Simultaneous Explanation of $R(D^{(*)})$ and $b\to s\mu^+\mu^-$: The Last Scalar Leptoquarks Standing

Over the past years, experiments accumulated intriguing hints for new physics (NP) in flavor observables, namely in the anomalous magnetic moment of the muon ($a_\mu$), in $R(D^{(*)})={\rm Br}(B\to D^{(*)}\tau\nu)/{\rm Br}(B\to D^{(*)}\ell\nu)$ and in $b\to s\mu^+\mu^-$ transitions, which are all at the $3-4\,\sigma$ level. In this article we point out that one can explain the $R(D^{(*)})$ anomaly using two scalar leptoquarks (LQs) with the same mass and coupling to fermions related via a discrete symmetry: an $SU(2)$ singlet and an $SU(2)$ triplet, both with hypercharge $Y=-2/3$. In this way, potentially dangerous contributions to $b\to s\nu\nu$ are avoided and non-CKM suppressed effects in $R(D^{(*)})$ can be generated. This allows for smaller overall couplings to fermions weakening the direct LHC bounds. In our model, $R(D^{(*)})$ is directly correlated to $b\to s\tau^+\tau^-$ transitions where an enhancement by orders of magnitude compared to the standard model (SM) is predicted, such that these decay modes are in the reach of LHCb and BELLE II. Furthermore, one can also naturally explain the $b\to s\mu^+\mu^-$ anomalies (including $R(K)$) by a $C_9=-C_{10}$ like contribution without spoiling $\mu-e$ universality in charged current decays. In this case sizable effects in $b\to s\tau\mu$ transitions are predicted which are again well within the experimental reach. One can even address the longstanding anomaly in $a_\mu$, generating a sizable decay rate for $\tau\to\mu\gamma$. However, we find that out of the three anomalies $R(D^{(*)})$, $b\to s\mu^+\mu^-$ and $a_{\mu}$ only two (but any two) can be explained simultaneously.

However, an explanation of R(D) and R(D * ) is getting more and more delicate. Since these processes are mediated in the SM already at tree-level, a rather large NP contribution is required to account for the 1: Feynman diagrams contributing to b → cτ ν, b → sνν and b → s processes. Both LQs contribute to b → cτ ν and b → sνν but only Φ3 to b → s . Note that with our assumption on the couplings to fermions, the LQs interfere constructively deviation. Therefore, new particles added to the SM for explaining R(D) and R(D * ) cannot be very heavy and must have sizable couplings. In the past, mainly three kinds of models with the following new particles have been proposed: 1. Charged Higgses [21,[48][49][50][51][52] 2. W gauge bosons [11,14,53,54] 3. Leptoquarks [9,10,12,13,15,16,19,20,42,[55][56][57][58][59][60] Models with charged Higgses lead to (too) large effects in the total B c lifetime [61] and, depending on the coupling structure, can also be disfavored by the q 2 distribution [62][63][64]. Interestingly, if the couplings of the charged Higgs are chosen in such a way that they are compatible with the measured q 2 distribution, these models are ruled out by direct searches [65]. Models with W gauge bosons are also delicate because they necessarily involve Z bosons due to SU (2) L gauge invariance. If the Z width is not unnaturally large, these models are again ruled out by direct searches [11,65].
In models with leptoquarks generating left-handed vector operators the coupling structure should be aligned to the bottom quark in order to avoid b → sνν bounds. However, in this case the effect in R(D) and R(D * ) is proportional to the small CKM element V cb and large third generation couplings are required to account for the anomalies. These large third generation couplings lead again to stringent bounds from direct LHC searches [65] and electroweak precision observables [66]. In principle, these constraints can be avoided with right-handed couplings [59] (including possibly right-handed neutrinos [16]). However, in such solutions no interference with the SM appears and very large couplings, close to nonperturbativity, are required.
As stated above, LHC bounds from τ τ searches can be avoided in case of non-CKM suppressed leptoquark contributions to R(D) and R(D * ). However, for single scalar leptoquark representations, this leads to unacceptably large effects in b → s transitions [59]. Therefore, we propose a novel solution to the R(D ( * ) ) problem in this article: we introduce two scalar leptoquarks with the same mass M and the same coupling strength to quarks and leptons; an SU (2) L singlet (Φ 1 ) and an SU (2) L triplet (Φ 3 ) both with hypercharge Y = −2/3. Here, the crucial observation is that Φ 1 and Φ 3 contribute with opposite relative sign to R(D ( * ) ) than to b → sνν processes such that the effect in R(D ( * ) ) is doubled while the contributions in B → K ( * ) νν cancel at tree-level (see Fig. 1). Therefore, the couplings to the second quark generation can be larger, non-CKM suppressed effects R(D ( * ) ) are possible and the required overall coupling strength is much smaller such that the direct LHC bounds from τ τ searches are significantly weakened and the remaining bounds from pair production of third generation LQs are still below the TeV scale [67,68]. Furthermore, this solution results in a simple rescaling of the SM contributions, predicts naturally R(D)/R(D) SM = R(D * )/R(D * ) SM and leaves the q 2 distribution unchanged. Adding couplings to muons, we can also address the b → sµµ anomalies with a C 9 = −C 10 like contribution. Finally, adding a (small) right-handed coupling of Φ 1 one can in principle explain a µ .
This article is structured as follows: in the next section we will present the contributions of our model to all relevant observables. Afterwards, we perform a phenomenological analysis in Sec. III before we conclude.

II. MODEL AND OBSERVABLES
The scalar leptoquark singlet Φ 1 and the triplet Φ 3 couple to fermions in the following way: As motivated in the introduction, we assume that both leptoquarks have the same mass M . In addition, to cancel their effect in b → sνν processes, we impose the discrete symmetry on the couplings to fermions. Note that for Φ 1 there is in principle an additional coupling λ R f i u c f i Φ † 1 allowed. We will assume that this coupling is zero and neglect its effect till the discussion of a µ where small values of λ R f i can be phenomenologically important due to m t /m µ enhanced effects. For our analysis we assume that the couplings λ L f i are given in the down-quark basis. I.e. after Note that already small couplings are sufficient to account for R(D) and R(D * ). Therefore, the bounds from LHC searches are weakened and the leptoquarks can also be easily heavier than 1 TeV and still explain the anomalies with couplings in the perturbative regime. Right: Prediction for Bs → τ τ (red) as a function of R(D ( * ) )/R(D ( * ) )SM.
Here we neglected small CKM suppressed contributions.
EW symmetry breaking the couplings to left-handed upquarks involve CKM elements: We will now discuss the various relevant processes to which our model contributes. As already noted above, our model is constructed in such a way that we do not get tree-level contributions to b → sνν transitions, which we therefore omit in the following. We also neglect couplings to the first generation of quarks and leptons. For quarks, this is only possible in the interaction basis since CKM rotations induce either couplings to up or down quarks. However, charged current decays of Kaons or D mesons involve large CKM angles in the SM, making the relative effects of LQs small.

A. R(D) and R(D * )
We define the effective Hamiltonian for b → c ν transitions as where in the SM C f i L = δ f i and the contribution of our model is given by With these conventions we have assuming vanishing contributions to the muon and electron channels. This has to be compared to the experimental values of R(D * ) EXP = 0.316 ± 0.016 ± 0.010 , and the corresponding SM predictions [69,70] R(D * ) SM = 0.252 ± 0.003 , (9) R(D) SM = 0.300 ± 0.008 .
Using the effective Hamiltonian we have The allowed range at the 2 σ level [2] (see also Ref. [3,4]) is given by at the (1 σ) 2 σ level. We will also need the process B s → τ + τ − . The current experimental limit is [71] Br The SM prediction is given by [72,73] Br and in our model we have with C SM 10 ≈ −4.3 [74,75]. For the analysis of B → K ( * ) τ µ we will use the results of Ref. [33].
Here we find which corresponds to an effect of the order of 1%. This can be easily understood as follows: Since we need an O(10%) effect in R(D ( * ) ) at the amplitude level, this effects gets squared for B 0 −B 0 and there are no enhancement factors, the final effect is around 1% and below the current sensitivity of approximately 10%.

D. aµ and τ → µγ
In order to aim at an explanation of a µ one needs a chirality enhanced effect. Therefore, let us add to Eq. (1) the following term In this case the numerically relevant m t enhanced contribution to a µ is given by with and C 23 R is obtained from C 23 L by L ↔ R. We will assume that λ R 32 is small compared to λ L 32 .

III. PHENOMENOLOGICAL ANALYSIS
A. R(D), R(D * ) and b → sτ + τ − Let us first consider the size of the couplings needed to explain R(D) and R(D * ). Here and in the following, we will assume them to be real. As we can see in the left plot of Fig. 2, we only need small couplings (of the order of 0.1 for 1 TeV leptoquarks) in order to explain R(D), R(D * ). This is possible because we avoid contributions to b → sνν and hence our effect in b → cτ ν does not need to be CKM suppressed. Therefore, the bounds from Ref. [65] do not apply to our model and we are not in conflict with LHC bounds, especially because the LQs can be much heavier than 1 TeV while still possessing perturbative couplings and explaining R(D ( * ) ).
Next, note that neglecting small CKM factors, the contributions to b → cτ ν and b → sτ τ depend on the same product of couplings λ L 23 λ L * 33 (modulus small CKM ratios). Therefore, we can express B s → τ + τ − in terms of the effect in R(D ( * ) ): The resulting numerical prediction for B s → τ + τ − is shown in Fig. 2. We can see that the branching ratio can be enhanced by up to three orders of magnitude compared to the SM prediction. Therefore, even though it is experimentally challenging to search for, our model can be tested with B s → τ + τ − measurements at LHCb. Also an enhancement of B → K ( * ) τ + τ − in the same ballpark is predicted which could be tested at BELLE II. = −0.5, i.e. assuming that C 22 9 takes the central value obtained from the b → sµ + µ − fit. The colored regions are allowed by the various processes. For R(D) and R(D * ) we used again the weighted average for R(D ( * ) )EXP/R(D ( * ) )SM. Right: The contour lines show Br[τ → µγ] × 10 8 . The gray region is excluded by the current upper limit and (light) red region is allowed by aµ at the (2 σ) 1 σ level. Note that both δaµ and τ → µγ are only a function of λ L 33 /λ L 32 and therefore independent of b → sµ + µ − transitions.
Let us now consider the effect of including b → sµ + µ − transitions in our analysis. In this case effects in B → D ( * ) µν/B → D ( * ) eν are predicted if still addressing R(D) and R(D * ) simultaneously. We checked that the effect is at the per-mill level which is compatible with BELLE and BABAR measurements 3 . However, interesting correlations with b → sτ µ processes appear. Here we find which depends only on the ratio λ L 33 /λ L 32 as a free parameter. Note that the dependence on C 22 9 is much weaker than on X D ( * ) . The resulting bounds and predictions are shown in the left plot of Fig. 3. We take the experimental limit [90] Br [B → Kτ µ] < 4.8 × 10 −5 .

C. aµ and τ → µγ
Considering only the couplings λ L the effect in τ → µγ is negligibly small. Things get much more interesting if we aim at a simultaneous explanation of the anomalous magnetic moment of the muon. In this case chirally enhanced effects also appear in τ → µγ. We have Here we set λ R 33 = 0. Note that Br (τ → µγ) can only be enhanced by allowing λ R 33 to be different from zero, resulting in the ≥ sign in Eq. (28). The result is shown in the right plot of Fig. 3. Note that a µ can only be explained for λ L 33 /λ L 32 < 0.65 (at the 2 σ level). This is opposite to the case of b → sµ + µ − which can only be explained for λ L 33 /λ L 32 > 1. Therefore, we conclude that our model can explain out of the three anomalies R(D ( * ) ), b → sµ + µ − and a µ only two simultaneously.

IV. CONCLUSIONS AND OUTLOOK
In this article we proposed a scalar leptoquark model which can give sizable effects on R(D) and R(D * ) without suffering from problems with b → sνν, q 2 distributions in R(D ( * ) ), from large couplings in the nonperturbative regime or from tensions with direct LHC searches as it is the case for nearly all other models on the market. Our model predicts sizable branching factions for b → sτ + τ − processes (of the order of 10 −3 ) being directly correlated to R(D ( * ) ). Furthermore, the model can naturally explain b → sµ + µ − (including R(K)) via a C 9 = −C 10 contribution and therefore also predicts R(K * ) to be significantly below the SM value. In case of a simultaneous explanation of R(D ( * ) ) with b → sµ + µ − we only get effects in B → D ( * ) µν/B → D ( * ) eν at the per-mill level, but sizable ones in b → sτ µ processes (depending on only one free parameter), making them potentially observable at LHCb or BELLE II in the near future.
The tension in a µ can be explained as well by adding a small right-handed coupling of the SU (2) L singlet LQ to tops and muons. As a consequence sizable rates for τ → µγ are predicted. Here the dependence on the remaining free parameter is opposite to b → sτ µ excluding a simultaneous explanation of all three anomalies, i.e. out of R(D ( * ) ), b → sµ + µ − and a µ our model can explain any two of them.
We stress that in general our approach of combining the SU (2) L singlet with the SU (2) L triplet is the only way of explaining R(D) and R(D * ), without violating b → sνν or direct LHC bounds, if the SM is extended by scalar LQs only. However, one can get the same phenomenology in B decays using a vector leptoquark SU (2) L singlet with hypercharge 2/3 and couplings to left-handed fermions. In this case, no tuning between couplings is necessary and effects in b → sνν are automatically avoided. However, the model is nonrenormalizable and while our model with scalar leptoquarks only predicts effects of the order of 1% in B s − B s mixing, this effect is much larger for the vector leptoquark [12]. Furthermore, adding right-handed couplings, the effect of the vector leptoquark in a µ is only enhanced by the bottom mass but not by the top one.
In our model we assumed a discrete symmetry between the couplings of the two leptoquarks in order to cancel exactly the effect in b → sνν. However, even if one disregards this assumption and allows for independent masses and couplings of Φ 1 and Φ 3 , a cancellation in b → sνν is still possible. In fact, as shown in Fig. 4 the constraints from b → sνν transitions still allow for an explanation of R(D ( * ) ) without severe fine-tuning. Therefore, our imposed symmetry does not to be exact in order to provide a valid explanation of the anomalies.
If one allows in addition for direct couplings (not only CKM induced couplings to up-quarks) to first generation quarks, a sizable effect in K → πµµ/K → πee is possible which could be tested at NA62 or KOTO [91]  and interesting correlations with b → d transitions occur. Therefore, a very detailed study of our model is important and promising in order to explore the many interesting effects which can be observed by ongoing and future experiments.