GRAND UNIFIED THEORIES

15. GRAND UNIFIED THEORIES
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15. Grand Unified Theories115. GRAND UNIFIED THEORIESRevised October 2005 by S. Raby (Ohio State University).15.1. Grand Unification15.1.1. Standard Model: An Introduction :In spite of all the successes of the Standard Model [SM], it is unlikely to be thefinal theory. It leaves many unanswered questions. Why the local gauge interactionsSU(3)C×SU(2)L×U(1)Y, and why 3 families of quarks and leptons? Moreover, why doesone family consist of the states [Q, uc,dc; L, ec] transforming as [(3, 2, 1/3), (¯3, 1, −4/3),(¯3, 1, 2/3); (1, 2, −1), (1, 1, 2)], where Q = (u, d), and L = (ν, e) are SU(2)Ldoublets, anduc, dc, ecare charge conjugate SU(2)Lsinglet fields with the U(1)Yquantum numbersgiven? [We use the convention that electric charge QEM= T3L+ Y/2 and all fields areleft-handed.] Note the SM gauge interactions of quarks and leptons are completely fixedby their gauge charges. Thus, if we understood the origin of this charge quantization,we would also understand why there are no fractionally charged hadrons. Finally, whatis the origin of quark and lepton masses, or the apparent hierarchy of family massesand quark mixing angles? Perhaps if we understood this, we would also know the originof CP violation, the solution to the strong CP problem, the origin of the cosmologicalmatter-antimatter asymmetry, or the nature of dark matter.The SM has 19 arbitrary parameters; their values are chosen to fit the data. Threearbitrary gauge couplings: g3, g, g (where g, g are the SU(2)L, U(1)Ycouplings,respectively) or equivalently, αs= (g23/4π), αEM= (e2/4π) (e = g sinθW), andsin2θW= (g )2/(g2+ (g )2). In addition, there are 13 parameters associated with the 9charged fermion masses and the four mixing angles in the CKM matrix. The remaining3 parameters are v, λ [the Higgs VEV (vacuum expectation value) and quartic coupling](or equivalently, MZ,m0h), and the QCD θ parameter. In addition, data from neutrinooscillation experiments provide convincing evidence for neutrino masses. With 3 lightMajorana neutrinos, there are at least 9 additional parameters in the neutrino sector;3 masses, 3 mixing angles and 3 phases. In summary, the SM has too many arbitraryparameters, and leaves open too many unresolved questions to be considered complete.These are the problems which grand unified theories hope to address.15.1.2. Charge Quantization :In the Standard Model, quarks and leptons are on an equal footing; both fundamentalparticles without substructure. It is now clear that they may be two faces of the samecoin; unified, for example, by extending QCD (or SU(3)C) to include leptons as the fourthcolor, SU(4)C[1]. The complete Pati-Salam gauge group is SU(4)C× SU(2)L× SU(2)R,with the states of one family [(Q, L), (Qc,Lc)] transforming as [(4, 2, 1), (¯4, 1,¯2)], whereQc= (dc,uc), Lc= (ec,νc) are doublets under SU(2)R. Electric charge is now givenby the relation QEM= T3L+ T3R+ 1/2(B – L), and SU(4)Ccontains the subgroupSU(3)C× (B –L) where B (L) is baryon (lepton) number. Note νchas no SM quantumnumbers and is thus completely “sterile.” It is introduced to complete the SU(2)Rleptondoublet. This additional state is desirable when considering neutrino masses.CITATION: W.-M. Yao et al., Journal of Physics G 33, 1 (2006)available on the PDG WWW pages (URL:http://pdg.lbl.gov/)July 14, 2006 10:37
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215. Grand Unified TheoriesAlthough quarks and leptons are unified with the states of one family forming twoirreducible representations of the gauge group, there are still 3 independent gaugecouplings (two if one also imposes parity, i.e., L ↔ R symmetry). As a result, the threelow-energy gauge couplings are still independent arbitrary parameters. This difficultyis resolved by embedding the SM gauge group into the simple unified gauge group,Georgi-Glashow SU(5), with one universal gauge coupling αGdefined at the grandunification scale MG[2]. Quarks and leptons still sit in two irreducible representations,as before, with a 10 = [Q, uc,ec] and¯5 = [dc,L]. Nevertheless, the three low energygauge couplings are now determined in terms of two independent parameters : αGandMG. Hence, there is one prediction.In order to break the electroweak symmetry at the weak scale and give mass to quarksand leptons, Higgs doublets are needed which can sit in either a 5Hor¯5H. The additional3 states are color triplet Higgs scalars. The couplings of these color triplets violate baryonand lepton number, and nucleons decay via the exchange of a single color triplet Higgsscalar. Hence, in order not to violently disagree with the non-observation of nucleondecay, their mass must be greater than ∼ 1010–11GeV. Moreover, in supersymmetricGUTs, in order to cancel anomalies, as well as give mass to both up and down quarks,both Higgs multiplets 5H,¯5Hare required. As we shall discuss later, nucleon decaynow constrains the color triplet Higgs states in a SUSY GUT to have mass significantlygreater than MG.Complete unification is possible with the symmetry group SO(10), with one universalgauge coupling αG, and one family of quarks and leptons sitting in the 16-dimensional-spinor representation 16 = [10 +¯5 + 1] [3]. The SU(5) singlet 1 is identified with νc. InTable 15.1 we present the states of one family of quarks and leptons, as they appear inthe 16. It is an amazing and perhaps even profound fact that all the states of a singlefamily of quarks and leptons can be represented digitally as a set of 5 zeros and/orones or equivalently as the tensor product of 5 “spin” 1/2 states with ± = | ±12>and with the condition that we have an even number of |− > spins. The first three“spins” correspond to SU(3)Ccolor quantum numbers, while the last two are SU(2)Lweak quantum numbers. In fact, an SU(3)Crotation just raises one color index andlowers another, thereby changing colors {r, b, y}. Similarly an SU(2)Lrotation raisesone weak index and lowers another, thereby flipping the weak isospin from up to downor vice versa. In this representation, weak hypercharge Y is given by the simple relationY = 2/3(∑color spins) –(∑weak spins). SU(5) rotations [not in the Standard Model]then raise (or lower) a color index, while at the same time lowering (or raising) a weakindex. It is easy to see that such rotations can mix the states {Q, uc, ec} and {dc,L}among themselves, and νcis a singlet. The new SO(10) rotations [not in SU(5)] are thengiven by either raising or lowering any two spins. For example, by lowering the two weakindices, νcrotates into ec, etc.SO(10) has two inequivalent maximal breaking patterns: SO(10) → SU(5) × U(1)Xand SO(10) → SU(4)C× SU(2)L× SU(2)R. In the first case, we obtain Georgi-GlashowSU(5) if QEMis given in terms of SU(5) generators alone, or so-called flipped SU(5) [4]if QEMis partly in U(1)X. In the latter case, we have the Pati-Salam symmetry. IfJuly 14, 2006 10:37
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15. Grand Unified Theories3Table 15.1: The quantum numbers of the 16 dimensional representation ofSO(10).StateYColorWeakνc0+ + +++ec2+ + +−−ur1/3− + ++−dr1/3− + +−+ub1/3+ − ++−db1/3+ − +−+uy1/3+ + −+−dy1/3+ + −−+ucr−4/3+ − −++ucb−4/3− + −++ucy−4/3− − +++dcr2/3+ − −−−dcb2/3− + −−−dcy2/3− − +−−ν−1− − −+−e−1− − −−+SO(10) breaks directly to the SM at MG, then we retain the prediction for gaugecoupling unification. However, more possibilities for breaking (hence more breaking scalesand more parameters) are available in SO(10). Nevertheless with one breaking patternSO(10) → SU(5) → SM, where the last breaking scale is MG, the predictions from gaugecoupling unification are preserved. The Higgs multiplets in minimal SO(10) are containedin the fundamental 10H= [5H,¯5H] representation. Note, only in SO(10) does the gaugesymmetry distinguish quark and lepton multiplets from Higgs multiplets.Finally, larger symmetry groups have been considered. For example, E(6) has afundamental representation 27, which under SO(10) transforms as a [16 + 10 + 1].The breaking pattern E(6) → SU(3)C× SU(3)L× SU(3)Ris also possible. With theadditional permutation symmetry Z(3) interchanging the three SU(3)s, we obtainso-called “trinification [5], ” with a universal gauge coupling. The latter breaking patternhas been used in phenomenological analyses of the heterotic string [6]. However, in largersymmetry groups, such as E(6), SU(6), etc., there are now many more states whichhave not been observed and must be removed from the effective low-energy theory. Inparticular, three families of 27s in E(6) contain three Higgs type multiplets transformingas 10s of SO(10). This makes these larger symmetry groups unattractive starting pointsJuly 14, 2006 10:37
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415. Grand Unified Theoriesfor model building.15.1.3. String Theory and Orbifold GUTs :Orbifold compactification of the heterotic string [7–9], and recent field theoreticconstructions known as orbifold GUTs [10], contain grand unified symmetries realized in5 and 6 dimensions. However, upon compactifying all but four of these extra dimensions,only the MSSM is recovered as a symmetry of the effective four dimensional field theory.1These theories can retain many of the nice features of four dimensional SUSY GUTs,such as charge quantization, gauge coupling unification and sometimes even Yukawaunification; while at the same time resolving some of the difficulties of 4d GUTs, inparticular problems with unwieldy Higgs sectors necessary for spontaneously breakingthe GUT symmetry, problems with doublet-triplet Higgs splitting or rapid proton decay.We will comment further on the corrections to the four dimensional GUT picture due toorbifold GUTs in the following sections.15.1.4. Gauge coupling unification :The biggest paradox of grand unification is to understand how it is possible to havea universal gauge coupling gGin a grand unified theory [GUT], and yet have threeunequal gauge couplings at the weak scale with g3> g > g . The solution is given interms of the concept of an effective field theory [EFT] [16]. The GUT symmetry isspontaneously broken at the scale MG, and all particles not in the SM obtain massof order MG. When calculating Green’s functions with external energies E ≫ MG,we can neglect the mass of all particles in the loop and hence all particles contributeto the renormalization group running of the universal gauge coupling. However, forE ≪ MG, one can consider an effective field theory including only the states withmass < E ≪ MG. The gauge symmetry of the EFT is SU(3)C× SU(2)L× U(1)Y, andthe three gauge couplings renormalize independently. The states of the EFT includeonly those of the SM; 12 gauge bosons, 3 families of quarks and leptons, and one or1Also, in recent years there has been a great deal of progress in constructing threeand four family models in Type IIA string theory with intersecting D6 branes [11]. Al-though these models can incorporate SU(5) or a Pati-Salam symmetry group in four di-mensions, they typically have problems with gauge coupling unification. In the formercase this is due to charged exotics which affect the RG running, while in the latter casethe SU(4)×SU(2)L×SU(2)Rsymmetry never unifies. Note, heterotic string theory modelsalso exist whose low energy effective 4d field theory is a SUSY GUT [12]. These modelshave all the virtues and problems of 4d GUTs. Finally, many heterotic string models havebeen constructed with the standard model gauge symmetry in 4d and no intermediateGUT symmetry in less than 10d. Recently some minimal 3 family supersymmetric modelshave been constructed [13,14].These theories may retain some of the symmetry rela-tions of GUTs, however the unification scale would typically be the string scale, of order5 × 1017GeV, which is inconsistent with low energy data. A way out of this problem wasdiscovered in the context of the strongly coupled heterotic string, defined in an effective 11dimensions [15]. In this case the 4d Planck scale (which controls the value of the stringscale) now unifies with the GUT scale.July 14, 2006 10:37
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15. Grand Unified Theories5more Higgs doublets. At MG, the two effective theories [the GUT itself is most likelythe EFT of a more fundamental theory defined at a higher scale] must give identicalresults; hence we have the boundary conditions g3= g2= g1≡ gG, where at any scaleµ < MG, we have g2≡ g and g1=√5/3 g . Then using two low-energy couplings, suchas αs(MZ), αEM(MZ), the two independent parameters αG, MGcan be fixed. Thethird gauge coupling, sin2θWin this case, is then predicted. This was the procedureup until about 1991 [17,18]. Subsequently, the uncertainties in sin2θWwere reducedtenfold. Since then, αEM(MZ), sin2θWhave been used as input to predict αG, MG,and αs(MZ) [19].We emphasize that the above boundary condition is only valid when using one-loop-renormalization group [RG] running. With precision electroweak data, however, itis necessary to use two-loop-RG running. Hence, one must include one-loop-thresholdcorrections to gauge coupling boundary conditions at both the weak and GUT scales.In this case, it is always possible to define the GUT scale as the point whereα1(MG) = α2(MG) ≡ ˜αGand α3(MG) = ˜αG(1 + ϵ3). The threshold correction ϵ3is alogarithmic function of all states with mass of order MGand ˜αG= αG+ ∆, where αGisthe GUT coupling constant above MG, and ∆ is a one-loop-threshold correction. To theextent that gauge coupling unification is perturbative, the GUT threshold corrections aresmall and calculable. This presumes that the GUT scale is sufficiently below the Planckscale or any other strong coupling extension of the GUT, such as a strongly coupledstring theory.Supersymmetric grand unified theories [SUSY GUTs] are an extension of non-SUSYGUTs [20]. The key difference between SUSY GUTs and non-SUSY GUTs is thelow-energy effective theory. The low-energy effective field theory in a SUSY GUT isassumed to satisfy N = 1 supersymmetry down to scales of order the weak scale, inaddition to the SM gauge symmetry. Hence, the spectrum includes all the SM states, plustheir supersymmetric partners. It also includes one pair (or more) of Higgs doublets; oneto give mass to up-type quarks, and the other to down-type quarks and charged leptons.Two doublets with opposite hypercharge Y are also needed to cancel fermionic triangleanomalies. Finally, it is important to recognize that a low-energy SUSY-breaking scale(the scale at which the SUSY partners of SM particles obtain mass) is necessary to solvethe gauge hierarchy problem.Simple non-SUSY SU(5) is ruled out, initially by the increased accuracy in themeasurement of sin2θW, and by early bounds on the proton lifetime (see below) [18].However, by now LEP data [19] has conclusively shown that SUSY GUTs is the newStandard Model; by which we mean the theory used to guide the search for new physicsbeyond the present SM (see Fig. 15.1). SUSY extensions of the SM have the propertythat their effects decouple as the effective SUSY-breaking scale is increased. Any theorybeyond the SM must have this property simply because the SM works so well. However,the SUSY-breaking scale cannot be increased with impunity, since this would reintroducea gauge hierarchy problem. Unfortunately there is no clear-cut answer to the question,“When is the SUSY-breaking scale too high?” A conservative bound would suggest thatthe third generation squarks and sleptons must be lighter than about 1 TeV, in orderthat the one-loop corrections to the Higgs mass from Yukawa interactions remain of orderJuly 14, 2006 10:37
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615. Grand Unified Theoriesthe Higgs mass bound itself.Figure 15.1: Gauge coupling unification in non-SUSY GUTs on the left vs. SUSYGUTs on the right using the LEP data as of 1991. Note, the difference in therunning for SUSY is the inclusion of supersymmetric partners of standard modelparticles at scales of order a TeV (Fig. taken from Ref. 21). Given the presentaccurate measurements of the three low energy couplings, in particular αs(MZ),GUT scale threshold corrections are now needed to precisely fit the low energy data.The dark blob in the plot on the right represents these model dependent corrections.At present, gauge coupling unification within SUSY GUTs works extremely well. Exactunification at MG, with two-loop-RG running from MGto MZ, and one-loop-thresholdcorrections at the weak scale, fits to within 3 σ of the present precise low-energy data. Asmall threshold correction at MG(ϵ3∼ −3 to − 4%) is sufficient to fit the low-energydata precisely [22–24].2This may be compared to non-SUSY GUTs, where the fitmisses by ∼ 12 σ, and a precise fit requires new weak-scale states in incomplete GUTmultiplets, or multiple GUT-breaking scales.32This result implicitly assumes universal GUT boundary conditions for soft SUSY-breaking parameters at MG. In the simplest case, we have a universal gaugino massM1/2, a universal mass for squarks and sleptons m16, and a universal Higgs mass m10, asmotivated by SO(10). In some cases, threshold corrections to gauge coupling unificationcan be exchanged for threshold corrections to soft SUSY parameters. See for example,Ref. 25 and references therein.3Non-SUSY GUTs with a more complicated breaking pattern can still fit the data.For example, non-SUSY SO(10) → SU(4)C× SU(2)L× SU(2)R→SM, with the secondbreaking scale of order an intermediate scale, determined by light neutrino masses usingthe see-saw mechanism, can fit the low-energy data for gauge couplings [26], and at thesame time survive nucleon decay bounds [27], discussed in the following section.July 14, 2006 10:37
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15. Grand Unified Theories7Following the analysis of Ref. 24 let us try to understand the need for the GUTthreshold correction and its order of magnitude. The renormalization group equationsrelate the low energy gauge coupling constants αi(MZ), i = 1, 2, 3 to the value of theunification scale ΛUand the GUT coupling αUby the expression1αi(MZ)=1αU+bi2πlog(ΛUMZ)+ δi(15.1)where ΛUis the GUT scale evaluated at one loop and the threshold corrections, δi, aregiven by δi= δ(2)i+ δ(l)i+ δ(g)iwith δ(2)irepresenting two loop running effects, δ(l)ithelight threshold corrections at the SUSY breaking scale and δ(g)i= δ(h)i+ δ(b)irepresentingGUT scale threshold corrections. Note, in this analysis, the two loop RG running istreated on the same footing as weak and GUT scale threshold corrections. One thenobtains the prediction(α3(MZ) − αLO3(MZ))/αLO3(MZ) = −αLO3(MZ) δs(15.2)where αLO3(MZ) is the leading order one loop RG result and δs=17(5δ1− 12δ2+ 7δ3) isthe net threshold correction. [A similar formula applies at the GUT scale with the GUTthreshold correction, ϵ3, given by ϵ3= −˜αGδ(g)s.] Given the experimental inputs [28]:α−1em(MZ) = 127.906 ± 0.019sin2θW(MZ) = 0.2312 ± 0.0002α3(MZ) = 0.1187 ± 0.0020(15.3)and taking into account the light threshold corrections, assuming an ensemble of 10 SUSYspectra [24]( corresponding to the Snowmass benchmark points), we haveαLO3(MZ) ≈ 0.118(15.4)andδ(2)s≈ −0.82δ(l)s≈ −0.50 +1928πlogMSUSYMZ.For MSUSY= 1 TeV, we have δ(2)s+ δ(l)s≈ −0.80. Since the one loop result αLO3(MZ)is very close to the experimental value, we need δs≈ 0 or equivalently, δ(g)s≈ 0.80. Thiscorresponds, at the GUT scale, to ϵ3≈ −3%.44In order to fit the low energy data for gauge coupling constants we require a relativeshift in α3(MG) of order 3% due to GUT scale threshold corrections. If these GUT scalecorrections were not present, however, weak scale threshold corrections of order 9% (dueto the larger value of α3at MZ) would be needed to resolve the discrepancy with the datafor exact gauge coupling unification at MG. Leaving out the fact that any consistent GUTnecessarily contributes threshold corrections at the GUT scale, it is much more difficult tofind the necessary larger corrections at the weak scale. For example, we need MSUSY≈ 40TeV for the necessary GUT scale threshold correction to vanish.July 14, 2006 10:37
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815. Grand Unified TheoriesIn four dimensional SUSY GUTs, the threshold correction ϵ3receives a positivecontribution from Higgs doublets and triplets.5Thus a larger, negative contribution mustcome from the GUT breaking sector of the theory. This is certainly possible in specificSO(10) [29] or SU(5) [30] models, but it is clearly a significant constraint on the 4dGUT sector of the theory. In five or six dimensional orbifold GUTs, on the other hand,the “GUT scale” threshold correction comes from the Kaluza-Klein modes between thecompactification scale, Mc, and the effective cutoff scale M∗.6Thus, in orbifold GUTs,gauge coupling unification at two loops is only consistent with the low energy data witha fixed value for Mcand M∗.7Typically, one finds Mc< MG= 3 × 1016GeV, whereMGis the 4d GUT scale. Since the grand unified gauge bosons, responsible for nucleondecay, get mass at the compactification scale, the result Mc< MGfor orbifold GUTs hassignificant consequences for nucleon decay.A few final comments are in order. We do not consider the scenario of splitsupersymmetry [33] in this review. In this scenario squarks and sleptons have mass at ascale ˜m ≫ MZ, while gauginos and Higgsinos have mass of order the weak scale. Gaugecoupling unification occurs at a scale of order 1016GeV, provided that the scale ˜m lies inthe range 103− 1011GeV [34]. A serious complaint concerning the split SUSY scenariois that it does not provide a solution to the gauge hierarchy problem. Moreover, it isonly consistent with grand unification if it also postulates an “intermediate” scale, ˜m, forscalar masses. In addition, it is in conflict with b − τ Yukawa unification, unless tanβ isfine-tuned to be close to 1 [34].8We have also neglected to discuss non-supersymmetric GUTs in four dimensions whichstill survive once one allows for several scales of GUT symmetry breaking [26]. Finally,it has been shown that non-supersymmetric GUTs in warped 5 dimensional orbifolds canbe consistent with gauge coupling unification, assuming that the right-handed top quarkand the Higgs doublets are composite-like objects with a compositeness scale of order a5Note, the Higgs contribution is given by ϵ3=3˜αG5πlog |˜MtγMG| where˜Mtis the effectivecolor triplet Higgs mass (setting the scale for dimension 5 baryon and lepton numberviolating operators) and γ = λb/λtat MG. Since˜Mtis necessarily greater than MG, theHiggs contribution to ϵ3is positive.6In string theory, the cutoff scale is the string scale.7It is interesting to note that a ratio M∗/Mc∼ 100, needed for gauge coupling unifi-cation to work in orbifold GUTs is typically the maximum value for this ratio consistentwith perturbativity [31].In addition, in orbifold GUTs brane-localized gauge kineticterms may destroy the successes of gauge coupling unification. However, for values ofM∗/Mc= M∗πR ≫ 1 the unified bulk gauge kinetic terms can dominate over the brane-localized terms [32].8b − τ Yukawa unification only works for ˜m < 104for tanβ ≥ 1.5. This is because theeffective theory between the gaugino mass scale and ˜m includes only one Higgs doublet, asin the standard model. In this case, the large top quark Yukawa coupling tends to increasethe ratio λb/λτas one runs down in energy below ˜m . This is opposite to what happensin MSSM where the large top quark Yukawa coupling decreases the ratio λb/λτ[35].July 14, 2006 10:37
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15. Grand Unified Theories9TeV [36].15.1.5. Nucleon Decay :Baryon number is necessarily violated in any GUT [37]. In SU(5), nucleons decayvia the exchange of gauge bosons with GUT scale masses, resulting in dimension-6baryon-number-violating operators suppressed by (1/M2G). The nucleon lifetime iscalculable and given by τN∝ M4G/(α2Gm5p). The dominant decay mode of the proton(and the baryon-violating decay mode of the neutron), via gauge exchange, is p → e+π0(n → e+π−). In any simple gauge symmetry, with one universal GUT coupling andscale (αG, MG), the nucleon lifetime from gauge exchange is calculable. Hence, the GUTscale may be directly observed via the extremely rare decay of the nucleon. Experimentalsearches for nucleon decay began with the Kolar Gold Mine, Homestake, Soudan, NUSEX,Frejus, HPW, and IMB detectors [17]. The present experimental bounds come fromSuper-Kamiokande and Soudan II. We discuss these results shortly. Non-SUSY GUTs arealso ruled out by the non-observation of nucleon decay [18]. In SUSY GUTs, the GUTscale is of order 3 × 1016GeV, as compared to the GUT scale in non-SUSY GUTs, whichis of order 1015GeV. Hence, the dimension-6 baryon-violating operators are significantlysuppressed in SUSY GUTs [20] with τp∼ 1034–38yrs.However, in SUSY GUTs, there are additional sources for baryon-number violation—dimension-4 and -5 operators [38]. Although the notation does not change, whendiscussing SUSY GUTs, all fields are implicitly bosonic superfields, and the operatorsconsidered are the so-called F terms, which contain two fermionic components, and therest scalars or products of scalars. Within the context of SU(5), the dimension-4 and -5operators have the form (10¯5¯5) ⊃ (ucdcdc) + (Q L dc) + (ecL L), and (10 10 10¯5)⊃ (Q Q Q L) + (ucucdcec) + B and L conserving terms, respectively. The dimension-4operators are renormalizable with dimensionless couplings; similar to Yukawa couplings.On the other hand, the dimension-5 operators have a dimensionful coupling of order(1/MG).The dimension-4 operators violate baryon number or lepton number, respectively,but not both. The nucleon lifetime is extremely short if both types of dimension-4operators are present in the low-energy theory. However, both types can be eliminatedby requiring R parity. In SU(5), the Higgs doublets reside in a 5H,¯5H, and R paritydistinguishes the¯5 (quarks and leptons) from¯5H(Higgs). R parity [39] (or moreprecisely, its cousin, family reflection symmetry) (see Dimopoulos and Georgi [20] andDRW [40]) takes F → −F, H → H with F = {10,¯5}, H = {¯5H, 5H}. This forbids thedimension-4 operator (10¯5¯5), but allows the Yukawa couplings of the form (10¯5¯5H)and (10 10 5H). It also forbids the dimension-3, lepton-number-violating operator (¯5 5H)⊃ (L Hu), with a coefficient with dimensions of mass which, like the µ parameter,could be of order the weak scale and the dimension-5, baryon-number-violating operator(10 10 10¯5H) ⊃ (Q Q Q Hd) + ···.Note, in the MSSM, it is possible to retain R-parity-violating operators at low energy,as long as they violate either baryon number or lepton number only, but not both. Suchschemes are natural if one assumes a low-energy symmetry, such as lepton number,baryon number, or a baryon parity [41]. However, these symmetries cannot be embeddedJuly 14, 2006 10:37
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1015. Grand Unified Theoriesin a GUT. Thus, in a SUSY GUT, only R parity can prevent unwanted dimensionfour operators. Hence, by naturalness arguments, R parity must be a symmetry in theeffective low-energy theory of any SUSY GUT. This does not mean to say that R parityis guaranteed to be satisfied in any GUT.Note also, R parity distinguishes Higgs multiplets from ordinary families. In SU(5),Higgs and quark/lepton multiplets have identical quantum numbers; while in E(6), Higgsand families are unified within the fundamental 27 representation. Only in SO(10) areHiggs and ordinary families distinguished by their gauge quantum numbers. Moreover,the Z(4) center of SO(10) distinguishes 10s from 16s, and can be associated withR parity [42].In SU(5), dimension-5 baryon-number-violating operators may be forbidden at treelevel by additional symmetries. These symmetries are typically broken, however, by theVEVs responsible for the color triplet Higgs masses. Consequently, these dimension-5operators are generically generated via color triplet Higgsino exchange. Hence, the colortriplet partners of Higgs doublets must necessarily obtain mass of order the GUT scale.The dominant decay modes from dimension-5 operators are p → K+ν (n → K0ν).This is due to a simple symmetry argument; the operators (QiQjQkLl), (uciucjdckecl)(where i, j, k, l = 1, 2, 3 are family indices, and color and weak indices are implicit)must be invariant under SU(3)Cand SU(2)L. As a result, their color and weak doubletindices must be anti-symmetrized. However, since these operators are given by bosonicsuperfields, they must be totally symmetric under interchange of all indices. Thus, thefirst operator vanishes for i = j = k, and the second vanishes for i = j. Hence, a secondor third generation member must exist in the final state [40].Recent Super-Kamiokande bounds on the proton lifetime severely constrain thesedimension-6 and dimension-5 operators with τ(p→e+π0)> 5.0 × 1033yrs (79.3 ktyrexposure), τ(n→e+π−)> 5 × 1033yrs (61 ktyr), and τ(p→K+ν)> 1.6 × 1033yrs (79.3ktyr), τ(n→K0ν)> 1.7×1032yrs (61 ktyr) at (90% CL) based on the listed exposures [43].These constraints are now sufficient to rule out minimal SUSY SU(5) [44].9Non-minimal Higgs sectors in SU(5) or SO(10) theories still survive [23,30]. The upperbound on the proton lifetime from these theories is approximately a factor of 5 above theexperimental bounds. They are, however, being pushed to their theoretical limits. Hence,if SUSY GUTs are correct, nucleon decay should be seen soon.Is there a way out of this conclusion? Orbifold GUTs and string theories, see Sect.15.1.3, contain grand unified symmetries realized in higher dimensions. In the processof compactification and GUT symmetry breaking, color triplet Higgs states are removed(projected out of the massless sector of the theory). In addition, the same projectionstypically rearrange the quark and lepton states so that the massless states which surviveemanate from different GUT multiplets. In these models, proton decay due to dimension9This conclusion relies on the mild assumption that the three-by-three matrices di-agonalizing squark and slepton mass matrices are not so different from their fermionicpartners. It has been shown that if this caveat is violated, then dimension five protondecay in minimal SUSY SU(5) may not be ruled out [45].July 14, 2006 10:37
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15. Grand Unified Theories115 operators can be severely suppressed or eliminated completely. However, proton decaydue to dimension 6 operators may be enhanced, since the gauge bosons mediating protondecay obtain mass at the compactification scale, Mc, which is less than the 4d GUTscale (see the discussion at the end of Section 15.1.4), or suppressed, if the states ofone family come from different irreducible representations. Which effect dominates is amodel dependent issue. In some complete 5d orbifold GUT models [47,24] the lifetimefor the decay τ(p → e+π0) can be near the excluded bound of 5 × 1033years with,however, large model dependent and/or theoretical uncertainties. In other cases, themodes p → K+¯ν and p → K0µ+may be dominant [24]. To summarize, in either 4d ororbifold string/field theories, nucleon decay remains a premier signature for SUSY GUTs.Moreover, the observation of nucleon decay may distinguish extra-dimensional orbifoldGUTs from four dimensional ones.Before concluding the topic of baryon-number violation, consider the status of ∆B = 2neutron- anti-neutron oscillations. Generically, the leading operator for this process isthe dimension-9 six-quark operator G(∆B=2)(ucdcdcucdcdc), with dimensionfulcoefficient G(∆B=2)∼ 1/M5. The present experimental bound τn–n≥ 0.86 × 108sec. at90% CL [48] probes only up to the scale M ≤ 106GeV. For M ∼ MG, n –n oscillationsappear to be unobservable for any GUT (for a recent discussion see Ref. 49).15.1.6. Yukawa coupling unification :15.1.6.1. 3rd generation, b–τ or t–b–τ unification:If quarks and leptons are two sides of the same coin, related by a new grand unifiedgauge symmetry, then that same symmetry relates the Yukawa couplings (and hencethe masses) of quarks and leptons. In SU(5), there are two independent renormalizableYukawa interactions given by λt(10 10 5H) + λ (10¯5¯5H). These contain the SMinteractions λt(Q ucHu) + λ (Q dcHd+ ecL Hd). Hence, at the GUT scale, wehave the tree-level relation, λb= λτ≡ λ [35]. In SO(10), there is only one independentrenormalizable Yukawa interaction given by λ (16 16 10H), which gives the tree-levelrelation, λt= λb= λτ≡ λ [50,51]. Note, in the discussion above, we assume theminimal Higgs content, with Higgs in 5,¯5 for SU(5) and 10 for SO(10). With Higgs inhigher-dimensional representations, there are more possible Yukawa couplings. [58–60]In order to make contact with the data, one now renormalizes the top, bottom, andτ Yukawa couplings, using two-loop-RG equations, from MGto MZ. One then obtainsthe running quark masses mt(MZ) = λt(MZ) vu,mb(MZ) = λb(MZ) vd, andmτ(MZ) = λτ(MZ) vd, where < H0u>≡ vu= sinβ v/√2, < H0d>≡ vd= cos β v/√2,vu/vd≡ tanβ, and v ∼ 246 GeV is fixed by the Fermi constant, Gµ.Including one-loop-threshold corrections at MZ, and additional RG running, one findsthe top, bottom, and τ-pole masses. In SUSY, b –τ unification has two possible solutions,with tanβ ∼ 1 or 40 –50. The small tanβ solution is now disfavored by the LEP limit,tanβ > 2.4 [52].10The large tanβ limit overlaps the SO(10) symmetry relation.10However, this bound disappears if one takes MSUSY= 2 TeV and mt= 180 GeV [53].July 14, 2006 10:37
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1215. Grand Unified TheoriesWhen tanβ is large, there are significant weak-scale threshold corrections to downquark and charged lepton masses, from either gluino and/or chargino loops [54]. Yukawaunification (consistent with low energy data) is only possible in a restricted region ofSUSY parameter space with important consequences for SUSY searches [55].15.1.6.2. Three families:Simple Yukawa unification is not possible for the first two generations, of quarksand leptons. Consider the SU(5) GUT scale relation λb= λτ. If extended to the firsttwo generations, one would have λs= λµ, λd= λe, which gives λs/λd= λµ/λe. Thelast relation is a renormalization group invariant, and is thus satisfied at any scale.In particular, at the weak scale, one obtains ms/md= mµ/me, which is in seriousdisagreement with the data, namely ms/md∼ 20 and mµ/me∼ 200. An elegant solutionto this problem was given by Georgi and Jarlskog [56]. Of course, a three-family modelmust also give the observed CKM mixing in the quark sector. Note, although there aretypically many more parameters in the GUT theory above MG, it is possible to obtaineffective low-energy theories with many fewer parameters making strong predictions forquark and lepton masses.It is important to note that grand unification alone is not sufficient to obtainpredictive theories of fermion masses and mixing angles. Other ingredients are needed.In one approach additional global family symmetries are introduced (non-abelian familysymmetries can significantly reduce the number of arbitrary parameters in the Yukawamatrices). These family symmetries constrain the set of effective higher dimensionalfermion mass operators. In addition, sequential breaking of the family symmetry iscorrelated with the hierarchy of fermion masses. Three-family models exist which fit allthe data, including neutrino masses and mixing [57]. In a completely separate approachfor SO(10) models, the Standard Model Higgs bosons are contained in the higherdimensional Higgs representations including the 10, 126 and/or 120. Such theories havebeen shown to make predictions for neutrino masses and mixing angles [58–60].15.1.7. Neutrino Masses :Atmospheric and solar neutrino oscillations require neutrino masses. Adding three“sterile” neutrinos νcwith the Yukawa coupling λν(νcL Hu), one easily obtains threemassive Dirac neutrinos with mass mν= λνvu.11However, in order to obtain a tauneutrino with mass of order 0.1 eV, one needs λντ/λτ≤ 10−10. The see-saw mechanism,on the other hand, can naturally explain such small neutrino masses [61,62]. Since νchas no SM quantum numbers, there is no symmetry (other than global lepton number)which prevents the mass term12νcM νc. Moreover, one might expect M ∼ MG. Heavy“sterile” neutrinos can be integrated out of the theory, defining an effective low-energytheory with only light active Majorana neutrinos, with the effective dimension-5 operator12(L Hu) λTνM−1λν(L Hu). This then leads to a 3×3 Majorana neutrino mass matrixm = mTνM−1mν.11Note, these “sterile” neutrinos are quite naturally identified with the right-handedneutrinos necessarily contained in complete families of SO(10) or Pati-Salam.July 14, 2006 10:37
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15. Grand Unified Theories13Atmospheric neutrino oscillations require neutrino masses with ∆m2ν∼ 3 × 10−3eV2with maximal mixing, in the simplest two-neutrino scenario. With hierarchicalneutrino masses, mντ=√∆m2ν∼ 0.055 eV. Moreover, via the “see-saw” mechanism,mντ= mt(mt)2/(3M). Hence, one finds M ∼ 2 × 1014GeV; remarkably close to theGUT scale. Note we have related the neutrino-Yukawa coupling to the top-quark-Yukawacoupling λντ= λtat MG, as given in SO(10) or SU(4) × SU(2)L× SU(2)R. However,at low energies they are no longer equal, and we have estimated this RG effect byλντ(MZ) ≈ λt(MZ)/√3.15.1.8. Selected Topics :15.1.8.1. Magnetic Monopoles:In the broken phase of a GUT, there are typically localized classical solutions carryingmagnetic charge under an unbroken U(1) symmetry [63]. These magnetic monopoleswith mass of order MG/αGare produced during the GUT phase transition in the earlyuniverse. The flux of magnetic monopoles is experimentally found to be less than ∼ 10−16cm−2s−1sr−1[64]. Many more are predicted however, hence the GUT monopoleproblem. In fact, one of the original motivations for an inflationary universe is to solvethe monopole problem by invoking an epoch of rapid inflation after the GUT phasetransition [65]. This would have the effect of diluting the monopole density as long as thereheat temperature is sufficiently below MG. Other possible solutions to the monopoleproblem include: sweeping them away by domain walls [66], U(1) electromagneticsymmetry breaking at high temperature [67] or GUT symmetry non-restoration [68].Parenthetically, it was also shown that GUT monopoles can catalyze nucleon decay [69].A significantly lower bound on the monopole flux can then be obtained by consideringX-ray emission from radio pulsars due to monopole capture and the subsequent nucleondecay catalysis [70].15.1.8.2. Baryogenesis via Leptogenesis:Baryon-number-violating operators in SU(5) or SO(10) preserve the global symmetryB – L. Hence, the value of the cosmological B – L density is an initial condition ofthe theory, and is typically assumed to be zero. On the other hand, anomalies ofthe electroweak symmetry violate B + L while also preserving B –L. Hence, thermalfluctuations in the early universe, via so-called sphaleron processes, can drive B + Lto zero, washing out any net baryon number generated in the early universe at GUTtemperatures [71].One way out of this dilemma is to generate a net B – L dynamically in the earlyuniverse. We have just seen that neutrino oscillations suggest a new scale of physicsof order 1014GeV. This scale is associated with heavy Majorana neutrinos with massM. If in the early universe, the decay of the heavy neutrinos is out of equilibrium andviolates both lepton number and CP, then a net lepton number may be generated.This lepton number will then be partially converted into baryon number via electroweakprocesses [72].July 14, 2006 10:37
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1415. Grand Unified Theories15.1.8.3. GUT symmetry breaking:The grand unification symmetry is necessarily broken spontaneously. Scalar potentials(or superpotentials) exist whose vacua spontaneously break SU(5) and SO(10). Thesepotentials are ad hoc (just like the Higgs potential in the SM), and, therefore it is hopedthat they may be replaced with better motivated sectors. Gauge coupling unificationnow tests GUT-breaking sectors, since it is one of the two dominant corrections tothe GUT threshold correction ϵ3. The other dominant correction comes from theHiggs sector and doublet-triplet splitting. This latter contribution is always positiveϵ3∝ ln(MT/MG) (where MTis an effective color triplet Higgs mass), while the low-energydata requires ϵ3< 0. Hence, the GUT-breaking sector must provide a significant (oforder −8%) contribution to ϵ3to be consistent with the Super-K bound on the protonlifetime [23,29,30,57].In string theory (and GUTs in extra-dimensions), GUT breaking may occur due toboundary conditions in the compactified dimensions [7,10]. This is still ad hoc. Themajor benefits are that it does not require complicated GUT-breaking sectors.15.1.8.4. Doublet-triplet splitting:The Minimal Supersymmetric Standard Model has a µ problem: why is the coefficientof the bilinear Higgs term in the superpotential µ (HuHd) of order the weak scale when,since it violates no low-energy symmetry, it could be as large as MG? In a SUSY GUT,the µ problem is replaced by the problem of doublet-triplet splitting—giving mass of orderMGto the color triplet Higgs, and mass µ to the Higgs doublets. Several mechanismsfor natural doublet-triplet splitting have been suggested, such as the sliding singlet,missing partner or missing VEV [73], and pseudo-Nambu-Goldstone boson mechanisms.Particular examples of the missing partner mechanism for SU(5) [30], the missing VEVmechanism for SO(10) [23,57], and the pseudo-Nambu-Goldstone boson mechanismfor SU(6) [74], have been shown to be consistent with gauge coupling unification andproton decay. There are also several mechanisms for explaining why µ is of order theSUSY-breaking scale [75]. Finally, for a recent review of the µ problem and somesuggested solutions in SUSY GUTs and string theory, see Refs. [76,9] and referencestherein.Once again, in string theory (and orbifold GUTs), the act of breaking the GUTsymmetry via orbifolding projects certain states out of the theory. It has been shownthat it is possible to remove the color triplet Higgs while retaining the Higgs doublets inthis process. Hence the doublet-triplet splitting problem is finessed. As discussed earlier(see Section 15.1.5), this has the effect of eliminating the contribution of dimension 5operators to nucleon decay.July 14, 2006 10:37
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15. Grand Unified Theories1515.2. ConclusionGrand unification of the strong and electroweak interactions requires that the threelow energy gauge couplings unify (up to small threshold corrections) at a unique scale,MG. Supersymmetric grand unified theories provide, by far, the most predictive andeconomical framework allowing for perturbative unification.The three pillars of SUSY GUTs are:• gauge coupling unification at MG∼ 3 × 1016GeV;• low-energy supersymmetry [with a large SUSY desert], and• nucleon decay.The first prediction has already been verified (see Fig. 15.1). Perhaps the next twowill soon be seen. Whether or not Yukawa couplings unify is more model dependent.Nevertheless, the “digital” 16-dimensional representation of quarks and leptons in SO(10)is very compelling, and may yet lead to an understanding of fermion masses and mixingangles.In any event, the experimental verification of the first three pillars of SUSY GUTswould forever change our view of Nature. Moreover, the concomitant evidence for a vastSUSY desert would expose a huge lever arm for discovery. For then it would becomeclear that experiments probing the TeV scale could reveal physics at the GUT scaleand perhaps beyond. Of course, some questions will still remain: Why do we have threefamilies of quarks and leptons? How is the grand unified symmetry and possible familysymmetries chosen by Nature? At what scale might stringy physics become relevant?Etc.References:1. J. Pati and A. Salam, Phys. Rev. D8, 1240 (1973);For more discussion on the standard charge assignments in this formalism, see A.Davidson, Phys. Rev. D20, 776 (1979); and R.N. Mohapatra and R.E. Marshak,Phys. Lett. B91, 222 (1980).2. H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32, 438 (1974).3. H. Georgi, Particles and Fields, Proceedings of the APS Div. of Particles and Fields,ed. C. Carlson, p. 575 (1975);H. Fritzsch and P. Minkowski, Ann. Phys. 93, 193 (1975).4. S.M. Barr, Phys. Lett. B112, 219 (1982).5. A. de Rujula et al., p. 88, 5th Workshop on Grand Unification, ed. K. Kang et al.,World Scientific, Singapore (1984);See also earlier paper by Y. Achiman and B. Stech, p. 303, “New Phenomena inLepton-Hadron Physics,” ed. D.E.C. Fries and J. Wess, Plenum, NY (1979).6. B.R. Greene et al., Nucl. Phys. B278, 667 (1986);ibid., Nucl. Phys. B292, 606 (1987);B.R. Greene et al., Nucl. Phys. B325, 101 (1989).7. P. Candelas et al., Nucl. Phys. B258, 46 (1985);L.J. Dixon et al., Nucl. Phys. B261, 678 (1985);ibid., Nucl. Phys. B274, 285 (1986);July 14, 2006 10:37
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