Revisiting the $B$-physics anomalies in $R$-parity violating MSSM

In recent years, several deviations from the Standard Model predictions in semileptonic decays of $B$-meson might suggest the existence of new physics which would break the lepton-flavour universality. In this work, we have explored the possibility of using muon sneutrinos and right-handed sbottoms to solve these $B$-physics anomalies simultaneously in $R$-parity violating minimal supersymmetric standard model. We find that the photonic penguin induced by exchanging sneutrino can provide sizable lepton flavour universal contribution due to the existence of logarithmic enhancement, for the first time. This prompts us to use the two-parameter scenario $(C^{\rm V}_9, \, C^{\rm U}_9)$ to explain $b \to s \ell^+ \ell^-$ anomaly. Finally, the numerical analyses show that the muon sneutrinos and right-handed sbottoms can explain $b \to s \ell^+ \ell^-$ and $R(D^{(\ast)})$ anomalies simultaneously, and satisfy the constraints of other related processes, such as $B \to K^{(\ast)} \nu \bar\nu$ decays, $B_s-\bar B_s$ mixing, $Z$ decays, as well as $D^0 \to \mu^+ \mu^-$, $\tau \to \mu \rho^0$, $B \to \tau \nu$, $D_s \to \tau \nu$, $\tau \to K \nu$, $\tau \to \mu \gamma$, and $\tau \to \mu\mu\mu$ decays.


I. INTRODUCTION
Recently, several flavour anomalies in semileptonic Bdecays have been reported, which have been attracting great interests. Among them, the observables R K ( * ) = B(B → K ( * ) µ + µ − )/B(B → K ( * ) e + e − ) in flavour-changing neutral current b → s + − ( = e, µ) transition and the observables R(D ( * ) ) = B(B → D ( * ) τ ν)/B(B → D ( * ) ν) in flavour-changing charged current b → cτ ν transition are particularly striking. The advantage of considering the ratios R K ( * ) and R(D ( * ) ) instead of the branching fractions themselves is that, apart from the significant reduction of the experimental systematic uncertainties, the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements cancel out and the dependence on the transition form factors become much weaker. These observables can be good probes to test the leptonflavour universality (LFU) held in the Standard Model (SM).
These deviations indicate the possible existence of new physics (NP) beyond the SM in b → s + − transition. This NP may break LFU. Many recent model-independent analyses [18][19][20][21][22][23][24][25] show that some scenarios can explain the b → s + − anomaly well. To express the fit results, we consider the low-energy effective weak Lagrangian governing the b → s + − transition where CKM factor η t ≡ V tb V * ts . We mainly concern the semileptonic operators where P L = (1 − γ 5 )/2 is the left-handed chirality projector. The Wilson coefficients C 9,10 = C SM 9,10 + C NP 9,10 . In this work, we try to explain the anomaly through a two-parameter scenario where the total NP effects are given by [26] The global analyses show that this scenario has the largest pull-value. The best-fit point performed by Ref. [20] is (C V 9 , C U 9 ) = (−0.30, −0.74), with the 2σ range being − 0.53 < C V 9 < −0.10, −1.15 < C U 9 < −0.25. (8) As we will see in the following discussion, this scenario can be implemented naturally in the R-parity violating minimal supersymmetric standard model (MSSM) [27].
There have been attempts to explain the b → s + − anomaly [48][49][50][51][52] or R(D ( * ) ) anomaly [53][54][55][56][57] or both of them [58][59][60] by R-parity violating interactions in the supersymmetric (SUSY) models. For example, based on the inspiration from the paper by Bauer and Neubert [61], the authors in Ref. [58] investigated the possibility of using right-handed down type squarks to explain the b → s + − and R(D ( * ) ) anomalies simultaneously, and found that this was impossible due to the severe constraints from B → K ( * ) νν decays. Considering that the parameter space obtained by using squarks to explain b → s + − anomaly is very small [49,50,58] due to the strict constraints from other related processes, such as B → K ( * ) νν decays and B s −B s mixing, the authors in Ref. [52] used sneutrinos to explain it and found that it is almost unconstrained by other related processes. Based on this knowledge, in this work, we will explore the possibility of using muon sneutrinosν µ and right-handed sbottomsb R to explain the b → s + − and R(D ( * ) ) anomalies simultaneously within the context of R-parity violating MSSM.
Our paper is organized as follows. In Sec. II, we scrutinize all the one-loop contributions of terms λ ijk L i Q j D c k to b → s + − processes in the framework of R-parity violating MSSM, and then give our scenario to explain the b → s + − anomaly. Discussions of R(D ( * ) ) anomaly and other related processes are included in Sec. III. The numerical analyses and results are shown in Sec. IV. Our conclusions are finally made in Sec. V.
The superpotential terms violating R-parity in the MSSM are [27] where the generation indices are denoted by i, j, k = 1, 2, 3 and the colour indices are suppressed. All repeated indices are assumed to be summed over throughout this paper unless otherwise stated (For example, repeated indices in both numerator and denominator are not automatically summed). H u , L and Q are SU (2) doublet chiral superfields while E c , D c and U c are SU (2) singlet chiral superfields.
In this work, we are mainly interested in the terms λ ijk L i Q j D c k which related to both quarks and leptons. This choice can also alleviate the constraint of sneutrino masses on the collider, because the lower limit of sneutrino masses will be as high as TeV scale [62][63][64][65] when there are non-zero λ and λ at the same time. The corresponding Lagrangian can be obtained by the chiral superfields composing of the fermions and sfermions as follows where the sparticles are denoted by "˜", and "c" indicates charge conjugated fields. Working in the mass eigenstates for the down type quarks and assuming sfermions are in their mass eigenstates, one replaces u Lj by (V † u L ) j in Eq. (13). These R-parity violating interactions can induce b → s + − processes by exchanging left-handed up squarksũ Lj at tree level, but resulting in the operators with right-handed quark current, which are unable to explain the b → s + − anomaly. This unwanted effect can be eliminated by assuming that the masses ofũ Lj are very large or/and by assuming that λ ij2 = 0. Assuming that λ ij2 = 0 also forbids the exchange ofl Li or/andd Lj in one loop level to affect the b → s + − processes 1 . In the following discussion, we should assume that λ ij1 = λ ij2 = 0.
Next, we will show the contributions of R-parity violating MSSM to b → s + − processes. All the Feynman diagrams include fourW − b box diagrams (Fig. 1a) (Fig. 1c), two 4λ box diagrams (Fig. 1d) and two γ-penguin diagrams (Fig. 2). Most of these results can be found in Refs [49,50,52,58], however, to our knowledge, the results of the diagram induced by exchanging charged Higgs H ± and right-handed sbottomb R in loop are the first to be given in this paper. The photonic penguin diagrams, which have been neglected in previous work, play an important role in our discussion, as we will explain in more detail later. We do not find sizable Z-penguin contributions to b → s + − processes. In this work, the contributions of γ/Z-penguin diagrams always include their supersymmetric counterparts unless otherwise specified. For convenience, the following Passarino-Veltman functions [67] D 0 and D 2 are defined as Fig. 1a shows an exampleW − b box diagram, Fig. 1b shows an example W −bR box diagram, Fig. 1c shows the H ± −bR box diagram, and Fig. 1d shows an example 4λ box diagram.
The contributions of box diagram are listed below. We eliminate the contributions of all box diagrams to b → se + e − processes by assuming λ 1j3 = 0.
where the winos engage these interactions with lefthand up type squarks and muon sneutrinos. The last term plays an important role in numerical analysis [52].
• The contributions of W −b R box diagram to b → sµ + µ − processes are given by The right-hand sbottomb R is the only NP particle here. In the limit mb which should be considered in the following numerical analysis. The tan β = v u /v d where v u and v d are the vacuum expectation values of two Higgs doublets respectively.
• The contributions of 4λ box diagram to b → sµ + µ − processes are given by FIG. 2. Photonic penguin diagrams studied in our scenario.
The contributions of photonic penguin diagrams are lepton flavour universal which naturally gives us a nonzero C U 9 As stated in Ref. [52], this result is consistent with that in Ref. [68], but it is a negative sign different from that in Ref. [50]. The first term in Eq. (20) comes from the contribution of Fig. 2b, like the photonic penguin induced by scalar leptoquark. We find this term give a negligible contribution, which is in agreement with Refs. [61,69]. However the second term in Eq. (20) has a significant contribution because of the logarithmic enhancement, which has never been addressed before. These photonic penguins also contribute new electromagnetic dipole operator O 7 = m b e (sσ αβ P R b)F αβ , which is strictly constrained by B → X s γ decay [9]. Fortunately, we find that the corresponding contribution can be ignored numerically because there such logarithmic enhancement absent [50,52,68].
We will discuss the possibility of using muon sneutrinosν µ and right-handed sbottomsb R to explain b → s + − anomaly, for which we set the mass of tauon sneutrinosν τ and three left-handed up type squarksũ Lj sufficiently large that the contributions of the loop diagrams containing them are ignored. The contribution from H ± −b R box diagram is usually positive, but we can simply suppress this effect by increasing parameter tan β. Thus, the contributions to only muon channel are , and the loop func-

III. R(D ( * ) ) ANOMALY AND OTHER CONSTRAINTS
In this section, we discuss the interpretation of R(D ( * ) ) anomaly and consider the constraints imposed by other related processes from B, D, K, τ , and Z decays.

III.1. R(D ( * ) ) anomaly
In R-parity violating MSSM, the charged current processes d j → u n l l ν i are induced by exchangingb R at tree level. The effective Lagrangian of these processes are given by Because taking λ 1j3 = 0 to eliminate the contributions of box diagrams to b → se + e − processes 2 , we have C nj1i = C njl1 = 0. It is useful to define the ratio and we have To obtain the allowed parameter region, we use the following best fit value in the R-parity violating scenario

III.2. Constraints from the tree-level processes
In the scenario we set up, some other processes receive tree level R-parity violating contributions. Here we mainly discuss the constraints from neutral current processes B → K ( * ) νν, B → πνν, K → πνν, D 0 → µ + µ − and τ → µρ 0 , as well as charged current processes B → τ ν, D s → τ ν and τ → Kν. These decays relate to The effective Lagrangian for B → K ( * ) νν, B → πνν and K → πνν decays are defined by is the SM one. The loop function X(x t ) ≡ xt(xt+2) The R-parity violating contributions are given by 2 In fact, by combining the assumptions λ 1j3 = 0 and λ ij1 = λ ij2 = 0, we can get λ 1jk = 0, which implies that the contribution of box diagrams of NP to the first generation leptons and sleptons is zero, because we only consider the terms λ ijk L i Q j D c k .

III.3. Constraints from the loop-level processes
First of all, the most important one-loop constraint comes from B s −B s mixing, which is governed by where the SM and NP Wilson coefficients are given respectively by At 2σ level, the UTf it collaboration [79] gives the bound 0.93 < |1 + C NP Bs /C SM Bs | < 1.29. Next, we investigate a series of Z decaying to two charged leptons with the same flavour like Z → µµ(τ τ ) and the different one like Z → µτ . The amplitude of these dia- here B 1 ij is the contribution from the diagram induced by ex- ij is the contribution from the diagram induced by exchangingb R − t − t in triangular loop. As shown in Ref. [50], for Z → µµ(τ τ ), demanding the interference term in the partial width between the SM tree-level contribution and the NP one-loop level ones is less than twice the experimental uncertainty on the partial width [75], there are the bounds | (B 22 )| < 0.32 and | (B 33 )| < 0.39 [50]. And the experimental upper limit B(Z → µτ ) < 1.2 × 10 −5 [75] makes the bound |B 23 | 2 + |B 32 | 2 < 2.1 [50].
In general, the effective Lagrangian leading to τ → µµµ decay is given by [80,81] This Lagrangian leads to [80,81] In our scenario, there are three different types of contributions, the photonic and Z penguins as well as box diagrams with four λ couplings, that can contribute to τ → µµµ decay. The nonzero Wilson coefficients are [50,68] where (53) and the off-shell effective coupling A L 1 is [68] The current experimental upper limit on the branching fraction for this decay is B(τ → µµµ) < 2.1 × 10 −8 at 90% CL [75].

IV. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we discuss how to interpret both b → s + − and R(D ( * ) ) anomalies and satisfy all these potential constraints simultaneously. The relevant model parameters in our scenario are the wino mass mW , the mass of muon sneutrino mν µ , the mass of right-handed sbottom mb R , as well as four nonzero couplings λ 223 , λ 233 , λ 323 , and λ 333 . We set mW = 270 GeV. It can be seen from Ref. [52] that a positive product λ 233 λ * 223 is needed to explain the b → s + − anomaly mainly through muon sneutrinos (the C V 9 part). Both λ 323 and λ 333 are positive to help solve R(D ( * ) ) anomaly by exchangingb R at tree level [56]. The combination of the choice of above couplings will naturally produce a negative C U 9 , which is in line with the conclusion of the global analysis [20]. Our numerical results are shown in Fig. 3. These results show that it is possible to explain b → s + − and R(D ( * ) ) anomalies simultaneously at 2σ level. The regions of NP parameters that can solve B-physics anomalies are most constrained by B → K ( * ) νν decays and B s −B s mixing. In addition, the processes of Z decays can provide a weak constraints. We find that other related processes, such as D 0 → µ + µ − , τ → µρ 0 , B → τ ν, D s → τ ν, τ → Kν, τ → µγ, and τ → µµµ decays, do not provide available constraints.
We show in Fig. 3a and Fig. 3b the allowed regions in the planes of coupling parameters (λ 233 , λ 333 ) and (λ 223 , λ 323 ) respectively when other parameters are fixed. These two subfigures show that in order to explain the B-physics anomalies, the coupling parameters need to satisfy the relation λ 333 > λ 233 > λ 223 λ 323 , and the required λ 223 and λ 323 are very small. Therefore, the next four subfigures in Fig. 3 mainly discuss the relationships between the coupling parameters λ 333 and λ 233 and the masses mb R and mν µ . From Fig. 3a, we can see that λ 333 is more constrained by R(D ( * ) ), B → K ( * ) νν decay and Z decays, but less affected by b → s + − processes and B s −B s mixing. On the contrary, λ 233 is greatly constrained by b → s + − processes and B s −B s mixing, but has little influence on R(D ( * ) ), B → K ( * ) νν decay and Z decays. As shown in Fig. 3c, after the variable parameter mb R is added, the constraints of λ 333 from R(D ( * ) ), B → K ( * ) νν decay and Z decays will be relaxed a lot. The parameters λ 333 and mb R are highly correlated. Because we choose a smaller mass of muon sneutrino, the B s −B s mixing is more sensitive to mν µ than to mb R , which can be seen by comparing Fig. 3c with Fig. 3e, or Fig. 3d with Fig. 3f. All subfigures contain parameter spaces (marked in purple) that can resolve b → s + − and R(D ( * ) ) anomalies, and satisfy the constraints from other related processes simultaneously.

V. CONCLUSIONS
The recent measurements on semileptonic decays of Bmeson suggest the existence of NP which breaks the LFU. Among them, the observables R K ( * ) and P 5 in b → s + − processes and the R(D ( * ) ) in B → D ( * ) τ ν decays are more striking. They are collectively called B-physics anomalies. In this work, we have explored the possibility of using muon sneutrinosν µ and right-handed sbottomsb R to solve these B-physics anomalies simultaneously in R-parity violating MSSM.
To explain the anomalies in b → s + − processes, we use a two-parameter scenario, where the total Wilson coefficients of NP are divided into two parts, one is the C V 9 (Noting C NP 10,µ = −C V 9 ) that only contributes the muon channel and the other is the C U 9 that contributes both the electron and the muon channels. First, we scrutinize all the one-loop contributions of the superpotential terms λ ijk L i Q j D c k to the b → s + − processes under the assumptions λ ij1 = λ ij2 = 0 and λ 1j3 = 0. We find that the contribution from the H ± −b R box diagram (Fig. 1c) is missed in the literature, this contribution is usually positive, and we can suppress it by increasing parameter tan β. The photonic penguin induced by exchanging sneutrino can provide important contribution due to the existence of logarithmic enhancement, which has never been addressed before. This contribution is lepton flavour universal due to the SM photon, so it is natural to contribute a nonzero C U 9 . Global analyses show that the sizable magnitude of C V 9 needed to explain b → s + − anomaly. However, C V 9 in the scenario with nonzero C U 9 is smaller than the one in the scenario without C U 9 . With the addition of the latest measurements from the Belle collaboration, the world averages of R(D ( * ) ) are closer to the predicted values of the SM. These changes make it possible to useν µ andb R to explain b → s + − and R(D ( * ) ) anomalies, simultaneously. We also consider the constraints of other related processes in our scenario. The strongest constraints come from B → K ( * ) νν decays and B s −B s mixing. Besides, the processes of Z decays can provide a few constraints. The other decays, such as D 0 → µ + µ − , τ → µρ 0 , B → τ ν, D s → τ ν, τ → Kν, τ → µγ, and τ → µµµ, do not provide available constraints.