Composite leptoquarks and anomalies in B -meson decays
CCavendish-HEP-14/14
Composite leptoquarks and anomalies in B -mesondecays Ben Gripaios a M. Nardecchia a,b
S. A. Renner b a Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge, CB3 0HE,UK b DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We attempt to explain recent anomalies in semileptonic B decays at LHCbvia a composite Higgs model, in which both the Higgs and an SU (2) L -triplet leptoquarkarise as pseudo-Goldstone bosons of the strong dynamics. Fermion masses are assumedto be generated via the mechanism of partial compositeness, which largely determines theleptoquark couplings and implies non-universal lepton interactions. The latter are neededto accommodate tensions in the b → sµµ dataset and to be consistent with a discrepancymeasured at LHCb in the ratio of B + → K + µ + µ − to B + → K + e + e − branching ratios.The data imply that the leptoquark should have a mass of around a TeV. We find that themodel is not in conflict with current flavour or direct production bounds, but we identifya few observables for which the new physics contributions are close to current limits andwhere the leptoquark is likely to show up in future measurements. The leptoquark willbe pair-produced at the LHC and decay predominantly to third-generation quarks andleptons, and LHC13 searches will provide further strong bounds. a r X i v : . [ h e p - ph ] J un ontents b → s(cid:96)(cid:96) fits and leptoquark quantum numbers 33 Details of the composite model 5 B decays 124.1.1 Fit to muonic ∆ B = ∆ S = 1 processes 124.1.2 R K b → sνν K + → π + νν µ → eγ and other radiative processes 174.2.5 Comments on other constraints and predictions 174.3 Direct searches at the LHC 18 The first run of the LHC brought us the long-awaited discovery of the Higgs boson, butno firm evidence for the physics beyond the Standard Model (SM) needed to avoid fine-tuning of the weak scale. This was perhaps not unexpected, given the plethora of indirectconstraints on new physics coming from, e.g. , flavour physics and electroweak precisiontests. Typically these point at scales of new physics way beyond a TeV; even when weinvoke all the dynamical dirty tricks that we know of, the best we can do is to lower thepossible scale of new physics to perhaps a few TeV. Therefore, there seems to be, nolensvolens , at least a small tuning in the weak scale.An unfortunate consequence of this is that, even if the electroweak scale is mostlynatural, we may struggle to probe the associated dynamics at the LHC. At best, we mighthope that one or two new states are anomalously light, such that we can either produce– 1 –hem on-shell, or see their effects indirectly in rare processes. It is clear that discoveryof such states will require painstaking work, including careful scrutiny of all discrepanciesbetween the data and SM predictions.In this work, we ask, in this vein, whether anomalies recently observed in semileptonicdecays of B -mesons at LHCb [4–6] can be explained by a model in which a scale of afew TeV arises naturally via strong dynamics. The necessary residual fine tuning requiredto generate the electroweak scale can be achieved by making the Higgs boson a pseudo-Goldstone boson (PGB) of global symmetries of the strong dynamics sector [7–10]. TheHiggs potential (and thus the electroweak scale) arises due to the breaking of the globalsymmetries by the SM gauging and by couplings to fermions, and one can hope thatthere is an accidental cancellation in the various contributions, whence a somewhat lowerelectroweak scale emerges. The Yukawa couplings of the SM are assumed to arise viathe mechanism of partial compositeness [11], which not only provides a rationale for thestructure of masses and mixings observed in the quark sector, but also provides a paradigmfor suppressing large flavour-violating effects in processes involving the light fermions, wherethe experimental constraints are strongest.The general framework of partial compositeness is an obvious choice for explaining theanomalies, which appear in processes involving second and third generation quarks, andwhich appear to require new physics in muonic, but not electronic processes. To fit thedetailed structure of the anomalies, we hypothesize that they are due to the presence in sucha model of an anomalously light ( c. TeV, as it turns out) leptoquark. As pointed out in [12], partial compositeness models necessarily feature a plethora of composite coloured fermionstates, namely the composite quarks, and so it would be something of a surprise if theydid not also feature composite coloured scalar states, which could couple as leptoquarks ordiquarks [13]. Moreover, one can easily arrange for a leptoquark state to be rather lighterthan the other resonances of the strong sector, by making it a PGB of the same symmetrybreaking that gives rise to the Higgs boson. A disadvantage of such models is that, being strongly coupled, we cannot calculate ad libitum . But we can use na¨ıve dimensional analysis (NDA) to compute and makepredictions modulo O (1) corrections. Using this framework, we find that the anomaliessingle out one among the possible SM irreps that allow leptoquark couplings, viz. a tripletunder both SU (3) c and SU (2) L . This leptoquark is one of those identified in a recentanalysis [14] of LHCb B meson anomalies, in which just two, non-vanishing leptoquarkcouplings (to b quarks, s quarks and muons) were invoked in an ad hoc fashion to fit theanomalies. In contrast, the model considered here is underpinned by a complete (albeitpresently uncalculable) framework for flavour physics, and all leptoquark couplings are Such light states might be present for a variety of reasons. For example, they might be desirablebecause they reduce fine-tuning (such as a light top squark and gluino in SUSY), or they might arisebecause of symmetries (such as additional pseudo-Goldstone boson states in composite Higgs models [1–3]). In fact, ref. [12], argued that evidence for such leptoquarks should first appear in b → sµµ processes,precisely where the anomalies are now observed. The leptoquark is nevertheless expected to be somewhat heavier than the Higgs [12], both because itreceives contributions to its potential from the QCD coupling and because we expect that the Higgs masshas been slightly tuned. – 2 –on-vanishing, with magnitude fixed by the degrees of compositeness of each of the SMfermion multiplets, giving 15 mixing parameters. In the quark sector, all but one of theseparameters is fixed by measurements of quark masses and the CKM matrix; there is moreambiguity in the lepton sector, but we find that everything can be fixed by assumingthat the mixings of the left and right-handed lepton multiplets are comparable. Thisassumption is a plausible one, from the point of view of the UV flavour dynamics, and hasthe additional benefit that new physics (NP) corrections to the most severely constrainedflavour-violating observable, µ → eγ , are minimized. As a result, we are left with just 3free parameters in the model: the mass, M , of the leptoquark, the coupling strength, g ρ , ofthe strong sector resonances, and the degree of compositeness, (cid:15) q , of the third generationquark doublet. Furthermore, all processes to which the leptoquark contributes (with theexception of meson mixing) result in constraints on the single combination x ≡ √ g ρ (cid:15) q /M .Thus the model is extremely predictive. We find that the preferred range of x correspondsto plausible values of the 3 underlying parameters of the strongly coupled theory (in whichthe weak scale is slightly tuned), namely g ρ ∼ π , M ∼ TeV, and (cid:15) q ∼
1. Thus, g ρ and (cid:15) q lie close to their maximal values, meaning that one cannot evade future direct searches atthe LHC by scaling up M and g ρ .As for the existing bounds, we find that there is no obvious conflict, but that there ispotential to see effects in µ → eγ , K + → π + νν , and B + → π + µ + µ − , in the near future.Moreover, the required mass range for the leptoquark is not far above that already excludedby LHC8, and so there is plenty of scope for discovery in direct production at LHC13.The outline is as follows. In the next Section, we describe the data anomalies andreview fits thereto using higher-dimensional SM operators. We also show that they can bedescribed by a leptoquark carrying the representation ( , , ) of the SU (3) × SU (2) × U (1)gauge group. In § &c. In §
4, we discuss important constraintson the model and describe the prospects for direct searches for the leptoquark at LHC13and indirect searches using flavour physics. b → s(cid:96)(cid:96) fits and leptoquark quantum numbers The anomalies that we wish to explain were observed at LHCb in semileptonic B mesondecays involving a b → s quark transition. These may be described via the low-energy,effective hamiltonian H eff = − G F √ V ∗ ts V tb ) (cid:88) i C (cid:96)i ( µ ) O (cid:96)i ( µ ) , (2.1)where O (cid:96)i are a basis of SU(3) C × U(1) Q -invariant dimension-six operators giving rise tothe flavour-changing transition. The superscript (cid:96) denotes the lepton flavour in the final– 3 –tate ( (cid:96) ∈ { e, µ, τ } ), and the operators O (cid:96)i are given in a standard basis by O ( (cid:48) )7 = e π m b (cid:0) ¯ sσ αβ P R ( L ) b (cid:1) F αβ , O (cid:96) ( (cid:48) )9 = α em π (cid:0) ¯ sγ α P L ( R ) b (cid:1) (¯ (cid:96)γ α (cid:96) ) , (2.2) O (cid:96) ( (cid:48) )10 = α em π (cid:0) ¯ sγ α P L ( R ) b (cid:1) (¯ (cid:96)γ α γ (cid:96) ) . We neglect possible (pseudo-)scalar and tensor operators, since these have been shown [14,15] to be constrained to be too small (in the absence of fine-tuning in the electron sector)to explain LHCb anomalies. In the SM, the operator coefficients are lepton universal andthe operators that have non-negligible coefficients are O , O (cid:96) , and O (cid:96) , with C SM = − . ,C SM = 4 . , (2.3) C SM = − . . at the scale m b [16].The first tension with the SM was observed last year in angular observables in thesemileptonic decay B → K ∗ µ + µ − [4, 5]. The rˆole of theoretical hadronic uncertainties inthe discrepancy is not yet clear [16–23]. Nevertheless, several model-independent analy-ses [19, 24–27] have been performed on the B → K ∗ µ + µ − decay data, as well as on other,relevant, semileptonic and leptonic processes, allowing for the possibility of new physicscontributions to the effective operators in eq. (2.2). There seems to be a consensus that,if only a single Wilson coefficient is allowed to be non-vanishing, then NP contributions tothe effective operator O µ are preferred, with the NP coefficient C NP of this operator beingnegative. A number of models of NP were proposed to explain this effect [28–33].Earlier this year LHCb measured another discrepancy in B decays. To wit, it wasfound that a certain ratio, R K , of branching ratios of B → Kµ + µ − to B → Ke + e − lay2.6 σ below the SM prediction [6]. Specifically, the observable is defined as R K = (cid:82) dq d Γ( B + → K + µ + µ − ) dq (cid:82) dq d Γ( B + → K + e + e − ) dq , (2.4)where q is the invariant mass of the di-lepton pair and the integral is performed overthe interval q ∈ [1 ,
6] GeV . Like the B → K ∗ µ + µ − decay, these processes proceed viaa b → s(cid:96)(cid:96) transition. The observable R K has the advantage of being theoretically well-understood, predicted to be almost exactly 1 in the SM [34] (specifically, 1 . ± . R K cannot be explained bylepton-flavour-universal NP, nor by any of the sources of theoretical uncertainty that mightunderlie the B → K ∗ µ + µ − anomalies. Analyses and fits including the R K data and otherrecent measurements were performed in [14, 22, 36, 37]. Due to the lepton non-universalityrequired by the R K data, these analyses allowed the electronic and muonic Wilson coef-ficients to differ. They found that a negative contribution to C µ remains favoured, whilecontributions to electronic Wilson coefficients C ei were found to be consistent with zero, but– 4 –ould have large deviations therefrom, due to larger experimental uncertainties in electronicmeasurements.One could argue that the ‘axial-vector’ basis, whilst convenient for studying physicsbelow the weak scale, is not the most natural choice in the context of models of NPabove the weak scale, which must respect the chiral gauge symmetries of the SM. In theabsence of multiple couplings or particles that have been somehow tuned (perhaps byadditional symmetries), NP is likely to generate operators that are coupled to a specificlepton chirality, and thus aligned with a ‘chiral basis’ in which C = − C , C = C , C (cid:48) = − C (cid:48) , C (cid:48) = C (cid:48) . Given this, the recent analyses have also made use of this basis [14,22, 36, 37] . They find that, when looking at NP contributions in a single Wilson coefficientat a time, the best fit in this basis is achieved by a negative contribution to C µ = − C µ .Therefore, of the possible scalar leptoquarks, which always generate contributionsto one Wilson coefficient in the chiral basis, the obvious choice to explain the anomaliesappears to be that with quantum numbers ( , , / C µ = − C µ at tree level. This leptoquark was already considered to explain R K in [14],in a scenario in which its only non-zero couplings were to bµ and to sµ . With the required quantum numbers of the leptoquark in hand, we now embed the lep-toquark in a composite Higgs model. We assume, then, the presence of a new strongsector and of an elementary sector. The strong sector is characterised by a mass scale m ρ and by a single coupling among the resonances, which is denoted by g ρ . We expect thestrong sector in isolation to have a global symmetry G which is spontaneously broken bythe strong dynamics to a subgroup H . The SM gauge interactions are introduced in thestrong sector by gauging a subgroup of H . We identify the Goldstone bosons coming fromthe breaking G / H with the Higgs boson H and the leptoquark state Π. So as to avoidlarge contributions to other flavour observables, we seek a model in which the coset spacecontains only H and Π.We postulate that the SM fermion Yukawa couplings are generated via the paradigmof partial compositeness [11]. The basic assumption is that elementary states f ai (where a ∈ { q, u, d, (cid:96), e } and i is the family index) couple linearly to fermionic operators O ai of thestrong sector. For example, the relevant lagrangian required to generate the masses of theup quarks is, schematically, L ⊃ g ρ (cid:15) q O q q + g ρ (cid:15) u O u u + m ρ (cid:0) O q O q + O u O u (cid:1) + g ρ O q H O u . (3.1) For a review see [38]. Vector leptoquarks with charges ( , , /
3) or ( , , /
3) also generate the required structure, butcannot be directly realized as Goldstone bosons. Another interpretation of both the R K anomaly and a deviation seen at CMS, in the context of R -parityviolating supersymmetry, was given in [39]. For a recent review see [40]. – 5 –fter electroweak symmetry breaking (EWSB), the resulting light mass eigenstates corre-spond to the SM fields and are given by linear combinations of the form f aSM = cos θ a f a + sin θ a O a , (3.2)with sin θ a = O ( (cid:15) a ). Thus, the parameters (cid:15) ai have a physical meaning: they measure thedegree of compositeness of the SM fields. If (cid:15) ai (cid:46)
1, we have that (at leading order in (cid:15) ) f SM ≈ f and the projections of the composite operators onto the SM fields are givenby ( O a ) SM ∼ (cid:15) a f SM . In this way, projecting operators such as g ρ O q H O u along the SMcomponents, we can read off the strength of the Yukawa interactions. In particular, for thethe up and down quarks, we have( Y u ) ij ∼ g ρ (cid:15) qi (cid:15) uj , ( Y d ) ij ∼ g ρ (cid:15) qi (cid:15) dj . (3.3)Throughout this Section, we use the symbol ∼ to mean a relation that holds up to anunknown O (1) coefficient whose value is fixed by the uncalculable strong sector dynamics.With an appropriate choice of the values of (cid:15) qi , (cid:15) ui , and (cid:15) di , it is possible to reproduce thehierarchy of the quark masses and the mixing angles of the CKM matrix. We find g ρ v(cid:15) qi (cid:15) ui ∼ m ui , g ρ v(cid:15) qi (cid:15) di ∼ m di (3.4) (cid:15) q (cid:15) q ∼ λ, (cid:15) q (cid:15) q ∼ λ , (cid:15) q (cid:15) q ∼ λ , where v is the Higgs VEV, λ = 0 .
23 is the Cabibbo angle and m ui and m di are the masses ofthe up- and down-type quarks, respectively. In our framework, then, the Yukawa sector isdescribed by 10 parameters ( g ρ , (cid:15) qi , (cid:15) ui , (cid:15) di ). The phenomenological relations (3.4) can be usedto reduce the number of free parameters that we can use to fit the anomalies. Indeed, thereare 8 independent relations in (3.4) and we choose to parametrize everything in terms of g ρ and (cid:15) q . In the lepton sector, there is more arbitrariness in the values of (cid:15) (cid:96)i and (cid:15) ei . This is dueto the fact that there are several mechanisms that can be envisaged for introducing massterms in the neutrino sector. In order to make progress, we shall assume the left and rightmixing parameters to be of the same order, (cid:15) ei ≈ (cid:15) (cid:96)i . This assumption about the unknownflavour dynamics at high scales is a plausible one, but it also has the phenomenologicaladvantage that it mitigates constraints on NP coming from lepton flavour violating (LFV)observables, such as µ → eγ , which are the most problematic flavour-violating observablesfor partial compositeness models [41, 42]. Indeed, physics at the scale m ρ generates acontribution to the radiative LFV decays of the form Γ( (cid:96) i → (cid:96) j γ ) ∼ (cid:12)(cid:12)(cid:12) (cid:15) (cid:96)i (cid:15) ej (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:15) (cid:96)j (cid:15) ei (cid:12)(cid:12)(cid:12) .Considering the mass constraints (cid:15) (cid:96)i (cid:15) ei = m ei g ρ v δ ij , it is easy to show that (cid:12)(cid:12)(cid:12) (cid:15) (cid:96)i (cid:15) ej (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:15) (cid:96)j (cid:15) ei (cid:12)(cid:12)(cid:12) isminimized when (cid:15) (cid:96)i (cid:15) (cid:96)j ∼ (cid:15) ei (cid:15) ej ∼ (cid:115) m ei m ej . (3.5)Evidently, this condition is implied by (but does not imply) our assumption that the leftand right leptonic mixings are equal. – 6 –ermion Mass e µ
103 MeV τ d . +1 . − . MeV s +14 − MeV b . ± .
08 GeV u . +0 . − . MeV c . ± .
07 GeV t . ± . Figure 1 . Values of running fermion masses at the scale µ = 1 TeV [43]. Mixing Parameter Value (cid:15) q = λ (cid:15) q . × − (cid:15) q (cid:15) q = λ (cid:15) q . × − (cid:15) q (cid:15) u = m u vg ρ λ (cid:15) q . × − / ( g ρ (cid:15) q ) (cid:15) u = m c vg ρ λ (cid:15) q . × − / ( g ρ (cid:15) q ) (cid:15) u = m t vg ρ (cid:15) q g ρ (cid:15) q ) (cid:15) d = m d vg ρ λ (cid:15) q . × − / ( g ρ (cid:15) q ) (cid:15) d = m s vg ρ λ (cid:15) q . × − / ( g ρ (cid:15) q ) (cid:15) d = m b vg ρ (cid:15) q . × − ( g ρ (cid:15) q ) (cid:15) (cid:96) = (cid:15) e = (cid:16) m e g ρ v (cid:17) / . × − /g / ρ (cid:15) (cid:96) = (cid:15) e = (cid:16) m µ g ρ v (cid:17) / . × − /g / ρ (cid:15) (cid:96) = (cid:15) e = (cid:16) m τ g ρ v (cid:17) / g / ρ Figure 2 . Partial compositeness mixing parameters and values.
In this way, we are able to fix all parameters in the lepton sector in terms of g ρ , and soall the NP effects of the model are parameterized by M , g ρ , and (cid:15) q . The phenomenologicalinputs and the expressions of the various mixing parameters are summarised in Figs. 1and 2.We may now determine the leptoquark couplings, as follows. Similarly to [44], below thescale of the strongly-coupled resonances we can describe the low energy physics by aneffective field theory (EFT) of the form L = m ρ g ρ L (0) (cid:32) g ρ (cid:15) ai f ai m / ρ , D µ m ρ , g ρ Hm ρ , g ρ Π m ρ (cid:33) . (3.6)– 7 – ij / ( c ij g / ρ (cid:15) q ) j = 1 j = 2 j = 3 i = 1 1 . × − . × − . × − i = 2 2 . × − . × − . × − i = 3 1 . × − . × − . Figure 3 . Values of leptoquark couplings, λ ij , where i denotes the lepton generation label and j the quark generation label. In the strongly-coupled, UV theory we expect the presence of an operator of the form g ρ Π O L O Q , where O Q (or O L ) is a composite operator with the same quantum numbers asa SM quark (or lepton). Below the scale m ρ , this operator generates a contribution to L of the form ∼ g ρ (cid:15) (cid:96)i (cid:15) qj Π (cid:96) i q j . At low energies, the renormalizable lagrangian of the model is L = L SM + ( D µ Π) † D µ Π − M Π † Π + λ ij q cLj iτ τ a (cid:96) Li Π + h.c. , (3.7)with λ ij = g ρ c ij (cid:15) (cid:96)i (cid:15) qj , where we have omitted quartic terms involving H and Π that are notrelevant to our discussion. Note that we have explicitly re-introduced the c ij parametersthat are expected to be of O (1), but are otherwise unknown. We summarise the values ofthe leptoquark couplings in Fig. 3. Here we supply a coset space construction that gives rise to the required SM quantumnumbers for the Higgs and leptoquark fields. First we describe the pattern of spontaneousbreaking of the symmetry of the strong sector G / H , and the embedding of the SM gaugegroup SU (3) C × SU (2) L × U (1) Y therein. We then discuss additional symmetry structurerequired to avoid constraints from nucleon decay and neutron-antineutron oscillations.To build a coset, we start from the minimal composite Higgs model [10], in whicha single SM Higgs doublet arises from the spontaneous breaking of SO (5) to SU (2) H × SU (2) R , with H transforming as a ( , ) of the unbroken subgroup. We must now enlargethe coset space somehow to include the leptoquark Π and its conjugate Π † . To see howthis may be achieved, consider first a model with just the leptoquark and no Higgs boson.This can be achieved using SO (9) broken to SU (4) × SU (2) Π . The 6 Goldstone bosons,(Π , Π † ), transform as ( , ).Now form the direct product of SO (5) and SO (9) and consider the coset space SO (9) × SO (5) SU (4) × SU (2) Π × SU (2) H × SU (2) R . (3.8)This has, of course, the same Goldstone boson content as the two models above. Thetrick is to somehow embed the SM gauge group in H so as to get the right charges for H and Π. To do so we embed SU (3) C into SU (4). Explicitly, SU (4) contains a maximalsubgroup SU (3) C × U (1) ψ , and the decomposition of the 6-d irrep of SU (4) under this groupgives = / + − / . We then embed SU (2) L as the diagonal subgroup of SU (2) H × – 8 – U (2) Π , while the hypercharge gauge group U (1) Y is embedded as T Y = − T ψ + T R + T X , where T ψ generates U (1) ψ , T R belongs to the SU (2) R algebra, and U (1) X is anadditional symmetry (under which the Higgs and the leptoquark are uncharged) which maybe required to reproduce the correct SM hypercharge assignments. It is now straightforwardto show that the SM quantum numbers of H and Π+Π † are respectively ( , , /
2) and( , , / , , − / H . In fact, a number of repre-sentations are available. One suitable assignment is O q ∼ (4 , , , +1 / , O u ∼ (4 , , , +1 / , O d ∼ (4 , , , +1 / , (3.9) O L ∼ (4 , , , − / , O e ∼ (4 , , , − / where the subscript denotes the charge under the U (1) X symmetry. It is straightforwardto check that this assignment permits tri-linear couplings between the fermionic resonancesand H and Π that yield the desired Yukawa and leptoquark couplings after mixing withthe elementary fermions.An advantage of this assignment is that we can use it to protect Γ( Z → bb ). This isdesirable since, with g ρ ∼ π , there would otherwise be sizable corrections to Γ( Z → bb ),even with m ρ ∼
10 TeV. The protection cannot be achieved in exactly the same way as in[45], because the semi-direct product ( SU (2) L × SU (2) R ) (cid:111) Z used there is not a subgroupof G . But we can instead use the symmetry ( SU (2) H × SU (2) R ) (cid:111) Z , with much thesame result. In a nutshell (for more details, see [46]), the group SO (5) ⊂ G contains notjust SU (2) H × SU (2) R , but also the larger subgroup K ≡ ( SU (2) H × SU (2) R ) (cid:111) Z . Werequire: (i) that this larger group be contained in H ; (ii) that b L couple to a resonance ofthe strong sector transforming as a (2 ,
2) under SU (2) H × SU (2) R ⊂ K and as either thetrivial irrep or the sign irrep under Z ⊂ K ; and (iii) that the coupling of b L to the strongsector respect the subgroup ( U (1) H × U (1) R ) (cid:111) Z ⊂ K . With these three requirements, astraightforward modification of the arguments given in [45, 46] shows that there can be nocorrections to Γ( Z → bb ).There is, however, a disadvantage with this assignment, in that the linear mixingbetween O q and q breaks the SU (2) L × SU (2) R custodial symmetry, which is often invokedto protect m W m Z . Since (cid:15) q = 1, these corrections are unsuppressed. Happily, we find thanksto m ρ ∼
10 TeV and to the presence of light custodians [47, 48], we are consistent with thebounds coming from EWPT observables. The global symmetry G is broken explicitly by the gauging of the SM group, as well asby the linear couplings between the elementary and composite sector. As a result of thesebreakings, the PGBs get a mass term. NDA suggests that the main contribution to the Or rather, strictly speaking, its universal cover Sp (2). Note that with the alternative assignment O q ∼ (4 , , , , O u ∼ (4 , , , , O d ∼ (4 , , , , O L ∼ (4 , , , , O e ∼ (4 , , , U (1) X is not necessary), the linear mixing between O q and q is SU (2) L × SU (2) R invariant, and corrections to m W m Z are suppressed by powers of ( (cid:15) u ) (cid:28)
1. Butthen one must relinquish custodial protection of Γ( Z → bb ). – 9 –ffective potential of the Higgs comes from from the top Yukawa coupling. This implies anegative contribution to the Higgs mass parameter [10], which can trigger EWSB, and theresulting Higgs mass is expected to be of order m H ∼ y t π m ρ . In contrast to the Higgsboson, the composite leptoquark gets its dominant mass term contribution from QCD. Theresulting leptoquark mass is of order m ∼ g s π m ρ and is positive-definite, avoiding thedanger of colour- and charge-breaking vacua.We now move on to discuss constraints from nucleon decay, &c . In models with TeVscale strong dynamics, we cannot expect the accidentally symmetries of the SM that leadto conservation of baryon and lepton number to be preserved. This problem is exacerbatedin our model with a light leptoquark state, since the SM gauge symmetry allows a ( , , )leptoquark to couple to both qq and q(cid:96) , and thus mediate proton decay.We now assess whether additional global symmetries can be imposed to prevent suchdecays. Our objective is to allow the coupling to q(cid:96) , but not that to qq . Evidently, then, q and (cid:96) must carry different charges, e q and e (cid:96) , say, under such a symmetry. We mustnow decide whether the leptoquark itself should carry charge or not.The easiest option to realise is for the leptoquark not to carry a charge. Then thecorresponding symmetry can lie outside of the SO (9) group of which the Π is a Goldstoneboson. Then the leptoquark coupling is allowed if e q + e (cid:96) = 0. A problem with any suchsymmetry is that it cannot forbid decays of 3 quarks to 3 anti-leptons. So, while the usualsuspects, like p → e + π are forbidden, decays such as p → e + ν and n → ν are not.In our framework the most stringent bound comes from searches for pp → µ + ν decays,where Γ < . × − GeV [49]. The leading contribution to this processes is generatedby the dimension-9 operator ( qqd c † )( (cid:96)(cid:96)e c † ), with τ neutrinos. A NDA estimate givesΓ( p → µ + ν τ ν τ ) NDA = m p (4 π ) (cid:18) g ρ (cid:15) d ( (cid:15) q ) (cid:15) (cid:96) ( (cid:15) (cid:96) ) M (cid:19) = 4 . × − GeV − . (3.10)It is then clear that the searches for such decays suffice to rule out a model with compos-iteness at multi-TeV scales. In comparing with the bound, we have used the values M = 1 TeV , g ρ = 4 π, and (cid:15) q = 1 , (3.11)and we shall continue to do so henceforth.We need, therefore, to explore the alternative option, which is to look for a symmetrythat lies (at least partly) within SO (9), such that the leptoquark is charged. A simpleexpedient is to use the Z ⊂ SO (9) symmetry whose non-trivial element in the definingrepresentation of SO (9) is the matrix (cid:32) − I I (cid:33) , where I n is the n × n identity matrix.This element commutes with SO (6) × SO (3) (and therefore is unbroken by the gauging ofthe SM subgroup) but anti-commutes with the broken generators in SO (9) /SO (6) × SO (3),meaning that the leptoquarks transform under Z as Π → − Π. Now, by insisting that the Z be unbroken by the strong dynamics and the couplings to elementary fermions, the We assume that all particles come in 1-d representations of the symmetry, so as not to have to introduceadditional states. – 10 –iquark coupling Π qq is forbidden. Provided, moreover, that the elementary q and (cid:96) areassigned opposite charges, the leptoquark coupling Π (cid:96)q is allowed. Yukawa couplings canbe retained by assigning the elementary ( u c , d c ) and e c to have the same charges as q and (cid:96) , respectively.Such a symmetry (which may be thought of as either a baryon or lepton parity)stabilizes nucleons completely, and so also solves potential problems from generic operatorsgenerated by the heavier resonances of the strong dynamics. Its drawback is that it cannotforbid neutron-antineutron oscillations, for which there are again strong experimentalconstraints. There are two dimension 9 operators in the EFT that could give a contributionto this process, namely qqqq ( d c d c ) † and u c d c d c u c d c d c . The low-energy effects of theseoperators are subject to large hadronic uncertainties; we estimate a rough bound on thenecessary scale as Λ (cid:38)
100 TeV.In our leptoquark model, we expect to generate the operator g ρ ( (cid:15) q ) ( (cid:15) d ) M qqqq ( d c d c ) † . (3.12)Using the nominal values in (3.11) and matching with the previous expression, we findΛ = 188 TeV. Given the high dimension of the operator, this scale comes with a largeuncertainty, but it would seem that we are safe.Finally, we remark that we could, of course, invoke both symmetries discussed above,in order to forbid both nucleon decay and oscillations absolutely. At tree level, the effects of the leptoquark on flavour physics observables can be studiedusing the effective lagrangian L effLQ = (cid:88) ij(cid:96)k λ ij ( λ (cid:96)k ) ∗ M (cid:2) ( q j τ a γ µ P L q k )( (cid:96) i τ a γ µ P L (cid:96) (cid:96) ) + 3( q j γ µ P L q k )( (cid:96) i γ µ P L (cid:96) (cid:96) ) (cid:3) , (4.1)where i, j, k, (cid:96) ∈ { , , } are generation indices. We work in a basis where the CKM matrixacts on the up sector such that q j is the quark doublet, q j = (cid:16) V † jkCKM u kL , d jL (cid:17) T , and (cid:96) i isthe lepton doublet, (cid:96) i = (cid:0) ν i , e iL (cid:1) T . We assume that the mass differences between thecomponents of the leptoquark triplet are small compared to the masses themselves, so thatthe components can be assumed to have a common mass, M . Therefore we may write L effLQ = (cid:88) ij(cid:96)k λ ij ( λ (cid:96)k ) ∗ M (cid:104) (cid:0) d L γ µ d L (cid:1) kj ( e L γ µ e L ) (cid:96)i + 2 ( u (cid:48) L γ µ u (cid:48) L ) kj ( ν L γ µ ν L ) (cid:96)i + (cid:0) d L γ µ d L (cid:1) kj ( ν L γ µ ν L ) (cid:96)i + ( u (cid:48) L γ µ u (cid:48) L ) kj ( e L γ µ e L ) (cid:96)i (4.2)+ ( u (cid:48) L γ µ d L ) kj ( e L γ µ ν L ) (cid:96)i + (cid:0) d L γ µ u (cid:48) L (cid:1) kj ( ν L γ µ e L ) (cid:96)i (cid:105) , where u (cid:48) jL = V † jkCKM u kL . All unprimed fields are mass eigenstates. For a review see [50]. We neglect neutrino masses. – 11 –e now comment briefly on the qualitative consequences of the various operators thatappear above.(i) Flavour changing neutral currents (FCNC) in the down quark sectorThese are generated by the operators (cid:0) d L γ µ d L (cid:1) kj ( e L γ µ e L ) (cid:96)i and (cid:0) d L γ µ d L (cid:1) kj ( ν L γ µ ν L ) (cid:96)i .They can mediate meson decays via the transitions b → s(cid:96)(cid:96) , b → sνν , s → d(cid:96)(cid:96) , s → dνν , b → d(cid:96)(cid:96) and b → dνν .The b → s(cid:96)(cid:96) transition is the main motivation for this work and will be discussed inmore detail below. The decays involving neutrinos can have large NP contributions,because couplings to tau neutrinos are large in the partial compositeness frameworkconsidered here. We provide a quantitative analysis of the decays B → K ( ∗ ) νν and K → πνν below. Constraints on leptoquark couplings from measurements of (lepton-flavour-conserving) K and B decays are summarized in Fig. 4 below, excluding b → s(cid:96)(cid:96) and b → sνν processes, which will be discussed in more detail in the text. Lepton-flavour-violating (LFV) processes, recently investigated in [51], are also possible inour set-up, but current bounds on these are weak. We will comment more on LFVprocesses in § u (cid:48) L γ µ u (cid:48) L ) kj ( ν L γ µ ν L ) (cid:96)i and ( u (cid:48) L γ µ u (cid:48) L ) kj ( e L γ µ e L ) (cid:96)i .They can mediate decays of charmed mesons via c → u(cid:96)(cid:96) and c → uνν transitions.Bounds on these processes are weak, and we know of no bounds for decays with τ leptons or neutrinos in the final state, which would receive the largest NP contribu-tions. These operators can also generate top decays into u or c quarks plus a pair ofcharged leptons or of neutrinos. The rates of these decays will be very small relativeto current limits on FCNC top quark decays [49] (which in any case search specificallyfor t → Zq , meaning they cannot be directly applied to leptoquarks). Since currentmeasurements of FCNC in the up sector do not provide strong constraints on ourmodel, we will not discuss them further.(iii) Charged currentsThese are generated by the operators ( u (cid:48) L γ µ d L ) kj ( e L γ µ ν L ) (cid:96)i and (cid:0) d L γ µ u (cid:48) L (cid:1) kj ( ν L γ µ e L ) (cid:96)i .Processes generated by these operators are also present at tree level in the SM, soNP contributions are not expected to be large relative to the SM predictions. Thelargest NP rates will occur in processes with τ or ν τ in the final state.With these considerations in mind, in the remainder of this Section we discuss thevalues of the model parameters that are needed to fit recent B -decay anomalies and thenlist important constraints on the model and predictions for its effects in other processes. B decays4.1.1 Fit to muonic ∆ B = ∆ S = 1 processes We consider recent results of [22], in which a fit to all available data on muonic (or lepton-universal) ∆ B = ∆ S = 1 processes is described. A part of that work involved allowing– 12 –ne Wilson Coefficient (or chiral combination thereof) to vary while assuming all othercoefficients are set to their SM values (for details of the fit please see [22]). The best fitvalue found in this way for the chiral combination relevant to our leptoquark is C NP µ = − C NP µ = − .
55, with 1 σ and 2 σ ranges C NP µ = − C NP µ ∈ [ − . , − .
36] (at 1 σ ) , (4.3) C NP µ = − C NP µ ∈ [ − . , − .
19] (at 2 σ ) . (4.4)It can be seen, by comparing the effective leptoquark lagrangian in (4.2) with the effectivehamiltonian in (2.1), that, for our model, C µNP = − C µNP = (cid:20) G F e ( V ∗ ts V tb )16 √ π (cid:21) − λ ∗ λ M = − . c ∗ c ( (cid:15) q ) (cid:18) M TeV (cid:19) − (cid:16) g ρ π (cid:17) , (4.5)and the requirements on the parameters areRe( c ∗ c ) = 2 . (cid:18) πg ρ (cid:19) (cid:18) (cid:15) q (cid:19) (cid:18) M TeV (cid:19) (Best fit) , (4.6)Re( c ∗ c ) ∈ [1 . , . (cid:18) πg ρ (cid:19) (cid:18) (cid:15) q (cid:19) (cid:18) M TeV (cid:19) (at 1 σ ) , (4.7)Re( c ∗ c ) ∈ [0 . , . (cid:18) πg ρ (cid:19) (cid:18) (cid:15) q (cid:19) (cid:18) M TeV (cid:19) (at 2 σ ) . (4.8). Thus, if this anomaly is to be explained, there are 3 immediate implications for theparameters of our model:1. the mass of the leptoquark states should be low enough, M (cid:46) (cid:15) q ∼ g ρ ∼ π .Indeed, if any one of these does not hold then we are forced to set Re( c ∗ c ) (cid:29)
1, implyingan inconsistency with the EFT paradigm described in the previous Section. R K R K , as defined in eq. (2.4), has been recently measured by LHCb to be R K = 0 . +0 . − . ± .
036 [6]. Roughly, adding errors in quadrature, we therefore take the measured value atthe 1 σ level to be within the range [0 . , . R K ≈ (cid:12)(cid:12)(cid:12) C SM + C µ NP10 + C µ (cid:48) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) C SM + C µ NP9 + C µ (cid:48) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) C SM + C e NP10 + C e (cid:48) (cid:12)(cid:12) + (cid:12)(cid:12) C SM + C e NP9 + C e (cid:48) (cid:12)(cid:12) , (4.9)– 13 –hich can be found from the full expression by neglecting the coefficient of the dipoleoperator, C . (In the SM C has a magnitude less than 10% that of C or C , and NPcontributions to it are constrained small by the measured branching ratio of B → X s γ ).The NP values of the Wilson coefficients are C µNP = − C µNP = − . c ∗ c ( (cid:15) q ) (cid:18) M TeV (cid:19) − (cid:16) g ρ π (cid:17) , (4.10) C eNP = − C eNP = − . × − c ∗ c ( (cid:15) q ) (cid:18) M TeV (cid:19) − (cid:16) g ρ π (cid:17) . (4.11)The values of C SM and C SM are given in eq. (2.3).We see that, due to the structure of partial compositeness, NP contributions in thedecay B + → K + e + e − are negligible. Neglecting these and the quadratic terms in C µNP , ,we obtain Re( c ∗ c ) ∈ [1 . , . (cid:18) πg ρ (cid:19) (cid:18) (cid:15) q (cid:19) (cid:18) M TeV (cid:19) (at 1 σ ) . (4.12)The allowed region thus has reasonable overlap with the 1 σ region found above usinga fit to muonic ∆ B = ∆ S = 1 observables. Therefore our model is able to fit the muonicdata and R K with no tension between the two. Of course, this is hardly surprising asseveral works [15, 22, 36] have pointed out the compatibility of the b → sµµ data with R K if the NP is predominantly in the muon sector, rather than the electron sector. Thisfeature is automatic in models with partial compositeness. The largest couplings of the composite leptoquark are to third generation quarks andleptons. Therefore, generically, the most important constraints and predictions will be inprocesses involving third generation quarks and fermions in initial or final states and alsoprocesses with third-generation fermions in a loop. This Section will look at some ofthese processes, discussing implications of current measurements on our model, as well ashighlighting promising channels for probing our scenario with future measurements. b → sνν Due to the SU (2) L structure of the leptoquark, it will couple to neutrinos as well ascharged leptons and thus induce b → sνν transitions. The importance of this channel ingeneral for pinning down NP has been recently emphasised in [52]. These B → K ∗ νν and B → Kνν decays are good channels to look for large effects from the composite leptoquarkwe consider. Indeed, since the identity of the neutrino cannot be determined in theseexperiments, large contributions from the processes involving tau neutrinos are expectedin our model. Thus our model predicts a much larger rate than that expected in modelswhere NP couples only to the second generation lepton doublet. Of course, this is only true generically, since the sensitivity depends not only on the size of the NPcontribution, but also on the experimental feasibility and also the size and nature of the competing SMcontributions. – 14 –urrent NP bounds from these decays can be found in [52], which are quoted in termsof ratios to Standard Model predictions. With a slight alteration of the notation of [52],so as not to cause confusion with the notation used here, the relevant quantities, and thelimits thereon, are R ∗ ννK ≡ B ( B → K ∗ νν ) B ( B → K ∗ νν ) SM < . , (4.13)and R ννK ≡ B ( B → Kνν ) B ( B → Kνν ) SM < . . (4.14)The leptoquark can in principle induce transitions involving any combination of neutrinoflavours, since it couples to all generations and also has flavour-violating couplings. Therewill be interference between NP and SM processes only in flavour-conserving transitions.The NP contributions to the ν τ ν τ and ν µ ν µ processes will induce a shift from unity in R ννK and R ( ∗ ) ννK given by∆( R ( ∗ ) ννK ) ττ = (cid:34) .
220 Re( c ∗ c ) + 0 . | c ∗ c | ( (cid:15) q ) (cid:18) M TeV (cid:19) − (cid:16) g ρ π (cid:17)(cid:35) ( (cid:15) q ) (cid:18) M TeV (cid:19) − (cid:16) g ρ π (cid:17) , ∆( R ( ∗ ) ννK ) µµ ≈ . × − Re( c ∗ c ) ( (cid:15) q ) (cid:18) M TeV (cid:19) − (cid:16) g ρ π (cid:17) . (The expression for ∆( R ( ∗ ) K ) µµ is approximate, because we have kept only the interferenceterm with the Standard Model, which is large compared to the term from purely NPcontributions.) The next biggest contribution comes from ν µ ν τ and ν τ ν µ final states. Inthese cases, there is no interference with the SM and the contribution is∆( R ( ∗ ) ννK ) µτ + ∆( R ( ∗ ) ννK ) τµ = 2 . × − (cid:0) | c ∗ c | + | c ∗ c | (cid:1) ( (cid:15) q ) (cid:18) M TeV (cid:19) − (cid:16) g ρ π (cid:17) . (4.15)As is clear from these equations, the most important contribution comes from the ν τ ν τ process. It is possible to pass the bound ∆( R ( ∗ ) ννK ) ττ < . R ννK and R ∗ ννK ( ∼
25% of the SM contribution)represent an interesting prediction of our composite leptoquarks scenario, which will betestable at the upcoming Belle II experiment [52, 53]. Our prediction can be comparedwith the case in which the leptoquark has only muonic couplings, in which the contributionsto ∆( R ( ∗ ) ννK ) are (cid:46)
5% (see section 4.5 of [52]). K + → π + νν Given that measurements involving neutrinos have the ability to probe some of the largestcouplings in our model – those involving third generation leptons – it is necessary to checkother rare meson decays with final state neutrinos.Following [54], (but rescaling the bound given there to match the slightly more recentmeasurement in [49]), the measurement of B ( K + → π + νν ) produces a bound (at 95%confidence level) on the real NP coefficient δC ν ¯ ν (defined in [54]) of δC ν ¯ ν ∈ [ − . , . . (4.16)– 15 –he branching ratio is given in terms of δC ν ¯ ν by B ( K + → π + νν ) = 8 . × − [1 + 0 . δC ν ¯ ν + 0 . δC ν ¯ ν ) ] . (4.17)Our leptoquark contributes to δC ν ¯ ν as δC ν ¯ ν = 0 .
62 Re( c c ∗ ) (cid:16) g ρ π (cid:17) ( (cid:15) q ) (cid:18) M TeV (cid:19) − , (4.18)via the dominant process involving a pair of tau neutrinos. So with c ∼ c ∼ O (1), and M ∼ TeV, our scenario passes current bounds.However the NA62 experiment, due to begin data-taking in 2015, will measure B ( K + → π + νν ) to an accuracy of 10% of the SM prediction [55]. This means it will be able to shrinkthe bounds on δC ν ¯ ν to δC ν ¯ ν ∈ [ − . , .
2] (4.19)at 95%. Thus, if c ∼ c ∼ O (1) and M ∼ TeV, measurements at NA62 will be sensitiveto our leptoquark.
The leptoquark we consider can mediate mixing between neutral mesons via box diagrams.This effect will be largest in B s mesons. From [38], the bound produced on the leptoquarkcouplings when both leptons exchanged in the box are taus (the dominant contribution inour scenario) is | λ λ ∗ | < π M ∆ m NPB s f B s m B s . (4.20)From [56], f B s = 0 .
231 GeV, and∆ m SMB s = (17 . ± . × (cid:126) s − = (1 . ± . × − MeV , (4.21)while from [49], the measured value of the mass splitting is∆ m B s = 17 . × (cid:126) s − = 1 . × − MeV . (4.22)Taking the uncertainty in the prediction to be roughly the size of the NP contribution, | ∆ m NPB s / ∆ m SMB s | < .
15 (as in [14]), then | λ λ ∗ | < . (cid:18) M TeV (cid:19) . (4.23)In terms of the parameters of our model this becomes | c c ∗ | < . (cid:18) πg ρ (cid:19) (cid:18) M TeV (cid:19) (cid:18) (cid:15) q (cid:19) . (4.24)We are able to pass this bound taking O (1) values for c and c and taking the otherparameters at values necessary to fit the anomalies as discussed above. The leptoquark willalso contribute to mixing of other neutral mesons. However bounds from the measurementof mixing observables are generally weaker than bounds from meson decays (see eg. [57]).– 16 – .2.4 µ → eγ and other radiative processes The leptoquark has only left handed couplings, meaning that we will not get chiral en-hancements to the branching ratio of µ → eγ . Nevertheless, the bound on B ( µ → eγ ) istight enough to be relevant for the model. The largest contributions come from diagramswith a loop containing either a top or a bottom quark, together with the leptoquark. Themost recent measurement was performed by the MEG collaboration [58], who found abound at 90% confidence level of B ( µ + → e + γ ) < . × − . Using the formula for therate given in [38], and neglecting all but the processes involving 3rd generation quarks inthe loop, | λ ∗ λ | < . × − (cid:18) M TeV (cid:19) , (4.25)which amounts to a bound on c ∗ c of | c ∗ c | < . (cid:18) πg ρ (cid:19) (cid:18) M TeV (cid:19) (cid:18) (cid:15) q (cid:19) . (4.26)This turns out to be a strong constraint for our model. Given that our EFT paradigmassumes c ij ∼ O (1), the bound is, roughly, saturated.Given our flavour structure we expect an even larger contribution to τ → µγ than to µ → eγ . However the current bound on the branching ratio of this process is B ( τ → µγ ) < . × − [49], which is several orders of magnitude larger than the model prediction.The process b → sγ can be generated via similar diagrams. Current bounds on thisprocess, which leave room for NP contributions up to about 30% of the SM prediction,lead to a bound on the combination | c ∗ c | of roughly | c ∗ c | (cid:46) (cid:16) πg ρ (cid:17) (cid:0) M TeV (cid:1) (cid:16) (cid:15) q (cid:17) . Despite the fact that contributions from leptoquark diagrams will be largest for processescontaining taus (or tau neutrinos) in the final state, we have not yet mentioned any boundsfrom meson decays with τ leptons in the final state. This is because existing bounds arevery weak due to the relative difficulty of tau measurements. The current bound [59] onthe decay B → Kτ + τ − from BaBar, B ( B → Kτ + τ − ) < . × − , is several ordersof magnitude larger than the NP prediction. Likewise the recent Belle measurement of B ( B + → τ + ν ) [60] has error bars much larger than the NP contribution (as does the SMprediction).We have discussed b → s(cid:96)(cid:96) processes and anomalies in previous subsections. Boundsfrom meson decays mediated by other FCNC processes in the down sector are summarisedin Fig. 4. The most constraining of these measurements is from the branching ratio of B + → π + µ + µ − , for which the bound is approximately saturated.Our leptoquark can appear in diagrams which contribute to the muon anomalousmagnetic moment, an observable which currently has a 2 . . σ discrepancy with SMcalculations [61]. However, as was pointed out in [62, 63], if a leptoquark couples only toone chirality of muon, as is the case for us, the couplings would need to be very large toexplain the measurement. Our scenario produces a prediction several orders of magnitude– 17 –oo small (for a mass of O (1 TeV)), and so does nothing to alleviate the current tensionbetween the SM and experiment.One hallmark of our model is that there should be only very small NP effects inthe electron sector. So decay measurements involving electrons should see no significantdeviations from the Standard Model in our scenario. A recent paper [22] contains a tablewith predictions of ratios of observables with muons in the final state versus those withelectrons for b → s(cid:96)(cid:96) processes. The predictions of our leptoquark model will, to a goodapproximation, coincide with those of the third column of their table, which contains thepredictions for a scenario with NP only in C µNP = − C µNP = − . B decays was recently discussed in detail in [51].There, the authors consider a model in which, similarly to our case, the NP contributionsto b → s(cid:96)(cid:96) decays arise in a V − A structure (ie. C (cid:96) = − C (cid:96) ) and the largest effectsare in the third generation of quarks and leptons. Interestingly, a special case of ourmodel can be made to fit into their framework, if we take all the O (1) coefficients c ij tobe equal (and for simplicity, equal to 1). Then the coupling denoted G in [51] is givenby G = ( g ρ /M ) ( (cid:15) q ) m τ v , and the mixing matrices U (cid:96)L i and U dL i therein are given by U dL = ( λ , λ ,
1) and U (cid:96)L i = (cid:112) m i /m τ . With these choices, we find that all bounds quotedin [51] are comfortably satisfied by the composite leptoquark model. More precise boundson LFV processes will certainly provide an interesting test of our model and other leptonnon-universal scenarios.Another recent paper [64] proposes double ratios of branching ratios as clean probes ofNP that is not lepton universal and couples to right-handed quarks. Since the leptoquarkwe consider has no couplings to right-handed quarks, measurements of these would be auseful test of the model if the B anomalies persist. If the leptoquark is light enough, as the arguments in § / , Π / and Π − / , with charges 4 /
3, 1 / − / c ij coefficients in the couplings to have a modulus equal to 1, their branching ratio to thirdgeneration quarks and leptons is around 94% or greater. So they predominantly decay asfollows: Π / → τ b, Π / → τ t or Π / → ν τ b, Π − / → ν τ t. The branching ratios are quite sensitive to the c ij coefficients, however, so other decaymodes ( eg. involving second generation leptons) may be important for different values of– 18 –ecay (ij)(kl) ∗ | λ ij λ ∗ kl | / (cid:0) M TeV (cid:1) | c ij c ∗ kl | (cid:0) g ρ π (cid:1) ( (cid:15) q ) / (cid:0) M TeV (cid:1) K S → e + e − (12)(11) ∗ < . < . × K L → e + e − (12)(11) ∗ < . × − < . × † K S → µ + µ − (22)(21) ∗ < . × − < . × K L → µ + µ − (22)(21) ∗ < . × − < . K + → π + e + e − (11)(12) ∗ < . × − < . × K L → π e + e − (11)(12) ∗ < . × − < . × K + → π + µ + µ − (21)(22) ∗ < . × − < . × K L → π ν ¯ ν (31)(32) ∗ < . × − < . † B d → µ + µ − (21)(23) ∗ < . × − < . B d → τ + τ − (31)(33) ∗ < . < . × † B + → π + e + e − (11)(13) ∗ < . × − < . × † B + → π + µ + µ − (21)(23) ∗ < . × − < . Figure 4 . 90% confidence level bounds [57] on leptoquark couplings from branching ratios of(semi-)leptonic meson decays involving b → d and s → d , rescaled to M = 1 TeV. A dagger denotesbounds that have been rescaled to newer measurements [49]. The final column gives bounds onpartial compositeness parameters in units of the nominal values in (3.11). c ij , even if they are all still O (1). The bounds and branching ratios in this section havebeen derived under the assumption that the modulus of all c ij coefficients be equal to 1, butwe will comment on the impact of lifting this assumption towards the end of the section.There will be electroweak mass splittings between the three leptoquark states, allowingthe heavier ones to decay to the lighter ones, but these decays will be subdominant to thosethrough the leptoquark couplings, if the mass splittings are small. Of the LHC leptoquarksearches, dedicated searches for third generation leptoquarks will put the strongest limitson our leptoquarks [65]. The Π − / leptoquark will decay to tops and missing energy, sostop searches, which look for the same signature, will apply. Likewise sbottom searcheswill apply to Π / . A recent CMS search [66] ruled out leptoquarks decaying wholly to τ and b up to a mass of 740 GeV. This bound roughly applies to the leptoquark Π / .This leptoquark’s branching ratio to τ and b is 0.94 (over the mass range of the search,the variation is only in higher decimal places), so the bound on it from [66] is roughly 720GeV. Another CMS search [67] puts bounds on leptoquarks decaying to either top and tauor bottom and neutrino with a combined branching ratio of 100%. Since the Π / statehas a combined branching ratio of 97% to these final states, to a good approximation theresults of this search should apply. This search implies a bound of 570 GeV on the mass ofthe Π / , which at this mass has a branching ratio of 0.40 to top and tau. A bound froman ATLAS stop search [68] can be applied to the remaining leptoquark state, Π − / . Inone scenario considered in the search, the stop is presumed to decay wholly to a top andthe lightest neutralino, and a 640 GeV bound on the mass of the stop is quoted, assumingthat the neutralino is massless. The production mechanism for the Π − / leptoquark isidentical to that for the stop, which is assumed in the search to be directly pair produced.– 19 –urthermore at a mass of 640 GeV, the branching ratio of Π − / to top and neutrino isgreater than 99.5%. We can hence take the 640 GeV bound to apply directly to the massof the leptoquark Π − / . Since we are assuming small mass splittings between the chargeeigenstates in the leptoquark multiplet, a bound on the mass of one eigenstate roughlycorresponds to a bound on them all. So we can apply the strongest of the bounds givenabove to the mass M ; we therefore conclude that M >
720 GeV.However all these bounds are found by assuming that the O (1) coefficients c ij in theleptoquark couplings all have a modulus equal to 1. The limits can change quite a lotif this is not the case. In particular, the ratio of λ j to λ j is ( c j √ m τ ) / ( c j √ m µ ), ie. ∼ c j /c j . So the branching ratio to e.g. top and muon can be larger than that totop and tau if c /c (cid:46) .
25. By contrast, the difference between the third and secondgenerations of quarks is harder to overcome by changes in the c ij coefficients, since thehierarchy in the mixing parameters is larger. A measured bound on leptoquarks decayingto third generation quarks and second generation leptons would be a useful measurementto cover a case where c were accidentally small. We have argued that current anomalies in semileptonic B decays seen at LHCb are con-sistent with a composite Higgs model featuring an additional, light leptoquark. This lep-toquark has quantum numbers ( ¯3 , , /
3) under the SM gauge group and couplings tothe SM fermions that are largely fixed by the partial compositeness paradigm. We haveidentified a possible coset structure that contains both the SM Higgs and the leptoquarkas pseudo-Goldstone bosons of the strong sector, which allows them to be rather lighterthan the resonances of the strong sector, and with a natural explanation of the sizes andsigns of their squared-mass parameters (bar a small, unavoidable, residual fine-tuning inthe electroweak scale).The partial compositeness framework automatically implies lepton non-universality inthe leptoquark coupings. In this way, the departure from unity of the value of the ratio ofthe branching ratio of B → Kµ + µ − to that of B → Ke + e − measured at LHCb this yearcan be accommodated, as can earlier anomalies in measurements of angular observables in B → K ∗ µ + µ − decays. The framework predicts large couplings to third-generation leptons,hence large deviations in observables involving tau leptons or neutrinos in the final state.These processes therefore provide a good check for the model, but the predictions are notin conflict with current bounds. In our scenario, with parameters chosen to fit the LHCb b → s(cid:96)(cid:96) anomalies, we predict deviations of ∼
25% from the SM value in B ( B → Kνν ),which will be testable at the Belle II experiment. And the NA62 experiment, starting in2015, will measure B ( K + → π + νν ) to sufficient accuracy that the effects of the modelshould be visible there.We find that the model is consistent with all other flavour constraints, however wehave identified a few processes for which the NP contributions are at or close to the cur-rent bounds. These are B ( µ → eγ ), the mass splitting ∆ m B s in B s meson mixing, and B ( B + → π + µ + µ − ). More precise measurements of these will also test the model. It should– 20 –e remembered, however, that the leptoquark couplings, λ ij , within partial compositenessare each only predicted up to an O (1) factor, c ij . Thus predictions can only be made to anaccuracy of order one or so, and even tight constraints could be evaded if, for a particularprocess, the combination of c ij factors involved is accidentally small. It should be notedthat, in particular, none of the processes listed in this paragraph get their dominant con-tributions from the same combination of couplings that are involved in the LHCb B decayanomalies. Thus, the various O (1) factors are not determined by fitting the anomalies.However, the fact that the framework can make predictions for a wide range of processes,due to non-zero couplings to all SM fermions, means that it is, nevertheless, falsifiable.If the composite leptoquark is the cause of the measured discrepancies in B decays,there are three implications for the model. Firstly, the composite sector must be stronglyinteracting, g ρ ∼ π . Secondly, the left handed doublet of the third quark generation mustbe highly composite (cid:15) q ∼
1. Thirdly, the leptoquark should have a mass of around a TeV,meaning that there is scope for its discovery at LHC13. Current LHC8 bounds, analysedin § M >
720 GeV (under an assumption onthe coefficients involved in the couplings). Large third generation couplings ensure that thethree charge eigenstates of the leptoquark triplet decay mostly to third generation quarksand leptons, so searches for third generation leptoquarks are effective for constraining theirmass.We finally comment on the physics associated with the strong sector at the scale m ρ .First of all let us estimate the value of that scale. According to the discussion in § M ∼ α s π m ρ . Given that we need M ∼ m ρ ∼
10 TeV. With such a scale for the composite sector it has been shown [69]that the structure of partial compositeness is enough to suppress dangerous contributionsto indirect search observables with the exception of the electron EDM and the radiativeLFV decay µ → eγ . The further suppression required in these channels might be obtainedby departing from the hypothesis of lepton flavour anarchy in the strong sector. Veryroughly, the amount of tuning needed to accommodate the right values of the EW scale andof the Higgs mass [77–81] is expected to be, at best, at or below the per cent level, whichis not much worse than the amount already required in generic supersymmetric extensionsof the SM, given current bounds. We feel that this is a not unreasonable price to pay,given the additional benefits of a motivated flavour paradigm and the power to explain theLHCb anomalies.
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