Explaining the Lepton Non-universality at the LHCb and CMS from a Unified Framework

The recent results from the LHCb in the context of $(B^+ \rightarrow K^+ l l)$ decay and the CMS analysis in the context of right handed $W$-boson ($W_R$) search show a $2.6\sigma$ and a $2.8\sigma$ deviations from the Standard Model expectations respectively. In this work, we address these two seemingly uncorrelated results in the context of \R-parity violating supersymmetry. We found that a particular combination of $LQD^c$-type operators which successfully explain the LHCb result, can also accommodate the CMS excess in the $eejj$ channel of the $W_R$ search.


Introduction
The recent LHCb measurement has observed [1] a significant deviation from the standard model (SM) expectation of the ratio R K , defined as R K = Br(B + → K + µ + µ − )/Br(B + → K + e + e − ) [2]. The measurement predicts a 2.6σ deviation from the SM prediction, in the low invariat mass region (1 GeV 2 ≤ M ≤ 6 GeV 2 ) of the di-lepton pair using a data set of 3 fb −1 integrated luminosity.
More interestingly, CMS analysis for the right-handed W -boson (W R ) search has also come up with a significant deviation from the standard model expectation [3]. The CMS search uses pp collision data at the Large Hadron Collider (LHC) at a center of mass energy of 8 TeV with 19.7 fb −1 integrated luminosity. The invariant mass distribution M eejj shows an excess around 2 TeV, with a local significance of 2.8σ [3]. The CMS collaboration has also reported a possible excess in the context of the di-leptoquark search [4]. The optimization of the data assuming a leptoquark mass ∼ 650 GeV yields a local significance of 2.4σ (2.6σ) in the eejj (eνjj) channel.
There have already been quite a few studies in an attempt to explain these results separately assuming different models. In Ref. [5], the authors studied the observed value of R K in the context of effective operator approach, illustrated with two leptoquark models. They also mentioned that the leptoquark couplings considered there, can correspond to certain R-parity violating (RPV) supersymmetric scenario. However the flavor structure of those couplings can not address the CMS eejj excesses. Certain constraints have been put on these effective operators in [6]. On the other hand, the CMS excess in the context of both W R and dileptoquark search is interpreted in [7] with a resonant coloron production and further decay of the coloron into a pair of leptoquark and in [8], with resonant production of vector like leptons via W /Z vector boson. A similar analysis [9] with W /Z have been performed in the context of W R search. Refs. [10,11] showed that the W R excess can be explained within the context of GUT models. Within the framework of R-parity violating supersymmetry (SUSY) an explanation via resonant slepton production has been provided in [12] in the context of CMS W R search. Ref. [13] the di-leptoquark excess is explained.
Though it is quite preliminary to jump into any conclusion before a more detailed analysis of the data and also the fact that the statistics is very low in the case of CMS analyses, one can still take these deviations at its face value in order to ensure a better search strategies either to claim a discovery or to put an exclusion limit. It is worth noting that while there are individual explanations for each of these results mentioned above, so far there has been no attempt to have explain both simultaneously. In this article we present an unified framework which can accommodate both the LHCb and the CMS W R search results in the context of the RPV minimal supersymmetric standard model (MSSM). Given the fact that the deviation in the measured value of R K can be consistent with new physics (NP) either in the electron or in the muon sector due to the large theoretical uncertainties present in the SM expectations, in this article, we focus on the former possibility motivated by the observed CMS excess.
The remaining of the paper is organized as follows: In section 2, we give a brief account of the RPV model. Section 3 describes both the B-physics and collider consequences of this model. Finally, we summarize our results and conclude in section 4.

Model: RPV SUSY
In this section, we will give a brief review of R-parity violating SUSY scenario. While R-parity conserving SUSY has many judicious features, which made it one of the most popular frameworks, R-parity violating SUSY [14][15][16][17][18] provides an alternative. It can relax the naturalness bound from LHC due to the absence of large missing transverse energy ( E T ) signature and at the same time provides rich collider phenomenology. In MSSM, the RPV interactions are generated through the following superpotential, where L i , E c i denote SU(2) L doublet and singlet superfields for leptons respectively, Q i , U c i and D c i represent the left-handed quark doublet, right-handed up-type quark singlet and righthanded down-type quark singlet superfield respectively and H u is the up-type Higgs superfield that gives mass to the up-type quarks. Here, W hTL are the trilinear terms which contains only dimensionless parameters, and W hBL denotes the holomorphic bilinear terms containing dimensionful couplings. The λ ijk 's and λ ijk 's are Lepton number violating and λ ijk 's are Baryon number violating Yukawa couplings.
In the context of the present study, we will work with only λ ijk -type of couplings, in particular, with λ 112 and λ 113 couplings, purely motivated by the observations of LHCb and CMS. We found that it is the only combination of RPV couplings which can be consistent with both of these observations. The coupling constants for the RPV operators are typically small due to the constraints from various observables including proton stability, neutrino mass and mixing, processes with flavor-changing neutral current and CP violation, cosmological baryon asymmetries, etc. (see e.g. Ref. [19] for a comprehensive review). Recently there has been some works for providing an organizing principle that explains why RPV couplings are typically small and hierarchical [20][21][22][23][24].
The choice of our RPV couplings λ 11k with k = 2 and 3, are constrained from various low energy observables such as, (i) charge-current universality, (ii) e−µ−τ universality, (iii) atomic parity violation etc. The most stringent bound on individual RPV coupling comes from (i) and (ii) [25,26] |λ 11k | 100 GeV md where, md kR is the mass of the right-handed down-type squark. The bounds on the product |λ 112 λ 113 | mainly come from charged B-meson decay B ± d → π ± K 0 , B s −B s mixing and B → X s γ transition. Assuming the mediator mass to be around 100 GeV these translate to [29,30]. (3) In addition to the known bounds on RPV couplings λ 11k listed above, we present new bounds obtained by analyzing the non-observation of "contact interactions" from collider searches in the following. The collider experiments at the LEP [31], HERA [32] and Tevatron [33,34] have put some bounds [35] on the cut-off scale of the four-fermion operator, 4π 2 TeV for q = s and Λ LR ∼ 2.8 TeV for q = b. This imposes the following bounds on the RPV couplings: Also from a global study of electron-quark contact interactions [36,37] through ZEUS [38], H1 [39], polarized e − on nuclei scattering experiments at SLAC [40], Mainz [41], and Bates [42], Drell-Yan production at the Tevatron [43], total hadronic cross section σ had at CERN LEP [44][45][46][47][48][49][50][51], and neutrino-nucleon scattering from CCFR [52], the highest fit value is found to be Λ LR ∼ 11.2 TeV, which translates into 3 Phenomenology

Lepton non-universality at the LHCb
We begin with analyzing the recent result on the measurement of R K reported by the LHCb collaboration. As we aim for finding an unified framework for explaining two seemingly uncorrelated measurements, we restrict our analysis for R K within the context of eejj excess reported by CMS in the context of W R search only. For this, the RPV LQD c -type λ 112 and λ 113 couplings are the most important parameters, 1 where the former plays a major role in determining the size of both observables. Therefore, in our B-physics analysis we will focus on finding a reasonable parameter space which would allow a sizable λ 112 coupling compatible with the CMS eejj data.
In the SM, b → s flavor changing neutral current transition is in general highly suppressed due to absence of tree level processes. The leading order contribution comes from electroweak loop processes. Therefore, it provides an important tools to test the flavor sector of the SM, as well as to probe and constrain its possible extensions. In this context, for the exclusive decay B + → K + ll with l = e, µ, one of the very useful observables is the ratio (R K ) of the branching fractions in the individual lepton flavor mode.
Theoretically, R K ≈ 1 from the lepton universality in the SM, which ensures that electron and muon couple to the gauge bosons with the same strength. Although the individual branching fractions of B + → K + ee and B + → K + µµ suffer from theoretical uncertainties of O(30%) [53], R K remains unaffected as the uncertainties cancel out while taking the ratio [2]. Hence, it is a clean and sensitive observable for probing the extension of the SM, specially for the flavor-non-universality.
The recent measurement of R K in the low di-lepton invariant mass squared region, 1 GeV 2 < q 2 < 6 GeV 2 , is found to be [1], This corresponds to a 2.6σ deviation from the SM prediction, R K = 1.0003 +0.00010 −0.00007 [2,54], indicating a possible hint of new physics. As discussed in the introduction, there are two possible explanations: it could be either due to the depletion in Br(B + → K + µµ) or an enhancement in Br(B + → K + ee). We focus on the latter, in order to explain the CMS eejj excess as well.
We begin with considering the following effective weak Hamiltonian forsbll transition, where α e , V ij , G F and µ are the fine structure constant, the CKM matrix elements, the Fermi constant, and the renormalization scale respectively. The relevant dimension sixsbll operators in our case are vector and axial-vector operators The corresponding chirality-flipped operators O are obtained by changing P L ↔ P R . It is convenient to divide the Wilson coefficients as where, C SM( ) (µ) is the SM contribution and C NP( ) (µ) is the NP contribution. We have C SM 9 (m b ) = −C SM 10 (m b ) = 4.2 for all leptons while rest of the semileptonic Wilson coefficients are negligible [55].
As described in section 1, in this work we consider the following R-parity violating term in Eq. (1) as a source of NP, In the context ofb →see transition (Fig. 1), we analyze the coupling between up-type squark u L , down-type quarks s and b, and electron, Integrating outũ L , we obtain the following effective Hamiltonian where, mũ L is the mass ofũ L , and the Wilson coefficients in terms of the RPV operators are given by Here from now onwards, we suppress the explicit µ dependencies of the Wilson coefficients for simplicity. Note that since we only have the vector and axial-vector operators, it is straightforward to obtain bounds on the relevant parameters from the experimental data. Following [5,56], the bound from R K at 1σ level is given by 2 where, X e = 2C e 9 and X µ = 0 in our case. The other important bound comes from theB s decay. In the absence of the scalar and pseudoscalar operator, the model independent constraint is given by [5], The corresponding experimental data [57] and SM value [58] of Br(B s → ee), and their ratio are given by From these, we obtain a bound on the Wilson coefficient C e 10 , Plugging in the values of input parameters [57], |V tb | = 0.999146 +0.000021 The above two equations sets the hierarchy between the two couplings λ 112 and λ 113 for a fixed value of mũ L .

Lepton non-universality at the CMS
We have considered resonant slepton production via λ 112 coupling in pp collision at the LHC at 8 TeV center of mass energy with 19.7 fb −1 integrated luminosity. The resonant slepton production at collider experiments has been studied extensively in [59][60][61][62][63][64][65][66]. The slepton thus produced can decay via both R-parity conserving and violating couplings. The branching ratio depends on the mass of the lightest SUSY particle (LSP) and the λ 112 coupling. Both the selectron and the sneutrino have a substantial decay branching fraction of decaying into eχ 0 1 and eχ + 1 respectively. The lightest neutralino and lighter chargino can further decay via the RPV coupling resulting in a eejj final state as studied in the context of the W R search at CMS [3], In Fig. 2 we show the Feynman diagrams leading to the above final state through the resonant production of selectron (left) and sneutrino (right). We have considered three different benchmark scenarios to take into account the model dependency in the branching ratio calculation: • Bino-like scenario: M 1 M 2 , the LSP is dominated by the bino-component, with heavy wino mass (> 2 TeV). In this scenario the branching ratio of the slepton decay via R-parity conserving coupling can be subdominant compared to the di-jet mode.
• Mixed inverted scenario: M 1 : M 2 = 3 : 2, i.e., the LSP is mostly wino-like with a small bino-admixture. In this case the slepton has a substantial branching ratio of decaying to the lightest neutralino and lighter chargino. In this scenario, however, both the lighter chargino and the lightest neutralino decay via RPV coupling. Hence, the lepton and jet multiplicity get enhanced in the final state compared to the above two cases.
• Wino-like scenario: M 2 M 1 , i.e., the LSP is purely wino-like. This scenario is similar to above (mixed-inverted) one with a slight enhancement in the effective˜ (ν) → eejj branching ratio.
The model spectrum and decay branching ratios have been calculated using SARAH-4.3.1 [67,68] and SPheno-3.3.2 [69,70]. In Fig. 3 we present the effectiveẽ L → eejj branching ratio vs. λ 112 . For the rest of the work we will assume the lighter slepton masses of the first generation mẽ L ,ν L = 2.1 TeV and dominantly left-chiral. Squark masses of first generation are ∼ 1.5 TeV We have simulated the resonant slepton production in pp collision at the LHC using Mad-Graph5 [72] and the subsequent decays, initial and final state radiation, parton showering and hadronization effects have been done using PYTHIA (v6.4) [73]. We have worked with CTEQ6L [74] parton distribution function. The factorization and the renormalization scales are set at the slepton mass µ F = µ R = mẽ L . To take into account the next-to-leading order QCD correction we multiply the tree-level cross-section by the K-factor 1.34 [65]. We have also used various resolution functions parametrized as in [75] for the final state objects to model the finite detector resolution effects. We have assumed the object definition described in [3] for the final state particles along with the following cuts, • Invariant mass of the electron pair, M ee > 200 GeV.
• Invariant mass of the two electrons and dijet system, M eejj > 600 GeV.

Results and Discussion
In this article, we have addressed the recent CMS and LHCb results from an unified framework. The results of our analysis are shown in Table 1 and Fig. 4 which show the comparison of signal, background and the corresponding data for a typical benchmark point of wino-like scenario. A more detailed analysis depicting the range of λ 112 coupling which can be compatible with the CMS result is presented in Fig. 5. From Fig. 5a   scenario. The color gradient in Fig. 5 signify the S/ √ S + B estimate 3 of the signal where, S is the signal event and B is the background event within 1.8 TeV ≤ M eejj ≤ 2.2 TeV. This is due to the fact that, the cross-section grows with |λ 112 | 2 , whereas the R-parity conserving decay branching ratios of the slepton falls with the increase in λ 112 (see Fig. 3). Thus two regions has been obtained which give equal event rates in the eejj analysis. For the bino-like scenario and the mixed scenario the contribution to eejj final state mainly comes from resonant selectron production (Fig. 2a). The contribution from the resonant sneutrino production is negligible due to the fact that the RPV decay of the chargino (see Fig. 2b) has a very small branching ratio.
We can see from Fig. 5b the allowed value of λ 112 is smaller compared to the bino-like case owing to the fact that additional contribution coming from resonant sneutrino production and also enhanced effective branching ratio of BR(ẽ L → eejj) compared to the bino-like scenario. Note that both the low and high values of the neutralino mass for a λ 112 ∼ 0. 21 Figure 5: The region of λ 112 -m χ 0 1 parameter space compatible with the CMS excess (W R search) for the (a) bino (left) and (b) wino-like (right) scenario. The color gradient signify the S/ √ S + B estimate of the signal where, S is the signal event and B is the background event within 1.8 TeV ≤ M eejj ≤ 2.2 TeV. It is not possible to explain the CMS excess assuming the mixed scenario due to a very small effective branching fraction BR(ẽ L → eejj). The mixed-inverted scenario also gives similar excess as in wino-like scenario.
branching ratio for m χ 0 1 ∼ 1 TeV is compensated with a higher cut-efficiency. We emphasize here that the CMS excess has reported the data summed over bins having total width of 400 GeV. The distribution of the data within this range is yet unknown. A fine binning of data is required at high luminosity run. In case the data is distributed over this wide range, a resonance explanation of a given mass may not be a good option. However, the wino and mixed-inverted scenario can be better suited in such a case. This requires a splitting O(10 2 GeV) between the selectron and the sneutrino which can be achieved by introducing large RPV soft-terms of the same type.
The CMS eejj excess constrains the λ 112 coupling independent of the LHCb result. We use the results discussed above to constrain the parameter space of B-physics analysis, namely, the λ 113 coupling and relevant mass parameter (mũ L ). Fig. 6 shows plot in the λ 113 -mũ L plane consistent with the experimental data coming from the measurements listed in section 3.1 assuming two fixed values of λ 112 = 0.2 and 0.4. The present LHC bound on mũ L (> 1 TeV) in presence of λ couplings comes from the di-leptoquark search analysis [4]. This leaves us with the choice of λ 113 as low as 0.006 (0.0125) for λ 112 = 0.4 (0.2).
In summary, our important observation is that the RPV SUSY operator which can explain the lepton non-universality hinted by the measurement of R K from LHCb, can easily accommodate the lepton non-universality observed by CMS in the context of W R search. In this analysis we do not address the CMS eejj and eνjj excesses in the context of di-leptoquark search. A dedicated analysis [76] is required to verify whether the CMS di-leptoquark result can be accommodated within this framework or not. We note that, future measurements in all these sectors can tell us with certainty whether the current deviations are robust or not. As an outlook, we also suggest few collider signatures such as, lepton charge asymmetry measurement in the eνjj-channel and ratio of same-sign di-lepton events to opposite-sign di-lepton events in the eejj channel to further discriminate our scenario at the LHC. Our result in the context of R K will be confronted with all the other B-physics observables, which might further constrain the allowed range of the parameter space of the model considered here. The effective operators considered here may also give rise to rare B-decays like b → sνν [77], which could be a promising channel for future B-physics experiments.