Explaining b→ s`+`− data by sneutrinos in the R-parity violating MSSM

The recent measurements on b→ s`+`− processes suggest the existence of lepton-flavouruniversality breaking new physics. In this work, we have explored the possibility of explaining these data by sneutrinos in the R-parity violating Minimal Supersymmetric Standard Model. We study the light sneutrinos, of order 1 TeV, and suppose that the rest of sfermions are much heavier than them. This setup can solve b→ sμ+μ− anomaly well, and it is almost unconstrained by other related processes, such as Bs − B̄s mixing, as well as B0 s → τ+τ−, B+ → K+τ+τ−, B0 s → τ±μ∓ and B+ → K+τ±μ∓ decays.


Introduction
The rare semileptonic b-hadron decays induced by the flavour-changing neutral current (FCNC) transition b → s + − do not arise at tree level and are highly suppressed at higher orders within the Standard Model (SM), due to the Glashow-Iliopoulos-Maiani (GIM) mechanism [1]. New TeV-scale particles in many extensions of the SM could lead to measurable effects in these rare decays. As a consequence, they play a crucial role in testing the SM and probing various new physics (NP) scenarios beyond it [2,3].
In recent years, several deviations from the SM predictions have been observed in b → s + − transition. Consider the ratios of the branching fractions R K ( * ) = B(B → K ( * ) µ + µ − )/B(B → K ( * ) e + e − ), which have negligible theoretical uncertainties. In the range 1.1 < q 2 < 6GeV 2 /c 4 , the latest experimental data by LHCb collaboration give R  [4,5], but the SM predicts it to be close to one [6]. The measurement of R K is 2.5σ smaller than the SM prediction. The measurements of R K * [7] by LHCb are R −0.069 ± 0.05, which are lower than the predicted values of the SM [6] about 2.1σ and 2.5σ, respectively. Belle collaboration also give the measurements of R K ( * ) [8,9], which are consistent with the SM predictions due to their large experimental errors. In addition to the tension with the SM in lepton-flavour-universality observables R K ( * ) , some other deviations have also been found in b → sµ + µ − transition. In particular, the form-factor-independent angular observable P 5 [10][11][12] in the B → K * µ + µ − decay was measured by LHCb [13,14], CMS [15], ATLAS [16] and Belle [17,18], showing a 2.6σ disagreement with the SM expectation [19].
Finally, LHCb has also observed a 3.3σ deficit in the B 0 s → φµ + µ − decay [20,21]. Motivated by these deviations and using the other available data on such rare b → s + − transitions, many global analyses have been carried out [19,[22][23][24][25][26][27], finding that a negative shift in a single Wilson coefficient of local operator like O µµ leads to a consistent description of the data, with the corresponding best-fit point can improve the fit to the data by more than 5σ compared to the SM. Furthermore, the operator O µµ LL performs better than O µµ 9 , mainly because there is now ∼ 2σ tension in the branching fraction of B s → µ + µ − [22,[28][29][30][31][32][33], which is not affected by O µµ 9 . In this paper, we work with the low-energy effective weak Lagrangian governing the b → sµ + µ − processes: where L SM eff represents contributions from the SM, and the remaining terms contain possible NP contributions. The CKM factor η t = V tb V * ts ≈ −0.04 [34]. The best-fit point performed by Ref. [22] is C µµ LL = −1.06, with the 2σ range being −1.38 < C µµ LL < −0.74. We find that such C µµ LL can be generated naturally in the R-parity violating Minimal Supersymmetric Standard Model (MSSM) [35] by exchanging smuon and winos.
However, they note that it is difficult to find a viable explanation due to the severe constraints from the upper limit on the branching fraction of B → K ( * ) νν decays. In addition tod R , the authors in Ref. [38] also consider the contribution to b → sµ + µ − transition from the box diagrams with a left-handed up type squarkũ L and sneutrinoν L in the loop. They find that this new contribution could help explain b → sµ + µ − anomaly, while satisfying the constraint from B → K ( * ) νν and D 0 → µ + µ − decays as well as B s −B s mixing. In Ref. [39], the authors focus on parameters for which diagrams involving winosW , which have not been considered before, make significant effects. They set the masses ofW and threeũ L to be light, of order 1 TeV, and at the same time, they consider heavyν L andd R , of order 10 TeV. In this scenario, the b → sµ + µ − anomaly may be explained by large values of λ , but the available parameter space is very small due to the constraints from relevant processes, such as τ → 3µ, B s −B s mixing and direct LHC searches. The restriction from B → K ( * ) νν decay is negligible because of the lagre mass ofd R .
There are two kinds of sfermions participating in theW box diagrams, namelyũ L andν L .
As an alternative, in this paper, we study the lightν L , of order 1 TeV, and suppose that the rest of sfermions are much heavier compared to it. This scenario can well produce the C µµ LL needed to explain b → sµ + µ − anomaly, and the corresponding parameter space is not constrained by other related processes, such as B s −B s mixing, as well as B 0 Our paper is organized as follows. In section 2, we first set up our scenario and then discuss 2 Contributions to b → sµ + µ − processes from R-parity

violating MSSM
The superpotential of the relevant R-parity violating terms in the MSSM is given by [35] where L, H u , E c , Q, D c , and U c are the chiral superfields for the MSSM multiplet, and we denote the generation indices by i, j, k = 1, 2, 3. The summation is applied for the repeated indices throughout this paper unless otherwise stated. The first three terms in Eq. (2.1) destroy the lepton number and the last term violates the baryon number. We will assume that λ coupling is zero to prevent rapid proton decay. In this work, we limit ourselves to consider the λ ijk L i Q j D c k term as the source of R-parity violating NP, because of the b → sµ + µ − processes involve both leptons and quarks. The effects of λ and λ terms simultaneously on b → sµ + µ − processes have been studied in such as Ref. [40,41]. Expanding the chiral superfields in terms of their fermions and sfermions, one has We will also assume that all sfermions are so heavy that they are decoupled, except for sneutrinosν Li of order 1 TeV. Under this assumption, only the λ ijkν LidRk d Lj term in Eq. (2.2) can lead to a valuable effect. In this paper we focus our attention on a parameter space where the λ ij3 couplings are large, i.e., keep λ ij1 = λ ij2 = 0 all the time. We will assume sneutrinos are in their mass eigenstate basis and nearly degenerate, and the degenerate mass is denoted as mν.
The b → sµ + µ − processes can occur at one-loop level by exchanging smuon and winos, see well as the corresponding Wilson coefficient given by where the loop function f (xν) ≡ 1−xν +log xν The corresponding parameter space is shown in Fig. 2.
There is also a contribution from the photonic penguin, which is shown in Fig. 1b e (sσ αβ P R b)F αβ after integrating out sneutrinos, and the corresponding Wilson coefficients given by Our results are consistent with those in Ref. [51]. Comparing with Ref. [39], we find that the result of C 7 is consistent, but the result of C 9 is different by a negative sign. All in all, we should suppress the effect of photonic penguin by setting λ i33 λ * i23 = 0 in order to take advantage of only nonzero C µµ LL scenario, which has the largest pull-value in single Wilson coefficient global analyses [22]. We also calculate the contribution of Z-penguin and find that it can be ignored due to the negligible down type quark masses.

Other possible constraints
In our scenario, several other processes may also obtain the effects of R-parity violating interactions, and the corresponding constraints should be taken into account. Next, we mainly study the constraints on λ i23 and λ i33 couplings, which play the key role in solving b → sµ + µ − anomaly.

Tree level decays
Exchanging sneutrinos and performing Fierz rearrangement, one obtains the following four fermion operators at tree level There is no valid constraint here. In addition, keeping λ i33 λ * i23 = 0 can prevent the occurrence of dangerous Υ − B s mixing.

Loop level decays
The potential constraint may come from B s −B s mixing, which can obtain the R-parity violating contributions by exchanging two sneutrinos in the loop. This NP contribution can lead to the same effective operator as the SM. The contributions of R-parity violating interactions are given by Because we keep λ i33 λ * i23 = 0, these contributions go away. In fact, in addition to muon channel, the nonzero λ i23 and λ i33 couplings can also induce b → s + i − j processes by exchanging sneutrinos and winos, as shown in Fig. 1a. The corresponding Wilson coefficients C ij LL can be obtained by replacing λ 233 λ * 223 with λ i33 λ * j23 in Eq. (2.3). In order for the NP to have no effect on b → se + e − processes we should keep C ee LL = λ 133 λ * 123 ≈ 0, which means λ 133 ≈ 0 or λ * 123 ≈ 0. Combining λ i33 λ * i23 = 0, we predict the same size of C µµ LL and C τ τ LL ∝ λ 333 λ * 323 , with similar result in the PS 3 model [52]. Such C τ τ LL satisfies the upper limit of B(B + → K + τ + τ − ) < 2.25 × 10 −3 [53] measured by BaBar at 90% confidence level (CL) and [54] measured by LHCb at 95% CL. The remaining potential constraints come from several lepton-flavour-violation decays B 0 s → τ ± µ ∓ and B + → K + τ ± µ ∓ . Those decays governed by the low-energy effective weak Lagrangian with i = j. The branching fractions of leptonic B 0 s → τ ± µ ∓ decays given by In our numerical analysis, we take as input the decay constant f Bs = 0.2272(34) GeV, the lifetime τ Bs = 1.510(4) ps, as well as the mass m Bs = 5.367 GeV, m τ = 1.777 GeV and m µ = 0.1057 GeV [34]. Lately, the upper limit on these branching fractions are measured by LHCb collaboration. At 95% CL one has [55] B(B 0 s → τ ± µ ± ) exp < 4.2 × 10 −5 . For semi-leptonic B + → K + τ ± µ ∓ decays, we can obtain

Conclusions
Recently, several deviations from the SM predictions in b → s + − data hint to exist leptonflavour-universality breaking NP. Many global analyses show that a negative shift in Wilson coefficient C µµ LL can explain these data well, and the corresponding best-fit point can improve the fit to the data by more than 5σ compared to the SM. This suggests that the NP primarily affects the b → sµ + µ − processes. Based on these knowledge, in this work we have explored the possibility of explaining b → sµ + µ − anomaly by sneutrinos in the R-parity violating MSSM.
After a brief introduction to the relevant terms in the superpotential of R-parity violating MSSM, we present our scenario, that is, we consider the lightν L of order 1 TeV and the other sfermions are so heavy that they are decoupled. We find that a positive product λ 233 λ * 223 can explain b → sµ + µ − anomaly, and the parameter space satisfied by λ 233 λ * 223 and mν is shown in Fig. 2. After that, we consider the other possible constraints, including tree level and one-loop level decays. Keeping λ i33 λ * i23 = 0 can inhibit the contribution of R-parity violating NP to B s −B s mixing and the photonic penguin of b → s + − processes, and prevents the emergence of dangerous Υ − B s mixing. We predict C τ τ LL ≈ −C µµ LL which satisfies the upper limit of the branching fractions of B + → K + τ + τ − and B 0 s → τ + τ − decays. Furthermore, we discuss the potential constraints come from several lepton-flavour-violation decays B 0 s → τ ± µ ∓ and B + → K + τ ± µ ∓ , and find that the experimental upper limit of these processes does not effectively exclude the parameter space needed to explain b → sµ + µ − anomaly.