Leptoquarks in Flavour Physics

While the LHC has not directly observed any new particle so far, experimental results from LHCb, BELLE and BABAR point towards the violation of lepton flavour universality in $b\to s\ell^{+}\ell^{-}$ and $b\to c\ell\nu$. In this context, also the discrepancy in the anomalous magnetic moment of the muon can be interpreted as a sign of lepton flavour universality violation. Here we discuss how these hints for new physics can also be explained by introducing leptoquarks as an extension of the Standard Model. Indeed, leptoquarks are good candidates to explain the anomaly in the anomalous magnetic moment of the muon because of an $m_{t}/m_{\mu}$ enhanced contribution giving correlated effects in $Z$ boson decays which is particularly interesting in the light of future precision experiments.


Introduction
So far the LHC has not directly observed any particles beyond the ones in the Standard Model of particle physics (SM). While in the SM the Higgs boson couplings to fermions are the only source of lepton flavour universality violation (LFUV), in the past years several hints for additional LFUV have been accumulated. The ratios of semi-leptonic B decays show sizeable deviations from the SM predictions. Considering b → s + − transitions, R(K) [1] and R(K * ) [2] show a combined significance for LFUV at the 4 σ level [3][4][5][6][7][8]. If we also take into account all other b → sµ + µ − observables, in several scenarios the global fit prefers new physics (NP) above the 5 σ level [9]. In R(D ( * ) ) BaBar [10], BELLE [11,12] and LHCb [13,14] found a significance for LFUV of about 4 σ [15]. In this context it is interesting to mention that LHCb recently measured a deviation of about 2 σ in R(J/Ψ) [16]. This is consistent with the anomaly in R(D) and R(D * ). The tension in the anomalous magnetic moment of the muon (a µ ) between measurement [17] and SM prediction is at the 3 σ level [18,19]. Also this discrepancy can be interpreted as a sign of LFUV even though the anomalous magnetic moments are already in the SM flavour nonuniversal: If we assume that NP coupled with the same a e-mail: dario.mueller@psi.ch strength to electrons and to muons, a e would be more sensitive for NP. Since no deviation from the SM prediction in a e has been observed so far, the tension in a µ can be interpreted as another sign for LFUV induced by NP. Motivated by these anomalies, we study leptoquarks (LQ) which provide a possible solution of them. Firstly we concentrate on each anomaly itself and discuss which LQ representations suit best for an explanation and consider correlated effects in other observables. Finally, we will investigate if we can explain the described anomalies simultaneously.

a µ
In order to explain the descrepancy in a µ , a rather large effect of the order of the SM weak interaction contribution is required. Among the set of new particles that can give the desired effect, LQ are interesting candidates [20][21][22][23][24]. Even though the LQ must be rather heavy due to LHC constraints (above 1 TeV), one can still get relevant effects in a µ . For some representations of LQ, the amplitude can be enhanced by a factor m t /m µ O(10 3 ) or m b /m µ O(10) compared to the SM. To achieve an enhancement of factor m t /m µ , the LQ must couple to left-and right-handed top or anti-top quarks simultaneously. However, not all LQ representations have this feature. Among the 10 scalar and vector LQ representations generating lepton-quark interaction terms that are invariant under the SM gauge group [25], only two scalars can possess this enhancement: An SU(2) singlet Φ 1 and an SU(2) doublet Φ 2 with quantum numbers  These representations couple to the fermions of the SM as follows Q and L are SU (2)  equal to λ L,R µ . Besides the contribution to a µ , we also obtain correlated effects in Z → µ + µ − (see the right Feynman diagram in Fig. 1 Fig. 2 shows the allowed regions in λ L µ -λ R µ parameter space for a LQ with mass of 1 TeV, to respect constraints from direct searches. For Z → µ + µ − the expected bounds from GigaZ [26] and TLEP [27] are shown as well. Additionally, for Φ 1 the expected BELLE II limit for B → K ( * ) νν is taken into account. Φ 2 gives a C 9 = C 10 -like contribution to b → s + − which provides a slight improvement of 1 σ in the global fit. If we also allow for couplings of the top quark to taus or electrons, lepton flavour violating processes as τ → µγ or Z → µe will be possible. A more detailed discussion can be found in Ref. [28].
Dealing with b → s + − transitions, one usually works with an effective Hamiltonian of the form Then, for several scenarios with NP in C 9 only, C 9 = −C 10 or C 9 = −C 9 , the global fit prefers NP above the 5 σ level [9]. We consider the case where NP contributes as C 9 = −C 10 .
While in the muon channel the preferred value deviates from the SM prediction with a tension of 5 σ, in the case with electrons the SM contribution is sufficient to give a good fit to the data. To reach the central value, an O(10%) NP effect in the muon channel is required. b → s + − is suppressed by a loop and a CKM factor because it is a flavour changing neutral current. Therefore, the effect of NP does not have to be very large to account for the data.
There are three LQ representations that give a C 9 = −C 10 - If we only allow for couplings to muons, these LQ will give tree-level effects in b → sµ + µ − but contribute at looplevel in other flavour observables. This is why LQ can explain the anomalies in b → sµ + µ − transitions but are not in conflict with other observables. Including couplings to electrons, we get additional effects in b → se + e − as well as in the lepton flavour violating processes µ → eγ and b → sµe. We can write where χ = y 32 /y 21 and γ = y 21 /y 22 , with y = λ for scalar LQ and y = κ for vector LQ. Note that one can avoid an effect in µ → eγ completely if C µµ 9 = −χ 2 C ee 9 . For real values of χ this means that sign[C ee 9 ] = −sign[C µµ 9 ]. In Fig. 3 you find a detailed analysis for the scalar triplet Φ 3 . For V µ 1 and V µ 3 the situation is quite similar, albeit the blue bands are more narrow. For more information see Ref. [43]. Note that if the expected sensitivity of TLEP was reached and no excess in the Zµμ-coupling was found, an explanation of the anomaly in a µ would strongly be disfavoured.

b → cτν
The transition b → cτν is a charged current process and is therefore generated at tree-level in the SM. Since an explanation of R(D) and R(D * ) requires an O(10%) effect, also the NP must contribute at tree-level. Charged Higgses [45][46][47][48][49], W bosons [32] and LQ [23,24,33,34,36,[50][51][52][53][54][55][56] are NP candidates. However, B c lifetime [57][58][59][60] and q 2 distributions [59] in combination with direct searches [61] strongly disfavour charged Higgses and W bosons. Hence, we are left with LQ. If we want to explain the excess in R(D) and R(D * ) by modifying the couplings to tau neutrinos and/or taus, we also get effects in b → sτ + τ − and b → sνν. It can be shown in a model independent way that these effects are of the order of 10 3 compared to the SM [34,62]. This is due to the fact that the processes b → sτ + τ − and b → sνν occur at loop-level in the SM, while LQ contribute at tree-level with considerably large couplings in order to explain R(D ( * ) ). An O(10 3 ) effect is in strong conflict with experimental results of B → K ( * ) νν. Hence, a model that explains R(D) and R(D * ) must avoid a major effect in b → sνν. One possibility is the vector LQ singlet V µ 1 (see Eq. (8)). Another model was proposed in Ref. [63], where instead of V µ 1 two scalar LQ (the SU(2) singlet Φ 1 in Eq. (2) and the SU(2) triplet Φ 3 in Eq. (7)) are required. If Fig. 4 we can see for which parameters we can explain R(D ( * ) ) without violating the limits of b → sνν. Besides explaining R(D ( * ) ), it also enhances B s → τ + τ − up to a factor 10 3 compared to the SM and therefore lies within experimental reach. The current limit is Br B s → τ + τ − < 6.8 × 10 −3 [64]. Fig. 5 shows the correlation of R(D ( * ) ) and Br B s → τ + τ − .

Simultaneous Explanation
We may ask if one can explain the three types of anomalies in a µ , b → sµ + µ − and R(D ( * ) ) simultaneously. If we take the model that we proposed in Sec. 4 and insert a discrete symmetry for the couplings λ 1L jk = e iπ j λ 3 jk ≡ λ L jk the effect in b → sνν will be cancelled exactly. If we include b → sµ + µ − , B → D ( * ) µν/B → D ( * ) eν will be affected. However, these observables remain compatible with experiments. Additionally the LFV process τ → µγ and b → sτµ appear. Both processes give constraints on our parameter space. On the left-hand side of Fig. 6 the allowed regions for M LQ = 1 TeV are shown, under the assumption that b → sµ + µ − matches the central value of the global fit. As we can see easily, the explanation of R(D ( * ) ) requires λ L 33 /λ L 32 > 1. On the right-hand side of Fig. 6 we need λ L 33 /λ L 32 < 0.6 to explain a µ without getting in conflict with experimental constraints of τ → µγ. With this contradiction, we cannot explain all three anomalies simultaneously. However, if we abandon anyone of them, it is possible to explain the other two anomalies. If we abandon the assumption λ 1L jk = e iπ j λ 3 jk one can avoid the discussed constraints and explain all three anomalies. Another good candidate is the vector LQ singlet V µ 1 because it does not affect the b → sνν transitions but gives a C 9 = −C 10 effect in b → s + − . It can also be used to explain R(D ( * ) ). In a µ we can obtain an m b /m µ contribution which is not as large as for the two scalar LQ Φ 1 and Φ 2 (see Sec.2), but still may be significant. However, one faces the problem that massive vector bosons without a Higgs mechanism are not renormalizable. For → γ in Sec. 3 this does not cause a problem because this process is finite in unitary gauge. Recently, models containing renormalizable massive vector LQ were proposed [65][66][67][68]. In Ref. [66] a Pati-Salam model in combination with vector-like fermions is phe- For real values of χ only the region with C µµ 9 < 0 and C ee 9 > 0 seems to be compatible with experimental limits. Indeed this is also where the central value of the global fit lies. However, if in the future the global fit prefers C µµ 9 < 0 and C ee 9 < 0, LQ with real couplings will not be able to explain b → s + − transitions without violating the constraints from µ → eγ.  nomenologically consistent and leads to a massive vector LQ. This vector LQ has the properties of V µ 1 in Eq. (8).

Conclusion
In these proceedings we reviewed which LQ are good candidates to explain flavour anomalies: For the deviation in a µ the scalar LQ Φ 1 and Φ 2 give an m t /m µ enhanced contribution. This is a useful feature to explain the rather large discrepancy. Considering b → s + − transitions, the three LQ representations Φ 3 , V µ 1 and V µ 3 are required by the fit, because they give C 9 = −C 10 . R(D ( * ) ) can be explained either by V µ 1 or, as we proposed, by a combination of a scalar LQ singlet Φ 1 and scalar LQ triplet Φ 3 without getting in conflict with b → sνν observables. We also showed that this model is able to explain any two of the three anomalies (a µ , b → sµ + µ − and R(D ( * ) ) simultaneously.