Quark-Lepton Symmetry and Quartification in Five Dimensions
aa r X i v : . [ h e p - ph ] J un October 29, 2018 6:18 WSPC - Proceedings Trim Size: 9.75in x 6.5in arx˙fest˙talk Quark-Lepton Symmetry and Quartification in Five Dimensions ∗ Kristian L. McDonald † Research Center for High Energy Physics, University of Melbourne,Victoria, 3010, Australia † E-mail: [email protected]
We outline some features of higher dimensional models possessing a Quark-Lepton (QL)symmetry. The QL symmetric model in five dimensions is discussed, with particularemphasis on the use of split fermions. An interesting fermionic geography which utilisesthe QL symmetry to suppress the proton decay rate and to motivate the flavor differencesin the quark and leptonic sectors is considered. We discuss the quartification model infive dimensions and contrast the features of this model with traditional four dimensionalconstructs.
Keywords : Extra dimensions, split fermions, quark-lepton symmetry, quartification.
1. Introduction
The Standard Model (SM) of particle physics displays a clear asymmetry betweenquarks and leptons. Quarks and leptons have different masses and charges and im-portantly quarks experience the strong force. Despite these differences a suggestivesimilarity exists between the family structure of quarks and leptons, leading oneto wonder if the SM may be an approximation to a more symmetric fundamentaltheory.The similar family structure of quarks and leptons is an automatic consequenceof the defining symmetry of the Quark-Lepton (QL) symmetric model. In thisframework the similarity between quarks and leptons is elevated to an exact sym-metry of nature. The model permits a complete interchange symmetry betweenquarks and a generalized set of leptons, with the SM resulting from the breaking ofan enlarged symmetry group.On an independent front, a number of model building tools which employ extraspatial dimensions have been developed in recent years. In particular new methodsof symmetry breaking have been uncovered. These methods allow one to reduce thescalar content required to achieve symmetry breaking in four dimensionsal models. Itis natural to investigate these methods within pre-existing frameworks to determine ∗ Talk given at the Festschrift in honour of Bruce McKellar and Girish Joshi at the University ofMelbourne, 14/12/2006. To appear in a special edition of IJMPA. The talk is based on the resultsof references . ctober 29, 2018 6:18 WSPC - Proceedings Trim Size: 9.75in x 6.5in arx˙fest˙talk the phenomenological distinctions between the new and traditional approaches.In this work we outline some recent investigations in five dimensional modelspossessing a QL symmetry. We focus our attention on two aspects of these investi-gations. First we consider the use of split fermions in models with a QL symmetry.We show that the symmetry constrains the implementation of split fermions andallows one to solve some outstanding problems in four dimensional with QL mod-els; namely how can one resolve the notion of a QL symmetry with the disimilaritybetween the observed masses of quark and lepton.We also consider the quartification model, a framework which extends the notionof a QL symmetry to permit gauge unification, in five dimensions. We show that thehigher dimensional quartification model allows one to remove many of the compli-cations which arise in four dimensional models. These complications result mainlyfrom the relatively large Higgs sector required to achieve the necessary symmetrybreaking. We show that an effective Higgsless limit may be obtained in the fivedimensional quartification model, thereby permitting considerable simplification.
2. Quark-Lepton Symmetry
How does one construct a quark-lepton symmetric model? Let us recall that theSM also displays a clear asymmetry between left and right handed fields; the lefthanded fields experience SU (2) L interactions whilst the right handed fields do not.One generation of SM fermions may be denoted as Q, L, u R , d R , e R , (1)revealing a further left-right asymmetry; namely the absence of ν R . However theleft-right asymmetry may be a purely low energy phenomenon and the constructionof a high energy left-right symmetric theory proceeds as follows. One must firstextend the SM particle content to include ν R and thereby equate the number of leftand right degrees of freedom. The SM gauge group must also be enlarged: SU (3) ⊗ SU (2) ⊗ U (1) → SU (3) ⊗ [ SU (2)] ⊗ U (1) , where the second SU (2) group acts on right-chiral fermion doublets. This enablesone to define a discrete symmetry interchanging all left and right handed fermionsin the Lagrangian, f L ↔ f R . Furthermore the model must be constructed suchthat the additional symmetries introduced to enable the left-right interchange aresuitably broken to reproduce the SM at low energies. In practise this means that asuitable extension of the Higgs sector is also required.One may construct a quark-lepton symmetric model by employing the samerecipe. First the fermion content of the SM must be extended. As quarks comein three colors one is required to introduce more leptons to equate the number ofquark and lepton degrees of freedom. For each SM lepton one includes two exoticleptons, e → E = ( e, e ′ , e ′′ ) ,ν → N = ( ν, ν ′ , ν ′′ ) , (2) ctober 29, 2018 6:18 WSPC - Proceedings Trim Size: 9.75in x 6.5in arx˙fest˙talk where the primed states are the exotics (known as liptons in the literature). Thegauge group must also be extended: SU (3) ⊗ SU (2) ⊗ U (1) → [ SU (3)] ⊗ SU (2) ⊗ U (1) , where the additional SU (3) gauge bosons induce transitions amongst the generalizedleptons in the same way that gluon exchange enables quarks to change color. Nowone may define a discrete symmetry interchanging all quarks and (generalized)leptons in the Lagrangian, Q ↔ L, u R ↔ E R , d R ↔ N R . (3)The gauge symmetry may be broken via the Higgs mechanism to reproduce thecharge differences between quarks and leptons and give heavy masses to the liptons.Phenomenologically consistent models can be obtained by breaking the lepton colorgroup SU (3) ℓ completely or by leaving an unbroken subgroup SU (2) ℓ ⊂ SU (3) ℓ (which serves to confine the liptons into exotic bound states). The QL symmetryimplies mass relations of the type m u = m e which are more difficult to rectify. In4D one may remove these by extending the scalar sector and thereby increasing thenumber of independent Yukawa couplings.
3. QL Symmetry in Five Dimensions
In recent years the study of models with extra dimensions has revealed a numberof new model building tools. The use of orbifolds provides a new means of reducingthe gauge symmetry operative at low energies by introducing a new mass scale,namely the compactification scale M c = 1 /R . In generic 5D models the mass ofexotic gauge bosons can be set by M c and the reduction of symmetries by orbifoldconstruction results in collider phenomenology which differs from that obtained inmodels employing the usual 4D Higgs symmetry breaking.The reduction of the QL symmetric gauge group via orbifold construction hasrecently been studied in 5D. One question that faces the model builder in extradimensional models is whether to place fermions on a brane or in the bulk. Weshall focus on the latter in what follows and demonstrate that the troublesomemass relations which arise in 4D QL models actually provide useful and interestingmodel building constraints in 5D models. We note that placing fermions in thebulk permits the unification of quark and lepton masses at TeV scales in more ageneral class of models.
In five dimensions bulk fermions lack chirality. However, chiral zero modefermions, which one may identify with SM fermions, may be obtained by employ-ing orbifold boundary conditions on a fifth dimension forming an S /Z orbifold.By coupling a bulk fermion to a bulk scalar field one may readily localise a chiralzero mode fermion at one of the orbifold fixed points. Denoting the bulk fermion(scalar) as ψ ( φ ) one has the following Lagrangian: L = ¯ ψ (Γ M ∂ M − f φ ) ψ − ∂ M φ∂ M φ − V ( φ ) , (4) ctober 29, 2018 6:18 WSPC - Proceedings Trim Size: 9.75in x 6.5in arx˙fest˙talk where Γ M are the Dirac matrices, f is a constant, M = 0 , ... V ( φ ) = λ φ − v ) , (5)is the usual quartic potential. If φ transforms trivially under Z its ground stateis given by h φ i = v . However if φ is odd under Z its ground state is required tovanish at the fixed points. This results in a kink vacuum profile for φ which servesto localise chiral zero mode fermions, ψ , at one of the orbifold fixed points. Thefixed point at which ψ is localised depends on the sign of f .In a QL symmetric model one must specify the transformation properties of thebulk scalar under the QL symmetry. An interesting choice is to make φ odd underthe QL symmetry, resulting in a Yukawa Lagrangian of the form L = − (cid:8) f Q ( Q − L ) + f u ( U − E ) + f d ( D − N ) (cid:9) φ, where Q = ¯ QQ , etc, and the SM fermions are identified with the chiral zero modesof the bulk fermions in an obvious fashion. Note that the choice f Q , f u , f d > Indeed the effective 4Dproton decay inducing non-renormalizable operator has the form O p = K Λ O Q O L (6)where O Q ( O L ) generically denotes a quark (lepton) operator and K ∼ exp (cid:8) − vL / (cid:9) represents the wavefunction overlap between quarks and leptons inthe extra dimension. L denotes the length of the fundamental domain of the orbifold.Observe that fermions related by the QL symmetry necessarily develop identicalwavefunction profiles in the extra dimension and consequently the troublesome massrelations of the type m e = m u persist in the effective 4D theory. With one bulk Fig. 1. The 5D wavefunctions for quarks and leptons in a QL symmetric model with one bulkscalar. ctober 29, 2018 6:18 WSPC - Proceedings Trim Size: 9.75in x 6.5in arx˙fest˙talk scalar it is only possible to localise fermions at the orbifold fixed points. Howevertwo bulk scalar models enable one to localise fermions within the bulk. This worksas follows. With one bulk scalar chiral zero mode fermions are found at oneof the orbifold fixed points, with the precise point of localisation determined bythe sign of the relevant Yukawa coupling. If a second bulk scalar is added with anopposite sign Yukawa coupling it will tend to drag the fermion towards the otherend of the extra dimension, thereby localising it in the bulk. Importantly thoughthe chiral fermion cannot be pulled very far into the bulk before it is dragged allthe way to the other end of the extra dimension.Let us add a second bulk scalar which is even under the QL symmetry, givingrise to the Lagrangian L = − (cid:8) h Q ( Q + L ) + h u ( U + E )+ h d ( D + N ) (cid:9) φ ′ . Let us again require h Q , h u , h d > φ and φ ′ will attempt to localize quarks at the same point in the extra dimen-sion and they will remain at their original point of localisation. This type of quarkgeography is precisely that recently advocated to allow one to construct quarkflavor without introducing large flavor changing neutral currents. However φ and φ ′ attempt to localise leptons at different fixed points and the resultant point oflocalisation for a given lepton depends on which scalar dominates (these statementswill be made numerically precise in a forthcoming publication ). An arrangementtypical of this setup is shown in Figure 2.This arrangement has a number of interesting features. Firstly note that shiftingleptons into the bulk significantly alters the degree of wavefunction overlap in thequark and leptonic sectors. When one obtains the effective 4D fermion masses thedifferent degree of overlap in the quark and lepton sectors will remove the unde-sirable QL mass relations. Furthermore the overlap between left and right chiralfermion wavefunctions is expected to be greater in the quark sector than in the Fig. 2. The 5D wavefunctions for quarks and leptons in a QL symmetric model with two bulkscalars. ctober 29, 2018 6:18 WSPC - Proceedings Trim Size: 9.75in x 6.5in arx˙fest˙talk lepton sector, leading to the generic expectation that quarks will be heavier thanthe lepton to which they’re related by the QL symmetry. Observe that dragging theright chiral neutrinos ν R all the way to the ‘quark end’ of the extra dimension signif-icantly suppresses the neutrino Dirac masses below the electroweak scale. Dragging ν R to the quark end of the extra dimension does not introduce rapid proton decay,provided that ν R has a large enough Majorana mass to kinematically preclude de-cays of the type p → πν R . Such a mass is expected to arise at the non-renormalizablelevel even if it is not induced by tree-level couplings in the theory. Thus the non-desirable Yukawa relations implied by 4D QL symmetric modelsturn out to be of interest in the 5D construct. They enable one to understandproton longevity within the split fermion framework in a less arbitrary fashion andinstead of inducing unwanted mass relations suggest an underlying motivation forthe flavour differences experimentally observed in the quark and lepton sectors.
4. Quartification in Five Dimensions
Adding a leptonic color group to the SM clearly renders the traditional approachesto gauge unification inapplicable. Recall that the simplest grand unified theory,namely SU (5), does not contain the left-right symmetric model. However there arelarger unifying groups which do admit the left-right symmetry, for example SO (10)and the trinification gauge group [ SU (3)] × Z . Similarly it is possible to constructa unified gauge theory admitting the QL symmetry by considering larger unificationgroups. It is natural to extend the notion of trinification to include a leptonic colorfactor, leading one to the so called quartification model. This possesses the gaugegroup G Q = [ SU (3)] × Z , where the additional SU (3) factor corresponds to leptoncolor and the Z symmetry cyclicly permutes the group factors to ensure a singlecoupling constant. It was been demonstrated that unification may be achieved within the quartifi-cation framework in 4D by enforcing additional symmetries upon the quartificationmodel. Subsequent work has shown that unification need not require additionalsymmetries, but does require multiple symmetry breaking scales between the unifi-cation scale and the electroweak scale. It was also shown that unification may beachieved via multiple symmetry breaking routes both with and without the remnantleptonic color symmetry SU (2) ℓ .The necessary symmetry breaking is accomplished with eight Higgs multiplets,giving rise to a complicated Higgs potential with a large number of free param-eters. The demand of multiple symmetry breaking scales also requires hierarchiesof vacuum expectation values (VEV’s) to exist within individual scalar multiplets,giving rise to a generalized version of the doublet-triplet splitting problem familiarfrom SU (5) unified theories. A large number of electroweak doublets also appear inthe 4D constructs. Thus many of the less satisfactory features of 4D quartificationmodels revolve around the Higgs sector required for symmetry breaking.Recently the quartification model has been studied in 5D, where intrinsically ctober 29, 2018 6:18 WSPC - Proceedings Trim Size: 9.75in x 6.5in arx˙fest˙talk higher dimensional symmetry breaking methods exist. By taking the fifth di-mension as an S /Z × Z ′ orbifold one may employ orbifold boundary conditions(OBC’s) on the bulk gauge sector to reduce the gauge symmetry operative at thezero mode level from G Q to SU (3) c ⊗ SU (2) L ⊗ SU (2) ℓ ⊗ SU (2) R ⊗ U (1) . (7)However the use of OBC’s does not reduce the rank of the gauge group so thatfurther symmetry breaking is required. Rank reducing symmetry breaking can beachieved in higher dimensional theories by employing a boundary scalar sector toalter the boundary conditions on the compactified space for gauge fields. Denotinga boundary scalar as χ and defining V ∝ h χ i one can show that V induces a shift inthe Kaluza-Klein mass spectrum of gauge fields which couple to χ . If such a gaugefield initially possessed a massless mode its tower receives a shift of the form M n ≈ M c (2 n + 1) (cid:18) M c πV + . . . (cid:19) , n = 0 , , , ... (8)giving a tower with the lowest-lying states M c , M c , M c , ... . This represents anoffset of M c relative to the V = 0 tower, with the field no longer retaining amassless zero mode. The association of V with the VEVs of the boundary scalarsector implies that the limit V → ∞ is attained when h χ i → ∞ . However, whenthe VEVs of the Higgs fields are taken to infinity, the shift in the KK masses ofthe gauge fields is finite, giving the exotic gauge fields masses dependent only uponthe compactification scale M c . Consequently, these fields remain as ingredients inthe effective theory while the boundary Higgs sector decouples entirely, and we canview our reduced symmetry theory in an effective Higgsless limit. Interestingly, inthis limit also, the high-energy behaviour of the massive gauge boson scatteringremains unspoilt as shown in. It was shown in that a unique set of OBC’s was required to ensure that quarkmasses could be generated and to prevent liptons from appearing at the electroweakscale. The inclusion of a boundary Higgs sector allows one to reduce the quartifi-cation gauge symmetry down to the SM gauge group G SM or to G SM ⊗ SU (2) ℓ at the zero mode level. In both cases fifth dimensional components of the SU (3) L gauge fields with the quantum numbers of the SM Higgs doublet retained a masslessmode, enabling one to use Wilson loops to reproduce the SM flavour structure.A surprising result however was that unification could only be achieved whenthe remnant lepton color symmetry SU (2) ℓ remained unbroken. Thus one arrivesat a unique minimal quartification model which unifies in 5D, a result to be con-trasted with the 4D case where a large number of symmetry breaking routes whichpermit unification have been uncovered. Unfortunately the unifying case requiresthe compactification scale to be greater than 10 GeV so that only a SM like Higgsfield is expected to appear at the LHC. ctober 29, 2018 6:18 WSPC - Proceedings Trim Size: 9.75in x 6.5in arx˙fest˙talk
5. Conclusion
Quark-lepton symmetric models in some sense unify the fermionic content of theSM and thereby motivate the similar family structures observed in the quark andlepton sectors. Recent investigations involving QL symmetries in 5D have uncov-ered a number of interesting results. In particular the QL symmetry provides usefulYukawa relationships in split fermion models and the quartification model is foundto be more constrained in 5D. A number of avenues for further investigation remain,including a detailed analysis of the fermionic geography of Figure 2 and the con-struction of a complete three generational model with combined QL and left-rightsymmetry in 5D. References
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