Proton Stability, Gauge Coupling Unification and a Light Z ′ in Heterotic-string Models
aa r X i v : . [ h e p - ph ] J un LTH–974
Proton Stability,Gauge Coupling Unificationand a Light Z ′ in Heterotic–string Models Alon E. Faraggi and Viraf M. Mehta Department of Mathematical SciencesUniversity of Liverpool, Liverpool, L69 7ZL, United Kingdom
Abstract
We explore the phenomenological viability of a light Z ′ in heterotic–stringmodels, whose existence has been motivated by proton stability arguments.A class of quasi–realistic string models that produce such a viable Z ′ are theLeft–Right Symmetric (LRS) heterotic–string models in the free fermionic for-mulation. A key feature of these models is that the matter charges under U (1) Z ′ do not admit an E embedding. The light Z ′ in the LRS heterotic–string models forbids baryon number violating operators, while allowing leptonnumber violating operators, hence suppressing proton decay yet allowing forsufficiently small neutrino masses via a seesaw mechanism. We show that theconstraints imposed by the gauge coupling data and heterotic–string couplingunification nullify the viability of a light Z ′ in these models. We further ar-gue that agreement with the gauge coupling data necessitates that the U (1) Z ′ charges admit an E embedding. We discuss how viable string models withthis property may be constructed. E-mail address: [email protected] E-mail address: [email protected]
Introduction
The discovery of the Higgs boson at the LHC lends further credence to the hypothesisthat the Standard Model (SM) provides a viable effective parametrisation of all sub-atomic interactions up to the GUT or heterotic–string unification scales. Support forthis possibility stems from: the matter gauge charges; proton longevity; suppressionof neutrino masses; and the logarithmic evolution of the SM parameters in its gaugeand matter sectors. Preservation of the logarithmic running in the SM scalar sectorentails that it must be augmented by a new symmetry. A concrete framework thatfulfils the task is given by supersymmetry.The supersymmetric extension of the SM introduces dimension four and fivebaryon and lepton number violating operators that mediate proton decay. This prob-lem is particularly acute in the context of heterotic–string derived constructions, inwhich one cannot assume the existence of global or local discrete symmetries thatsimply forbid the undesired operators. Indeed, the issue has been examined in thepast by a number of authors [1]. The avenues explored range from the existence ofmatter parity at special points in the moduli space of specific models, to the emer-gence of non–abelian custodial symmetries in some compactifications. However, acaveat to these arguments is that in addition to suppressing the proton decay medi-ating operators, one must also ensure that the mass terms of left–handed neutrinosare sufficiently suppressed. That is, while baryon number should be conserved toensure proton longevity, lepton number must be broken to allow for suppression ofleft–handed neutrino masses. In heterotic–string constructions, due to the absenceof higher–order representations of the Grand Unified Theory [2], one typically hasto break lepton number by one unit, which generically results in both lepton andbaryon number violation. An alternative solution to this conundrum is obtained ifan additional U (1) gauge symmetry, beyond the SM gauge group, remains unbrokendown to low scales. An additional abelian gauge symmetry, which is broken near theTeV scale, may also explain the suppression of the µ –term in the supersymmetricpotential [3].The possibility of a low scale Z ′ arising from heterotic–string inspired models has along history and continues to attract wide interest [4]. Surprisingly, however, keepinga Z ′ in explicit string derived constructions, unbroken down to the low scale, turnsout to be notoriously difficult, as such an extra symmetry must satisfy a varietyof phenomenological constraints. Obviously, to play a role in the suppression ofproton decay mediating operators (PDMOs) implies that the SM matter states arecharged under this symmetry. While forbidding baryon number violation, it shouldallow for lepton number violation, required for the suppression of neutrino masses.Furthermore, it should be family universal, otherwise there is a danger of generatingFlavour Changing Neutral Currents (FCNC), or of generating the PDMOs via mixing.The additional symmetry should also allow for the fermion Yukawa couplings toelectroweak Higgs doublets and must be anomaly free. Explicit string models that2o give rise to an extra U (1) symmetry with the required properties are the left–right symmetric models of [5, 6]. The existence of the required symmetry in explicitstring constructions ensures that, in these examples, the extra U (1) is free of anygauge and gravitational anomalies. In [7] we constructed toy string–inspired modelsthat are compatible with the charge assignments in the string derived models. Inthese models, the proton lifeguarding extra U (1) symmetry can, in principle, remainunbroken down to low scales.An additional constraint that must be imposed on the extra gauge and mat-ter states that arise in the Z ′ models, is compatibility with the gauge parameters,sin θ W ( M Z ) and α ( M Z ). The perturbative heterotic–string predicts that all thegauge couplings are unified at the string unification scale, M S , which is of the order5 · GeV. Nonperturbatively, the heterotic–string can be pushed to the GUT unifi-cation scale, M GUT , of the order 2 · GeV [8]. In this paper we study the constraintsthat are imposed on the string inspired Z ′ models by gauge coupling unification andshow that the gauge coupling data is not in agreement with the left–right symmetricheterotic–string models. The origin for the disagreement lies in the specific U (1) Z ′ charges, which do not admit an E embedding. For comparison we also performthe analysis for U (1) Z ′ charges that maintain the E embedding and show that, inthis case, agreement with the data is achieved. We discuss how viable string derivedmodels that preserve the E embedding may be constructed. U (1) s in free fermionic models In this section we review the structure of the free fermionic models. We focus on theextra U (1) symmetries that arise in the models and the charges of the matter states.We elaborate on the gauge symmetry breaking patterns induced by the GeneralisedGSO (GGSO) projections but concentrate here on the group theory structure and thematter charges. Further details of the free fermionic models and their construction arefound in earlier literature [5, 9–16]. The free fermionic models correspond to Z × Z orbifold compactifications at special points in the moduli space [17]. It should beemphasized that our results are applicable to the wider range of orbifold modelsbecause they merely depend on the symmetry breaking patterns of the observablegauge symmetry.Free fermionic heterotic–string models are constructed by specifying a consistentset of boundary condition basis vectors and the associated one–loop GGSO phases [9].These basis vectors span a finite additive group, Ξ, where the physical states of agiven sector, α ∈ Ξ, are obtained by acting on the vacuum with bosonic and fermionicoperators and by applying the GGSO projections. The U (1) charges, with respect tothe unbroken Cartan generators of the four dimensional gauge group, are given by: Q ( f ) = 12 α ( f ) + F ( f ) , (2.1)3here α ( f ) is the boundary condition of the complex world–sheet fermion f in thesector α , and F α ( f ) is a fermion number operator counting each mode of f once ( f ∗ minus once). For periodic fermions with α ( f ) = 1, the vacuum is a spinor representingthe Clifford algebra of the zero modes. For each periodic complex fermion, f , thereare two degenerate vacua, | + i and |−i , annihilated by the zero modes, f and f ∗ , withfermion numbers F ( f ) = 0 , − { , S, b , b , b } , are fixed; b , b and b correspond to the three twisted sectors of the Z × Z orbifold and S is the spacetime supersymmetry generator. The gauge symme-try at the level of the NAHE set is SO (10) × SO (6) × E with N = 1 spacetime super-symmetry. The second stage of the construction consists of adding three additionalbasis vectors to the NAHE set. The additional vectors reduce the number of gener-ations to three and simultaneously break the four dimensional group. The SO (10)symmetry is broken to one of its maximal subgroups: SU (5) × U (1) (FSU5) [10]; SU (3) × SU (2) × U (1) (SLM) [11]; SO (6) × SO (4) (PS) [12]; SU (3) × U (1) × SU (2) (LRS) [5]; and SU (4) × SU (2) × U (1) (SU421) [14].An important distinction between the last two cases and the first three is in regardto the anomalous U (1) A symmetry that arises in these models [5, 14, 18]. The Cartansubalgebra of the observable rank eight gauge group is generated by eight complexfermions, denoted by { ¯ ψ , ··· , , ¯ η , , } , where ¯ ψ , ··· , are the Cartan generators of the SO (10) group and ¯ η , , generate three U (1) symmetries, denoted by U (1) , , . In theFSU5, PS and SLM cases the U (1) , , , as well as their linear combination, U (1) ζ = U (1) + U (1) + U (1) , (2.2)are anomalous, whereas in the LRS and SU421 models they are anomaly free. Thedistinction can be seen to arise from the symmetry breaking patterns induced in thetwo cases from the underlying N = 4 toroidal model in four dimensions. Startingfrom the E × E , in the first case the symmetry is broken to SO (16) × SO (16)by the choice of GGSO projection phases in the fermionic models, or equivalentlyby a Wilson line in the corresponding orbifold models. The basis vectors b and b break the symmetry further to SO (10) × U (1) × SO (16). Alternatively, we canimplement the b and b twists in the E × E vacuum, which break the gaugesymmetry to E × U (1) × E . The Wilson line breaking then reduces the symmetryto SO (10) × U (1) ζ × U (1) × SO (16). It is then clear that the U (1) ζ becomes anomalousbecause of the E symmetry breaking to SO (10) × U (1) ζ and the projection of somestates from the spectrum by the GGSO projections [18]. On the other hand, the LRS4nd SU421 heterotic–vacua arise from an N = 4 vacuum with E × E × SO (16)gauge symmetry [5, 14]. In this case, one of the E factors produces the observablegauge symmetry and the second is hidden. The important point here is that thesemodels circumvent the E embedding. Hence, in these cases, the U (1) ζ does not havean E embedding and therefore remains anomaly free.The case of the symmetric orbifolds studied in [15] only allows for models withan E embedding of U (1) ζ . Thus, in these models U (1) ζ is, generically, anomalous.There is, however, a class of models in which it is anomaly free. This is the casein the self–dual models under the spinor–vector duality of [19]. In these models thenumber of SO (10) spinorial representations and the number of vectorial rep-resentations, arising from the twisted sectors is identical, although the E symmetryis broken. This situation occurs when the spinorial and vectorial representations areobtained from different fixed points of the Z × Z toroidal orbifold. A self–dual, threegeneration model with unbroken SO (10) symmetry is given in ref. [15], however, aviable model, of this type, with broken SO (10) symmetry has not been constructedto date.Alternatively, we may construct U (1) ζ ⊂ E as an anomaly free combination byfollowing a different symmetry breaking pattern to the E → SO (10) × U (1) discussedabove. Originally, the E → SO (10) × U (1) breaking is achieved by projecting thevector bosons that arise in the spinorial representation of SO (16) and enhancethe SO (16) symmetry to E . We may construct models in which these vector bosonsare not projected and, thus, the E symmetry is broken to a different subgroup.Examples of such models include the three generation SU (6) × SU (2) models of [20].In this case, the U (1) ζ is anomaly free by virtue of its embedding in the enhancedsymmetry. In this section we present a comparative analysis of the two classes mentioned above.It will be instructive to specify a model in each class: • Model I: This model was first presented in [7]. In this case the extra U (1) ζ doesnot admit an E embedding, i.e. SO (10) × U (1) ζ E . • Model II: This model preserves the E embedding of the U (1) ζ and is akin to Z ′ models arising in string inspired E models [4].Before proceeding with the gauge coupling analysis, it is instructive to detail thesymmetry breaking patterns applicable to both models. The SM gauge group willbe embedded, for our analysis, in SO (10). As previously mentioned, this is brokento the LRS gauge group via the addition of basis vectors, α, β, and γ at the stringscale, M S . The SU (2) R is then broken at some intermediate scale, M R . An anomaly5ree U (1) combination that remains is the U (1) Z ′ which is required to survive to lowenergies to preserve proton longevity [6, 7].In our analysis we vary the unification scale in the range 2 · − · GeV.The lower scale is the natural MSSM unification scale [21], M X , whereas the higherscale corresponds to the heterotic–string unification scale [22], M S . This factor of20 discrepancy was discussed in [23] and it was concluded that intermediate matterthresholds contributed enough to overcome the difference, allowing coupling unifica-tion in a wide class of realistic free–fermionic string models [24]. From the spectra ofour models, we will see that it is natural to include intermediate matter thresholds toachieve string unification. It has also been demonstrated that nonperturbative effectsarising in heterotic M–theory [25] can push the unification scale down to the MSSMunification scale [8]. Our aim here is to study, qualitatively, the question of gaugecoupling unification in the LRS heterotic–string models. In particular, to demon-strate that a low scale Z ′ in these models is incompatible with the gauge couplingdata at the electroweak scale. The novel feature of the LRS models is the U (1) Z ′ charge assignments. These admit an E embedding and therefore similar charge as-signments also arise in heterotic M–theory and so we take the unification scale to varybetween M X and M S to allow for the possible nonperturbative effects. We contrastthe analysis in the LRS heterotic–string models with the models that admit the E embedding of the U (1) Z ′ charges. In both models there are four intermediate scalesbetween M S and M Z , corresponding to: M R : SU (2) R breaking scale. The neutral components of H R + ¯ H R acquire a VEV tobreak the SU (2) R symmetry and leave the U (1) Z ′ unbroken. M D : Colour triplet scale.
The additional colour triplets in our model acquire amass at this scale. This will also resolve the discrepancy between the MSSMunification scale and string scale unification. M Z ′ : U (1) Z ′ breaking scale. The U (1) Z ′ is broken at this scale by singlets acquiringVEVs. The anomaly cancelling doublets also acquire mass at this scale andonly the MSSM spectrum survives to lower scales. M SUSY : Supersymmetry breaking scale.
The current bounds from the LHC will be in-cluded here to get a phenomenologically viable supersymmetry scale. Only theSM states remain down to the M Z –scale, at which the gauge data is extracted.Threshold corrections for the top quark and Higgs boson are included in theanalysis.In addition, due to the extra abelian gauge symmetry acting as our proton protector, M Z ′ should be sufficiently low in order for adequate suppression of induced PDMOs[6, 7] . By starting from the string scale and evolving the couplings down to M Z ,our analysis may test whether the predictions of these models are in accordance withlow–energy experimental data. 6 ow–energy inputs For our analysis, we take the following values for the masses and couplings [26]: M Z = 91 . ± . α − ≡ α − e.m. ( M Z ) = 127 . ± .
014 sin θ W ( M Z ) (cid:12)(cid:12) MS = 0 . ± . α ( M Z ) = 0 . ± . . (3.1)We also include the top quark mass of M t ∼ . M H ∼
125 GeV [27] in our analysis.
Renormalization Group Equations
For the analyses of both models, we follow [23]. String unification implies that theSM gauge couplings are unified at the heterotic–string scale. The one–loop renor-malization group equations (RGEs) for the couplings are given by4 πα i ( µ ) = k i πα string + β i log M string µ + ∆ ( total ) i , (3.2)where β i are the one–loop beta–function coefficients, and ∆ ( total ) i represents possiblecorrections from the additional gauge or matter states. By solving the one–loop RGEswe obtain expressions for sin θ W ( M Z ) and α ( M Z ). In each model, we initiallyassume the MSSM spectrum between the string scale, M S , and the Z scale, M Z , andtreat all perturbations as effective correction terms. At the string unification scalewe have α S ≡ α ( M S ) = α ( M S ) = k α Y ( M S ) , (3.3)where k = 5 / SO (10) normalisation. Thus, the expression forsin θ W ( M Z ) (cid:12)(cid:12) MS takes the general form [23]sin θ W ( M Z ) (cid:12)(cid:12) MS = ∆ sin θ W MSSM + ∆ sin θ W I.M. + ∆ sin θ W L.S. + ∆ sin θ W I.G. + ∆ sin θ W T.C. (3.4)with α ( M Z ) | MS taking similar form with corresponding ∆ α corrections. Here∆ MSSM represents the one–loop contributions from the spectrum of the MSSM be-tween the unification scale and the Z scale. The following three ∆ terms correspondto corrections from the intermediate matter thresholds, the light SUSY thresholds,and the intermediate vector bosons corresponding to the SU (2) R symmetry breaking.The last term, ∆ sin θ W T.C. = ∆ sin θ W H.S. + ∆ sin θ W Yuk. + ∆ sin θ W + ∆ sin θ W Conv. , (3.5)includes the corrections due to heavy string thresholds, and those arising from Yukawacouplings, two–loops and scheme conversion. These corrections are small and areneglected for this demonstrative analysis.7or sin θ W ( M Z ) we obtain∆ sin θ W MSSM = 11 + k (cid:20) − α π (11 − k ) log M S M Z (cid:21) ;∆ sin θ W I.M. = 12 π X i k α (1 + k ) (cid:18) β i − β i (cid:19) log M S M i ;∆ sin θ W L.S. = 12 π k α (1 + k ) (cid:18) β L.S. − β L.S. (cid:19) log M SUSY M Z , (3.6)where α = α e.m. ( M Z ) and M i are the intermediate gauge and matter scales discussedearlier. Similarly for α ( M Z ), we have:∆ α = 11 + k (cid:20) α − π (cid:18)
15 + 3 k (cid:19) log M S M Z (cid:21) ;∆ α = 12 π k ) X i (cid:20) (1 + k ) β i − ( β i + k β i ) (cid:21) log M S M i ;∆ α = − π k ) (cid:20) (1 + k ) β L.S. − ( β L.S. + k β L.S. ) (cid:21) log M SUSY M Z . (3.7)A subtle issue in the analysis of gauge coupling unification in string models is thenormalisation of the U (1) generators. In GUTs the normalisation of abelian genera-tors is fixed by their embedding in non–abelian groups. However, in string theory thenon–abelian symmetry is not manifest, and the proper normalisation of the U (1) cur-rents is obscured. The U (1) normalisation in string models that utilise a world–sheetconformal field theory construction is fixed by their contribution to the conformal di-mensions of physical states. The procedure for fixing the normalisation was outlinedin [23, 28] and we repeat it here for completeness.In the free fermionic heterotic–string models, the Kaˇc–Moody level of non–abeliangroup factors is always one. In general, a given U (1) current, U , in the Cartansubalgebra of the four dimensional gauge group, is a combination of the simple world–sheet currents U (1) f ≡ f ∗ f , corresponding to individual world–sheet fermions, f . U then takes the form U = P f a f U (1) f , where the a f are model dependent coefficients.Each U (1) f is normalised to one, so that h U (1) f , U (1) f i = 1, and each of the linearcombinations must also be normalised to one. The proper normalisation coefficientfor the linear combination U is given by N = ( P f a f ) − , and the properly normalised U (1) current is, thus, given by ˆ U (1) = N · U .In general, the Kaˇc–Moody level, k , of a U (1) generator can be deduced from theoperator product expansion between two of the U (1) currents, and is given by k = 2 N − = 2 X f a f . (3.8)8he result is generalised to k = P i a i k i when the U (1) is a combination of several U (1)s with different normalisations. This procedure is used to determine the Kaˇc–Moody level, k , of the weak–hypercharge generator, as well as that of any other U (1)combination in the effective low–energy field theory.In the LRS heterotic–string models, the SO (10) symmetry is broken to SU (3) C × U (1) C × SU (2) L × SU (2) R , where the combinations of world–sheet currents13 (cid:0) ¯ ψ ∗ ¯ ψ + ¯ ψ ∗ ¯ ψ + ¯ ψ ∗ ¯ ψ (cid:1) (3.9)and 12 (cid:0) ¯ ψ ∗ ¯ ψ + ¯ ψ ∗ ¯ ψ (cid:1) (3.10)generate U (1) C and T R , respectively, where the latter is the diagonal generator of SU (2) R . The weak–hypercharge is then given by U (1) Y = T R + 13 U (1) C . (3.11)The symmetry of SU (2) R is incorporated in the analysis at the M R scale, whereabove this scale the multiplets are in representations of the LRS gauge group andbelow the M R scale they are in SM representations. The weak-hypercharge couplingrelation is given by1 α ( M R ) = 1 α R ( M R ) + k C α ˆ C ( M R ) = 1 α R ( M R ) + 23 1 α ˆ C ( M R ) . (3.12)Here we have used (3.8) to find that the Kaˇc–Moody level of U (1) C is k C = 6. Againusing (3.8) we find that k = as expected. This reproduces the expected result atthe unification scale sin θ W ( M S ) = 11 + k ≡ . (3.13) This model is an example of a three generation, free fermionic model that yields anunbroken, anomaly free U (1) symmetry. Heterotic–string models with this propertybreak the SO (10) symmetry to the left–right symmetric subgroup [5] and are thereforesupersymmetric and completely free of gauge and gravitational anomalies. The U (1) ζ symmetry in the string models is an anomaly free, family universal symmetry thatforbids the dimension four, five and six PDMOs, while allowing for the SM fermionmass terms. A combination of U (1) ζ , U (1) B − L and U (1) T R remains unbroken downto low energies and forbids baryon number violation while allowing for lepton numberviolation. Hence, it allows for the generation of small left–handed neutrino masses via9 seesaw mechanism, specifically an extended seesaw with the singlets, φ [5,7]. Protondecay mediating operators are only generated when the U (1) Z ′ is broken. Thus, thescale of the U (1) Z ′ breaking is constrained by proton lifetime limits and can be withinreach of the contemporary experiments. A field theory model demonstrating theseproperties was presented in [7]. Spectrum
Field SU (3) C × SU (2) L × SU (2) R U (1) C U (1) ζ β β L β Y Q iL −
32 16 Q iR ¯3 1 2 − + L iL − −
12 12 L iR + H H ijL +
12 12 H ′ ijL − +
12 12 H ijR − − H ′ ijR − D n ¯ D n ¯3 1 1 − H R − H R − + S i − S i φ a Table 1:
High scale spectrum and SU (3) C × SU (2) L × SU (2) R × U (1) C × U (1) E quantum numbers, with i = 1 , , for the three light generations, j = 1 , for thenumber of doublets required by anomaly cancellation, n = 1 , ..., k , and a = 1 , ..., p .The β i show the contributions for each state, relevant for the RGE analysis later. The spectrum of our model above the left–right symmetry breaking scale is sum-marised in Table 1. The spectrum below the intermediate symmetry breaking scaleis shown in Table 2. The spectra above and below the SU (2) R breaking scale are10oth free of all gauge and gravitational anomalies. Hence, the U (1) Z ′ combinationgiven in equation (3.14) is viable to low energies. Field SU (3) C × SU (2) L T R U (1) Y U (1) Z ′ β β L β Y Q iL −
32 16 u c iL ¯3 1 − − +
35 12 d c iL ¯3 1 + + +
15 12 L iL − −
12 12 e c iL − +1 + ν c iL H u + −
12 12 H d − − +
12 12 H iL +
32 32 H ′ iL − +
32 32 E iR − − − N iR − E ′ iR +1 − N ′ iR − D n +
15 12 ¯ D n ¯3 1 0 − −
15 12 S i − S i φ a Table 2:
Low scale matter spectrum and SU (3) C × SU (2) L × U (1) Y × U (1) Z ′ quantumnumbers with β i contributions. The heavy Higgs’, H kR + ¯ H kR that break the SU (2) R × U (1) C → U (1) Y , along aflat direction, leave the orthogonal combination U (1) Z ′ = 15 U C − T R + U ζ (3.14)unbroken. Here, the index k allows for the possibility that the heavy Higgs sectorcontains more than two fields, as is typically the case in the string constructions.11urther discussion of this model, including a trilinear level superpotential, can befound in [7]. Here we notice that the incomplete representations added to the MSSMmay cause problems with gauge coupling unification. The induced gauge anomaliesin the SU (2) L/R × U (1) ζ diagrams require the addition of H ijL , H ′ ijL H ijR , H ′ ijR , whichdiffer from the E case. The addition of triplets may help subdue any adverse effectsand will also give scope for the inclusion of intermediate matter scales. Renormalization group analysis
The properly normalised β –function coefficients are shown in Tables 1 and 2. Thenumerical output of equation (3.6) and (3.7) is generated subject to the variation ofthe scales and is displayed in Figure 1. The intermediate scales are varied to findphenomenologically viable areas of the parameter space. The scales and ranges ofsin θ W ( M Z ) and α ( M Z ) were first restricted to the experimentally allowed regionsand then also allowed to take values outside this range. The hierarchy of scales wasconstrained to be M S & M R > M D & M Z ′ & M SUSY > M Z . (3.15)To this end, we restricted the allowed range of sin θ W ( M Z ) and α ( M Z ) to fivesigma deviations from the central values shown in eq (3.1). The RGEs were runin Mathematica. Restricting the output to the experimentally constrained intervalproduced no phenomenologically viable results. Allowing the values of sin θ W ( M Z )and α ( M Z ) to run freely and restricting the relevant mass scales to (in GeV)2 · ≤ M S ≤ · ;10 ≤ M R ≤ · ; 10 ≤ M D ≤ ;10 ≤ M Z ′ , M SUSY ≤ , (3.16)also produced no phenomenologically viable results, as shown in Figure 1. Contrasting analysis with E embedding of U (1) ζ To further elucidate the constraints on the LRS heterotic–string models arising fromcoupling unification, we contrast the outcome with the corresponding results whenthe U (1) ζ charges are embedded in E representations. For models that allow the E embedding of the U (1) Z ′ charges, the spectrum consists of three generations of sthat decompose under SO (10) as: i → i + i − + i . (3.17)12 sin θ W ( M Z ) α ( M Z ) Figure 1: Freely running sin θ W ( M Z ) and α ( M Z ): sin θ W ( M Z ) vs. α ( M Z ) with0 . . α string . . SU (3) C × SU (2) L × SU (2) R × U (1) C × U (1) ζ , this results in a similar spectrumto the LRS model. The decomposes exactly as for the LRS model, Q iL ∼ (cid:18) , , , + 12 , + 12 (cid:19) ; Q iR ∼ (cid:18) ¯3 , , , − , + 12 (cid:19) ; L iL ∼ (cid:18) , , , − , + 12 (cid:19) ; L iR ∼ (cid:18) , , , + 32 , + 12 (cid:19) , (3.18)with the proviso that the charges under U (1) ζ take the same sign. The decomposesas H i ∼ (1 , , , , −
1) ; D i ∼ (3 , , , +1 , −
1) ; ¯ D i ∼ (¯3 , , , − , − . (3.19)The remaining singlets are neutral under the SM gauge group and are used to breakthe U (1) Z ′ . In addition to the complete SO (10) representations above, the E spec-trum includes a bidoublet, H ∼ (1 , , , , − , (3.20)that facilitates gauge coupling unification. The model also contains the pair of heavyHiggs right–handed doublets, H R + ¯ H R = (cid:18) , , , , (cid:19) + (cid:18) , , , − , − (cid:19) , (3.21)13 .1140.1160.1180.120.1220.1240.1260.1280.13 0.231 0.232 0.233 sin θ W ( M Z ) α ( M Z ) Figure 2: Freely running sin θ W ( M Z ) and α ( M Z ): sin θ W ( M Z ) vs. α ( M Z ) with0 . . α string . . SU (2) R symmetry. We run the RGEs in exactly thesame way as shown for the LRS model, constraining the mass scales to the hierarchy M S & M R & M D = M Z ′ & M SUSY ≫ M Z . (3.22)In this model we find that unification does occur, as found in previous literature. Wenote that the phenomenologically viable results (see Figure 2) required M S ∼ M X ∼ · GeV as expected. The intermediate scales were found to be (in GeV)1 · ≤ M R ≤ · ; 1 · ≤ M D ≤ · ; 1 · ≤ M SUSY ≤ · , (3.23)with M Z ′ between 1 − TeV. In this case we have taken the mass of the vector–likedoublets, M Z ′ , and triplets, M D to be degenerate, which is the case in E inspiredmodels, as they are generated by the same singlet VEV. String models afford moreflexibility that we do not make use of in our analysis here. Fine–tuning the M SUSY allows for M Z ′ to be in agreement with current experimental bounds.The contrast between the two cases can be elucidated further by examiningmore closely the contributions of the intermediate gauge and matter thresholds tosin θ W ( M Z ) and α ( M Z ). Using the general expressions in equations (3.6) and (3.7)we find that, in the case of the spectrum and charge assignments in the LRS heterotic–string model, shown in Tables 1 and 2, the threshold corrections from intermediate14auge and matter scales are given by δ (cid:0) sin θ W ( M Z ) (cid:1) I.T. = 12 π k α k (cid:18)
125 log M S M R −
245 log M S M Z ′ − n D M S M D (cid:19) ,δ ( α ( M Z )) I.T. = 12 π (cid:18)
32 log M S M R − M S M Z ′ + 3 n D M S M D (cid:19) . (3.24)In the case of models that admit an E embedding of the charges, the same thresholdcorrections are given by δ (cid:0) sin θ W ( M Z ) (cid:1) I.T. = 12 π k α k (cid:18)
125 log M S M R + 65 log M S M H −
65 log M S M D (cid:19) ,δ ( α ( M Z )) I.T. = 12 π (cid:18)
32 log M S M R −
94 log M S M H + 94 log M S M D (cid:19) . (3.25)If we take M S to coincide with the MSSM unification scale and with M R as well,then the first lines in equations (3.6) and (3.7), which only contain the MSSM con-tributions, are in good agreement with the observable data. The corrections arisingfrom the intermediate gauge and matter thresholds in equations (3.24) and (3.25)then have to cancel. We see from equation (3.24) that the corrections from the in-termediate doublet and triplet thresholds contribute with equal sign in sin θ W ( M Z ).For α ( M Z ), the corrections from these thresholds contribute with opposite sign, butthe contribution of the doublets outweigh the contribution of the triplets. We maycompensate for the negative contribution from the extra doublets by lowering the SU (2) R breaking scale. Requiring that m ν τ . M R ≥ GeV.Keeping the extra triplets at the GUT scale, and the Z ′ scale at 10 GeV then yieldsrough agreement with sin θ W ( M Z ) but gross disagreement with α ( M Z ). Loweringthe triplet scale improves the agreement with α ( M Z ) but conflicts with the data forsin θ W ( M Z ). We therefore conclude that a low scale Z ′ in the LRS heterotic–stringmodels is incompatible with the gauge data at the Z –boson scale. In contrast, fromequation (3.25) we see that the corresponding corrections cancel each other, providedthat M H = M Z ′ = M D . This is the case as both are generated by the Z ′ break-ing VEV. This cancellation is, of course, the well known cancellation that occurswhen the representations fall into SU (5) multiplets. Allowing M R to be at 10 GeVthen compensates for the SUSY threshold at 1TeV, enabling accommodations of thelow–energy data, as illustrated in Figure 2. E embedding The low scale Z ′ in the string models is, in essence, a combination of the Cartangenerators, U (1) , , , that are generated by the right–moving complex world–sheetfermions ¯ η , , , together with a U (1) symmetry, embedded in the SO (10) GUT, and is15rthogonal to the weak hypercharge. Whether, or not, the symmetry is anomaly freedepends on the specific symmetry breaking pattern induced by the GGSO projections.As we discussed above, in the FSU5, PS and SLM the symmetry is anomalous,whereas in the LRS models it is anomaly free. The difference stems from the factthat in the former cases the combination for U (1) ζ admits the E embedding butin the latter it does not. On the other hand, as we have seen in Section 3, the E embedding allows for compatibility with the low scale gauge coupling data. The Z ′ in the LRS models, which do not admit the E embedding, is constrained to beheavier than at least 10 GeV. Gauge coupling data, therefore, seems to indicate thatthe E embedding of the charges is necessary. We emphasize that the indication isthat the charges must admit an E embedding and not that the E symmetry isactually realised. An illustration of this phenomenon is the existence of self–dualmodels under the spinor–vector duality without E enhancement [19]. The questionthen arises as to how one constructs heterotic–string models with anomaly free U (1) ζ ,which admit an E embedding. Here we discuss how viable heterotic–string modelswith E embedding of the U (1) Z ′ charges may be obtained. The main constraintbeing that the extra U (1) symmetry has to be anomaly free. For this purpose, wefirst give a general overview as to how the gauge symmetry is generated in the stringmodels.The vector bosons that generate the four dimensional gauge group in the stringmodels arise from two principal sectors: the untwisted sector and the sector x = { ¯ ψ , ··· , , ¯ η , , } . In the x –sector the complex right–moving world–sheet fermions, thatgenerate the Cartan subalgebra of the observable gauge group, are all periodic. Atthe level of the E × E heterotic–string in ten dimensions, the vector bosons ofthe observable E are obtained from the untwisted sector and from the x –sector.Under the decomposition E → SO (16), the adjoint representation decomposes as → + , where the adjoint representation is obtained from the un-twisted sector and the spinorial representation is obtained from the x –sector.The set { , S, x, ζ } produces a model with N = 4 spacetime supersymmetry in fourdimensions. The gauge symmetry arising in this model, at a generic point in thecompactified space, is either E × E or SO (16) × SO (16) depending on the GGSOphase c ( xζ ) = ± b and b reduces the spacetime supersymmetry to N =1. The observable gauge symmetry reduces from E to E × U (1) or SO (16) → SO (10) × U (1) . Additional vectors reduce the gauge symmetry further. Asidefrom the model of [20], all the quasi–realistic free fermionic models follow the secondsymmetry breaking pattern. That is, in all these models, the vector bosons arisingfrom the x –sector are projected out.We consider, then, the symmetry breaking pattern induced by the following16oundary condition assignments in two separate basis vectors1 . b { ¯ ψ ··· } = {
12 12 12 12 12 } ⇒ SU (5) × U (1) , (4.1)2 . b { ¯ ψ ··· } = { } ⇒ SO (6) × SO (4) . (4.2)The assignment in equation (4.1) reduces the untwisted SO (10) gauge symmetry to SU (5) × U (1), however the assignment in eq. (4.2) reduces it to SO (6) × SO (4). Thus,the inclusion of equations (4.1) and (4.2) in two separate boundary condition basisvectors reduces the SO (10) gauge symmetry to SU (3) C × SU (2) L × U (1) C × U (1) L ,where 2 U (1) C = 3 U (1) B − L and U (1) L = 2 U (1) T R . For appropriate choices of theGGSO projection coefficients, the vector bosons arising from the x –sector enhancethe SU (3) × SU (2) × U (1) × U (1) ζ arising from the untwisted sector to SU (4) C × SU (2) L × SU (2) R × U (1) ζ ′ , where U (1) = U (1) C + 3 U (1) L − U (1) ζ ; (4.3) U (1) = U (1) C + U (1) L + U (1) ζ ; (4.4) U (1) ζ ′ = − U (1) C + 3 U (1) L + U (1) ζ . (4.5) U (1) and U (1) are embedded in SU (4) C and SU (2) R , respectively, and U (1) ζ isgiven by equation (2.2). The matter representations charged under this group arisefrom the sectors b j and are complemented by states from b j + x to form the ordinaryrepresentations of the Pati–Salam model. The difference, as compared to the Pati–Salam string models of [12], is that U (1) ζ ′ is anomaly free. The reason is that allthe states of the representation of E are retained in the spectrum, whereasin the Pati–Salam models of [12] the corresponding states are projected out. Thesymmetry breaking of the Pati–Salam SU (4) C × SU (2) R group is induced by theVEV of the heavy Higgs in the (¯4 , , − ⊕ (4 , , + representation of SU (4) C × SU (2) L × SU (2) R × U (1) ζ ′ . In addition to the weak–hypercharge, this VEV leavesthe unbroken combination U (1) Z ′ = 12 U (1) B − L − U (1) T R + 53 U (1) ζ ′ , (4.6)which is anomaly free and admits the E embedding of the charges. In this paper we examined the gauge coupling unification constraints imposed on a lowscale Z ′ arising in LRS heterotic–string derived models. The existence of a low–scale Z ′ in these models guarantees that PDMOs are sufficiently suppressed. However, wehave shown that the hypothesis of a low scale Z ′ in these models is incompatiblewith the gauge coupling data at the electroweak scale. We contrasted this result17ith the corresponding result in string models that admit an E embedding of the U (1) charges. In the latter case the possibility of a low scale Z ′ is viable. We furtherdiscussed how heterotic–string models that admit the E embedding may be obtainedin the free fermionic formulation, though an explicit three generation viable model isyet to be constructed. Similarly, a more complete analysis of the phenomenologicalrealisation of this U (1) symmetry in heterotic–string models is warranted and will bereported in future publications. We also remark that other U (1) symmetries that havebeen proposed in the literature to suppress proton decay mediating operators [4, 29]have also been invalidated due to neutrino masses and other constraints [6]. Theenigma of the proton lifetime in heterotic–string unification continues to serve as animportant guide in the search for viable string vacua. Acknowledgements
AEF thanks Subir Sarkar and Theoretical Physics Department at the Univer-sity of Oxford for hospitality. This work was supported in part by the STFC(PP/D000416/1).
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