotential that contains only the Yukawa couplings is given as follows: ^ f 3 Y u ®¯²ijklm10ij F®10kl F¯5mH + Y d ®¯10ij F®5Fi¯5Hj : (1.50) The mass matrices generated by the VEVs of the the SU(2)L doublets in both 5H and 5H then read Md = ML = Y dh5Hi; Mu = Y uh5Hi: (1.51) Since the ¯rst term in Eq.(1.50) contains two identical 10s, the upquark Yukawa couplings are symmetric in the generation indices, i.e., Mu = M> u . Diagonalization of the down quarks and charged leptons mass matrix leads to me = md m¹ = ms m¿ = mb: (1.52) Note that the above mass relations are only valid at mass scales where the SU(5) is a good symmetry. But the light fermion masses are observed at low energy scale of order (25) GeV. Therefore, the above mass relations should be extrapolated to low energy scale. The results are the following: the ¯rst two mass relations in Eq.(1.52) are violated by experiment, while the third one is considered as a successful prediction of minimal SUSY SU(5). One way to correct the bad mass relations for the ¯rst and second generations is to employ the 45H [18]. In this case, the price that we have to pay is including several Higgs multiplets. It is obvious that the Higgs multiplets 5H and 5H do not break SU(5) to SU(3)c£ SU(2)L £ U(1)Y since they do not contain a SM singlet. The smallest dimensional Higgs representation that contains the SM singlet is the adjoint of SU(5). The adjoint Higgs representation 24H decomposes under SU(3)c £ SU(2)L £ U(1)Y to 24H = (1; 1; 0) © (8; 1; 0) © (1; 3; 0) © (3; 2;¡5=6) © (3; 2; +5=6); (1.53) 19 and the (1,1,0) component can acquire a GUTscale VEV. Equivalently, one can show [19] h24Hi = ¾ 0 BBBBBBBBBBBBBB@ 2 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 ¡3 0 0 0 0 0 ¡3 1 CCCCCCCCCCCCCCA : (1.54) The two Higgs ¯elds 24H and 5H develop hugely di®erent VEVs (i.e., h24Hi of order MGUT ¼ 1016 GeV and h5Hi of order MW ¼ 102 GeV). Consequently, this leads to a huge hierarchy of the gauge symmetry. In nonSUSY model, the param eters at tree level of the Higgs potential should be ¯netuned in order to maintain this huge hierarchy. On the other hand, this ¯netuning gets worse via radiative cor rections. However, in the minimal SUSY SU(5), once the parameters of the Higgs superpotential ^ f 3 m55H5H + m24Tr[24H24H] + ¸1Tr[24H24H24H] + ¸25H24H5H (1.55) are ¯netuned properly at tree level, the SUSY nonrenormalization theorem of Gris aru, Rocek and Siegel [67] ensures that it does not get upset by radiative corrections, since according to this theorem these parameters do not receive either ¯nite or in¯nite corrections. 1.4.3 Gauge Sector of Minimal SU(5) The adjoint representation of SU(5) has the dimension 52 ¡1 = 24. Hence, there are 24 gauge bosons associated with SU(5). They decompose under SU(3)c £ SU(2)L £ U(1)Y as given in Eq.(1.53). The gauge bosons of SM are contained within 24 gauge bosons of SU(5) as follows: (8; 1; 0) are SU(3)c gluons , (1; 3; 0) are the three SU(2)L vector ¯elds W, and (1; 1; 0) is the U(1) B¯eld. The remaining 12 gauge bosons, 20 which transform under the SM gauge group as (3; 2; 5 3 ) and (3¤; 2;¡5 3 ) are called leptoquark gauge bosons denoted respectively by X and Y . These gauge bosons can be collectively described by a 5 £ 5 matrix form, A¹ = Aa¸a=2, where ¸a are the SU(5) generators (a runs from 1 to 24) and the summation over index a is implied. As we have discussed before, the Higgs phenomenon can provide masses to the gauge bosons by developing a VEV to the Higgs ¯eld. This can be seen by writing down the invariant kinetic term of the Higgs ¯elds as follows LKE = Tr[(D¹24H)(D¹24H)¤]: (1.56) Here the covariant derivative of the adjoint representation 24H is de¯ned as follows: D¹24H = @¹24H + ig5[A¹; 24H]; (1.57) where [A¹; 24H] = A¹24H ¡ 24HA¹, and g5 is the SU(5) gauge coupling. The factor g2 5Tr[A¹; h24Hi]2 contains the mass term for the gauge bosons. Since 24H commutes with the generators of the SM gauge group, the gauge bosons of the SM (Wr, B, G®¯ ) do not pick up mass, while the X and Y gauge bosons acquire masses according to MX = MY = 5 p 2g5¾ (1.58) 1.5 Minimal SUSYSO(10) We have seen that the SM fermions can be accommodated within two irreducible representations of the simplest uni¯ed model based on SU(5) gauge symmetry. This leads to the uni¯cation of the Yukawa couplings of the down quarks and charged lep tons. On the other hand, a single 16dimensional chiral spinor of SO(10) is enough to accommodate all the SM model fermions of one generation. This brings the follow ing bene¯ts: First, the righthanded neutrino is automatically accommodated within the same multiplet. Second, the number of independent parameters of the e®ective fermion masses and mixing matrices can be reduced considerably. These observations motivate us to consider the SO(10) gauge symmetry. 21 1.5.1 Matter Fields in SO(10) GUTs The reducible spinorial representation of SO(10) splits into a pair of spinorial repre sentations 16 and 16 under a chiral projection operator, for details see Ref. [21]. All the femions reside in only one chirality of a SO(10) spinorial representation (i.e, 16 dimensional representation of SO(10)). In order to see how the SM fermions can be ¯tted within a 16dimensional irrep of SO(10), let us write down its decompositions under SU(3)c £ SU(2)L £ U(1)Y : 16 = (3; 2; +1=3) © (1; 2;¡1) © (3; 1;¡4=3) © (3; 1; +2=3) © (1; 1; +2) © (1; 1; 0); (1.59) where the quantum numbers on the righthand side (except the last one) are those for the SM fermions (see Table 1), while the last one is the righthanded neutrino. Equivalently, the 16dimensional irrep of SO(10) can be written in terms of the SU(5) basis as follows: 16 = 5 © 10 © 1; (1.60) where the matrix representations of the irreducible representations of SU(5) (5 and 10) are given in Eq.(1.46). The righthanded neutrino (or equivalently ºc) is assigned to the singlet of SU(5). 1.5.2 The Higgs Fields and Yukawa Couplings in SO(10) GUTs The Higgs sector of any realistic SO(10) model should be chosen appropriately in order to satisfy the following requirements. First, the Yukawa couplings should be invariant under SO(10) and compatible with the current data on the quark and the lepton masses and mixings. Second, the Higgs sector should lead to the proper spontaneous symmetry breaking of SO(10) gauge symmetry down to the SU(3)c £ SU(2)L £ U(1)Y of the MSSM. The invariant Yukawa couplings follow 22 from the decomposition of 16 16 = 10 © 126 © 120: (1.61) Thus, there are three types of SO(10) Higgs multiplets that can give masses to the matter fermions: the 10dimensional vector representation 10H, the 126dimensional 5index antisymmetric tensor 126H and the 120dimensional threeindex antisymmet ric tensor 120H. Then, the most general Yukawa couplings are WY = Y ®¯ 10 16F®16F¯10H + Y ®¯ 12016F®16F¯120H + Y ®¯ 12616F®16F¯126H: (1.62) The good feature of the 10dimensional Higgs multiplet of SUSYSO(10) is that 10H contains the SUSYSU(5) Higgs multiplets 5H and 5H that give masses to the uptype and the downtype quarks respectively. The fermion masses are generated by giving VEVs to the Higgs ¯elds in Eq.(1.62). The fermion masses with Higgs ¯eld belonging to the 10dimensional irrep can be calculated by writing the irreps of SO(10) matter and Higgs ¯elds in terms of SU(5) £ U1 basis as [22]: 10 = 5(2) + 5(¡2); 16 = 1(¡5) + 5(3) + 10(¡1): (1.63) where the numbers in the bracket are quantum numbers of U1. Then we construct the invariant combinations of SU(5) £ U1 multiplets as Y ®¯ 10 1F®(¡5)5F¯i(3)5j H(2) + Y ®¯ 10 ²ijklm10ij F®(¡1)10kl F¯(¡1)5mH (2) + Y ®¯ 10 5F®i(3)10ij F¯(¡1)5Hj(¡2): (1.64) We remind the reader that 5F and 10F are the usual SU(5) representations of Georgi and Glashow given in Eq.(1.46). The ¯rst line in Eq.(1.64) shows that the Dirac neutrinos and upquarks couple with the same Higgs multiplets 5H while the second line tell us that the charged leptons and down quarks couple with the other Higgs muliplets 5H. Thus, M®¯ d = M®¯ e = Y ®¯ 10 h5Hi M®¯ u = M®¯ º = Y ®¯ 10 h5Hi: (1.65) 23 The above fermion mass matrices are symmetric. Since the up and down quark mass matrices in Eq.(1.65) can be diagonalized by the same unitary matrix, the quark mixing matrix is an identity matrix. This can be considered as a zeroth order ap proximation for the CKM mixing matrix. The 120dimensional Higgs representation is antisymmetric under the °avor index, however it contributes to mixings between various generations. On the other hand, the 126dimensional is symmetric under the °avor index and by itself would lead to the following mass relations [21]: Me = ¡3Y126v126 d = ¡3Md; Mº = ¡3Yºv126 d = ¡3Mu: (1.66) A realistic Higgs spectrum would include, for example, 10H © 126H. In order to achieve the spontaneous symmetry breaking of SO(10) gauge symmetry down to SU(3)c£SU(2)L£U(1)Y a (GSM) of the MSSM, we need to consider all possible Higgs ¯elds that contain GSM singlet in their decomposition under the SM gauge group such as 45H, 54H, 210H and 126H. Since SO(10) is a rank 5 group, there are many symmetry breaking chains leading to the rank4 GSM. The most common breaking chains and the Higgs representation that has been used to break the intermediate symmetries at each step are represented in Fig1.4. In any SO(10) breaking chain, there must be a Higgs multiplet capable to break the considered symmetry down to the subsequent one by giving a VEV to the com ponent that transforms as a singlet under the lower intermediate symmetry group. Being a rank 5 group, there should be at least two Higgs ¯elds to break SO(10) down to the SM. One is needed to break the rank of SO(10) from 5 to 4 while the other breaks the remnant symmetry down to the SM gauge group. There are two simple choices of the Higgs ¯elds that not only break the rank of SO(10) but also give a superlarge mass to the right handed neutrino as shown in section 1.5.3. The choices are an antisymmetric ¯ve index tensor 126H or a spinor 16H. In either case, there 24 16 + 16 or 126 + 126 16 + 16 or 126 + 126 16 + 16 or 126 + 126 45H or 54H 45H 54H or 210H SU(5) U(1) SU(4)c SU(2) L R SU(2) (1) (2) SO(10) (1) (2) GSM (1) (2) SO(10) (1) B L SU(3)c SU(2) U(1) _ L R SU(2) (2) GSM (1) (2) SO(10) (1) (2) GSM Figure 1.4: The most common breaking chains of SO(10) gauge group to the SM gauge group (GSM) should be a Higgs ¯eld in the conjugate representation, 126H or 16H, to go along with it, in order to obtain Dterm cancelation and consequently maintain the invariance of supersymmetry down to the electroweak scale. Breaking the rank of SO(10) by either 16H or 126H leaves SU(5) unbroken because both 16H and 126H contain a SU(5) singlet in their decomposition under SU(5) as shown below [22]: 126H = 1 © 5 © 10 © 15 © 45 © 50; 16H = 1 © 5 © 10: (1.67) Therefore, a second Higgs ¯eld is needed to break SU(5) down to the SM. The appropriate Higgs multiplets of SO(10), that can break SU(5), should contain a 24dimensional representation with neutral U(1) charge in their SU(5) £ U(1) com ponents (recall that the adjoint of SU(5) (24H) is used to break SU(5) to G321 of SM). For example, the decomposition of the following Higgs multiplets 45H , 54H, 25 and 210H under SU(5) £ U(1) [22] 45H = 1(0) © 10(4) © 10(¡4) © 24(0); 54H = 15(4) © 15(¡4) © 24(0); 210H = 1(0) © 5(¡8) © 5(8) © 10(4) © 10(¡4) ©24(0) © 40(¡4) © 40(¡4) © 75(0) (1.68) makes them capable of breaking SU(5) down to the SM. There are two approaches that have been adopted so far in order to break the SO(10) gauge group to the SM gauge group. One uses large Higgs representations such as 210H, 126H, and 126H [23]. Although this approach has the advantage that R parity is automatic, the uni¯ed gauge coupling diverges in this case just above the GUT scale. On the other hand, the other approach uses only small Higgs representations [24, 25]. This choice of Higgs representations guarantees that the theory is perturbative up to the Planck scale [26] and also has the potential to arise from string theory. Therefore, we shall adopt the simplest breaking scheme; a pair of spinors 16H and 16H is used to break the rank of SO(10) and only one adjoint 45H is used to break SU(5). The general VEV direction of 45H required to break SU(5) gauge symmetry is given by [19] h45Hi = diag(b; b; a; a; a) i¿2: (1.69) The h45Hi is proportional to the generator of B¡L when b = 0 and it is proportional to T3R when a = 0. The former VEV direction is preferred in the DimopoulosWilczek (DW) [27] mechanism in order to solve the doublettriplet splitting problem. 1.5.3 Neutrino Masses The existence of righthanded neutrinos is important to understand the smallness of the neutrino mass as we have seen in the seesaw mechanism in the context of SM. The accommodation of righthanded neutrinos within the 16dimensional irreps of SO(10) 26 indicates that the seesaw mechanism can be implemented in SO(10) models. In order to see this, let us assume that the only source for the quark and lepton masses is the 10dimensional Higgs representation of SO(10), causing Mu = Mº. The following coupling Y12616F 16F 126H; (1.70) can be used to generate a Majorana mass term for righthanded neutrinos by giving VEV to the SU(5) singlet component of 126H, so the combination of the Dirac and Majorana neutrino mass terms are given by L = ºcMDº + 1 2 MRºcº; (1.71) Here MR = Y126h1(126H)i = Y126v126 and the notation p(q) refers to p of SU(5) contained in q of SO(10). This can be written in a 2£2 mass matrix for the (º,ºc) system as given in Eq.(1.25). If we ignore the mixing among generations, the light neutrino masses for the three generations are given by mºe ¼ m2 u MR1 ; mº¹ ¼ m2c MR2 ; mº¿ ¼ m2t MR3 ; (1.72) where we have used MD = Mu. The magnitude of the scale h1(126H)i is model dependent. For example, if the MSSM is a valid symmetry all the way until the GUT scale, then v126 = MU ¼ 2£1016. It is important to point out that the assumption we have made that the fermion masses arise only from 10H is not good, because it leads to the undesirable relation md=ms = me=m¹. Therefore, we need additional ¯elds, in order to have a realistic SO(10) GUT model. Another way to give Majorana masses to righthanded neutrinos is by using a bilinear product of 16H. The relevant interaction is the e®ective nonrenormalizable 27 interaction fij16i16j16H16H=M which may arise from integrating out a heavy state with mass M. Several realistic models were published along these lines [30]. By giving a VEV to the component of 16 in the SU(5) singlet direction, the righthanded neutrino mass matrix is generated as follows: MRij = fij h16Hi2 M : (1.73) If we assume that both 16H and 16H break the rank of SO(10) at the GUT scale, then h16Hi ¼ 2 £ 1016 GeV. In order to obtain the heaviest right handed neutrino mass to be of order 2 £ 1014 GeV, the mass of the heavy state should be around the Planck scale (2 £ 1018 GeV) [31] One advantage of 126H is that it leads to a theory that conserves R parity auto matically. This is because 126H breaks B¡L by two units. Plugging B¡L = 2 back into the R formula in Eq (1.36), one can see that R parity remains invariant even after symmetry breaking. While in the case of 16H, B ¡ L is broken by one unit, then R parity is not conserved after symmetry breaking. However, the superpotential terms that contain 16H and break B ¡L by one unit can be avoided by imposing a discrete symmetry. Besides, as we mentioned in the previous section, the choice of 16H and 16H is inspired by string theory, and the fact that using small Higgs representations leads to make the uni¯ed gauge coupling perturbative up to the Planck scale. 28 CHAPTER 2 Fermion Masses and Mixings in a Minimal SO(10) £ A4 SUSY GUT We have seen that the GUT models unify the strong and electroweak interactions into a simple group. The simplest GUT model is based on SU(5) gauge symmetry. The minimal SU(5) model predicts a good mass relation for the third generation (i.e., m0b = m0¿ at GUT scale). However, it gives bad prediction for the ¯rst and second generation masses (i.e., m0s = m0 ¹, m0 d = m0e at the GUT scale). In addi tion, SU(5) does not naturally accommodate the righthanded neutrino. On the other hand, SO(10) models accommodate all chiral fermions of one generation plus a right handedneutrino within a 16dimensional irreducible representation (irrep). Also, minimal SO(10) with only 10H involved in Yukawa couplings leads to the up quark mass matrix being proportional to the down quark mass matrix, so it is consid ered a good zeroth order approximation for CKM mixings. Models based on SO(10) symmetry, without including any family symmetry, were proposed to explain most of the features of quarks and leptons [32, 33]. However, one is not really fully satis¯ed with only producing the fermion masses and mixing angles without explaining why we have three generations and without understanding the relation among generations, such as the mass hierarchy and features of the mixing angles. For example, the °a vor symmetry A4 [34] can be employed to explain why the observed neutrino mixing matrix is in very good agreement with the so called tribimaximal (TBM) mixing structure given by Eq(1.35). Thus, it may be important to consider the underlying family symmetry. One of the best candidates for °avor symmetry is the nonAbelian discrete symmetry A4, for the following reasons. First, it is the smallest group that 29 has a 3dimensional irrep. Second, SUSYSO(10)£ A4 symmetry solves the FCNC problem since the scalar fermions, which belong to the 16irrep of SO(10) and trans form as a triplet under A4, have degenerate masses. Finally, it was shown that the TBM mixing structure for the neutrinos can be obtained by imposing A4 symmetry [34]. Several models based on the SO(10)£A4 group have been studied [14, 15, 16]. In these models, large Higgs representations are employed. For example, in Ref.[16], the authors employed a (126H,3) representation, where the ¯rst (second) entry indicates the transformation under SO(10) (A4), in order to produce the fermion masses and mixing angles for both normal and inverted neutrino mass spectra. Besides employ ing the large Higgs representation 126H, the models in Refs.[14, 15] contain more than one adjoint 45H representation. It has been shown that only one adjoint Higgs ¯eld is required to break SO(10) while preserving the gauge coupling uni¯cation [35]. Also, using large Higgs representations like 126H leads to the uni¯ed gauge coupling being nonperturbative before the Planck scale, which might be hard to obtain from superstring theory [36]. Therefore, the purpose of this chapter is to construct an SO(10)£A4 model in which SO(10) is broken to the standard model (SM) group in the minimal breaking scheme. This means using only a spinorantispinor (16H,16H) to break the rank of SO(10) from ¯ve to four, and the righthanded neutrino gets a heavy mass from the antispinor Higgs ¯eld (16H). Then one adjoint representation 45H is used to break the group all the way to the SM group. Recently, a numerical analysis for quark and charged lepton masses and mixings based on nonsupersymmet ric SO(10) without °avor symmetry was done [33]. The authors did not include the neutrino sector in the numerical ¯tting. Their result for the atmospheric angle was sin µatm = 0:89. However, as this work shows, when the neutrino sector is included, not only is the result a better ¯t for the atmospheric angle sin µatm = 0:776, but the known light neutrino mass di®erences are also accommodated. 30 This chapter is organized as follows. In section 2.1, a general structure of the fermion mass matrices for the second and third generations is constructed. Then, based on that structure, the fermion mass hierarchy and relations are explained. In section 2.2, it is shown that introducing several 10plets of matter ¯elds to the model leads to the doubly lopsided structure which produces large neutrino mixing angles and small quark mixing angles simultaneously [37]. Then, some analytical expressions for quark masses and mixing angles at the GUT scale are derived in a certain approximation on the model parameters. In Sec 2.3, an exact numerical analysis is done to ¯nd the outputs at the GUT scale. To get predictions of fermion masses and mixings at low scale, the quark masses and mixings at the GUT scale will be run to the low scale by using renormalization group equations. section 2.4 shows how to get a suitable righthanded neutrino mass structure that gives the correct ¯ts for the atmospheric angle after adding the charged lepton contribution. 2.1 Fermion Mass Structure in SO(10) £ A4 Symmetry In this section, the renormalizable Yukawa couplings of the SM fermions with the extra spinorantispinor matter ¯elds are considered as a concrete example of the model. The known matter ¯elds of the SM (quarks and leptons) plus the right handed neutrino are contained in the three spinors (16,3). The ordinary fermions, 16i, do not couple with 45H in the minimal SO(10). As a result, some of the predictions of the minimal SO(10) such as m¹ = ms and mc=mt = ms=mb will follow; these are badly broken in nature. Therefore, extra heavy fermion ¯elds must be introduced in order to allow the 45H to couple directly with the quarks and leptons of the standard model. The transformation of the ordinary fermions and the extra matter ¯elds under A4 and the additional symmetry Z2 £ Z4 £ Z2 are summarized in Table 2.1. Let us consider ¯rst the invariant superpotential W1 under the assigned symmetry that contains the 31 16 161 16 16 16 16 16 16 1 45 1’ H H 10 i 3 3 2 2 j Hi Hj M1 M3 M2 1 Figure 2.1: This ¯gure shows a diagrammatic representation of the couplings in the superpotential W1. coupling of ordinary fermions with the spinorantispinor matter ¯elds. W1 = b116i1611Hi + b216i16210 Hi + 16116345H + a16316210H +M1161161 +M2162162 +M3163163: (2.1) Table 2.2 summarizes the transformation of the Higgs ¯elds that are needed to achieve a minimum breaking scheme as well as the Higgs singlets that are needed to break the A4 symmetry. Although in this model, the structure in Eq.(2.1) does not include the Yukawa term 16i16i10H which is forbidden by the discrete symmetry Z2 £ Z4 £ Z2, the ordinary standard model fermions get their masses through their coupling with heavy extra ¯elds. This is similar to how the light neutrinos get their masses through coupling with the heavy righthanded neutrinos in the known seesaw mechanism. The coupling terms in the superpotential W1 can be represented diagrammatically as shown in Fig.2.1. After integrating out the heavy states, the approximate e®ective operators can be read from the diagram, i.e., Wij ¼ X ij 16i16jh45Hih10Hih1Hiih10 Hji M1M2M3 : (2.2) The VEVs of the Higgs ¯elds can be written down in a general form as h45Hi = Q; (2.3) h1Hii = 0 BBBBBB@ ²1 ²2 ²3 1 CCCCCCA ; (2.4) 32 SO(10) 16i 161,161 162,162 163,163 1ci A4 3 1 1 1 3 Z2 £ Z4 £ Z2 +,+,+ +,,+ ,+,+ +,+, +,+,+ SO(10) 10i 100 i 1000 i 10000 i 1i A4 3 3 3 3 3 Z2 £ Z4 £ Z2 +,i,+ +,¡i,+ +,i, +,¡i, +,¡i,+ Table 2.1: The transformation of the matter ¯elds under SO(10)£A4 and Z2£Z4£Z2. h10 Hii = 0 BBBBBB@ s1 s2 s3 1 CCCCCCA ; (2.5) h5(10)i = vu; h5(10)i = vd: (2.6) Here the notation hp(q)i refers to a p of SU(5) contained in a q of SO(10). The Q from Eq.(2.3) is a linear combination of SO(10) generators. One can rede¯ne, without loss of generality, the light fermion states as 161²1 + 162²2 + 163²3 = ²160 3; 161s1 + 162s2 + 163s3 = S(160 2sµ + 160 3cµ); (2.7) where ² = q ²21 + ²22 + ²23 and S = q s21 + s22 + s23 . In terms of the rede¯ned light fermion states, after dropping the prime notation and plugging in the VEVs, one gets W0 ¼ ²Sh10Hi M1M2M3 (163162Q(163)sµ + 163163Q(163)cµ): (2.8) In general, the above e®ective operator can be written in terms of quark and lepton ¯elds as WF ¼ ²Sh10Hi M1M2M3 (F3Fc 2QF sµ + Fc 3F2QFcsµ + F3Fc 3 (QF + QFc)cµ): (2.9) Here F is a general notation for up quarks (U), neutrinos (N), charged leptons (L), and down quarks (D). The quantity QF (QFc) refers to the assigned charge of the 33 lefthanded fermion (charge conjugate of the righthanded fermions) after breaking the SO(10) group down to the SM group. The unbroken charge Q can be written as a linear combination of two generators that commute with SU(3)c £SU(2)L £U(1)Y as: Q = 2I3R + 6 5 ±( Y 2 ); (2.10) where I3R is the third generator of SU(2)R and Y is the hypercharge of the Abelian U(1) group. The charge Q for di®erent quarks and leptons is given by. Qu = Qd = 1 5 ±; Quc = ¡1 ¡ 4 5 ±; Qdc = 1 + 2 5 ±; Ql = Q¹ = ¡ 3 5 ±; Qlc = 1 + 6 5 ±; Qºc = ¡1: (2.11) Eq.(2.9) can be expressed in the following matrix form: WF ¼ µ Fc 1 Fc 2 Fc 3 ¶ ( ²Sh10Hi M1M2M3 ) 0 BBBBBB@ 0 0 0 0 0 QF sµ 0 QFcsµ (QF + QFc)cµ 1 CCCCCCA 0 BBBBBB@ F1 F2 F3 1 CCCCCCA : (2.12) Some factors that arise from doing the algebra exactly should be included in the above mass matrix as we are going to see later. Finding these factors that we have assumed to be of order one is important in the °avor violation analysis. The ¯rst feature of the general mass matrix of the light fermions in Eq.(2.12) is an explanation for the mass hierarchy between the second and third generations in the limit sµ ! 0. It is remarkable that a relation among generations is related to the vacuum alignment of the A4 Higgs. Another feature of the above light fermion mass matrix m0b = m0¿ is obtained through MD33 = ML33, which follows from the relation Qdc + Qd = Qlc + Ql. This relation occurs because both down quarks and charged leptons get their masses from the same Higgs. A further consequence of the light fermion mass structure is that m0s 6= m0 ¹. This inequality relation follows from m0 ¹=m0s = L32L23=D32D23 = QlcQl=QdcQd, which 34 SO(10) 10H 45H 16H 16H 1Hi 10 Hi 100 Hi 1000 Hi A4 1 1 1 1 3 3 3 3 Z2 £ Z4 £ Z2 ,+, +,, +,¡i,+ +,¡i,+ +,,+ ,+,+ +,+, +,i,+ Table 2.2: The transformation of the Higgs ¯elds under SO(10)£A4 and Z2£Z4£Z2. is not necessarily equal to 1. This leads to the following question: What VEV direction should be given to 45H in order to obtain the GeorgiJarlskog relation jm0 ¹j = 3jm0s j? There are two choices, either ± ! 0 or ± ! ¡1:25. The former choice gives the unwanted relation (m0c =m0t )=(m0s =m0¿ ) ! 1, while the latter leads to (m0c =m0t )=(m0s =m0¿ ) ! 0. Thus, a good ¯t for ± should be around ¡1:25. 2.2 Extension to the First Generation and Doubly Lopsided Structure In this section, vector 10plet fermions are added to the model to generate masses and mixings of the ¯rst generation. These vector multiplets do not contribute to the upquark mass matrix since 10plets do not contain a charge of (§2=3). Therefore, the upquark matrix is still rank 2, and this is consistent with m0 u m0t ¼ 10¡5 being much smaller than m0 d m0b ¼ 10¡3 and m0e m0¿ ¼ 0:3£10¡5. First, I will show how the model leads to the doubly lopsided structure by employing these vector multiplets; then some analytical expressions for masses and mixing angles of fermions at the GUT scale will be derived. Let us ¯rst consider the invariant couplings under the assigned symmetry, which can be read from the Feynman diagram in Fig.2.2. The allowed couplings in the superpotential W2 are W2 = 16i10i16H +M1010i100 i + h0 ijk100 i100 j1Hk + hijk10i10j1Hk: (2.13) The important point is that Fig.2.2 gives a °avorsymmetric contribution to the down quark and charged lepton mass matrices. In order to understand this, recall that the general product of three triplets(a1, a2, a3), (b1, b2, b3), and (c1, c2, c3)that 35 16 16 16 10 10’ 10’ 10 16 1 i i j j H Hk H i j M M 10 10 Figure 2.2: This ¯gure leads to the °avor symmetric contribution to the down quarks and charged leptons. transform as a singlet under A4 is given by h1(a2b3c1 + a3b1c2 + a1b2c3) + h2(a3b2c1 + a1b3c2 + a2b1c3): (2.14) The third term of Eq.(2.13) gives a symmetric contribution since there are two iden tical 10plets. The last term in Eq.(2.13) has been ignored by assuming the Yukawa couplings hijk to be very small. The contribution of Fig.2.2 to the mass matrices of the down quarks and charged leptons, after integrating out the extra vector multiplets is then MsL = MsD / 0 BBBBBB@ 0 c12 c13 c12 0 c23 c13 c23 0 1 CCCCCCA; (2.15) where c12, c13, and c23 are proportional to ²1, ²2, ²3, respectively. To obtain the desired fermion mass structure (the doubly lopsided structure, which is going to be explained later in this section), other couplings need to be included by employing four vector 10plets plus adding another Higgs singlet 100 iH to the model (their transformations under the assigned symmetry are shown in Tables 2.1 and 2.2). The purpose of these couplings is to give a °avorantisymmetric contribution to the downquark and charged lepton mass matrices. Since the adjoint of SO(10) (45H) is an antisymmetric tensor which changes its sign under the interchange 100 i $ 10000 i , one can consider employing the Yukawa coupling 10000 i 100 i45H. Also, due to the fact that when we write 36 16 16 10 10 10’ 10’’ 10’’’ 10’ 1 45 16 M 10 M 10 m i i i j j j j j H Hk H H 16 Figure 2.3: This ¯gure leads to the °avorantisymmetric contribution to the down quarks and charged leptons. the SO(10)vectors in the SU(5) basis such as 10i = 5i + 5i, the charged lepton and down quark contents of 5i or 5i have di®erent chiralities, the structures of matrices ML and MD therefore have opposite signs [look at the mass structures in Eqs.(2.182.19). It is important to emphasize that the minimum Higgs breaking scheme assumption does not allow us to add another adjoint to the model. Therefore, the same adjoint 45H Higgs representation that breaks the SO(10) group to the SM group is going to be used. Additional couplings to the previous superpotential can be read from Fig.2.3, i.e., W3 = 100 i1000 j 100 Hk + m1000 i 10000 i + 10000 i 100 i45H; (2.16) where h45Hi has been de¯ned previously. The VEV of the Higgs singlet 100 H is given below: h100 Hi = 0 BBBBBB@ ±1 ±2 ±3 1 CCCCCCA : (2.17) After integrating out the heavy states, the following contribution to the ML and MD is obtained: MA L / 0 BBBBBB@ 0 ¡±3Ql ±2Ql ±3Ql 0 ¡±1Ql ¡±2Ql ±1Ql 0 1 CCCCCCA ; (2.18) 37 MA D / 0 BBBBBB@ 0 ±3Qdc ¡±2Qdc ¡±3Qdc 0 ±1Qdc ±2Qdc ¡±1Qdc 0 1 CCCCCCA ; (2.19) where the overall constant has been absorbed in the rede¯nition of ±1, ±2, and ±3. Equations (2.182.19) show that the o®diagonal elements of MA D (MA L ) are propor tional to Qdc (Ql). This is because 5i(10) contains, in its representation, the charge conjugation of a color triplet of the lefthanded down quarks dc Li and the lefthanded charged leptons eLi. The full treelevel mass matrices, which are obtained by adding the three superpotentials W1 +W2 +W3, have the following forms: ML = m0 d 0 BBBBBBBBBBBBBBBBBBBBBB@ 0 c12 + 3±3(¡1+® 5 ) ¡±2® + ³ c12 ¡ 3±3(¡1+® 5 ) 0 ±1® + ¯ ¡3s(¡1+® 5 ) ³ ¡ ±2 6¡® 5 ±1( 6¡® 5 ) + ¯ 1 +s(¡1+6® 5 ) 1 CCCCCCCCCCCCCCCCCCCCCCA ; (2.20) MD = m0 d 0 BBBBBBBBBBBBBBBBBB@ 0 c12 + ±3(3+2® 5 ) ¡2±2(3+2® 5 ) + ³ c12 ¡ ±3(3+2® 5 ) 0 2±1(3+2® 5 ) + ¯ +s(¡1+® 5 ) ³ s(3+2® 5 ) + ¯ 1 1 CCCCCCCCCCCCCCCCCCA ; (2.21) MU = m0 u 0 BBBBBB@ 0 0 0 0 0 ( 1¡® 5 )s 0 (1+4® 5 )s 1 1 CCCCCCA ; (2.22) 38 MN = m0 u 0 BBBBBB@ 0 0 0 0 0 (¡3+3® 5 )s 0 s 1 1 CCCCCCA ; (2.23) the convention being used here is the lefthanded fermions multiplied from the right. The parameters of the model have been de¯ned as follows: ³ = c13 + ±2Qdc ; ¯ = c23 + ±1Qdc ; ± = ¡1 + ®; (2.24) s = sµ ( 3 5± + 1)cµ : The above fermion mass structure has eight parameters. If ® goes to zero, the fermion mass matrices in Eqs.(2.202.21) go to the SU(5) limit (m0b = m0¿ , m0s = m0 ¹, m0 d = m0e ). To avoid the bad prediction of SU(5) for lighter generations, a good numerical ¯tting for ® should deviate from zero. On the other hand, to keep the good SU(5) prediction for the third generation, the parameter ® should satisfy ® << 1. If ±1 and ±2 are of order 1 and the other model parameters are very small (¯; ³; ®; ±3; c12; s << ±1; ±2), the model leads to the doubly lopsided structure. To see this clearly, let us go to the limit where the small parameters are zero (except s). So the MD and ML go to the following form: ML = MTD = m0 d 0 BBBBBB@ 0 0 0 0 0 ( 3s 5 ) ¡±2 6 5 (¡s 5 ) + ±1( 6 5 ) 1 1 CCCCCCA : (2.25) In diagonalizing ML of Eq.(2.25), the large o®diagonal elements ±1 and ±2 that appear asymmetrically in MD and ML must be eliminated from the right by a large left handed rotation angle µsol in the 12 plane, where tan(µsol) = ¡±2 ±1 . The next step of diagonalization is to remove the large element ¾ ¼ (±2 1 +±2 2) 1 2 that has been produced 39 after doing the ¯rst diagonalization, where the (3,2) element of the matrix in Eq.(2.25) is replaced by ¾. This can be done by a rotation acting from the right by a large lefthanded angle µ23 in the 23 plane, where tan(µ23) ¼ ¡¾. On the other hand, there are no corresponding large lefthanded rotation angles in diagonalizing MD since ML = MTD . However, the large o®diagonal elements in MD can be eliminated by large righthanded rotation angles acting from the left on the MD in Eq.(2.25), while the lefthanded rotation angles are small. This explains how the doubly lopsided structure leads to small CKM mixing angles and large neutrino mixing angles simultaneously. If the parameters c12, ±3, and ³ are zero, analytical expressions can be written down for the ratios of quark and lepton masses of the second and third generations, Vcb, and neutrino mixing angles (tan µ12 and tan µ23) in terms of ±1, ±2, s , ®, and ¯: m0c m0t = s2(1 ¡ ®)(1 + 4®) 25 ; m0s m0b = ¡2(3 + 2®)(¯ + s(3+2® 5 )) q ±2 1 + ±2 2 5(1 + 4 25(3 + 2®)2(±2 1 + ±2 2)) ; m0 ¹ m0¿ = q (¡3s 5 (¡1 + ®) + ±1® + ¯)2 + ±2 2®2 q (±2 1 + ±2 2)(6 ¡ ®) 5(1 + (6¡®)2 25 (±2 1 + ±2 2)) ; (2.26) V D cb = ¯ + s(3+2®) 5 (1 + 4 25(3 + 2®)2(±2 1 + ±2 2)) ; V U cb = ¡s(1 + 4®) 5 ; tan µ12 = ±2( 6¡® 5 ) ±1( 6¡® 5 ) + s(¡1+6® 5 ) + ¯ ; tan µ23 = ¡( 6 ¡ ® 5 ) q ±2 2 + ±2 1: These expressions are derived by using the approximation ®; s; ¯ << ±1; ±2, and are useful for ¯tting the data. The best ¯t for the data is obtained by setting tan µ23 = ¡2 and tan µ12 = 0:68, which correspond to µ23 = ¡63o and µ12 = 34o. The central value of the atmospheric angle is around 45o. In order to bring 63o close to the central value, the neutrino sector is required to be included as shown in Sec 2.5. Also, it will be shown that the contribution of the neutrino sector to the solar angle is small. 40 2.3 Numerical Results The model can be shown to be concrete by giving numerical values to the parameters of the model, and producing the six mass ratios of quarks and leptons, CKM mixing angles (Vus, Vub, and Vcb), the CP violation parameter ´ = ¡Im(VubVcs=VusVcb), and neutrino mixing angles (sin µ12, and sin µ13). The ten parameters (±1, ±2, ±3, ®, ¯, s, ³, c12, m0 d, and m0 u) appearing in Eqs(2.202.23) are in general complex. Five phases of the complex parameters can be removed by rede¯ning the phases of the quark and lepton ¯elds. Then, we have ten real parameters and ¯ve phases in order to ¯t the 16 quantities appearing in Table 2.3. However, the best numerical ¯t is obtained when two parameters (±3; c12) are complex while the others are real. If ±1 = ¡1:302, ±2 = 1:0142, ±3 = 0:015 £ e4:95i, ® = ¡0:05801, s = 0:29, ³ = 0:0105, c12 = ¡0:00153e1:1126i, and ¯ = ¡0:12303, the following excellent ¯t at the GUT scale is obtained : m0c m0t = 0:002717, m0b m0¿ = 0:958, m0e m0 ¹ = 0:00473, m0 ¹ m0¿ = 0:0585, m0 d m0e = 3:63, m0s m0 ¹ = 0:302, ´ = 0:357, Vus = 0:2264, Vub = 0:0037, Vcb = 0:0362, sin µ12 = 0:569, and sin µ13 = 0:0653. The above numerical ¯ttings lead to sin µL 23 = 0:904, which is not close to the central value sin µatm 23 = 0:707. One can see from the superscript L that the mixing angle µL 23 comes only from the charged lepton contribution. To obtain close to the expected atmospheric angle and the correct neutrino mass di®erences, it is important to include the neutrino sector contribution to the atmospheric angle by ¯nding out a suitable righthanded neutrino structure which respects the assigned symmetry of the model. In order to compare with experiment, the predicted fermion masses and mixing angles at the low energy scale need to be found. The above numerical values of the fermion masses and mixing angles which are obtained at the GUT scale have been evolved to the low scale in two steps. First, the running from the GUT scale to MSUSY = 1 TeV is done by using the twoloop MSSM beta function. The running factors denoted by ´i depend on the value of tan ¯. The known fermion masses and 41 mixing data are best ¯tted with tan ¯ = 10. The running factors for tan ¯ = 10 are (´s=b, ´¹=¿ , ´b=¿ , ´c=t, ´cb= ´ub)=(0.8736, 0.9968, 0.5207, 0.73986, 0.910335), where ´i=j = (m0i =m0j )=(mi(1TeV)=mj(1TeV)) and ´cb;ub = V 0 cb;ub=Vcb;ub(1TeV). The second step is to evolve the fermion masses and mixing angles from MSUSY = 1 TeV to the low scale. The renormalization factors ´i that run fermion masses from their respective masses up to the supersymmetric scale MSUSY = 1 TeV are computed using threeloop QCD and oneloop QED, or the electroweak renormalization group equation with inputs ®s(MZ) = 0:118, ®(MZ) = 1=127:9, and sin µw(MZ) = 0:2315. The relevant renormalization equations can be found in [38][39]. The results are (´c, ´b, ´e, ´¹, ´¿ , ´t, ´ub=´cb)=(0.4456, 0.5309, 0.8188, 0.83606, 0.8454, 0.98833, 1.0151). By using the above renormalization factors, m¿ = 1776 MeV, and mt = 172:5 GeV, the following predictions at the low scale can be obtained: mc(mc) = 1:4 GeV, mb(mb) = 5:2 GeV, me(me) = 0:511 MeV, m¹(m¹) = 105:6 MeV, md(2 GeV) = 7:5 MeV, ms(2 GeV) = 132 MeV, ´ = 0:357, Vus = 0:2264, Vub = 0:004, Vcb = 0:0392, sin µ12 = 0:569, and sin µ13 = 0:0653. Note that the numerical value of mb is not in perfect agreement with the exper imental value mb = 4:20+0:17 ¡0:07 GeV [40]. In order to ¯x this, the ¯nite gluino and chargino loop corrections [41] are required to be included in the downtype quark masses (md, ms, mb). The total contributions are denoted as (1+¢d), (1+¢s), and (1+¢b). These corrections are proportional to the supersymmetric particle spectrum: ¢b ¼ tan ¯ Ã 2®3 3¼ ¹M~g m2 ~b L ¡m2 ~b R h f(m2 ~b L =M2 ~g ) ¡f(m2 ~b R =M2 ~g ) i + ¸2t 16¼2 ¹At m2 ~tL ¡m2 ~tR h f(m2 ~tL =¹2)¡ f(m2 ~tR =¹2) i´ , where f(x) = ln(x)=(1 ¡ x) and the ¯rst (second) term refers to the gluino (chargino) correction. Similar expressions exist for ¢s and ¢d, but without the chargino contribution and ~b ! ~s; ~ d. If the chargino loop corrections are neg ligible and m~ d, m~s, and m~b are degenerate, the equality relation ¢d = ¢s = ¢b is approximately satis¯ed. In order to get a better ¯tting for downtype quark masses, let us take ¢d = ¢s = ¢b = ¡0:17, which gives md(2 GeV) = 6:24 MeV, 42 m0s (2 GeV) = 109:65 MeV, and mb(mb) = 4:31 GeV. The comparison of the model predictions and experimental data at the low scale is summarized in Table 2.3, where the quark and charged lepton masses, the CKM mixing angles (Vub, Vus, Vcb), the neutrino mixing angles (sin µsol, sin µatm, sin µ13), and the CP violation parameter (´) are taken from [40]. The masses are all in GeV. Although the model here predicts mu(GUT) = 0, the quantity mud = (mu + md)=2 is considered in Table 2.3, where it is assumed that the tiny up quark mass at GUT scale may be generated either by including the coupling 16i16i10H into the model or by considering higher dimensional operators. If mu(2 GeV) = 2:4 MeV, the model predictions of the quantities mud and ms mud , which are wellknown from lattice calculations [42], are given in Table 2.3. The asterisks in Table 2.3 indicate that the model predictions of neutrino mixing angles are obtained after including the neutrino sector in section 2.5. 2.4 Right Handed Neutrino Mass Structure So far, the model gives excellent agreement with the known values for the CKM mixings, the quark masses, the charged lepton masses, the CP violation parameter, and the neutrino mixing angles (sin µ12 and sin µ13). However, the whole picture is still not complete and the following question arises. What is the appropriate light neutrino mass matrix (Mº = ¡MTN M¡1 R MN) that gives not only the correct contribution to the atmospheric angle, but also the correct neutrino mass di®erences: ¢m2 21 = (7:59 § 0:2)£10¡5eV2, j¢m2 32j = (2:43§0:13)£10¡3eV2 [37]? In other words, we are looking for a suitable structure of righthanded neutrino mass matrix MR since MN is ¯xed. Recall that the MNS mixing matrix is given by UMNS = Uy LUº; (2.27) 43 where UL and Uº are the unitary matrices needed to diagonalize the Hermitian lepton matrix My LML and the light neutrino matrix Mº, respectively. Mdiagy L Mdiag L = Uy LMy LMLUL; Mdiag º = UT º MºUº; (2.28) where Mº is assumed to be real and symmetric. The Dirac neutrino mass matrix MN in Eq. (2.23) has vanishing ¯rst row and column, and the same is true for Mº. So the matrix required to diagonalize Mº is simply a rotation in the 23 plane by an angle µº, while Uy L is determined numerically from the charged lepton mass matrix. Thus, the mixing matrix of neutrinos is given by UMNS = 0 BBBBB@ ¡0:14 ¡ 0:81i 0:13 + 0:55i 0:065 0:25 + 0:06i 0:34 ¡ 0:04i 0:90 ¡0:51 ¡0:75 0:42 1 CCCCCA 0 BBBBB@ 1 0 0 0 cos µº sin µº 0 ¡sin µº cos µº 1 CCCCCA : (2.29) One can conclude that the correct contribution of the neutrino sector to the atmo spheric angle is around µº=¡20o. For example, if we take µº=¡20o, the neutrino mixing angles (sin µatm, sin µsol, sin µ13) become (0.707, 0.53, 0.21). In order to ¯nd the suitable righthanded neutrino mass structure, one can easily prove the inverse of the seesaw relation, MR = ¡MNUº(Mdiag º )¡1UT º MT º : (2.30) A similar technique was used in Ref [43]. Note that one of the eigenvalues of Mº is zero (i.e. Mdiag º is singular), so the inverse of Mdiag º does not exist. To overcome this problem, one can generally de¯ne Mdiag º =diag( m1, m2, m3 ), and m1 will not appear in MR. By using the numerical result of MN, µº=¡20o, and m2/m3=0.178, the righthanded mass structure can be presented numerically. 0 BBBBBB@ 0 0 0 0 0:0186 ¡0:13 0 ¡0:13 1 1 CCCCCCA : (2.31) 44 16i 1 1 1 1 16j c l k Hm H m1 16H H 16 i j c l 1’’’ 1’’’ Figure 2.4: This ¯gure leads to the righthanded neutrino mass matrix. From the above numerical mass matrix, one concludes (MR)23 £ (MR)23 ¼ (MR)22, so to a good approximation, the above numerical structure can be represented ana lytically as follows: 0 BBBBBB@ 0 0 0 0 r2 ar 0 ar 1 1 CCCCCCA : (2.32) The constant a should not be equal to 1 because then MR would be singular. Now our mission is to ¯nd the Yukawa couplings that respect the symmetry of the model and lead to an analytical structure similar to Eq.(2.32). This can be accomplished by considering the following Yukawa couplings represented by the Feynman diagram in Fig.2.4, i.e., W4 = 16i16H1i + hijk1i1cj 1000 Hk + m11ci 1ci ; (2.33) where two fermion singlets 1i and 1ci , which couple with the singlet Higgs 1000 iH, have been introduced (their transformation under SO(10)£A4 and the additional symme try are shown in Tables 2.12.2). The product of the three triplets of the second term in Eq. (2.33) that transform as a singlet under A4 is given by h1(N1Nc 2®3 + N2Nc 3®1 + N3Nc 1®2) + h2(N1Nc 3®2 + N3Nc 2®1 + N2Nc 1®3), where ®1, ®2, and ®3 are the VEV's components of 1000 iH. By assuming h1=h2, Fig.2.4 leads to the desired right 45 handedneutrino mass structure. MR = ¤ 0 BBBBBB@ ®21 ®23 ®1®2(¡1 ®23 + 2 ®21 +®22 +®23 ) ¡®1(®21 ¡®22 +®23 ) ®3(®21 +®22 +®23 ) ®1®2(¡1 ®23 + 2 ®21 +®22 +®23 ) ®22 ®23 ¡®2(¡®21 +®22 +®23 ) ®3(®21 +®22 +®23 ) ¡®1(®21 ¡®22 +®23 ) ®3(®21 +®22 +®23 ) ¡®2(¡®21 +®22 +®23 ) ®3(®21 +®22 +®23 ) 1 1 CCCCCCA : (2.34) By comparing the 23 block of the above structure with the mass structure in Eq. (2.32), one can see the constant a is equivalent to the quantity ((¡®2 1 + ®2 2 + ®2 3)=(®2 1 + ®2 2 + ®2 3)), which is equal to 1 in the limit ®1 ! 0. So, let us expand the eigenvalues of the right handed neutrino mass structure in Eq(2.34) around ®1. MR1 = 1 + ®2 2 ®2 3 + ®2 1(®4 2 ¡ 6®2 2®2 3 + ®4 3) ®2 3(®2 2 + ®2 3)2 + O(®4 1); MR2 = 4®2 1®2 2 (®2 2 + ®2 3)2 ¡ 8®3 1®3 2®3 (®2 2 + ®2 3)7=2 + O(®4 1); (2.35) MR3 = 4®2 1®2 2 (®2 2 + ®2 3)2 + 8®3 1®3 2®3 (®2 2 + ®2 3)7=2 + O(®4 1): One can see that two of the righthanded neutrino masses are approximately de generate for small values of ®1 (i.e. MR2 ¼ MR3). By setting (®1, ®2, ®3, ¤)=(¡0:05, 0.125, 0.994, 8:42 £ 1015), the numerical ¯t for the neutrino mixing angles, the light neutrino masses, and the right handedneutrino masses are obtained as follows: m1 = 0 eV; sin µsol = 0:551; MR1 = 8:57 £ 1015 GeV; m2 = 0:01 eV; sin µatm = 0:776; MR2 = 1:3 £ 1012 GeV; m3 = 0:056 eV; sin µ13 = 0:154; MR3 = 1:28 £ 1012 GeV: As can be seen from Table 2.3, the masses and mixing angles of the quarks and leptons after including the neutrino sector are predicted in this model to be within 2¾ error bars of their experimental values. 46 Model predictions Experiment Pull me(me) 0.511£10¡3 0.511£10¡3 ... m¹(m¹) 105.6£10¡3 105.6£10¡3 ... m¿ (m¿ ) 1.776 1.776 ... mud 4:32 £ 10¡3 (3:85 § 0:52)£10¡3 0.9 mc(mc) 1.4 1:27+0:07 ¡0:11 1.85 mt(mt) 172.5 171.3§2.3 0.52 ms mud 25.36 27:3 § 1:5 1.29 ms(2Gev) 109.6£10¡3 105+25 ¡35 £ 10¡3 0.184 mb(mb) 4.31 4:2+0:17 ¡0:07 0.58 Vus 0.2264 0.2255§0.0019 0.473 Vcb 39.2£10¡3 (41.2§1.1)£10¡3 1.82 Vub 4.00£10¡3 (3.93§0.36)£10¡3 0.194 ´ 0.3569 0:349+0:015 ¡0:017 0.526 sin µsol 12 0.551 0.566§0.018 0.83 sin µatm 23 0.776 0.707§0.108 0.63 sin µ13 0.154 < 0:22  Table 2.3: This Table shows the comparison of the model predictions at low scale and the experimental data. 47 CHAPTER 3 Flavor Violation in a Minimal SO(10) £ A4 SUSY GUT Flavor changing neutral current (FCNC) processes impose severe constraints on the soft supersymmetric breaking (SSB) sector of the minimal supersymmetric standard model (MSSM). The simplest way to satisfy the FCNC constraints is to adopt univer sality in the scalar masses at a high energy scale where the e®ects of supersymmetry (SUSY) breaking in the hidden sector is communicated to the scalar masses of MSSM via gravitational interactions. For example, in the the minimal supergravity model (mSUGRA) [44] the MSSM is a valid symmetry between the weak scale and grand uni¯cation scale (MGUT) at which the universality conditions are assumed to hold. In this case, the leptonic °avor violation (LFV) is not induced. However, in a di®erent class of models studied in Refs [45, 46, 47, 48, 49, 50] the universality of the scalar masses will be broken by radiative corrections. Consequently, FCNC will be induced in these models as discussed below. If the universality conditions hold at the grand uni¯cation scale MGUT, the LFV is induced below GUT scale by radiative corrections in the MSSM with righthanded neutrino [45, 46, 47] or SUSYSU(5) [48] models. Unfortunately, it is di±cult to predict LFV decay rates in these models because the Dirac neutrino Yukawa couplings are arbitrary within MSSM. However, in an SO(10) GUT model, we can predict the LFV decay rates below the GUT scale because the Dirac neutrino couplings are related to the uptype quark Yukawa couplings and are thus ¯xed. The FCNC could also be induced above the GUT scale by radiative corrections. 48 It was shown that as a consequence of the large top Yukawa coupling at the uni¯ cation scale, SUSY GUTs with universality conditions valid at the scale M¤, where MGUT < M¤ · MPlanck, predict lepton °avor violating processes with observable rates [49, 50]. The experimental search for these processes provides a signi¯cant test for supersymmetric grand uni¯cation theory (SUSY GUT). Both contributions of FCNC that are induced above and below MGUT will be studied in our model. In this chapter, the °avor violation processes for charged lepton and quark sectors are investigated in the framework of a realistic SUSY GUT model based on the gauge group SO(10) and a discrete nonabelian A4 °avor symmetry [51]. This model is realistic because it successfully describes the fermion masses, CKM mixings and neutrino mixing angles. This work di®ers from other studies in several aspects. First, it is di®erent from those based on MSSM with righthanded neutrino masses or SUSY SU(5) in the sense that the Dirac neutrino Yukawa couplings are determined from the fermion masses and mixing ¯t of the SO(10) £ A4 model. Thus, this model predicts the lepton °avor violation arising from the renormalization group (RG) running from MGUT to the righthanded neutrino mass scales. Second, it is di®erent from those based on SUSY SO(10) studied in [52] in the sense that the FCNC processes are closely tied to fermion masses and mixings. Finally, in the SO(10)£A4 model °avor violation is induced at the GUT scale at which A4 symmetry is broken due to large (order one) mixing of the third generation of MSSM ¯elds (Ã3) with the exotic heavy ¯elds (Âi, i runs from 1 to 3). This large mixing arises when the A4 °avor symmetry is broken at the GUT scale. This is di®erent from the case where the °avor violation is induced due to large top Yukawa coupling at the GUT scale [49, 50]. The reason for introducing the exotic heavy fermion ¯elds in our model is to obtain the correct fermion mass relations at the GUT scale as we shall see in section 1. The mass scales of these exotic ¯elds range from 1014 GeV to 1018 GeV depending on the values of the Yukawa couplings and the scale of A4 °avor symmetry breaking. 49 In this chapter we study °avor violation of the hadronic and leptonic processes by calculating the °avor violating scalar fermion mass insertion parameters (±AB)ij = (m2 AB)ij ~m2 , for (A;B) = (L;R), with ~m being the average mass of the relevant scalar partner of standard model fermions (sfermions). All the °avor violation sources are included in our calculations. The sfermion mass insertions, ±LL;RR;LR, arise from the large mixing between the Ã3 and Âi and the mass insertions, (±ij LL)RHN, arise from RG running from MGUT to the righthanded neutrino mass scales. These scalar mass insertion parameters are analyzed in the framework of our model; then they are compared with their experimental upper bounds. We found that the most stringent constraint on °avor violation comes from the ¹ ! e° process. This constraint requires a high degree of degeneracy of the soft masses of MSSM ¯elds and the exotic ¯elds. Therefore, in this model we assume that these soft masses are universal at the scale M¤ with M¤ > MGUT, then we run them down to the GUT scale. The branching ratio Br(¹ ! e°) close to experimental bound (i.e. Br(¹ ! e°)=1:2 £ 10¡11) is obtained when the slepton masses of order 1 TeV , while the Yukawa couplings remain perturbative at the scale M¤. We also found in the framework of our model that once the constraint from Br(¹ ! e°) is satis¯ed, all the FCNC processes will be automatically consistent with experiments. This chapter is organized as follows. In section 1, we show how the fermion mass matrices are constructed in SO(10)£A4 model. In section 2, we discuss the sources of °avor violation by ¯nding the sfermion mass insertion parameters ±ij LL;RR at the GUT scale at which A4 symmetry is assumed to be broken as well as below the GUT scale. The results of the SO(10)£A4 model regarding °avor violation analysis are presented in section 4. Section 5 has our conclusion. The derivation of the light fermion mass matrices and the light neutrino mass matrix after disentangling the exotic fermions is shown in appendix A. In appendix B, we list the renormalization group equations (RGEs) for various SUSY preserving and breaking parameters between MGUT and 50 M¤ relevant for FCNC analysis. 3.1 A Brief Review of Minimal SO(10) £ A4 SUSY GUT In the SO(10) gauge group, all the quarks and leptons of the SM are naturally accommodated within a 16dimensional irreducible representation. However, minimal SO(10) (i.e., with only one 10dimensional Higgs representation) leads to fermion mass relations at the GUT scale, such as m0c m0t = m0s m0b and m0 ¹ = m0s , that are inconsistent with experiment. This can be ¯xed by introducing exotic 16 + 16 fermions and by coupling 16i with these exotic ¯elds via 45H, which is used for SO(10) symmetry breaking. The nonabelian discrete A4 symmetry is chosen in our model because it is the smallest group that has a 3dimensional representation, so the three generations of SM ¯elds transform as triplet under A4. Besides, FCNC is not induced in the SUSY SO(10) £ A4 as long as A4 symmetry is preserved. However, as we will see later, the breaking of A4 symmetry at the GUT scale will reintroduce the FCNC via large mixing between the exotic and light ¯elds. Based on the above reasons, a SO(10)£A4 model is proposed in [51]. In this model, a minimal set of Higgs representations are used to break the SO(10) gauge group to the SM gauge group so the uni¯ed gauge coupling remains perturbative all the way to the Planck scale. Employing this minimal Higgs representation and A4 symmetry, our model successfully accommodates small mixings of the quark sector and large mixings of the neutrino sector in the uni¯ed framework as shown summarized below. The fermion mass matrices of the model proposed in [51] were constructed approx imately. In this section, we construct these matrices by doing the algebra exactly and show that the excellent ¯t for fermion masses and mixings is obtained by slightly modifying the numerical values of the input parameters of Ref.[51]. There are two superpotentials of the model. The ¯rst one (Wspin:) describes the couplings of the standard model ¯elds (Ãi(16i), i runs from 13) with the exotic heavy spinorantispinor 51 SO(10) Ãi Â1,Â1 Â2,Â2 Â3,Â3 Zc i A4 3 1 1 1 3 Z2 £ Z4 £ Z2 +,+,+ +,,+ ,+,+ +,+, +,+,+ SO(10) Ái Á0 i Á00 i Á000 i Zi A4 3 3 3 3 3 Z2 £ Z4 £ Z2 +,i,+ +,¡i,+ +,i, +,¡i, +,¡i,+ Table 3.1: The transformation of the matter ¯elds under SO(10)£A4 and Z2£Z4£Z2. SO(10) 10H 45H 16H 16H 1Hi 10 Hi 100 Hi 1000 Hi A4 1 1 1 1 3 3 3 3 Z2 £ Z4 £ Z2 ,+, +,, +,¡i,+ +,¡i,+ +,,+ ,+,+ +,+, +,i,+ Table 3.2: The transformation of the Higgs ¯elds under SO(10)£A4 and Z2£Z4£Z2. ¯elds (Âi(16i), Âi(16i), i runs from 1 to 3), while the second one (Wvect:) describes the couplings of Ãi with the exotic 10vector ¯elds (Ái, Á0 i, Á00 i , Á000 i , i runs from 1 to 3) as given below: Wspin: = b1ÃiÂ11Hi + b2ÃiÂ210 Hi + k1Â1Â345H + aÂ3Â210H +M®Â®Â®; (3.1) Wvect: = b3ÃiÁi16H +M10ÁiÁ0 i + h0 ijkÁ0 iÁ0 j1Hk + hijkÁiÁj1Hk +AijkÁ0 iÁ00 j 100 Hk + mÁ00 i Á000 i + k2Á000 i Á0 i45H: (3.2) The above superpotentials are invariant under A4 and the additional symmetry Z2 £ Z4 £ Z2. The transformations of the matter ¯elds (i.e., the ordinary and exotic fermion ¯elds) and the Higgs ¯elds under the assigned symmetry are given in Table 3.1 and 3.2. The general fermion mass matrix structure that results from integrating out the 52 exotic heavy spinorantispinor ¯elds in Wspin: is: MF (spin:) = Ã aT1T2T3f2h10Hi rF rFc ! 0 BBBBBB@ 0 0 0 0 0 QF sµ rFc f 0 QFcsµ rF f (QF + QFc)cµ 1 CCCCCCA ; (3.3) where we have made the following transformation: Ã1²1 + Ã2²2 + Ã3²3 = ²Ã03 and Ã1s1 + Ã2s2 + Ã3s3 = S(Ã02 sµ + Ã03 cµ). Here ²i and si are VEVcomponents of h1Hi and h10 Hi respectively and sµ(cµ) is sin µ(cos µ). f = (1 + T2 2 + T2 1 (1 + s2µ T2 2 ))¡1=2 and rF = (1 + Q2 FT2 3 T2 1 (1 + s2µ T2 2 )f2)1=2 are factors that come from doing the algebra exactly (see appendix A). Here T1 = b1² M1 , T2 = b2S M2 , T3 = k1 M3 and Q = 2I3R + 6 5±(Y 2 ) is the unbroken charge that results from breaking SO(10) to the SM gauge group by giving a VEV to 45H, where h45Hi = Q. The charge Q for di®erent quarks and leptons is given as. Qu = Qd = 1 5 ±; Quc = ¡1 ¡ 4 5 ±; Qdc = 1 + 2 5 ±; Ql = Q¹ = ¡ 3 5 ±; Qlc = 1 + 6 5 ±; Qºc = ¡1: (3.4) The above general structure of fermion mass matrix has the following interesting features: (1) The relation m0b = m0¿ automatically follows from Qd +Qdc = Qe +Qec , (2) The hierarchy of the the second and third masses generation is obtained by taking the limit sµ ! 0, and (3) The approximate GeorgiJarlskog relation m0 ¹ = 3m0s leads to two possible values for ±, either ± ! 0 or ± ! ¡1:25, (4) the former possibility is excluded by experiment since it leads to (m0c =m0t )=(m0s =m0b ) ! 1 at the GUT scale, while the latter possibility leads to (m0c =m0t )=(m0s =m0b ) ! 0 which is closer to experiments. Let us de¯ne ± = 1+®. The masses and mixings of the ¯rst generation arise from Wvector. The full mass matrices arising from Wspinor and Wvector have the 53 following form: MD = m0 d 0 BBBBBBBBBBBBBBBBB@ 0 (c12 + ±3( 3+2® 5 ))rdrdc (¡2±2( 3+2® 5 ) + ³)rdc (c12 0 (2±1( 3+2® 5 ) ¡±3( 3+2® 5 ))rdrdc +s(¡1+® 5 ) + ¯)rdc ³rd (s( 3+2® 5 ) + ¯)rd 1 ¡2(¯ + 3+2® 5 ±1)fcµsµT2 2 1 CCCCCCCCCCCCCCCCCA ; MU = m0 u 0 BBBB@ 0 0 0 0 0 ( 1¡® 5 )sruc 0 ( 1+4® 5 )sru 1 1 CCCCA ; (3.5) ML = m0 d 0 BBBBBBBBBBBBBBBBB@ 0 (c12 + 3±3(¡1+® 5 ))rerec (¡±2® + ³)rec (c12 0 (±1® ¡3±3(¡1+® 5 ))rerec ¡3s(¡1+® 5 ) + ¯)rec (³ (s(¡1+6® 5 ) + ±1( 6¡® 5 ) 1 ¡±2 6¡® 5 )re +¯)re ¡2(¯ + 3+2® 5 ±1)fcµsµT2 2 1 CCCCCCCCCCCCCCCCCA ; MN = m0 u 0 BBBB@ 0 0 0 0 0 (¡3+3® 5 )srºc 0 srº 1 1 CCCCA ; where the parameters are de¯ned in terms of the Yukawa couplings of the super potential (Wspin: + Wvect:) and the VEVs of the Higgs ¯elds as shown in appendix A. These matrices are multiplied by lefthanded fermions on the right and right handed fermions on the left. A doubly lopsided structure for the charged lepton and down quark mass matrices of Eq.(3.5) can be obtained by going to the limit ¯; ³; ®; ±3; c12; s ¿ 1 and ±1; ±2 are of order one. This doubly lopsided form leads simultaneously to large neutrino mixing angles and to small quark mixing angles. Based only on the above fermion mass matrices in Eq.(3.5), an excellent ¯t is found for fermion masses (except for the neutrino masses), quark mixing angles and neu 54 trino mixing angles (except the atmospheric angle) by giving the input parame ters, appearing in Eq.(3.5), the following numerical values: ±1 = ¡1:28, ±2 = 1:01, ±3 = 0:015 £ e4:95i, ® = ¡0:0668, s = 0:2897, ³ = 0:0126, c12 = ¡0:0011e1:124i, and ¯ = ¡0:11218. The above numerical values lead to sin µL 23 = 0:92 which is not close to the experimental central value of atmospheric angle sin µatm 23 = 0:707 [7]. This con tribution to the atmospheric angle is only from the charged lepton sector. Therefore, the neutrino sector should be included by considering the following superpotential: WN = b4ÃiZi16H + hijkZiZc j 1000 Hk + m1Zc i Zc i ; (3.6) where two fermion singlets Zi and Zc i that couple with the Higgs singlet 1000 Hk have been introduced. The full neutrino mass matrix is constructed in Appendix B. The Higgs singlet 1000 Hk has the VEVcomponents (®1, ®2, ®3). The light neutrino mass matrix is obtained by employing the seesaw mechanism. The numerical values (®1 = 0:075, ®2 = 0:07, ®3 = 0:9, and ¸ = 0:0465 eV), where ¸ is de¯ned in appendix B, lead to not only the correct contribution to the atmospheric angles (sin µatm 23 = 0:811) but also to the correct light neutrino mass di®erences. The predictions of the fermion masses and mixings are slightly altered by doing the algebra exactly compared to the analysis of Ref.[51]. These predictions and their updated experimental values obtained from [7] are shown in Table 3.3. The right handedneutrino masses arise from integrating out the exotic fermion singlets Zi and Zc i in Eq.(3.6). The right handedneutrino mass matrix is MR = ¤ 0 BBBBBB@ ®21 ®23 ®1®2(¡1 ®23 + 2 ®21 +®22 +®23 ) ¡®1(®21 ¡®22 +®23 ) ®3(®21 +®22 +®23 ) ®1®2(¡1 ®23 + 2 ®21 +®22 +®23 ) ®22 ®23 ¡®2(¡®21 +®22 +®23 ) ®3(®21 +®22 +®23 ) ¡®1(®21 ¡®22 +®23 ) ®3(®21 +®22 +®23 ) ¡®2(¡®21 +®22 +®23 ) ®3(®21 +®22 +®23 ) 1 1 CCCCCCA ; (3.7) where ¤ = 8:45 £ 1015 GeV and the righthanded neutrino masses are given by MR1 ¼ MR2 ¼ 1:4 £ 1012 GeV and MR3 = 8:5 £ 1015 GeV. 55 Another interesting feature of this model is that it contains a minimal set of Higgs ¯elds needed to break SO(10) to the SM gauge group. Consequently, the uni¯ed gauge coupling remains perturbative all the way up to the Planck scale. This can be understood from the running of the uni¯ed gauge coupling with energy scale ¹ > MGUT as 1 ® = 1 ®G ¡ bG 2¼ log( ¹ MGUT ); (3.8) where ® = g2=(4¼) and bG = S(R) ¡ 3C(G). Here C(G) is the quadratic Casimir invariant and S(R) is the Dynkin index summed over all chiral multiplets of the model. The uni¯ed gauge coupling stays perturbative at the Planck scale (i.e g(MP ) < p 2) as long as bG < 26. Employing large Higgs representations might lead to bG ¸ 26. For example, using 126H+126H gives bG = 46. On the other hand, the SO(10)£A4model gives bG = 19 which is consistent with the uni¯ed gauge coupling being perturbative till the Planck scale. We will use the same ¯t for fermion masses and mixings to calculate the mass insertion parameters ±ij LL;RR, and ±ij LR;RL in the quark and lepton sectors and conse quently investigate the FCNC in this model. The charged lepton and down quark mass matrices in Eq.(3.5) are diagonalized at the GUT scale by biunitary transfor mation: Mdiag: d;l = V yd;l R MD;LV d;l L ; (3.9) where V u;d;l R;L are known numerically. Now, we discuss the sources of FCNC in this model. 3.2 Sources of Flavor Violation in SO(10) £ A4 Model We assume in our °avor violation analysis that A4 °avor symmetry is preserved above GUT scale and it is only broken at GUT scale. In this case °avor violation is induced 56 at GUT scale where A4 symmetry is broken. In this section we discuss the °avor violation induced at the GUT scale by studying the sfermion mass insertion parameter ±ij LL;RR and the chirality °ipping mass insertion (Aterms) parameter ±ij LR;RL. We will see that these °avor violation sources arise from large mixing of the light ¯elds with the heavy ¯elds. This large mixing is due to the breaking of A4 symmetry. In addition, we discuss the induced °avor violation arising below GUT scale through the RG running from MGUT to the righthanded neutrino mass scales. 3.2.1 The Scalar Mass Insertion Parameters Let us assume the soft supersymmetry breaking terms originate at the messenger scale M¤, where MGUT < M¤ · MPlanck. The quadratic soft mass terms of the matter super¯elds that appear in the superpotential Wspin: are ¡L = ~m2 ÃÃy iÃi + ~m2 ÂiÂy iÂi + ~m2 Âi Ây iÂi: (3.10) The MSSM scalar fermions that reside in Ãi transform as triplets under the non abelian A4 symmetry. Since the A4 symmetry is intact, they have common mass ( ~m2 Ã) at the scale M¤. On the other hand, the exotic ¯elds each of which transforms as singlet under A4 symmetry have di®erent masses ( ~m2 Âi , ~m2Â i , i runs 13) at the scale M¤. The MSSM scalars remain degenerate above the GUT scale where the A4 symme try is broken. In order to ¯nd the scalar masses in the fermion mass eigenstates, two transformations are required. The ¯rst transformation is needed to blockdiagonalize the fermion mass matrix into a light and a heavy blocks as shown in Appendix A. The upper left corner represents the 3 £ 3 light fermions mass matrix. The second trans formation is the complete diagonalization of the light fermion mass matrix. Applying the ¯rst transformation to the quadratic soft mass terms of Eq.(3.10) by going to the new orthogonal basis (L2, L3, H1, H2, H3) as de¯ned in appendix A, the quadratic 57 soft mass matrix of the light states is transformed as follows: ~m2 ÃI ! ~m2 ÃI + ± ~m2 Ã; (3.11) where, ± ~m2 Ã = 0 BBBBBB@ 0 0 0 0 0 ² 0 ² ± 1 CCCCCCA ; (3.12) ² = f rF T2 2 sµ( ~m2 Â2 ¡ ~m2 Ã), ± = (( f rF )2 ¡ 1) ~m2 Ã + ( f rF )2( ~m2 Â1T2 1 + ~m2 Â2T2 2 + ~m2 Â3Q2T2 1 T2 3 ), and we have safely ignored the terms that contain s2µ ¿ 1. It is obvious that the ¯rst two generations of the light scalars are almost degenerate because the mixing of the second light generation (L2) with the heavy states is proportional to sµ ¿ 1. On the other hand, since the mixing of the third light generation (L3) with the heavy states is of order one, its mass splits from those of the ¯rst two generations. The top Yukawa coupling is given in terms of T1, T2, and T3 as: Yt = af2(Qu + Quc)T1T2T3 rucru : (3.13) The numerical values of T1 = 0:0305, T2 = 2, T3 = 100 and a » 1:2 are consistent with the top Yukawa coupling at the GUT scale to be of order ¸GUT t » 0:5 and ru;uc to be of order one. Plugging these numerical values and sµ = 0:0465 into the expressions for ² and ± gives us: (±d; ±dc ; ±e; ±ec) = (0:81; 0:87; 0:88; 0:82)( ~m2 Â ¡ ~m2 Ã); (²d; ²dc ; ²e; ²ec) = (0:061; 0:05; 0:048; 0:06)( ~m2 Â ¡ ~m2 Ã): (3.14) Here we have dropped ~m2 Â1 terms because their coe±cients are negligible. Also, the RGE expressions of ~m2 Â2 and ~m2 Â3 are the same (see Eq.(B.13)), so we have assumed that ~m2 Â2 = ~m2 Â3 = ~m2 Â. The next step is to apply the second transformation by evaluating V yd;l L ±m2 ÃV d;l L and similarly for L ! R. The unitary matrices V d;l L are numerically known from the 58 ¯tting for fermion masses and mixings. So, the mass insertion parameters for charged leptons and down quarks are given respectively by (±d;e LL;RR)ij = (V yd;l L;R ± ~m2 d;lV d;l L;R)ij= ~m2 d;l: (3.15) The above mass insertion analysis without including the superpotential Wvect: is good enough because we assumed in our analysis that the mixing of the 10 vector multiplets with the ordinary spinor ¯elds is small. 3.2.2 The Chirality Flipping Mass Insertion (Aterms) The FV processes are also induced from the o®diagonal entries of the chirality °ipping mass matrix ~M RL. The chirality °ipping soft terms are divided into two parts Lspin and Lvect: ¡Lspin = ~b 1b1 ~ Ãi~Â 11Hi +~b 2b2 ~ Ãi~Â 210 Hi + ~k1k1 ~Â1~Â 345H +~aa~Â3 ~Â210H + ~G iMi ~Âi~Â i; (3.16) ¡Lvect = ~b 3b3 ~ Ãi ~Ái16H + ~B 10M10 ~Ái ~ Á0 i + ~ h0 ijkh0 ijk ~ Á0 i ~ Á0 j1Hk +~h ijkhijk ~Ái ~Áj1Hk +A~ijkAijk ~ Á0 i Á~00 j100 Hk + g~mÁ~00 i Á~000 i + ~k2k2 Á~000 i ~ Á0 i45H: (3.17) The fourth term of Eq.(3.16) induces the o®diagonal elements of the chirality °ipping mass matrix, if it is written in terms of the new orthogonal basis de¯ned in Eqs.(A.1). This transformation can be represented by ~M 2R L(spin:) ! ~aMF (spin:); (3.18) where MF (spin:) is de¯ned in Eq.(3.3). The entire chirality °ipping mass matrix in the new orthogonal basis is obtained by including ¡Lvect. The biunitary transfor mations that blockdiagonalize the full fermion mass matrix is applied on the entire chirality °ipping mass matrix (see Appendix A). Accordingly, the 3 £ 3 quadratic mass matrix ( ~M 2 LR) associated with the light states is transformed as follows: ~M 2R L ! ~aMF (spin:) +~b 3MF (vector); (3.19) 59 where MF (vect:) = ¡mM¡1M0 (see Eq.(A.6)) and we have assumed for simplicity that the soft parameters appearing in Eq.(3.17) are all of the same order. Then, the M2 LR matrix is written in the fermion mass eigenstate basis as: ~M 2R L ! V y R(~aMF (spin:) +~b 3MF (vect:))VL: (3.20) It is straightforward to show that the chirality mass insertion parameters are given by: (±RL)ij = ~b 3 ~m2f Mdiag: Fi ±ij + (~zV y RMF (spinor)VL)ij ; (3.21) where Mdiag: F = V y RMFVL and ~z = ~a¡~b 3 ~m2f . The induced FV arises only from the second term of Eq.(3.21). 3.2.3 Mass Insertion Parameters Induced Below MGUT The Dirac neutrino Yukawa couplings (YN)ij induce °avor violating o®diagonal ele ments in the lefthanded slepton mass matrix through the RG running from MGUT to the righthanded neutrino mass scales. The RGEs for MSSM with righthanded neutrinos are given in Ref.[46]. The righthanded neutrinos MRi are determined in the SO(10) £ A4 model. In this case, the induced mass insertion parameters for lefthanded sleptons are given by [50], (±l LL)RHN ij = ¡ 3m2 Ã + ~a2 8m2 Ã¼2 X3 k=1 (YN)ik(Y ¤ N)jkln MGUT MRk ; (3.22) where the matrix YN is written in the mass eigenstates of charged leptons and right handed neutrinos. The total LL contribution for the charged leptons is given by (±l LL)Tot ij = (±l LL)RHN ij + (±l LL)ij : (3.23) 3.3 Results In this section, we investigate the °avor violating processes by calculating the mass insertion parameters ±LL, ±RR, and ±LR;RL, then we compare them with their exper 60 imental bounds. These bounds in the quark and lepton sectors were obtained by comparing the hadronic and leptonic °avor changing processes to their experimental values/limits [54, 55]. Eq.(3.12), Eq.(3.14) and Eq.(3.15) are used to calculate ±LL;RR and Eq.(3.21) is used to calculate ±LR;RL for both charged leptons and down quarks. The result of mass insertion calculations and their experimental bounds are presented in Table 3.4. In this table, we have de¯ned ¾ = ~m2 Â2 ¡~m2 Ãi ~m2 Ãi and ~k = ~zmb;¿ The stringent bounds on leptonic ±12, ±13, and ±23 in Table 3.4 come only from the decay rates li ! lj°. The experimental bounds on the mass insertion parameters listed in column 3 were obtained by making a scan of m0 and M1=2 over the ranges m0 < 380 GeV and M1=2 < 160 GeV , where m0 and M1=2 are the scalar universal mass and the gaugino mass respectively [55]. Glancing at Table 3.4, we note that the stringent constraint on leptonic °avor violation arises from ±l 12 which corresponds to the decay rate of ¹ ! e°. On the other hand, there is a weaker constraint that arises from ±d 12 on the quark sector. One can do an arrangement such that ~a ¡~b 3 = 200 GeV and ~mf = 800 GeV (equivalent to ~k = 2:6£10¡4) so that all the chirality °ipping mass insertions will be within their experimental bounds. This arrangement is possible if the trilinear soft terms vanish at the scale M¤. Since the stringent constraint comes from the ¹ ! e° process, let us discuss the branching ratio of this process in more details. In general, the branching ratio of li ! lj° is given by BR(li ! lj°) BR(li ! ljºi ¹ ºj) = 48¼3® G2 F (jAij L j2 + jAij Rj2): (3.24) We have used the general expressions for the amplitudes Aij L;R given by Ref.[57] where the contributions from both chargino and the neutralino loops are included. These expressions are written in terms of mass insertion parameters. The correct suppression of the decay rate ¡(¹ ! e°) requires a high degree of degeneracy of the soft mass terms of MSSM ¯elds and the exotic ¯elds. For example, 61 ¾ ¼ 0:01, as can be seen from Table 3.4. In order to obtain high degree of degeneracy, let us assume that the SSB terms which are generated at the messenger scale M¤ satisfy the universality boundary conditions at the scale M¤ given by ~m2 Ãi = ~m2 Âi = ~m2 Âi = ~m2 10H = ~m21 H = ~m21 0 H = m0; M¸ = M0; ~a = ~b 1 =~b 2 = 0; (3.25) where M¸ is the gaugino mass of SO(10) gauge group. Solving the RGE listed in Appendix C with the boundary conditions given by Eq.(3.25) determines the value of ¾. In Table 3.5 we give the branching ratio of the process ¹ ! e° predicted by the SO(10) £ A4 model for di®erent choices of the input parameters a, b1, b2, ~mÃ and M1=2 at the GUT scale. The experimental searches have put the upper limit on the branching ratio of ¹ ! e° as Br(¹ ! e°) · 1:2 £ 10¡11 [56]. Note that ~mÃ and M1=2 originate respectively from m0 and M0 through RGEs. In this Table we consider ln M¤ MGUT = 1 and ln M¤ MGUT = 4:6 that correspond respectively to M¤ ¼ 3MGUT and M¤ ¼ MPlanck. Let us analyze the four cases in the Table 3.5. In the cases (I, II and III), the chosen values of the parameters a are consistent with the top Yukawa coupling of order 0:5 at the GUT scale and with the ¯tting for fermion masses and mixing. On the other hand, the choice of a = 0:68 in Case IV is not consistent with the ¯t. Although the medium slepton masses of order 550 GeV are obtained in Case I, the choice b1 = b2 = 1:9 corresponds to nonperturbative Yukawa couplings at the scale M¤ (i.e. b1 = b2 = 4 at M¤). In this case, the solutions of the 1loop RGEs are not trusted since the Yukawa couplings b1 and b2 go nonperturbative above the GUT scale. Also, it is important to point out that the °avor violation constraint on ¹ ! e° in Case III requires heavy slepton masses (¸ 3 TeV) while it requires slepton masses of order » 900 GeV in Case II. In other words, Case II is preferred in our model 62 in the sense that the decay rate of ¹ ! e° is close to the experimental limit with a reasonable supersymmetric mass spectrum, so it might be tested in the ongoing MEG experiment[58]. Besides, the Yukawa couplings remain perturbative at the messenger scale M¤. Figure 3.1 shows the allowed values of mÃ that correspond to the graphs below the xaxis for the cases I and II. 63 Predictions Expt. Pull mc(mc) 1.4 1:27+0:07 ¡0:11 1.85 mt(mt) 172.5 171.3§2.3 0.52 ms=md 19.4 19:5 § 2:5 0.04 ms(2Gev) 109.6£10¡3 105+25 ¡35 £ 10¡3 0.184 mb(mb) 4.31 4:2+0:17 ¡0:07 0.58 Vus 0.223 0.2255§0.0019 1.3 Vcb 38.9£10¡3 (41.2§1.1)£10¡3 2 Vub 4.00£10¡3 (3.93§0.36)£10¡3 0.7 ´ 0.319 0:349+0:015 ¡0:017 1.7 me(me) 0.511£10¡3 0.511£10¡3  m¹(m¹) 105.6£10¡3 105.6£10¡3  m¿ (m¿ ) 1.776 1.776  ¢m2 21 7:69 £ 10¡3eV2 (7:59 § 0:2) £ 10¡3eV2 0.5 ¢m2 32 2:36 £ 10¡3eV2 (2:43 § 0:13) £ 10¡3eV2 0.5 sin µsol 12 0.555 0.566§0.018 0.61 sin µl 23 0.811 0.707§0.108 0.96 sin µ13 0.141 < 0:22 Table 3.3: The fermion masses and mixings and their experimental values. The fermion masses, except the neutrino masses, are in GeV. 64 500 1000 1500 myHGeVL 0.5 0.5 1.0 1.5 2.0 2.5 log10@ Br Hm egL Expt limit D 175 GeV 437 GeV 787 GeV Case II 500 1000 1500 myHGeVL 6 4 2 log10@ Br Hm egL Expt limit D 175 GeV 437 GeV 787 GeV Case I Figure 3.1: The above graphs show the plot of Log of Br(¹ ! e°) divided by exper imental bound (1:2 £ 10¡11) versus mÃ for two cases I and II with M1=2=787 GeV, 437 GeV and 175 GeV. 65 Mass Insertion (±) Model Predictions Exp. Upper Bounds (±l 12)LL 0.062 ¾+(±l 12)RHN LL 6 £ 10¡4 (±l 12)RR 6.1 £ 10¡4 ¾ 0.09 (±l 12)RL;LR (0.084, 0.0096) ~k 10¡5 (±l 13)LL 0.022 ¾+(±l 13)RHN LL 0.15 (±l 13)RR 0.028 ¾  (±l 13)RL;LR (0.0335, 0.076) ~k 0.04 (±l 23)LL 0.27 ¾+(±l 13)RHN LL 0.12 (±l 23)RR 0.034 ¾  (±l 23)RL;LR (0.055, 0.899) ~k 0.03 (±d 12)LL 1.9 £ 10¡4 ¾ 0.014 (±d 12)RR 0.15 ¾ 0.009 (±d 12)LR;RL (0.029, 0.035) ~k 9 £ 10¡5 (±d 13)LL 0.014 ¾ 0.09 (±d 13)RR 0.061 ¾ 0.07 (±d 13)LR;RL (0.173, 0.016) ~k 1:7 £ 10¡2 (±d 23)LL 0.054 ¾ 0.16 (±d 23)RR 0.29 ¾ 0.22 (±d 23)LR;RL (0.875, 0.064) ~k (0.006, 0.0045) Table 3.4: The mass insertion parameters predicted by SO(10)£A4 model and their experimental upper bounds obtained from [55]. 66 I II III IV a 1.14 1.07 1.14 0.62 b1 1.9 1.5 1.24 1.24 b2 1.9 1.5 1.24 1.24 ~mÃi 542 886 2932 675 M1=2 350 787 1924 350 BR(¹ ! e°) 1:4 £ 10¡13 1:16 £ 10¡11 1:2 £ 10¡11 2:2 £ 10¡12 Table 3.5: Branching ratio of ¹ ! e° for di®erent choices of input parameters at the GUT scale. Cases I and II correspond to ln M¤ MGUT = 1 and cases III and IV correspond to ln M¤ MGUT = 4:6. ~mÃi and M1=2 are given in GeV 67 CHAPTER 4 Higgs Boson Mass in GaugeMediating Supersymmetry Breaking with MessengerMatter Mixing Supersymmetric (SUSY) grand uni¯cation theories (GUTs) are promising candidates for physics beyond the standard model (SM). However, supersymmetry is not an exact symmetry at the lowenergy scale and it must be broken somehow to be relevant to nature. SUSY can not be broken at tree level since the supertrace theorem leads to nonphenomenological particle spectra. Therefore, it is assumed that SUSY breaking occurs in the hidden sector which has no renormalizable tree level couplings with the observable sector. SUSY breaking is transmitted to the visible sector either via gravitational interactions as inspired by supergravity models (SUGRA)[44], or by SM gauge interactions as in theories with gaugemediated SUSY breaking (GMSB)[59, 60, 61]. In the ¯rst scenario, the soft terms are generated at the Planck scale. In general, these soft terms are not °avorinvariant. The gravitymediated scenario can only give realistic models if the universality or an approximate alignment between particle and sparticle masses is imposed in order to suppress the °avor violation processes. On the other hand, the universality condition is naturally satis¯ed in the GMSB where the soft terms are generated at the messenger scale, below the GUT scale, from radiative corrections. In GMSB theories, messenger ¯elds communicate the SUSY breaking from the hidden sector to the visible sector. In addition to the observable sector, at least one gauge singlet super¯eld (Z) is needed in order to give mass to the messenger ¯elds and break SUSY by giving vacuum expectation values (VEVs) to its scalar 68 component (hZi) and to its auxiliary Fcomponent (hFZi) respectively. The SUSY breaking factor (i.e. hFZi) that appears in the mass splitting between the fermionic and scalar components of the messenger ¯eld is communicated to the MSSM particles through radiative corrections. For example, the gauginos and the scalars of MSSM get their masses at the messenger scale Mmess from oneloop and twoloop Feynman diagrams respectively as fellows: M¸r = gNmess ®r 4¼ ¤; (4.1) ~m2 = 2f X3 r=1 NmessC ~ f r ®2 r (4¼)2¤2; (4.2) where Nmess is called the messenger index. For example, Nmess = 1 (Nmess = 3) for messenger ¯elds belong to 5 + 5 (10 + 10) of SU(5). Here, ¤ = hFZi hZi is the e®ective SUSY breaking scale, C ~ f r are the quadratic Casimir invariants for the scalar ¯elds, and ®r are the gauge coupling constants at the scale Mmess. These gauge couplings are all equal at the GUT scale. In Eqs.(4.1) and (4.2), f and g are the 1loop and 2loop functions whose exact expressions can be found e.g. in Ref.[61]. The universal scalar masses in Eq.(4.2) are obtained when the messenger and matter ¯elds are completely separated. There are additional contributions to universal masses if messengermatter mixing is allowed. Two interesting features of GMSB are concluded from Eqs.(4.1) and (4.2). Firstly, the scalar masses are only functions of gauge quantum number so scalar masses with the same gauge quantum number are degenerate. As a result, the supersymmetric °avor problem is solved. Secondly, GMSB is highly predictive since all soft terms at the messenger scale are determined by only two parameters ¤ and Nmess. In order to preserve the successful gauge coupling uni¯cation of MSSM, the messenger ¯elds should reside in complete SU(5) multiplets. In this chapter, we consider two cases when the messenger ¯elds belong to 5 + 5 and 10 + 10 of SU(5). In both cases the perturbative uni¯cation is still maintained, as shown in Fig. 4.1. 69 a3 1 a2  1 a1  1 2 4 6 8 10 12 14 16 10 20 30 40 50 60 Log10 Hm GeVL a 1 Figure 4.1: The evolutions of the gauge couplings with Mmess = 108 GeV and tan ¯ = 10. Solid lines correspond to MSSM. Dashed lines are for MSSM+10+10 and dotted lines are for MSSM+5 + 5. The complete separation of messenger sector and visible sector is problematic in cosmology because this leads to models possessing stable particles [62]. Besides, messengermatter couplings are allowed by gauge symmetry and they can only be forbidden by imposing discrete °avor symmetry. If one allows these couplings, ad ditional contributions to the universal scalar mass given by Eq.(4.1) and (4.2) are obtained [63, 64, 65]. These new contributions reintroduce °avor violation either in the leptonic or the quark sector depending on the structure of the messenger ¯elds. In this chapter, we have shown that the induced °avor violation from messengermatter mixing that occurs mainly with the third generation is still su±ciently suppressed. Another advantage of the messengermatter mixingthe main result of this chapter is that it might increase the lightest Higgs mass to value as large as 125 GeV, which is di±cult to realize without such mixing. In order to reproduce the known qualitative features of quark and lepton masses and mixings, we consider the FroggattNielsen mechanism [66]. This mechanism leads to the lopsided structure of downquark and charged lepton mass matrix. It 70 was shown that in this kind of structure the ¹ ! e° decay rate is generally large by adopting gravity mediated SUSY breaking and it is consistent with the experimental limit of Br(¹ ! e°) only with a heavy SUSY spectrum [67]. On the other hand, the lopsided structure works well in the GMSB regarding the °avor violation processes even with light SUSY spectra as we show in this chapter. This chapter is organized as fellows: In section 4.1 the Higgs mass bounds are considered in two models. The ¯rst is 5+5 model in which the messenger ¯elds belong to the 5 + 5 representation of SU(5) while the second is 10 + 10 model in which the messenger ¯elds belong to the 10 + 10 representation of SU(5). In both models, the messengermatter couplings (i.e. the exotic couplings) are allowed. We investigate the e®ect of these couplings on the lightest Higgs mass of MSSM. In section 4.2, we construct the general structure of the superpotential of both models by employing the U(1) °avor symmetry of the FroggattNielsen mechanism as discussed in section 4.2.1. We ¯nd that the FCNC processes that are induced by the exotic Yukawa couplings are in agreement with experimental bounds. The Yukawa RGEs between messenger and GUT scales for both models are listed in Appendix C. The soft terms which are induced by the exotic Yukawa couplings are evaluated in Appendix D. 4.1 Higgs Mass Bounds One of the interesting features of MSSM is setting upper bounds on the lightest Higgs mass. The tree level bound on the lightest Higgs mass equal to Mz has been already excluded by the LEP2 lower bound mh > 114:4 [68]. However, radiative corrections push this mass above the LEP2 bound. The leading 1 and 2 loop contributions to the CPeven Higgs boson mass in the MSSM are given by [70, 71] m2 h = M2 z cos2 2¯(1 ¡ 3 8¼2 m2t v2 t) + 3 4¼2 m4t v2 [ 1 2 Ât + t + 1 16¼2 ( 3 2 m2t v2 ¡ 32¼®3)(Âtt + t2)]; (4.3) 71 where v2 = v2 d + v2u , t = log( M2 s M2 t ); Ât = 2 ~ A2t M2 s (1 ¡ ~ A2t 12M2 s ): (4.4) Here the scale Ms has been de¯ned in terms of the stop mass eigenvalues as M2 s = m~ t1m~ t2 ; (4.5) A~t = At¡¹ cot ¯, where At denotes the stop left and stop right soft mixing parameter. The upper bound on the lightest Higgs mass depends crucially on the soft su persymmetry breaking terms. For example, the upper bound of around 125 GeV corresponds to the maximal mixing condition, A~t = p 6Ms. Since there are restric tions on these soft terms from GMSB, it will be interesting to study the e®ect of these restrictions on the lightest CPeven Higgs mass. In the following subsections we will investigate the e®ect of allowing messengermatter couplings on the soft terms of MSSM and consequently on the lightest CPeven Higgs mass. In the ordinary GMSB (i.e. without messengermatter mixing), both Aterms and the soft breaking param eter B vanish at the messenger scale. However, B can be induced in the process of running. By using the following equations that result from minimizing the Higgs potential, M2 z 2 = ¡¹2 ¡ m2 Hu tan2 ¯ ¡ m2 Hd tan2 ¯ ¡ 1 ; (4.6) sin 2¯ = 2B¹ 2¹2 + m2 Hu + m2 Hd ; (4.7) one can solve for the parameters tan ¯ and ¹. Then tan ¯ turns out to be large (around 3545) when the messenger scale is close to the e®ective SUSY breaking scale ¤. On the other hand, by allowing messengermatter couplings B is induced signi¯cantly at low energy scale. This can be understood from the following RGE for the parameter B: dB dt = 1 2¼ (3®tAt + 3®2M2 + 3 5 ®1M1); (4.8) 72 ¸0 0 mh(GeV) ¤(105GeV) M(1013GeV) ~mt1(GeV) ~mt2(GeV) 0 117 2 1.78 1634 2012 0.8 118 2 10 1590 1857 1.2 119 2 10 1065 2788 Table 4.1: We show the values of the minimal GMSB input parameters, ¤, ¸ex and Mmess that lead to the highest mh values at tan ¯ = 10. where ®t = ¸t 4¼ and ¸t is the top Yukawa coupling. Since At does not vanish in the presence of messengermatter mixing as shown in Eqs.(4.13) and (4.21), the ¯rst term of Eq.(4.8) that pushes B to large values becomes more signi¯cant than in the case when At is zero. This leads to small tan ¯. For example in the 10 + 10 model, the range 1:64 · tan ¯ · 7 corresponds to 105 GeV ·Mmess ·1014 GeV. In the subsequent analysis, we will give the scalar mass spectrum that leads to the highest mh for two cases. The ¯rst case is to assume a nonvanishing B is somehow generated at the messenger scale such that tan ¯ = 10 is obtained by using Eqs.(4.6) and( 4.7). The potential solution to the ¹ problem based on °avor symmetries was suggested by Ref. [72]. The authors of Ref. [72] gave an example of B¹ » ¹2 that leads to unconstrained values on tan ¯ by introducing three singlets that are charged under U(1) °avor symmetry. The second case is having a vanishing B at the messenger scale as predicted by both 5 + 5 and 10 + 10 models. In this case tan ¯ is determined by Eqs.(4.6) and (4.7) where B at low energy scales is obtained by solving the RGE with the boundary condition of vanishing B at the messenger scale. 4.1.1 Higgs Mass Bounds in the 5 + 5 Model The messenger ¯elds belonging to 5 + 5 of SU(5) decompose to downquark singlets dc m and dc m, and to lepton doublets Lm and Lm. The additional contributions to the 73 0 2 4 6 8 10 0 0.5 1 1.5 2 ~m2 tc /105 (GeV2) l’0 Mmess=107 GeV Mmess=1014 GeV 0 2 4 6 8 10 0 0.5 1 1.5 2 m~2 tc /105 (GeV2) l’0 Mmess=107 GeV Mmess=1014 GeV Figure 4.2: The left graph is ~m2¿ c versus ¸0 0 at the scale Mmess for two di®erent messenger scales. The right graph is ~m2t c versus ¸0 0 at the low energy scale for two di®erent messenger scales. MSSM superpotential due to messengermatter couplings is W5+5 = fddc mdc mZ + ¸0 bQ3dc mHd + feLmLmZ + ¸0 ¿cLmec 3Hd: (4.9) We assume the messenger ¯elds couple only with the third generation of MSSM. We will show later that the superpotential W5+5 can be obtained by imposing the U(1) °avor symmetry of the FroggattNielsen mechanism. Also, we have assumed that the exotic Yukawa couplings ¸0 b and ¸0 ¿c (fd and fe) are obtained from one uni¯ed coupling ¸0 0(f0) at the GUT scale by solving the RGEs listed in the Appendix C.1 between the messenger scale and the GUT scale. In the universal case (i.e. without including messengermatter couplings), the scalar masses are obtained by employing Eqs.(4.1) and (4.2), while the trilinear soft terms (Aterms) vanish at the scale Mmess. There are new contributions to the u 



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