See-Saw Masses for Quarks and Leptons in SU(5)
TTUM-HEP-773/10revised version
See-Saw Massesfor Quarks and Leptons in SU (5) Thorsten Feldmann ∗ Physik Department, Technische Universit¨at M¨unchen,James-Franck-Straße, D-85748 Garching, Germany
Abstract
We build on a recent paper by Grinstein, Redi and Villadoro, where a see-saw like mechanism for quark masses was derived in the context of spontaneouslybroken gauged flavour symmetries. The see-saw mechanism is induced by heavyDirac fermions which are added to the Standard Model spectrum in order to renderthe flavour symmetries anomaly-free. In this letter we report on the embeddingof these fermions into multiplets of an SU (5) grand unified theory and discuss anumber of interesting consequences. ∗ Address since January 2011:IPPP, Department of Physics, University of Durham, Durham DH1 3LE, UK.email:[email protected] a r X i v : . [ h e p - ph ] F e b Introduction
The phenomenological motivations for considering extensions of the Standard Model(SM) which are based on grand unified theories (GUTs; see, for instance, [1–5], or [6–9]and references therein) are two-fold: • First, the embedding of the SM gauge group SU (3) × SU (2) × U (1) into simpleor semi-simple gauge groups like SU (4) × SU (2) × SU (2), SU (5), or SO (10), toname a few popular examples, allows for a reduction of independent gauge couplingconstants. Indeed, from the renormalization-group (RG) running of the measuredSM couplings, there is evidence that a unification around scales of ∼ GeVis possible [10], although for this idea to work quantitatively, additional degreesof freedom at high energies should contribute to the RG running. A well-knownexample which could exhibit the unification of coupling constants is, of course, theminimally super-symmetric SM (MSSM). • Second, the embedding of the SM fermions into a few (smaller) GUT represen-tations appears attractive, both, for minimalistic reasons as well as for a possiblereduction of independent mass and mixing parameters in the flavour sector. Exceptfor the similar masses of the τ -lepton and the b -quark, there is, however, no directevidence for such simplifications. On the other hand, with a suitable Higgs sec-tor – sometimes also supplied with additional constraints from postulated discreteflavour symmetries – reasonable quark and lepton spectra can be obtained fromGUT models. Often GUT models also allow for naturally tiny neutrino massesthrough a see-saw mechanism, e.g. induced by heavy Majorana neutrinos.In a slightly different context, Grinstein, Redi and Villadoro [11] have recently pro-posed an approach to generate the SM Yukawa couplings in the quark sector from asee-saw mechanism, which is induced by heavy Dirac fermions that appear in construc-tions with gauged flavour symmetries at (relatively) low energies (for related work, seealso [12, 13] and references therein).In this letter, we are going to show that the mechanism in [11] can be modifiedand extended in such a way that up-type and down-type quarks, charged leptons andneutrinos, as well as their heavy partners can be embedded into multiplets of SU (5). Thefermion spectrum below the GUT scale can be represented in an anomaly-free way withrespect to a flavour symmetry group of the SM gauge interactions. In contrast to theoriginal approach in [11], our modification also introduces new heavy SU (2) L doublets,and therefore, in principle, offers the potential to adjust the unification of SM couplingconstants via the RG running and threshold corrections induced by the new fermions.Furthermore, it can lead to a natural suppression of neutrino masses with respect tothe masses of the charged leptons. Finally, the SU (5) fermion spectrum can be madeanomaly-free with respect to the flavour symmetry of the GUT theory by introducinga set of heavy fermions that are distinguished from the SM matter by a postulated Z -symmetry and contain a SM singlet component which may serve as a dark mattercandidate. 1 See-Saw mechanism with Heavy Dirac Fermions
The inclusion of heavy Dirac partners for the SM particles is motivated by the require-ment that the chiral anomalies related to the flavour symmetries of the SM gauge sectorare canceled between SM quarks and their heavy partners, which would allow for a pro-motion of the global flavour symmetries to local transformations of a renormalizablegauge theory. This, in particular, removes the Goldstone modes that appear when theflavour symmetries are to be spontaneously broken by the vacuum expectation values(VEVs) of some scalar fields. The choice of quantum numbers for such fermion fieldsis not completely fixed: The model in [11] contains the minimal set of fermions neededto reproduce the SM quark Yukawa terms from renormalizable dim-4 operators, only.The approach discussed in [12] already starts from an effective theory where the heavyfermions decouple from the SM fermions. The model that we are going to discuss inthis work represents a modification of [11], based on dim-4 terms with different fermioncontent and couplings.Let us first summarize the relevant expressions in the Lagrangian relevant for energies below the GUT scale.
The down-quark sector of our model coincides with the construction in [11]. It featuresa dim-4 Lagrangian
L (cid:51) ¯ Q L H ψ d R + ¯ ψ d S D ψ d R + M D ¯ ψ d D R + h.c. (2.1)The notation for the new fermion fields ψ d R and ψ d follows [11]. Here and in the following,we will not specify O (1) coupling constants for simplicity. The explicit Dirac mass M D controls the coupling between the right-handed SM quarks with their heavy partners. S D is a matrix-valued scalar field, which we will refer to as a spurion field as in thecontext of minimal flavour violation [14]. It will achieve a VEV from an (unspecified)potential. Finally, the VEV of the SM Higgs doublet, (cid:104) H (cid:105) = v SM , as usual, sets the scaleof electro-weak symmetry breaking (EWSB).The different mass scales in (2.1) are to obey v SM (cid:28) M D (cid:28) |(cid:104) S D (cid:105)| . To derive theeffective theory at the electro-weak scale, we may then simply integrate out the heavyfermions by solving the classical e.o.m. for ψ d (or, equivalently, ψ d R ), ∂ L ∂ψ d = (cid:104) S D (cid:105) ψ d R + M D D R ! = 0 ⇒ ψ d R = − M D (cid:104) S D (cid:105) − D R . (2.2)Inserting the e.o.m. back into L generates effective Yukawa terms at energies below (cid:104) S D (cid:105) , L eff = − M D ¯ Q L H (cid:104) S D (cid:105) − D R + h.c. (2.3)One therefore encounters an inverted relation between the down-quark Yukawa matrixand the spurion VEV (cid:104) S D (cid:105) , Y D = M D (cid:104) S D (cid:105) − . (2.4)2igure 1: Illustration of the see-saw mechanism in the down-quark sector (see text). The flavourspurion S D breaks the SU (3) Q L × SU (3) D R flavour symmetry and gives masses to the heavy partners ψ d , ψ d R of the SM quarks ( Q L , D R ). This new kind of see-saw mechanism is also illustrated in Fig. 1: (i) The left-handed SMquark doublet Q L and a new right-handed quark singlet ψ d R transform in the fundamentalrepresentation of an SU (3) Q L flavour symmetry. They couple via the SM Higgs doubletin a gauge-invariant way under both, the SM symmetry group and the local flavoursymmetry. (ii) The right-handed SM down-quark singlet D R and a new left-handedquark singlet ψ d transform in the fundamental representation of an SU (3) D R flavoursymmetry. They allow for a gauge-invariant Dirac mass term parameterized by M D . (iii)The two sectors of the flavour symmetry group are connected via the flavour spurion field S D ∼ (¯3 ,
3) which transforms in a bi-fundamental representation of SU (3) Q L × SU (3) D R and is a singlet under the SM gauge group. It thus couples to the new fermion fields,only. The VEV of S D breaks the SU (3) Q L × SU (3) D R flavour symmetry and gives massesto ψ D R and ψ d . (iv) Upon integrating out the heavy partners of the down-type quarks,one generates an effective Yukawa term for the SM down-quark fields with a Yukawamatrix inversely proportional to (cid:104) S D (cid:105) . In [11], it has been shown that an anomaly-free fermion spectrum for the SU (3) Q L × SU (3) U R × SU (3) D R flavour symmetry of the SM can be obtained if the up-quark sector istreated analogously to the down-quark sector, adding another set of heavy singlet fields ψ u R , ψ u and a spurion field S U transforming as a bi-doublet under SU (3) Q L × SU (3) U R .In our model, we propose a modification, • which also contains new heavy quark doublets , allowing for an embedding into10-plets of SU (5), • which is based on a different flavour symmetry group in the quark sector, SU (3) Q L = U cR × SU (3) D R , Illustration of the see-saw mechanism in the up-quark sector (see text). The flavourspurion T U = ( T U ) T breaks the SU (3) Q L = U cR flavour symmetry and gives masses to the heavypartners of the SM quarks. The fermions in the lower left (upper right) part of the figure are SU (2) L doublets (singlets). • and where, in the up-quark sector, the flavour symmetry is broken by a spurion fieldtransforming as a 6-plet of SU (3) Q L = U cR (reflecting a symmetric up-quark Yukawamatrix in SU (5)).In this case, a see-saw mechanism can be achieved by the following terms in the dimension-4 Lagrangian, L (cid:51) (cid:16) ¯ Q L ˜ H ψ u R + ¯ ψ Q ˜ H U R (cid:17) + T U (cid:0) ¯ ψ u ψ u R + ¯ ψ Q ψ Q R (cid:1) + M U (cid:0) ¯ ψ u U R + ¯ Q L ψ Q R (cid:1) + h.c. (2.5)Here, ψ Q and ψ Q R are heavy quark doublets, while ψ u and ψ u R are heavy quark singlets.Letting aside, for the moment, the case of the top quark, where M U ∼ (cid:104) T U (cid:105) should beconsidered, we may again integrate out the heavy fermions by solving the classical e.o.m.to obtain the effective up-quark Yukawa matrix. Similarly as before, it is proportionalto M U (cid:104) T U (cid:105) − , weighted by the coupling constants that have not been shown in (2.5).The situation in the up-quark sector is illustrated in Fig. 2: (i) The SM quark doublet Q L , a new heavy right-handed doublet ψ Q R and a new singlet ψ u R transform in thefundamental representation of SU (3) Q L = U cR . The field ψ u R couples to Q L via the SMHiggs doublet, while the right-handed doublet ψ Q R has a Dirac mass term with Q L . (ii)Similarly, the right-handed SM up-quark singlet U R , a new left-handed doublet ψ Q anda new singlet ψ u transform in the anti-fundamental representation of SU (3) Q L = U cR , withanalogous couplings as in (i). (iii) The SU (3) Q L = U cR flavour symmetry is broken by theVEV of the symmetric spurion matrix T U ∼ ¯6 which couples the new fermion fields ψ Q R The case of the top quark in models with spontaneously broken flavour symmetry is special [15].In the see-saw construction the mass eigenstates for the top quark and its heavy partner will be linearcombinations with a mixing angle controlled by M U / (cid:104) T U (cid:105) , see [11] for details. ψ Q , or ψ u R and ψ u , respectively, and makes them massive. (iv) Upon integratingout the heavy partners of the up-type quarks, one generates an effective Yukawa termfor the SM up-quark fields with a Yukawa matrix inversely proportional to (cid:104) T U (cid:105) .It is instructive to derive the number of light quark fields (in the limit (cid:104) H (cid:105) → S D , T U ,leaving one light doublet and two light singlets which are identified as Q L , U R , D R in theSM. In the lepton sector, we consider the flavour symmetry group SU (3) (cid:96) L × SU (3) E R × SU (3) ν R , where the first two groups refer to the left-handed lepton doublet and right-handed singlet in the SM, and the last factor corresponds to an additional right-handedDirac neutrino ν R for each family. The masses for the SM leptons and their heavypartners descend from the dim-4 Lagrangian L (cid:96) = (cid:16) ¯ E R H † + ¯ ψ ν R ˜ H † (cid:17) ψ (cid:96) + ¯ ψ (cid:96) R S E ψ (cid:96) + ¯ ψ ν R S ν ψ ν + M E ¯ ψ (cid:96) R (cid:96) L + M ν ¯ ν R ψ ν + h.c.(2.6)Here, ψ (cid:96) and ψ (cid:96) R are heavy lepton doublets which, together with the heavy quark singlets ψ d , ψ d R in the down-quark sector, can be combined into 5-plets of SU (5). Furthermore, ψ ν and ψ ν R are heavy singlet partners for the neutrinos. We also introduced two newmatrix-valued scalar spurion fields, transforming as S E ∼ (3 , ¯3 ,
1) and S ν ∼ (1 , , ¯3)under the flavour-symmetry group in the lepton sector.Solving the e.o.m. for ¯ ψ (cid:96) R and ψ ν in the limit (cid:104) S E,ν (cid:105) (cid:29) M E,ν , one obtains ψ (cid:96) (cid:39) − M E S − E (cid:96) L , ¯ ψ ν R (cid:39) − M ν ¯ ν R S − ν . (2.7)Inserting this back into L (cid:96) , yields the effective low-energy mass terms L eff (cid:39) − M E (cid:16) ¯ E R H † − M ν ¯ ν R S − ν ˜ H † (cid:17) S − E (cid:96) L , (2.8)from which we read off the Yukawa matrices responsible for the Dirac masses of chargedleptons and neutrinos, Y E (cid:39) M E (cid:104) S E (cid:105) − , Y ν (cid:39) − M ν (cid:104) S ν (cid:105) − Y E (cid:39) − M ν M E (cid:104) S E S ν (cid:105) − . (2.9)Interestingly, the neutrino Dirac mass matrix in this case is doubly suppressed byboth, M ν / (cid:104) S ν (cid:105) and M E / (cid:104) S E (cid:105) , which provides a natural explanation for the smallness ofthe neutrino masses compared to the masses of charged leptons. Notice, that for thispurpose we did not have to introduce Majorana neutrinos.The situation in the lepton sector is illustrated in Fig. 3: (i) The SM lepton doublet (cid:96) L and a new doublet ψ (cid:96) R transform in the fundamental representation of SU (3) (cid:96) L andcouple via a fundamental Dirac mass term, M E . (ii) The SM singlet E R , a new doublet5igure 3: Illustration of the see-saw mechanism in the lepton sector (see text): The flavourspurion fields S E and S ν break the SU (3) (cid:96) L × SU (3) E R × SU (3) ν R symmetry and give massesto the heavy partners of the SM leptons. The neutrino Dirac masses get doubly suppressed by M E / (cid:104) S E (cid:105) and M ν / (cid:104) S ν (cid:105) . The fermions on the left-hand (right-hand) side are SU (2) L doublets(singlets). ψ (cid:96) and a new singlet ψ ν R transform in the fundamental representation of SU (3) E R andcouple via the SM Higgs field. (iii) The right-handed neutrino ν R and its partner ψ ν transform in the fundamental representation of SU (3) ν R and couple via a fundamentalmass term M ν . (iv) The flavour symmetry in the lepton sector is broken by the VEVs ofthe spurion matrices S E and S ν which couple the new fermion fields from the differentflavour sectors and make them massive. (v) Upon integrating out the heavy partnersof the leptons, one generates an effective Yukawa term for the charged leptons with aYukawa matrix inversely proportional to (cid:104) S E (cid:105) . The neutrinos receive a Dirac mass whichis doubly suppressed and stems from a Yukawa matrix proportional to (cid:104) S E S ν (cid:105) − .The counting of light and heavy lepton degrees of freedom works analogously as inthe quark case: From the three doublet (four singlet) Weyl spinors for each family, two(two) Weyl spinors become massive due to the Dirac masses generated from the scalarspurions S E , S ν , leaving one light doublet and two light singlets which are identified as (cid:96) L , E R , ν R . The transformation properties of the SM quark fields and its heavy partners under the SU (2) L group of the SM and under the SU (3) Q L = U cR × SU (3) D R flavour symmetry in thequark sector are as follows, Q L ∼ L (3 , + y Q ψ cQ R ∼ L (¯3 , − y Q ψ Q ∼ L (¯3 , + y Q U cR ∼ L (3 , − y uR ψ u ∼ L (¯3 , + y uR ψ cu R ∼ L (¯3 , − y uR D cR ∼ L (1 , ¯3) − y dR ψ d ∼ L (1 , + y dR ψ cd R ∼ L (¯3 , − y dR (2.10)where we have also indicated the appropriate hyper-charges. Concerning anomalies, itis easy to see that: 6 The chiral anomalies from the new heavy quarks with respect to the SM gaugegroup cancel between the two last columns in (2.10). • Concerning the mixed anomalies between the flavour symmetries and hyper-charge,we observe that the first two columns trivially cancel, while within the third-columnthe mixed anomalies cancel because 2 y Q = y u R + y d R . • Chiral anomalies related to the SU (3) D R cancel, as we have one left-handed tripletand one left-handed anti-triplet. Chiral anomalies of the SU (3) Q L = U cR flavour sym-metry group do not cancel by counting the number of left-handed triplets (2+1)and anti-triplets (2+2+1+1+1).The anomaly-free representation in [11] could be recovered by dropping the heavydoublets ψ Q and ψ Q R . The alternative option that we are pursuing here is to addtwo more doublet fields χ L ∼ L (3 , − y Q , χ cR ∼ L (3 , +5 y Q . (2.11)The quoted hyper-charges follow from the embedding into a 24-plet in SU (5), seebelow. Furthermore, in order to prevent χ L,R from coupling to the other fermions,we introduce a discrete Z -symmetry, dubbed “ F -parity” in this work, under which χ L,R are assumed to be odd, while the SM fermions and their see-saw partners areeven. The mass term for the additional fields is given by¯ χ R T U χ L + h.c. (2.12)The lightest of these states would have a mass of the order (cid:104) T U (cid:105) ∼ M U and maybe detectable as a heavy quark doublet with exotic quantum numbers.In the lepton sector, a similar analysis for the SU (3) (cid:96) L × SU (3) E R × SU (3) ν R symmetryyields (cid:96) L ∼ L (3 , , + y (cid:96) ψ (cid:96) cR ∼ L (¯3 , , − y (cid:96) ψ (cid:96) ∼ L (1 , , + y (cid:96) E cR ∼ L (1 , ¯3 , − y eR (2.13)and ν cR ∼ L (1 , , ¯3) ψ ν ∼ L (1 , , ψ cν R ∼ L (1 , ¯3 , (2.14)where the cancellation of SM anomalies from the new leptons is obvious (one new vectorrepresentation of charged leptons, trivial representations for right-handed neutrinos),while the cancellation of anomalies related to each flavour sub-group can be seen bycounting the number of triplets and anti-triplets.We have thus shown that the flavour symmetry SU (3) Q L = U cR × SU (3) D R × SU (3) (cid:96) L × SU (3) E R × SU (3) ν R can be represented in an anomaly-free manner, and therefore weare free to introduce the corresponding gauge bosons which will become massive by theusual Higgs mechanism as soon as the flavour spurions get their VEVs. The following discussion differs from the one presented in an earlier preprint version of this paperand aims to resolve some of its problematic issues. .5 MFV Perspective Compared to the standard set-up of MFV [14], the up-quark sector is modified as theflavour symmetry is broken by a symmetric complex matrix (cid:104) T U (cid:105) instead of a generic com-plex matrix Y U . Counting the respective flavour parameters, we encounter 12 instead of18 degrees of freedom in the spurion matrix, while the number of flavour symmetry gener-ators is reduced by 8 compared to the standard case. This leaves two additional flavourparameters which can be traced back to relative phases between the heavy fermionicpartners of the up-quarks, similar as for the Majorana phases in the PMNS matrix forthe neutrino sector. In the low-energy effective Hamiltonian for B -, D -, and K -mesontransitions one thus encounters new sources for CP violation in addition to the SMCKM phase. This can be viewed as a particular example for next-to-minimal flavourviolation as defined in [16]. Similarly, MFV in the neutrino sector [17], is modified asthe fundamental sources of flavour violation now transform as (cid:104) S E (cid:105) − ∼ (3 , ¯3 ,
1) and (cid:104) S ν (cid:105) − ∼ (1 , , ¯3), and not as the neutrino Yukawa matrix Y ν . Still, the spurion fieldsobey consistency relations (here between (cid:104) S E S ν (cid:105) − and Y ν ) as required in [16].We emphasize that, at this point, the above considerations apply independently ofa possible embedding into a grand unified framework. From the low-energy (i.e. belowthe GUT scale) point of view, our set-up provides an alternative realization of the ideaproposed in [11] with a different heavy fermion spectrum, different new sources for flavourphenomenology in the quark sector, and an interesting explanation for small Dirac -neutrino masses together with new sources for lepton-flavour violation. In addition, ourset-up introduces new heavy lepton and quark doublets, which will contribute to therunning of the SM coupling constant α ( µ ) at energies between the scale set by thecorresponding spurion VEVs and the UV cut-off of the theory (e.g. M GUT ). SU (5) Embedding
Our aim is to find an embedding of the heavy-fermion spectrum identified in the previoussections in the context of a grand-unified theory based on SU (5) fermion multiplets. Thequestions we want to address are: • Do the fermions fit in multiplets of SU (5) ? • How do the various mass terms descend from an SU (5)-invariant Lagrangian? • How are the flavour symmetry groups below and above the GUT scale related?For simplicity, we will use a symbolic notation in terms of left-handed Weyl spinorsand Higgs fields, and again suppress couplings of O (1). The coupling terms in theLagrangian, allowed by SU (5) symmetry, and their decomposition into SM multipletscan, for instance, be found with the help of [7]. We may assume in the following a minimalHiggs sector with a (cid:104) H (cid:105) breaking SU (5) to the SM group, and a 5 H containing the SM8iggs doublet responsible for EWSB. As the flavour symmetry group above the GUTscale, we consider G GUT F = SU (3) × SU (3) × SU (3) , which is defined with respect to the GUT multiplets, 10 L (3 , , L (1 , ¯3 , L (1 , , S ∼ (¯3 , , , T ∼ (¯6 , , , S ∼ (¯3 , , . (3.15)Looking at the fermion representations of the low-energy theory, as defined in (2.10)and (2.13,2.14), we observe that • The down-quark singlets and lepton doublets, together with their respective heavypartners, can be combined into three 5-plets of SU (5) in a straight-forward manner,SM Matter: ¯5 L (1 , ¯3 , (cid:51) D cR , (cid:96) L , Heavy Partners: ¯5 L (¯3 , , (cid:51) ψ cd R , ψ (cid:96) , L (1 , , (cid:51) ψ d , ψ c(cid:96) R , (3.16)where in brackets, we have given the transformation under G GUT F and identified SU (3) = SU (3) (cid:96) cL = D R and SU (3) = SU (3) Q L = U cR = E cR . • Similarly, the right-handed neutrinos are represented by the SU (5) singletsSM Matter: 1 L (1 , ,
3) = ν cR , Heavy Partners: 1 L (3 , ,
1) = ψ cν R , L (1 , , ¯3) = ψ ν , (3.17)with SU (3) = SU (3) ν cR . • The up-quark sector, on the other hand, is special and – at least in the set-upidentified in the last section – prevents a straight-forward embedding into SU (5).While the SM fermions form the usual SU (5) 10-plets,SM Matter: 10 L (3 , , (cid:51) Q L , U cR , E cR , (3.18)in (2.10,2.13), we only encounter heavy partners for the quark doublets and for theup-quark singlets, but not for the lepton singlets.However, completing the multiplets byHeavy Partners: 10 L (¯3 , , (cid:51) ψ Q , ψ cu R , X ce R , L (¯3 , , (cid:51) ψ cQ R , ψ u , X e , (3.19)we will see below that the additional fields X e and X e R do not contribute to theYukawa terms in the low-energy Lagrangian, as long as the triplet componentof the 5 H Higgs representation in SU (5) is heavy (which requires that the usualdoublet-triplet splitting problem has been solved in one or the other way).9ith this, one can write invariant mass terms for the down-type quarks and leptons(in symbolic notation) as L D + E + ν = (cid:16) L (3 , ,
1) 5 † H + 1 L (3 , ,
1) 5 H (cid:17) ¯5 L (¯3 , , (cid:16) ¯5 L (¯3 , , S † + M ¯5 L (1 , ¯3 , (cid:17) L (1 , , L (3 , , S + M L (1 , , L (1 , , ¯3) + h.c. (3.20)which reproduces the Lagrangians in (2.1,2.6) when decomposed into its SM components.Integrating out the heavy fermion fields, the see-saw mechanism generates the effectiveYukawa matrices Y D = Y E = M (cid:104) S (cid:105) − , Y ν = − M (cid:104) S † (cid:105) − Y E (3.21)which generates the Dirac masses for down-type quarks, charged leptons and neutrinosas indicated.As a consequence of the unification of the charged lepton singlets with the quarkdoublets and the up-quark singlets, we may write down an additional term, L ν R = 12 1 L (3 , , T L (3 , ,
1) + h.c. (3.22)which gives rise to a Majorana mass term for the right-handed neutrino M ν R = M ( S − ) T T ( S ) − . (3.23)The masses for the up-quarks are realized by the following expressions, L U = 12 10 L (3 , ,
1) 10 L (¯3 , ,
1) 5 H + 10 L (¯3 , , (cid:16) T † L (¯3 , ,
1) + M L (3 , , (cid:17) + h.c. (3.24)which reproduces (2.5) together with the additional terms involving X e and X e R ,∆ L = ¯ X e R U cR Φ + ¯ E R X cu R Φ + ¯ X e R T † X e + M ¯ E R X e + h.c. (3.25)When the triplet Higgs component Φ in 5 H is set to zero, the fields X e and X e R can beintegrated out without contributing to the effective SM Yukawa couplings. Integratingout the heavy fermions from (3.24), we obtain the effective SM Yukawa matrix in theup-quark sector, as before, Y U = M (cid:104) T (cid:105) − . (3.26)Finally, we may embed the fields χ L,R , which we have introduced in (2.12) in orderto compensate for the chiral anomalies in SU (3) Q L = U cR , in a 24-plet of SU (5) which is F -odd. This also fixes the hyper-charges quoted in (2.12),24 L (3 , , (cid:51) χ L (3 , , , χ cR (¯3 , , , . . . ( F -odd) (3.27)10s indicated, this introduces a number of additional fermion fields, including a generationof SM singlets. The 24-plet allows for a unique mass term,24 L (3 , , T L (3 , , , such that the lightest generation would receive a mass contribution of order (cid:104) T (cid:105) . If –after renormalization-group running to the weak scale – the lightest of the F -odd statesturns out to be the SM singlet, and F -parity is unbroken, it may provide a dark mattercandidate. Let us again discuss the anomalies associated to the fermion representations of SU (5) ×G GUT F . • The SU (5) anomalies in the F -even sector cancel, because we have the same num-ber of left-handed 10-plets (10-plets) and 5-plets (5-plets). Notice also, that to-gether with the three singlets they form complete 16-dimensional representationof SO (10). The 24-plet in the F -odd sector is a real representation of SU (5), anddoes not contribute to the anomaly. • The SU (3) flavour symmetry has a vector-like representation with one 5 L (1 , , L (1 , ¯3 , SU (3) which contains a 1 L (1 , ,
3) and one 1 L (1 , , ¯3) fermion representation. • The case with the SU (3) again is more involved: Counting the triplet (anti-triplet) representations in the F -even sector, we obtain 10 + 1 (10 + 10 + 5), leavinga mismatch of 14. The F -odd sector overcompensates this number by adding 24flavour triplets. In order to obtain an anomaly-free spectrum with complete SU (5)multiplets, the simplest option is to add SU (3) -triplets which are F -odd andcome in a vector-like representation of SU (5),5 L (¯3 , ,
1) and ¯5 L (¯3 , ,
1) ( F -odd) . Being F -odd, these representations can couple only among themselves via a Diracmass term, 5 L (¯3 , , T † ¯5 L (¯3 , , , or to the 24-plet via the 5 H Higgs. The fermions in the F -odd sector thus allreceive masses of order (cid:104) T U (cid:105) .The complete fermion spectrum and the various couplings are summarized in Fig. 4. This suggests that, as an alternative to the ad-hoc F-parity, we may also consider an SU (5) × U (1) X ⊂ SO (10) embedding, where the F -odd fermions and the Higgs fields have even U (1) X charges,and the F -even fermions have odd charges. -even F -oddFigure 4: SU (5) fermion spectrum and realization of the see-saw mechanism. On the left-hand side we show the fermion representations which are even under the postulated F -parity.The numbers in brackets indicate the transformations under the GUT flavour symmetry group SU (3) × SU (3) × SU (3) . The multiplets in frames contain the light SM quarks and leptons.The corresponding couplings in the dim-4 Lagrangian (3.20,3.24) via Higgs fields H , funda-mental mass parameters M , , , or scalar flavour spurions field S , , T are indicated by thearrows. On the right-hand side we show the representations of the F -even fermions which areadded to render the flavour symmetry group anomaly-free (see text). Our little exercise has shown that the see-saw mechanism from SU (5) multiplets hassome interesting consequences for the potential fermion spectrum in GUT theories whichgoes beyond the minimalistic approach mentioned in the introduction. However, we shallpoint out a potential problem of the SU (5) embedding: The mass parameters M and M for the 5- and 10-plets, in principle, can also receive contributions from the VEV ofthe 24-plet Higgs field which is usually considered responsible for SU (5) breaking to theSM gauge group. This amounts to replacing M → M + κ (cid:104) H (cid:105) , M → M + κ (cid:104) H (cid:105) , (3.28)and the natural scale for these parameters appears to be the GUT scale. The phe-nomenology with heavy fermion masses at or way above the GUT scale would be quitedifferent from (and less interesting than) the one discussed in [11], where the masses ofthe lightest of the heavy fermions and gauge bosons can be close to the TeV scale. Thiscalls for an alternative realization of SU (5) breaking which protects M and M againstlarge contributions from the GUT scale in the presence of a 24 H representation.12 Summary
We have studied local flavour symmetries in the context of SU (5) unification of theStandard Model (SM) fermion spectrum. We have shown that, below the GUT scale, thechiral anomalies of a SM flavour symmetry group can be canceled by heavy Dirac fermionswhich receive masses from the vacuum expectation values (VEVs) of scalar spurion fieldsthat break the flavour group spontaneously. The quark and lepton Yukawa couplings atlow energies follow from a see-saw mechanism which – compared to the original idea ofGrinstein, Redi and Villadoro [11] – is extended to the lepton sector and modified in theup-quark sector, in order to allow for unification of the light and heavy fermions into SU (5) multiplets.At low energies, our construction leads to a modification of the standard scenarioconsidered in the context of minimal flavour violation: Flavour transitions in the up-quark sector are induced by a symmetric matrix T U = ( T U ) T which transform as a6-plet of the flavour symmetry SU (3) Q L = U cR . Similarly, in the neutrino sector, the massmatrix is generated from the product of (small) flavour matrices, which leads to a naturalsuppression of neutrino masses compared to the charged lepton masses, without the needto introduce heavy Majorana neutrinos. In contrast to the original model, also electro-weak doublets appear in the heavy fermion spectrum, and therefore the unification of SMcoupling constants at the GUT scale, potentially, may be adjusted by the contributionsof the new fermions to the renormalization-group running and threshold corrections.In order to cancel the chiral anomalies related to the flavour-symmetry group, wehave introduced a number of new fermion fields that are odd under a postulated new Z -symmetry (dubbed “ F -parity in this work). From the SU (5) perspective they canbe organized into a 24-plet and two 5-plets. If the lightest of these fermions is the SMsinglet contained in the 24-plet, it may serve as a dark matter candidate. On a morespeculative level, we have pointed out that our scenario requires an alternative to theconventional mechanism to break SU (5) via a (cid:104) H (cid:105) Higgs field, in order to protect thefundamental Dirac mass parameters against large contributions from the GUT scale.Answering this question goes beyond the scope of this work.
Acknowledgements
I would like to thank Thomas Mannel for a critical reading of the manuscript and forhelpful discussions. I also thank Ben Grinstein for useful comments on the issue of mixedanomalies. 13
See-Saw with the Pati-Salam Group
For completeness, we also describe the embedding of the see-saw mechanism in a GUTbased on the Pati-Salam (PS) Group [1], SU (4) C × SU (2) L × SU (2) R . In this case, thebasic matter multiplets are 4 L ∼ (4 , , , R ∼ (4 , , , where each multiplet contains a doublet of up-, down-type quarks and leptons with thequoted chirality. The two multiplets define the flavour symmetry group of the PS model, G PS F = SU (3) L × SU (3) R . (A.29)Considering again the simplest embedding of the SM Higgs within a bi-doublet h ∼ (1 , , L (cid:51) ¯4 R (¯3 , h L (3 ,
1) + (¯4 R (¯3 , S L + M L ¯4 R (1 , ¯3)) 4 L (1 ,
3) + h.c. (A.30)Decomposing the above expression into its SM components, we recover the original resultfor the quark sector from [11], but now with two Higgs doublets and all flavour spurionsequal to S L at the unification scale. Analogous terms appear in the lepton sector, L (cid:51) ¯ (cid:96) L H ψ ν R + ¯ ψ ν S ν ψ ν R + M ν ¯ ψ ν ν R + ¯ (cid:96) L H ψ e R + ¯ ψ e S E ψ e R + M E ¯ ψ e E R + h.c. (A.31)with new heavy fermions singlets ψ ν , ψ ν R , ψ e , ψ e R . In this case, the low-energy Yukawamatrices for charged leptons and neutrinos are simply given by Y E ∼ M E (cid:104) S E (cid:105) − and Y ν ∼ M ν (cid:104) S ν (cid:105) − , and we do not obtain a natural suppression of the neutrino masses asin (2.6). Contrary to the SU (5) case, we do not need additional F -odd fermions to getan anomaly-free representation of the PS flavour group G PS F . References [1] J. C. Pati and A. Salam, “Unified Lepton-Hadron Symmetry And A Gauge TheoryOf The Basic Interactions,” Phys. Rev. D , 1240 (1973).[2] H. Georgi and S. L. Glashow, “Unity Of All Elementary Particle Forces,” Phys.Rev. Lett. (1974) 438.[3] H. Georgi, in Particles and Fields, 1974 (APS/DPF Williamsburg), ed. C. E. Carl-son (AIP, New York, 1975) p. 575. Notice, that in (A.30) we have treated the left- and right-handed multiplet in an asymmetric way.For an explicitly left-right symmetric version, we can simply symmetrize the effective Lagrangian, in-troducing another set of heavy fermions. (1975) 193.[5] R. N. Mohapatra and J. C. Pati, “A Natural Left-Right Symmetry,” Phys. Rev. D (1975) 2558.[6] A. J. Buras, J. R. Ellis, M. K. Gaillard and D. V. Nanopoulos, “Aspects Of TheGrand Unification Of Strong, Weak And Electromagnetic Interactions,” Nucl. Phys.B (1978) 66.[7] R. Slansky, “Group Theory For Unified Model Building,” Phys. Rept. , 1 (1981).[8] G. Senjanovic, “ SO (10): A theory of fermion masses and mixings,” arXiv:hep-ph/0612312.[9] S. Raby, “Grand Unified Theories,” arXiv:hep-ph/0608183.[10] U. Amaldi, W. de Boer and H. Furstenau, “Comparison of grand unified theorieswith electroweak and strong coupling constants measured at LEP,” Phys. Lett. B (1991) 447.[11] B. Grinstein, M. Redi, G. Villadoro, JHEP , 067 (2010). [arXiv:1009.2049[hep-ph]].[12] M. E. Albrecht, Th. Feldmann, Th. Mannel, “Goldstone Bosons in EffectiveTheories with Spontaneously Broken Flavour Symmetry,” JHEP (2010) 089[arXiv:1002.4798 [hep-ph]]; M. E. Albrecht, “Two Approaches towards the FlavourPuzzle”, PhD Thesis, Techn. Univ. Munich (2010).[13] Th. Feldmann, M. Jung and Th. Mannel, “Sequential Flavour Symmetry Breaking,”Phys. Rev. D , 033003 (2009) [arXiv:0906.1523 [hep-ph]].[14] G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, “Minimal flavour viola-tion: An effective field theory approach,” Nucl. Phys. B (2002) 155 [arXiv:hep-ph/0207036].[15] Th. Feldmann and Th. Mannel, “Large Top Mass and Non-Linear Representationof Flavour Symmetry,” Phys. Rev. Lett. , 171601 (2008) [arXiv:0801.1802 [hep-ph]].[16] Th. Feldmann and T. Mannel, “Minimal Flavour Violation and Beyond,” JHEP , 067 (2007) [arXiv:hep-ph/0611095].[17] V. Cirigliano, B. Grinstein, G. Isidori and M. B. Wise, “Minimal flavor violation inthe lepton sector,” Nucl. Phys. B728