A unified leptoquark model confronted with lepton non-universality in B -meson decays
Thomas Faber, Matěj Hudec, Michal Malinský, Peter Meinzinger, Werner Porod, Florian Staub
KKA-TP-21-2018
A unified leptoquark modelconfronted with lepton non-universality in B -meson decays T. Faber, ∗ P. Meinzinger, † and W. Porod ‡ Institut f¨ur Theoretische Physik und Astrophysik, Uni W¨urzburg, Germany
M. Hudec § and M. Malinsk´y ¶ Institute for Particle and Nuclear Physics, Charles University, Czech Republic
F. Staub ∗∗ Institute for Theoretical Physics (ITP), Karlsruhe Institute of Technology,Engesserstraße 7, D-76128 Karlsruhe, Germany andInstitute for Nuclear Physics (IKP), Karlsruhe Institute of Technology,Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldshafen, Germany
The anomalies in the B -meson sector, in particular R K ( ∗ ) and R D ( ∗ ) , are often interpreted ashints for physics beyond the Standard Model. To this end, leptoquarks or a heavy Z (cid:48) representthe most popular SM extensions which can explain the observations. However, adding these fieldsby hand is not very satisfactory as it does not address the big questions like a possible embeddinginto a unified gauge theory. On the other hand, light leptoquarks within a unified framework arechallenging due to additional constraints such as lepton flavor violation. The existing accountstypically deal with this issue by providing estimates on the relevant couplings. In this letter weconsider a complete model based on the SU (4) C ⊗ SU (2) L ⊗ U (1) R gauge symmetry, a subgroupof SO (10), featuring both scalar and vector leptoquarks. We demonstrate that this setup has, inprinciple, all the potential to accommodate R K ( ∗ ) and R D ( ∗ ) while respecting bounds from othersectors usually checked in this context. However, it turns out that K L → e ± µ ∓ severely constraintsnot only the vector but also the scalar leptoquarks and, consequently, also the room for any sizeabledeviations of R K ( ∗ ) from 1. We briefly comment on the options for extending the model in orderto conform this constraint. Moreover, we present a simple criterion for all-orders proton stabilitywithin this class of models. I. INTRODUCTION
In recent years a few anomalies in the B-meson sectorhave been observed by different experiments. The moststriking one is a 3.5- σ deviation in the ratios R D ( ∗ ) = Γ( ¯ B → D ( ∗ ) τ ¯ ν )Γ( ¯ B → D ( ∗ ) l ¯ ν ) ( l = e, µ ) , with R exp D = 0 . ± . , R exp D ∗ = 0 . ± . , (1)from the Standard Model (SM) lepton universality ex-pectations R SM D = 0 . ± . , R SM D ∗ = 0 . ± . . (2)This was first reported by BaBar [1, 2] consistent withmeasurements by Belle [3–5]. Recently this has been con- ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected]ff.cuni.cz ¶ E-mail: [email protected]ff.cuni.cz ∗∗ E-mail: fl[email protected] firmed by LHCb in case of R D ∗ [6] at the 2.1- σ level. Ad-ditional deviations from lepton universality have recentlybeen reported by LHCb in the ratio R K = Γ( ¯ B → ¯ Kµ + µ − )Γ( ¯ B → ¯ Ke + e − ) = 0 . +0 . − . ± . , (3) R K ∗ = Γ( ¯ B → ¯ K ∗ µ + µ − )Γ( ¯ B → ¯ K ∗ e + e − ) = 0 . +0 . − . ± .
05 (4)in the dilepton invariant mass bin 1 GeV ≤ q ≤ [7, 8]. These ratios are predicted to be 1 within the SMand are practically free from theoretical uncertainties.Equally intriguing is a discrepancy in the angular observ-ables in the rare ¯ B → ¯ Kµ + µ − decays measured by LHCb[9] which, however, is subject to significant hadronic un-certainties [10, 11]. While the individual discrepanciesare between 2 and 3 σ , they all point in the same di-rection and amount to more than 4.5- σ deviations oncecombined in a fit [12, 13].In Refs. [14, 15] it has been shown that the deviationsin R D and R K can be explained by an effective modeladding one generation of scalar leptoquarks (LQs) withthe quantum numbers of the right-handed d -quark and anadditional scalar gauge singlet which couples to the LQs.However, it has been shown that this leads to a too largerate for b → sνν [16]. In Ref. [17] another model with twodifferent LQs, one with gauge quantum numbers of the a r X i v : . [ h e p - ph ] A ug right-handed d -quark and one with charge 4 /
3, has beenpresented which explains also neutrino masses at the 2-loop level. As has been shown in Refs. [18–25], anotherpossibility to successfully accommodate the data is to usevector LQs. A somewhat more complete model contain-ing two types of vector LQs to explain the two-photon ex-cess, based on a Froggatt-Nielsen ansatz for the requiredcoupling structures, has been presented in [26]. Besidethe above mentioned violations of lepton-universality thismodel is also compatible with the neutrino data. Anotherpossibility is that the required leptoquarks are boundstates of strongly interacting fermions [27].Most of these settings are effective models containingjust the pieces required to account for the discussed ex-perimental observations, which is clearly the first logicalstep to make when a new signal shows up. However,eventually one would like to understand the observationsfrom a more fundamental perspective. Several attemptsin this direction exist already in the literature [27–38].Of course, the most attractive scenario would be a UVcompletion compatible with theoretical requirements likegauge-coupling unification with the potential to explainalso the observed dark matter relic density.From the GUT perspective the Pati-Salam (PS) model[39] emerges as the first and very natural candidate for alow-energy gauge framework featuring vector as well asscalar leptoquarks within a simple dynamical and renor-malizable scheme. However, the Kibble-Zurek mecha-nism of the early-Universe monopole creation [40] sug-gests that the PS-breaking should occur above the in-flation scale [41]. It is therefore advisable to choose SU (4) C ⊗ SU (2) L ⊗ U (1) R instead as a gauge group of apotentially viable model (as in Ref. [42]) which, indeed,does not suffer from the monopole issue.The structure of this letter is as follows: in SectionII we present the model and discuss the possibilities toobtain leptoquarks with masses in the TeV range. InSection III we discuss in which parts of the parameterspace the B anomalies could be accounted for and whatare the constraints from the existing low energy data. InSection IV we draw our conclusions. In Appendix A wedemonstrate that in this class of models proton remainsstable to all orders in perturbation theory. II. MODEL DESCRIPTION
In what follows we consider the model proposed byPerez and Wise in Ref. [42]. For convenience, we brieflyoutline it here, focusing on the features related to flavorphysics.The model is based on the gauge group G = SU (4) C ⊗ SU (2) L ⊗ U (1) R , where the first factor unifies the threecolors of quarks with the lepton number. This groupis spontaneously broken to G SM = SU (3) c ⊗ SU (2) L ⊗ U (1) Y and further down to G vac = SU (3) c ⊗ U (1) Q , fol-lowing the branching rules Y = R + / [ B − L ] , Q = T + Y, (5) where [ B − L ] = (cid:112) / T = diag (cid:0) + , + , + , − (cid:1) . (6)The matching condition for the QCD coupling at thescale, where SU (4) C is broken, is simply g = g .The entire field content of the model is summarised inTable I. We also include information about other U (1)charges which we need in Appendix A where the detailsof the baryon number conservation and lepton numberviolation are discussed.The SM fermions together with the right-handed neu-trinos are combined into three quadruplets under SU (4) C appearing in three copies representing different genera-tions. On top of that, three fermionic gauge singlets N necessary to generate the correct neutrino masses via in-verse seesaw [43] are added.The gauge field sector corresponding to SU (4) C con-sists of the gluons, Z (cid:48) and a vector leptoquark X ∼ (3 , , + ) which mediates flavor violating processes suchas K L → e ± µ ∓ . This tight constraint implies that, forstandard-size couplings, the mass of the vector lepto-quark has to be at least 1600 TeV.The scalar sector consists of three multiplets χ, Φ and H , see Table I. The most general renormalizable scalarpotential for these fields reads V = m H | H | + m χ | χ | + m Tr( | Φ | ) + λ | H | | χ | + λ | H | Tr( | Φ | ) + λ | χ | Tr( | Φ | ) + ( λ H † i χ † Φ i χ + h . c . )+ λ H † i Tr(Φ † j Φ i ) H j + λ χ † Φ i Φ † i χ + λ | H | + λ | χ | + λ Tr( | Φ | ) + λ (Tr | Φ | ) + (cid:16) λ H † i Tr(Φ i Φ j ) H † j + λ H † i Tr(Φ i Φ j Φ † j ) + λ H † i Tr(Φ i Φ † j Φ j ) + h . c . (cid:17) + λ χ † | Φ | χ + λ Tr(Φ † i Φ j Φ † j Φ i )+ λ Tr(Φ † i Φ j ) Tr(Φ † j Φ i ) + λ Tr(Φ † i Φ † j )Tr(Φ i Φ j )+ λ Tr(Φ † i Φ † j Φ i Φ j ) + λ Tr(Φ † i Φ † j Φ j Φ i ) (7)with | H | = H † i H i , | χ | = χ † χ , | Φ | = Φ † i Φ i where i, j are the SU (2) indices. The trace is taken over the SU (4) C indices only. Notice that the terms proportionalto λ , . . . , λ have been omitted in the original paper[42]; we include them here for completeness. However,for simplicity, in the numerical analysis in Sect. III westick to the original setting [42] with the extra couplingsdeliberately set to zero.The breaking of the SU (4) C group as well as the elec-troweak symmetry breaking is triggered by the corre- We use the square brackets here in order to indicate an indivisiblesymbol. G PS G SM G vac [ B − L ] F M B L, L (cid:48)
Fermions F L = (cid:18) QL (cid:19) (4 , , Q (3 , , / ) L (1 , , − / ) u (3 , / ) d (3 , − / ) ν (1 , e (1 , − (cid:18) + 1 / − (cid:19) +1 +1 (cid:18) + 1 / (cid:19) (cid:18) (cid:19) f cu = (cid:0) u c ν c (cid:1) (4 , , − / ) u c (3 , , − / ) ν c (1 , ,
0) (3 , − / )(1 , (cid:0) − / (cid:1) − − (cid:0) − / (cid:1) (cid:0) − (cid:1) f cd = (cid:0) d c e c (cid:1) (4 , , / ) d c (3 , , / ) e c (1 , ,
1) (3 , / )(1 , (cid:0) − / (cid:1) − − (cid:0) − / (cid:1) (cid:0) − (cid:1) N (1 , ,
0) (1 , ,
0) (1 ,
0) 0 +1 0 0 0 , +1 Scalars χ = (cid:18) ¯ S † χ (cid:19) (4 , , / ) ¯ S † (3 , , / ) χ (1 , ,
0) (3 , / )(1 , (cid:18) + 1 / − (cid:19) (cid:18) + 1 / (cid:19) (cid:18) (cid:19) , (cid:18) − (cid:19) H (1 , , / ) (1 , , / ) H +1 (1 , H (1 ,
0) 0 0 0 0 0Φ = (cid:18)
G R ˜ R † (cid:19) + √ T H (15 , , / ) R (3 , , / )˜ R † (3 , , − / ) G (8 , , / ) H (1 , , / ) R / (3 , / ) R / (3 , / )˜ R − / † (3 , / )˜ R / † (3 , − / ) G + (8 , G (8 , H +2 (1 , H (1 , (cid:18) + 4 / − / (cid:19) (cid:18) + 1 / − / (cid:19) (cid:18) − (cid:19) Gauge Bosons A µ = (cid:18) G µ X µ X ∗ µ (cid:19) + T B (cid:48) µ (15 , , G µ (8 , , X µ (3 , , / ) B (cid:48) µ (1 , ,
0) (8 , , / )(1 , (cid:18) + 4 / − / (cid:19) (cid:18) + 1 / − / (cid:19) (cid:18) − (cid:19) W µ (1 , ,
0) (1 , , W ± µ (1 , ± W µ (1 ,
0) 0 0 0 0 0 B µ (1 , ,
0) (1 , ,
0) (1 ,
0) 0 0 0 0 0TABLE I. The field content of the model together with all gauge quantum numbers for the different regimes as well as chargesunder several other global U (1)’s. Whenever L = L (cid:48) only the common value is displayed. sponding vacuum expectation values (VEVs) (cid:104) χ (cid:105) = 1 √ (cid:18) v χ (cid:19) , (cid:104) H (cid:105) = 1 √ (cid:20) v (cid:21) , (cid:104) Φ (cid:105) = 12 √ (cid:18) − (cid:19) ⊗ (cid:20) v (cid:21) , (8)which are parametrised by v = v ew sin β , v = v ew cos β ,with v ew (cid:39)
246 GeV. The SM-like Higgs h is a superpo-sition of the fields Re( { H , H , χ } ).The fermion masses are generated by the following in-teractions between the scalars and fermions −L Y = f cu Y H F L + f cu Y Φ F L + f cd Y H † F L + f cd Y Φ † F L + f cu Y χN + 12 N µN + h.c. (9)In the broken phase, this leads to the following relationsbetween the mass matrices for the SM fermions and the The round and square brackets are used to distinguish betweenthe SU (4) C and SU (2) L multiplets, respectively. underlying Yukawa matrices: U † u ˆ M u V u = v √ Y + v √ Y , (10) U † d ˆ M d V d = v √ Y + v √ Y , (11)ˆ M e = v √ Y − v √ Y . (12)Here ˆ M are diagonal and U , V are unitary matrices de-scribing the relation between the gauge and mass eigen-states. We work in a basis where the lepton mass matrixis flavor-diagonal. The only constraints on U ’s and V ’sare that V CKM = V † u V d must be reproduced. In the cur-rent study, we shall assume that all the Yukawa matri-ces in (9) are symmetric in the flavor space and, hence, U d = V ∗ d and U u = V ∗ u ; besides simplicity, this is mo-tivated by the idea that this model might eventually beembedded in a variant of the minimal SO (10) framework(such as proposed in Ref. [44]). Thus, we are left withjust one mixing matrix V d which we can choose freely.As we will see, this freedom is crucial for accommodat-ing the B anomalies without violating other constraintsfrom lepton flavor violating observables. R bl − sl + FIG. 1. Tree-level contribution to R K via leptoquarks R (cid:48) . In the neutrino sector we have a 9 × ν, ν c , N ) which should yield thelight-neutrino PMNS matrix as well as their measuredmass differences.The scalar leptoquarks in this model reside in Φ and χ . In total, there are four physical scalars together withthe would-be-Goldstone boson associated to the vectorleptoquark. However, just the mixture of R / , ˜ R / and¯ S † is relevant to addressing the B -meson anomalies; themasses of the two physical eigenstates (to be called R (cid:48) , )are approximately given by m R (cid:48) , = (cid:32) √ λ tan β − λ (cid:33) v χ ∓ λ v χ + O ( v ) . (13)It is well known that for a proper explanation of R K ( ∗ ) thelighter one should have dominant R / component [45]. III. LOW ENERGY OBSERVABLES
We turn now to a discussion of the relevant low-energyobservables in the current model. First of all, we wantto explain the observed deviation from the lepton uni-versality in the B -meson decays. The Feynman diagramresponsible for the tree-level contributions to R K via thescalar leptoquarks R (cid:48) is depicted in Fig. 1. It is im-portant to notice, however, that the same leptoquarkwhich should explain the B -meson anomalies would alsocontribute to other observables. At the tree level, onecan expect an impact on other meson observables like B → X s νν , B s → l ¯ l or K L → eµ . Moreover, thereare important loop contributions to B → X s γ , l → l (cid:48) γ and l → l (cid:48) . An example of the responsible Feynmandiagrams is shown in Fig. 2.For our numerical study we used the Mathematica package
SARAH [46–50] and extended it to support themodel under consideration. In the first step, we usedthe model files to produce a spectrum generator based Details about the new feature to support unbroken subgroups in
SARAH will be given in [51]. R d dR µ e ee FIG. 2. One-loop contribution to µ → e via leptoquarks R (cid:48) .Numerical input values Y diag(10 − , − , − ) Y diag(10 − , · − , − ) θ , θ , θ π / , , π / v χ · GeV m A , m R (cid:48) · GeV, 900 GeVtan β O ( m A ). on SPheno [52, 53].
SPheno calculates the mass spec-trum providing the option to include all one-loop and theimportant two-loop corrections to neutral scalar masses[54–56] in the DR or the MS scheme. However, we areassuming here a full on-shell calculation of all masses,i.e., all shifts can be absorbed into counter-terms of thecouplings leaving the mass spectrum unchanged.In addition,
SPheno provides an interface to
HiggsBounds [57–59] which we used to check the con-straints on the neutral scalars. Moreover,
SPheno cal-culates electroweak precision as well as flavor observ-ables. The calculation of flavor observables is based onthe
FlavorKit functionality presented in Ref. [60]. Weused this feature in order to calculate the values for allnecessary lepton flavor violating observables including K L → eµ . Moreover, the values of the Wilson coeffi-cients relevant for the B -physics calculated by SPheno were passed to flavio [61] to obtain predictions for the B -meson observables.In Table II we collect the input parameters for thisstudy. The remaining parameters affect the heavy stateswhich do not contribute to the observables discussed be-low. For the fermions we take as input the known quarkand lepton masses, the CKM and the PMNS matricesusing the best fit values reported in [62], the Yukawacouplings Y , Y and tan β . For an explanation of the B -physics observables we need an off-diagonal structurein Y . We therefore take V d as input, parametrizing it as V d = c s − s c · c s e iδ − s e iδ c · c s − s c
00 0 1 , (14)denoting c ij = cos φ ij and s ij = sin φ ij , and vary all threeangles in the range [0 , π ] with δ = 0 for simplicity. In thescalar sector we take the SU (4) C -breaking VEV v χ andthe overall scale ( m A ) of the heavy integer-charge Higgsbosons as input and set λ such that the R (cid:48) (cid:39) R lep-toquark remains light. With R (cid:48) (cid:39) ˜ R heavy, the mixingbetween the charge-2 / . λ accordingly. As we takethe heavy Higgs bosons to be in the multi-TeV range, weare in the decoupling limit and fulfil automatically theexperimental constraints on the observed Higgs boson.As long as R D and R D ∗ is concerned, there are in prin-ciple two ways to accommodate the data in the currentscenario: (i) The model automatically contains a vectorleptoquark of a suitable type [32]; however, its potentialeffects in R D ( ∗ ) are strongly suppressed by the need tobe compatible with K L → µe which pushes its allowedmass above 1600 TeV. (ii) A way larger effect than (i) isexpected if the two scalar SU (2)-doublet leptoquarks inthe spectrum remain light and, at the same time, theircharge +2 / µ → eγ ) is driven wayabove the experimental limit. Hence, the simple modelat stakes has serious issues with accommodating the ex-isting B → D ( ∗ ) (cid:96) ¯ ν data.By contrast, it is fairly easy to explain the currently ob-served values for R K and R K ∗ as demonstrated in Fig. 3.Here we have fixed the input parameters as given in Ta-ble II and varied one of the relevant angles. The newphysics contributions to the Wilson coefficients are givenby C NP ij ,R = C NP ij ,R = −
14 1 m R (cid:48) (cid:16) Y V † d (cid:17) i (cid:16) V d Y † (cid:17) j , (15)where i = j = 1 , b → see and b → sµµ , respectively. Using Eqs. (11) and (12) we cal-culate the Yukawa couplings Y and Y in terms of thelepton and quark masses as well as V d . Exploiting the hi-erarchy in the fermion mass spectrum we get, to a goodapproximation, C NP ee ,R (cid:39) − m b m s m R (cid:48) v (1 + tan β ) f − , (16) C NP µµ ,R (cid:39) m b ( m s − m µ )128 m R (cid:48) v (1 + tan β ) f + , (17) FIG. 3. R K (first row) and R K ∗ (second row) as a function ofthe mixing angles in the down-quark sector. The remainingparameters are given in Table II. with f ± = 4 cos 2 φ sin 2 φ sin φ + sin 2 φ (cid:16) φ ± ± φ
13 sin φ cos φ (cid:17) . (18)Combining this with R K (cid:39) | C SM9 | (cid:16) | C NP µµ ,R | − | C NP ee ,R | (cid:17) (19)from [63] one obtains an excellent analytic approximationto the numerical results shown in Fig. 3.In Fig. 4 we display the contour lines R K and R K ∗ inthe m R (cid:48) -tan β plane. The shaded regions indicate the1- σ regions consistent with present data. The increase oftan β with increasing m R (cid:48) can easily be understood fromEqs. (16)–(17) as the new physics contributions scale as Y m R (cid:48) ∝ tan βm R (cid:48) . (20)We recall that the leptoquarks in general also con-tribute to lepton-flavor violating decays of the muon suchas µ → eγ , see for instance [64]. There are two main FIG. 4. Contour plots of R K and R K ∗ as a function of theleptoquark mass and tan β . The shaded regions indicate therange preferred by current data. The remaining parametersare given in Table II.FIG. 5. BR( µ → eγ ) as a function of s which rescales thevalues of Y in Table II. The other parameters are fixed asbefore. contributions to this observable in the current settingcoming, namely, from heavy-neutrino and W -boson loopsas well as leptoquark and quark loops. As an exam-ple, in Fig. 5 we show BR( µ → eγ ) as a function of anextra factor s rescaling the Y eigenvalues in Table IIinto Y → sY . For s > ∼ s ≤ . In principle, s (and, hence, Y ) may be even larger than that FIG. 6. Correlation between BR( µ → eγ ), BR( µ → e ) and R K obtained by varying φ , φ , φ in equal steps of π/
27 inthe range [0 , π ] each. The other parameters are fixed as givenin Table II. The color code indicates the value of R K ∗ .FIG. 7. Correlation between BR( K L → eµ ) and R K obtainedby varying φ ij as in Fig. 6. The other parameters are fixed asgiven in Table II. The color code indicates the value of R K ∗ for each point. Let us point out that in the current model the boundsfrom µ → e are in general stronger than those from µ → eγ in the range interesting for R K and R K ∗ , seeFig. 6. The main reason for this is the negative interfer- indicated in Fig. 5 if off-diagonal elements of Y were invokedtogether with this negative interference. However, as Y doesnot enter the calculation of R K ( ∗ ) at the lowest order, we do notinvestigate this further. ence in µ → eγ discussed above which does neither takeplace in the Z -penguins nor in the box-contributions to µ → e (see Fig. 2) for the same set of parameters.In addition, we have checked that rare τ decays do notimpose any constraints in the R K ( ∗ ) -interesting regions.The same holds for rare b -decays, such as B s,d → µ + µ − , B → sγ , B → X s νν . Taking only the couplings of thescalar sector given in ref. [42] one would get a tight cor-relation between the masses of the scalar leptoquark ofinterest (13) and the scalar gluons leading to too largecontributions to ∆ M K and ∆ M B . However, this rela-tionship gets broken by λ in Eq. (7) implying that alsothese bounds can be avoided.In any case, there is another stringent constraint to beconsidered, namely the bound on K L → eµ . This mode isusually used as a limit on the mass of vector leptoquarksbut it is typically not being taken into account for thescalar ones. As can be seen in Fig. 7, in the interestingregion for R K the bound is violated by several orders ofmagnitude, thus ruling out this model even in those tunedparts of the parameter space where all other constraintscan be satisfied.It has been argued in [32] that additional fermions invector-like representations of the gauge group can reducethe couplings of the SM-fermions to vector leptoquarkswhich, in turn, may be used for lowering the generic ex-perimental limits for their masses. The same mechanismcan in principle work also for the couplings of the scalarleptoquarks such that the constraints due to K L → eµ can be satisfied. However, a detailed exploration of thisaspect is beyond the scope of this paper and will be elab-orated on elsewhere [51].We note for completeness that leptoquarks with massesof about 1 TeV are already constrained by the LHCsearches. These, however, typically focus on the situationwhen the decays are dominated by one channel; in thescenarios where the LQs interact through multiple cou-plings the corresponding bounds must be re-evaluated.We would also like to point out that the “LQ beta-decay modes”, i.e., the decays of a heavier LQ into itslighter SU (2) L companion and W have not been con-sidered so far within the collider searches. Nevertheless,depending on the exact mass splitting, they may be ofsignificant interest, especially if the on-shell W produc-tion is kinematically allowed . IV. DISCUSSION AND CONCLUSIONS
Motived by the successful attempts to explain the ob-served values of R D ( ∗ ) and R K ( ∗ ) by leptoquarks we studya unified SU (4) C ⊗ SU (2) L ⊗ U (1) R gauge model in order Let us note that in the model of our interest the LQ-doubletmass splitting would be below m W if we were to consider onlythe potential given in [42] but can be larger once the additionalterms as in Eq. (7) are included. to demonstrate the challenges one faces when all addi-tional relations inherent to a unified scenario are takeninto account. Among these, the dominant role is typicallyplayed by the constraints on the Yukawa couplings fromthe quark and lepton masses and mixing data and/orthe tight connection between the relevant gauge couplingand α s .The model under consideration contains three differ-ent types of leptoquarks: a hypercharge-2/3 vector lepto-quark and a pair of scalar leptoquarks with hypercharges1 / /
6. In its minimal version, with the SM fermionsector extended such that it supports the inverse-seesawmechanism for neutrinos, one finds [42] that the kaonphysics constrains the mass of the vector leptoquark tosuch an extent that it cannot significantly impact the B -physics observables.We have shown that in the setting under considera-tion one can get the hypercharge-7 / / R K ( ∗ ) but not R D ( ∗ ) . Furthermore, the allowed parame-ter space gets severely constrained by the bounds on raremuon decays; in particular, µ → e is more importantthan µ → eγ . Neither lepton flavor violating τ -decaysnor other B -physics observables lead to additional con-straints. However, it turns out that no points in theavailable parameter space are compatible with K L → eµ .On the other hand, this does not imply that this kindof a leptoquark model is ruled out straight away as anexplanation of R K ( ∗ ) because one can always enlarge thefermion sector by vector-like representations. In this wayone may in principle reduce the couplings to the d -quarkby mixing effects and, thus, avoid the bound due to K L → eµ . This goes beyond the scope of this letterand will be elaborated on in a future study. ACKNOWLEDGEMENTS
We thank H. Koleˇsov´a for discussion in the initialphase of this project. T.F., P.M. and W.P. have beensupported by the DFG, project nr. PO-1337/7-1. FS issupported by the ERC Recognition Award ERC-RA-0008of the Helmholtz Association. M.H. and M.M. acknowl-edge the support from the Grant Agency of the CzechRepublic, Project No. 17-04902S.
Appendix A: Baryon number conservation
It has been noted in [42] that the model has an ap-proximate extra U (1) F symmetry corresponding to thefermion number F , which is explicitly broken by the Ma-jorana mass term for N . However, there is another inde-pendent accidental global symmetry U (1) M , the chargesof which are M ( F Lα ) = M ( χ α ) = +1 ,M ( f cuα ) = M ( f cdα ) = − ,M (Φ αβ ) = M ( H ) = M ( N ) = 0 ,M ( A µαβ ) = M ( W µ ) = M ( B µ ) = 0 . (A1)Here α, β denote SU (4) vector-like indices. Notice thatwe can obtain the M -charges of all the field multipletsby the prescription M = (cid:0) SU (4) indices (cid:1) − (cid:0) SU (4) indices (cid:1) . (A2)The U (1) M symmetry of each term in the Lagrangianis then guaranteed by the fact that every upper SU (4)index is contracted to a lower one, all carried by thedynamical fields. It is also clear that any hypothetical M –violating but SU (4)–preserving term necessarily con-tains the antisymmetric tensor ε αβγδ . Hence, such typeof a symmetry is realized in any SU (4) model whose fieldcontent does not allow for the SU (4) Levi-Civita symbolto occur in the interaction Lagrangian at the renormaliz-able level; for example, in [65], the corresponding numberis called B (cid:48) .Having the M -charge at hand, we can combine it withthe gauge charge (6) as B = 14 ( M + [ B − L ]) , (A3)which obviously yields the baryon number (see Table I). As one can verify readily, in the model under consid-eration both [ B − L ] and M are spontaneously brokenby (cid:104) χ (cid:105) whilst their sum (A3) remains a good symmetryeven in the asymmetric phase.On a more general ground, one can rephrase the sameargument as follows: If M defined as (A2) is a good sym-metry of the unbroken-phase theory, there is no SU (4) C -Levi-Civita tensor in its Lagrangian L . This means thatthere is no SU (3) c -Levi-Civita tensor in the broken-phaseLagrangian L either. Consequently, a global charge de-fined as3 B = (cid:0) SU (3) indices (cid:1) − (cid:0) SU (3) indices (cid:1) (A4)generates a good symmetry of L and, hence, B – theusual SM baryon number – is perturbatively conserved.Let us also note that there are two slightly differentcandidates for the lepton number, none of which is, how-ever, related to a fully conserved quantity in our model.The first option is intuitive, L = B − [ B − L ] . (A5)Here, U (1) L is a good symmetry of the classical actionbut it is spontaneously broken, together with [ B − L ], by (cid:104) χ (cid:105) . The alternative, L (cid:48) = F − B, (A6)is, on the other hand, preserved by the vacuum but ex-plicitly broken by the Majorana mass term because F isso. [1] BaBar, J. P. Lees et al., Phys. Rev. Lett. , 101802(2012), arXiv:1205.5442.[2] BaBar, J. P. Lees et al., Phys. Rev. D88 , 072012 (2013),arXiv:1303.0571.[3] Belle, A. Matyja et al., Phys. Rev. Lett. , 191807(2007), arXiv:0706.4429.[4] Belle, A. Bozek et al., Phys. Rev. D82 , 072005 (2010),arXiv:1005.2302.[5] Belle, M. Huschle et al., Phys. Rev.
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