Unified Flavor Symmetry from warped dimensions
Mariana Frank, Cherif Hamzaoui, Nima Pourtolami, Manuel Toharia
CCUMQ/HEP 182, UQAM-PHE/02-2014
Unified Flavor Symmetry from warped dimensions
Mariana Frank (1) , Cherif Hamzaoui (2) , Nima Pourtolami (1) , and Manuel Toharia (1) (1)
Department of Physics, Concordia University, 7141 Sherbrooke St. West, Montreal, Quebec, Canada, H4B 1R6, and (2)
Groupe de Physique Th´eorique des Particules, D´epartement des Sciences de la Terre etde L’Atmosph`ere, Universit´e du Qu´ebec `a Montr´eal, Case Postale 8888,Succ. Centre-Ville, Montr´eal, Qu´ebec, Canada, H3C 3P8. (Dated: October 3, 2018)We propose a scenario which accommodates all the masses and mixings of the SM fermions in amodel of warped extra-dimensions with all matter fields in the bulk. In this scenario, the same flavorsymmetric structure is imposed on all the fermions of the Standard Model (SM), including neutrinos.Due to the exponential sensitivity on bulk fermion masses, a small breaking of the symmetry canbe greatly enhanced and produce seemingly un-symmetric hierarchical masses and small mixingangles among the charged fermion zero-modes (SM quarks and charged leptons) and wash-out theobvious effects of the symmetry. With the Higgs field leaking into the bulk, and Dirac neutrinossufficiently localized towards the UV boundary, the neutrino mass hierarchy and flavor structurewill still be largely dominated by the fundamental flavor structure. The neutrino sector would thenreflect the fundamental flavor structure, whereas the quark sector would probe the effects of theflavor symmetry breaking sector. As an example, we explore these features in the context of a familypermutation symmetry imposed in both quark and lepton sectors.
The original motivation for warped extra-dimensions,or Randall-Sundrum models (RS), was to address thehierarchy problem. In RS the fundamental scale of grav-ity is exponentially reduced from the Planck mass scaleto a TeV size due to a Higgs sector localized near theboundary of the extra dimension [1]. If SM fermions areallowed to propagate in the extra dimension [2], and be-come localized towards either boundary, the scenario alsoaddresses the flavor problem of the SM and suppressesgeneric flavor-violating higher-order operators present inthe original RS setup. However, KK-mediated processesstill generate dangerous contributions to electroweak andflavor observables [3–5], pushing the KK scale to 5 − O (1) values for these5D parameters already generate viable masses and mix-ings. The neutrino sector must behave differently, firstdue to the possibility of Majorana mass terms, and sec-ond because this setup generates large mass hierarchiesand small mixing angles, at odds with neutrino observa-tions. An interesting property of warped scenarios wasinvestigated in [7], for the case of a bulk Higgs wave func-tion leaking into the extra dimension. There one wouldobtain small mixing angles and hierarchical masses for allcharged fermions, and at the same time very small Diracmasses, with large mixing angles and negligible mass hi-erarchy for neutrinos. Thus the flavor anarchy paradigmcould still work in these scenarios.Here we present a scenario where instead of adoptingflavor anarchy, we propose that all fermions share the same flavor symmetry. We assume that the flavor vio-lating effects in the 5D Lagrangian can be parametrizedby a small coefficient whose size is controlled by a ra-tio of scales, <φ> n Λ n , with < φ > the vacuum expectationvalue (VEV) of some flavon field, and Λ some cut-off massscale, or the KK mass of some other flavon fields. Thissmall breaking of the flavor symmetry is enough to repro-duce correctly the flavor structure of the SM in both thequark and lepton sectors. The (stable) static spacetimebackground is: ds = e − A ( y ) η µν dx µ dx ν − dy , (1)where the extra coordinate y ranges between the twoboundaries at y = 0 and y = y TeV , and where A ( y ) isthe warp factor responsible for exponentially suppressingmass scales at different slices of the extra dimension. Inthe original RS scenario A ( y ) = ky , with k the curva-ture scale of the AdS interval, while in general warpedscenarios A ( y ) is a more general (growing) function of y .The appeal of more complicated metrics lies on the pos-sibility of having light KK resonances ( ∼ A ( y ) = ky , unless other-wise specified. Assuming invariance under the usual SMgauge group, the 5D quark Lagrangian is L q = L kinetic + M q i ¯ Q i Q i + M u i ¯ U i U i + M d i ¯ D i D i +( Y Dij u H ¯ Q i U j + h.c. ) + ( Y Dij d H ¯ Q i D j + h.c. ) , (2)where Q i , U i and D i are 5D quarks (doublets and singletsunder SU (2)). In the lepton sector, we assume that Ma-jorana mass terms are forbidden, and so the Lagrangian a r X i v : . [ h e p - ph ] J un can be trivially obtained from the previous one by substi-tuting Q i by L i , U i by N i and D i by E i , where L i are lep-ton doublets, and N i and E i are neutrino and lepton sin-glets, respectively. The Higgs field H is a bulk scalar thatacquires a nontrivial VEV v ( y ) = v e aky , exponentiallylocalized towards the TeV boundary, with delocalizationcontrolled by the parameter a . Such nontrivial exponen-tial VEV-s appear naturally in warped backgrounds withsimple scalar potentials and appropriate boundary con-ditions. This extra dimensional scenario has two sourcesof flavor. One is the usual Yukawa couplings Y uij , Y dij , Y eij and Y νij (dimensionless parameters defined in units of thecurvature out the dimension-full 5D Yukawa couplings as Y Dij = √ kY ij ). The other comes from the fermion bulkmass terms, diagonal in flavor space, taken to be con-stant bulk terms written in units of the curvature k , i.e. M i = c i k ( M i = M q i , M u i , M d i , M L i , M ν i , M e i ).As noted in [7], whenever the bulk Higgs localizationparameter a is small enough in comparison with the c i pa-rameters, (i.e., for the Higgs sufficiently delocalized fromthe TeV brane), the 4D effective masses depend expo-nentially on a rather than on the c i parameters. Theeffective 4D masses for all the SM fermions become m t = v ˜ Y c q , c u < / m f ) ij = v(cid:15) ( c Li − ) (cid:15) ( c Rj − ) ˜ Y ij a > c L i + c R j (4)( m ν ) ij = v(cid:15) a − ˜ Y ij a < c l i + c ν j , (5)where m t is the top quark mass, ( m f ) ij are mass ma-trices for light quarks and charged leptons, and ( m ν ) ij is the Dirac neutrino mass matrix. The parameters c L i ≡ c q i , c l i correspond to the SU (2) doublets, and c R j ≡ c u j , c d j , c e j , c ν j are for the SU (2) singlets. Thewarp factor (cid:15) defined by the background parameters as (cid:15) = e − ky TeV ∼ − encapsulates the hierarchy betweenthe UV (gravity) brane and the TeV (SM) brane. The c -parameters dependence of masses is shown in Fig. 1 forthe case of a brane localized Higgs VEV ( a = 30) and fora delocalized Higgs VEV ( a = 1 . c i values in that limit. A source of tensionarises since, in order to generate viable neutrino massesfrom equation (5), one requires that a ∼ . − . a < a = 1 .
95, someindependent parameters of the 5D Higgs potential mustbe fixed to be equal to within about 1%. However, thissame tuning will also be responsible for generating a lightenough Higgs mode compared to the KK scale [8]. Inmore general warped backgrounds this tension can easilydisappear due to an enlarged parameter space, justifyingthe choice a = 1 . top light quarkscharged leptons bulk Higgsneutrinosbrane Higgsneutrinos (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) c y SM FIG. 1. Effective 4D Yukawa couplings for fermions as a func-tion of the fermion bulk mass parameter c . For simplicity, wehave taken the c -parameters for the doublet and the singletto be equal. structure broken by small terms i.e. c f = c f + δ c f (6) Y Y = Y Y + δY Y , (7)for all fermions of the model with Y Y = Y u , Y d , Y e , Y ν and c f = c q , c u , c d , c l , c e , c ν . The matrices Y Y and c f are flavor symmetric, while the small corrections δ c f and δY Y do not have a priori any flavor structure.From Eqs. (3)– (5), the fermion masses receive differ-ent corrections due to flavor violating terms: m t = m t + δm t c q , c u < / m f ) ij = ( m f ) ij (cid:15) ( δc Li + δc Rj ) a > c L i + c R j (9)( m ν ) ij = ( m ν ) ij + δ ( m ν ) ij a < c l i + c ν j . (10)The exponential sensitivity on the c -parameters is re-sponsible for an exponential sensitivity of symmetrybreaking terms. Since (cid:15) ∼ − , the correctionsto the mass matrices caused by these are of order ∼ − δc i + δc j ) , which means they could account for theobserved hierarchies in the quark and charged leptonmasses, as long as the symmetry breaking corrections δc i remain between − . .
1. The mixing angles arealso exponentially sensitive to small symmetry breakingterms so that the mixing angles diagonalizing the massmatrices from the left will be V ij ∼ (cid:15) ( δc Li − δc Lj ) for ( i < j )for quarks and charged leptons. Thus effects of the orig-inal symmetry are washed out in the quark and chargedlepton sectors, while in the Dirac neutrino sector the sen-sitivity to the symmetry breaking is linear (i.e. small).To qualify our assertions, we impose a simple structurefor all the flavor parameters of the model, namely onewhich remains invariant under family permutations [10].This leads to a flavor structure where the 5D Yukawa cou-plings are invariant under S × S , while the 5D fermionbulk mass matrices are invariant under S . This leadsto democratic 5D Yukawa couplings and to 5D fermionbulk mass matrices parametrized as Y DY ∝ and c Df = A f B f B f B f A f B f B f B f A f . (11)Since all flavor structure is described by Eq. (11), wesimultaneously diagonalize all matrices and obtain Y Y = y Y and c f = c f c f
00 0 c f , (12)where y Y = y u , y d , y e , y ν are complex Yukawa couplingsin the up, down, charged lepton and neutrino Yukawasectors. The matrix c f is in its diagonal basis with realentries and c f . Democratic mass matrices produce twomassless fermions and one massive one. The 0-th orderCKM and PMNS matrices can be parametrized as V i = cos θ i sin θ i − sin θ i cos θ i
00 0 1 , (13)where i =CKM, PMNS. Both matrices contain an angle θ i , not fixed by the S × S symmetry, but by the sym-metry breaking terms, implemented in Eqs. (6) and (7).The small breaking of the symmetry is responsible for asmall lift of the zero masses, yielding a viable neutrinospectrum, with one heavier and two lighter eigenstateswith similar masses, and a normal hierarchy ordering: m ∼ m ν (cid:112) | (cid:15) (2 c l − − | , (14) m ∼ δY ν m ν m ∼ δY ν m ν , (15)where m ν = v (cid:15) a − . We have kept the term in c l sinceit can have an effect when c l < ∼ . Neutrino mass datarequire m ν ∼ (0 . − .
1) eV, which in turn fixes thesize of the Higgs localization parameter a . The depen-dence on δY ν is evident while the δc i ’s are basically free(even the 0-th order c νi are relatively free, as long asthey satisfy a < c ν + c l ). For example, one finds that δY ν < ∼ √ r ∼ .
17, to generate a viable neutrino mass hi-erarchy ratio r = ( | m | − | m | ) / ( | m | − | m | ) ∼ . δY . Inaddition, the exponential dependence on the symmetrybreaking parameters δc f i creates a hierarchy among all the masses. The third generation charged fermion massesare m t ∼ m t , (16) m b ∼ m b (cid:15) δc e , (17) m τ ∼ m τ (cid:15) ( δc l + δc e ) , (18)with the 0-th order masses m t = y u v , m b = y d v (cid:15) c d − / and m τ = y e v (cid:15) ( c l + c e − . The lighterfermion masses are m f ∼ δY Y m f (cid:15) ( δc L + δc R ) , (19) m f ∼ δY Y m f (cid:15) ( δc L + δc R ) , (20)where m f ≡ m c , m s , m µ , m f ≡ m u , m d , m e and m f = v (cid:15) ( c L + c R − . Note that in the flavor symmetric limit,the electron and muon, the down and strange quarks, andthe up and charm quarks, are massless. The symmetrybreaking produces non-zero masses proportional to thegeneric size of δY among these fermions , with an addedsource of hierarchy due to the exponential dependence onthe δc i . It is quite simple in this scenario to obtain phe-nomenologically viable quark and lepton masses by ap-propriately fixing the different δc i within the constraint | δc i | < ∼ .
1. The hierarchies between fermion masses oc-cur naturally and are under control since they depend ex-ponentially on small numbers (they are hierarchical butnot too hierarchical). Some masses and mixings still de-pend linearly on δY so that the typical size of these termscannot be too small since, for instance, δY > ∼ m t /m c inthe (extreme) limit where the charm quark c-parametersare top-like.The observed mixing angles in the CKM and PMNSmatrices can also be generated in this unified scenario.The CKM entries become V us ∼ (cid:15) ( δc q − δc q ) , (21) V cb ∼ δY (cid:15) ( δc q − δc q ) (22) V ub ∼ δY (cid:15) ( δc q − δc q ) . (23)With respect to the 0-th order CKM matrix fromEq. (13), V us receives a suppression exponentially sen-sitive to the difference between two small terms withrespect to the CKM, which can easily reproduce the Assuming that c q , c u < and c d , c e , c l > . The value of the light quark wavefunctions on the TeV brane isslightly higher than in usual RS scenarios to overcome the δY suppression. This induces stronger couplings with KK gluonsand thus generically enhance dangerous FCNC processes. Thesedangerous effects can be addressed by invoking additional flavorconstrains such as enforcing exactly c d = c d [11], or going tomore general warped backgrounds where flavor bounds can bemuch milder [8]. V cb and V ub receive an extra parametric suppression linear inthe small Yukawa perturbations δY , caused by the broken S symmetry. Cabibbo angle. The angles V cb and V ub , lifted fromthe initial zero value, acquire a double suppression, oneexponential, and one proportional to δY ∼ .
1, withthe ratio V cb /V us of the correct order of magnitude forthe typical size for δY and δc , assuming an ordering δc q > δc q > δc q . The expected order of the ratio V ub /V cb ∼ V us also remains realistic, up to order one fac-tors not taken into account in the estimates. This lastfeature is generic in usual RS scenarios.The parametric dependence of the PMNS entries isdifferent: V e ∼ sin θ ν (24) V e ∼ δY ν (cid:113) | (cid:15) (2 c l − − | (25) V µ ∼ δY ν (cid:113) | (cid:15) (2 c l − − | . (26)Contrary to the quark sector, the value of V e is notsuppressed and remains generically of O (1), fixed by thestructure of the neutrino Yukawa flavor violating matrix δY νij . The entries V e and V µ are lifted from zero, bothdepend on δY and, not only are they not further sup-pressed by exponential terms, but can actually be en-hanced by exponential terms (as long as the approxi-mation remains valid). In particular if c l (cid:46) /
2, it ispossible to lift the values of the mixing angles as shownin Eqs. (25) and (26). This feature is specific to the case a < c l + c ν j and c l < /
2, and is not generic in usualRS scenarios. More precise (and less compact) formulaewill be presented in a companion long paper.The observed mixing angles in the neutrino sector aremost sensitive to the flavor structure of the neutrinoYukawa matrix δY ν , but not much to the charged leptonYukawa matrix δY l or to the δc i . The bulk mass pa-rameter of the third family lepton doublet should satisfy c l < / δY ∼ . V exp µ ∼ .
65 and V exp e ∼ . c q (cid:46) /
2, to obtain a large topquark mass, which could be a hint of an additional familysymmetry among the SU (2) doublets of the third fam-ily. Comparing expressions for the V P MNS mixing anglesand the neutrino masses, the element δY must be largerthat the rest of δY so that δY ∼ δY ∼ δY .In this scenario it is easy to find a set of 0-th orderbulk parameters that reproduce the SM and that showthe features described above. For example, a workingpoint for which the SM is a small perturbation (of order10% around an S symmetric set of parameters) awayis shown in Table I. Charged fermions results are nottoo sensitive to small deviations in the Yukawa couplings( (cid:46) . δc i ’s are fixed, the δY ’s can even betaken randomly as long as they remain at around 10%.One then obtains generically charged fermion masses andmixings consistent with the SM and any level of precisionis possible by tuning these values. f q u d l ν ec f ( ≡ c f ) 0.55 0.60 0.60 0.55 5.00 0.60 c f c -parameters. For sim-plicity, we also set all the 0-th order Yukawa coefficients to beuniversal y u = y d = y ν = y e = 4 . a = 1 . In conclusion, we have proposed a general frameworkin warped extra dimensions where the SM flavor struc-ture is unified in all fermion sectors. Small breakingterms are introduced for the 5D bulk mass and Yukawaparameters. The quark and charged lepton sectors aredominated by the small flavor breaking in the bulk c -parameters whereas the (Dirac) neutrino sector is dom-inated by flavor symmetry breaking Yukawa couplings.The main difference between these stems from allowingthe Higgs field leak sufficiently out of the TeV brane sothat the neutrino sector looses sensitivity on the 5D bulkmass parameters. A permutation symmetry was stud-ied to illustrate the idea, but other symmetries can beinvoked and explored within this framework, as will befurther studied in a companion paper.We thank NSERC for partial financial support undergrant number SAP105354. [1] L. Randall and R. Sundrum, Phys. Rev. Lett. , 3370(1999); ibid. , Phys. Rev. Lett. , 4690 (1999).[2] T. Gherghetta and A. Pomarol, Nucl. Phys. B , 141(2000); Y. Grossman and M. Neubert, Phys. Lett. B ,361 (2000).[3] K. Agashe, A. Delgado, M. J. May and R. Sundrum,JHEP , 050 (2003).[4] G. Burdman, Phys. Rev. D , 076003 (2002); G. Bur-dman, Phys. Lett. B , 86 (2004); S. J. Huber, Nucl.Phys. B , 269 (2003); M. S. Carena, A. Delgado,E. Ponton, T. M. P. Tait and C. E. M. Wagner, Phys.Rev. D , 015010 (2005); ibid. , Phys. Rev. D , 035010(2003).[5] K. Agashe, G. Perez and A. Soni, Phys. Rev. D ,016002 (2005).[6] C. Csaki, A. Falkowski and A. Weiler, JHEP , 008(2008); K. Agashe, A. Azatov and L. Zhu, Phys. Rev. D , 056006 (2009).[7] K. Agashe, T. Okui and R. Sundrum, Phys. Rev. Lett. , 101801 (2009).[8] J. A. Cabrer, G. von Gersdorff and M. Quiros, New J.Phys. , 075012 (2010); ibid. , Phys. Lett. B , 208(2011); ibid. , JHEP , 083 (2011); ibid. Phys. Rev.D , 035024 (2011); ibid. , JHEP , 033 (2012);M. Quiros, arXiv:1311.2824 [hep-ph].[9] A. Carmona, E. Ponton and J. Santiago, JHEP ,137 (2011); S. Mert Aybat and J. Santiago, Phys. Rev.D , 035005 (2009); A. Delgado and D. Diego, Phys.Rev. D , 024030 (2009).[10] Among the earliest works on flavor permutation symme- tries are: H. Harari, H. Haut and J. Weyers, Phys. Lett.B , 459 (1978). S. Pakvasa and H. Sugawara, Phys.Lett. B , 61 (1978); ibid. , Phys. Lett. B , 105 (1979);E. Derman and H.S.Tsao, Phys. Rev. D , 1207 (1979); Y. Yamanaka, H. Sugawara and S. Pakvasa, Phys. Rev.D , 1895 (1982).[11] J. Santiago, JHEP0812