Studying the $b\rightarrow s \ell^+\ell^-$ anomalies and $(g-2)_{\mu}$ in $R$-parity violating MSSM framework with the inverse seesaw mechanism

Inspired by the recent experimental results which show deviations from the standard model (SM) predictions of $b\rightarrow s \ell^+\ell^-$ transitions, we study the $R$-parity violating minimal supersymmetric standard model (RPV-MSSM) extended by the inverse seesaw mechanism. The trilinear $R$-parity violating terms, together with the chiral mixing of sneutrinos, induce the loop contributions to the $b\rightarrow s \ell^+\ell^-$ anomaly. We study the parameter space of the single-parameter scenario $C^{\rm NP}_{9,\mu}=-C^{\rm NP}_{10,\mu}=C_{\rm V}$ and the double-parameter scenario $(C_{\rm V},C_{\rm U})$, respectively, constrained by other experimental data such as $B_s-\bar{B}_s$ mixing, $B\rightarrow X_s \gamma$ decay, the lepton flavour violating decays, etc. Both the single-parameter and the double-parameter scenario can resolve the long existing muon anomalous magnetic moment problem as well, and allow the anomalous $t\rightarrow cg$ process to reach the sensitivity at the Future Circular hadron-hadron Collider (FCC-hh).


Introduction
In recent years, several hints of new physics (NP) beyond the SM have shown up, such as, R K ( * ) = B(B → K ( * ) µ + µ − )/B(B → K ( * ) e + e − ) on the transitions of b → s + − ( = e, µ), which exhibits the very attractive anomalies. Especially, the measurement of R K by the LHCb Collaboration has just been updated with the full run II data as R K = 0.846 +0.042 +0.013 −0.039 −0.012 in the q 2 bin [1.1, 6] GeV 2 [1], which is much more precise than the previous one R K = 0.846 +0.060 +0.016 −0.054 −0.014 [2], giving rise to the discrepancy with the SM value changing from preceding 2.5σ to 3.1σ. The recent measurements of R K * by LHCb give R K * = 0.66 +0. 11 −0.07 ± 0.03 at [0.045, 1.1] GeV 2 bin and R K * = 0.69 +0. 11 −0.07 ± 0.05 at [1.1, 6] GeV 2 bin, showing 2.1σ deviation at low q 2 region and 2.5σ deviation at high region, respectively [3]. The R K ( * ) results by the Belle Collaboration [4,5] show the consistency with the SM predictions although the results have sizable experimental error bars. Besides, there are also other anomalies in the b → s + − transition, for instance, the angular observable P 5 anomaly of B → K * µ + µ − decay persists with the new data [6] compared with the run I results [7][8][9][10][11][12]. All these anomalies may indicate the NP that breaks lepton flavour universality (LFU).
The first scenario, called scenario A here, requires C U = 0 to realize the single-parameter scenario C NP 9,µ = −C NP 10,µ in fact. We adopt the fit result −0.46 < C V < −0.32 that conforms to the rare B-meson decays at 1σ level in Ref. [28]. Except the new R K measurement [1], authors in Ref. [28] have also considered other series of new experimental results, such as the new angular analyses of B 0 → K * 0 µ + µ − [6] and B ± → K * ± µ + µ − [37], the updated branching ratio measurement of the B s → φµ + µ − [38] that confirms the previous tension [39] with the SM prediction, as well as the recent results of B s → µ + µ − from CMS [40] and LHCb [41,42].
For the case of C U = 0, named as scenario B, we also utilize the fit regions in Ref. [28] with the best fit point (C V , C U ) ≈ (−0.34, −0.32).
After these results of the model-independent analyses are obtained, imperative works are to find the concrete NP models which can conform to them. Both the scenario A and B have been implemented in RPV-MSSM [43][44][45][46][47][48]. When masses of sneutrinos/sd-quarks are sufficiently heavy or there is a cancellation in the penguin contribution [46], the scenario B turns into the scenario A.
The clues of LFU violation exist not only in B-meson decays but also in other processes, such as, the muon anomalous magnetic moment problem which has existed for several decades.
The combined deviation average of the two experiments, ∆a µ = a exp µ −a SM µ = (2.51±0.59)×10 −9 shows the increased tension at the significant 4.2σ level and this is a growing motivation of NP.
For the electron anomalous magnetic moment, there is a negative 2.4σ discrepancy between the measurement [115] and the SM prediction [116], ∆a e = a exp e − a SM e = (−8.7 ± 3.6) × 10 −13 , due to the new measurement of the fine structure constant in Ref. [117] 2 . There are plentiful articles discussing the (g − 2) µ problem in the SUSY framework (e.g., see Refs. [48,).
In this work, we will investigate whether the parameter space for the explanation of b → s + − anomalies can be in accord with the deviation ∆a µ , and then discuss the NP effects on a e . Our paper is organized as follows. At first, the new model in this work is introduced in section 2. Then in section 3, the whole one-loop contributions to the b → s + − transition in this model are scrutinized and we emphasize the main contributions to explain the b → s + − anomaly in our parameter scheme. We discuss NP contributions to (g − 2) and other related constraints in section 4 followed by the numerical results and discussions in section 5. Our conclusions are presented in section 6.
1 A recent calculation of the hadronic vacuum polarization (HVP) [105] weakens the discrepancy between the experiment and SM prediction of a µ while it shows the tension with the R-ratio determinations [106][107][108][109]. And even the large HVP contribution can account for the measurement of a µ , there exists the tension within the electroweak (EW) fit [110][111][112][113]. We do not consider this HVP result here but consider the preceding review of various SM results [104]. 2 Another new measurement of the fine structure constant [118] differs by more than 5σ to the previous one [117] and affects the deviation ∆a e to positive 1.6σ level [119]. The NP hint search in a e still expects more experimental researches and we focus on a µ anomaly explanations in this work.

The Model
The model considered in this work is RPV-MSSM with inverse seesaw mechanisms called RPV-MSSMIS here and the superpotential is expressed by where the generation indices i, j, k = 1, 2, 3 while the colour ones are omitted. All the repeated indices are defaulted to be summed over unless otherwise stated. Here the superpotential of MSSM [170,171] is expressed as In RPV-MSSMIS, MSSM superfields are extended by three generations of pairs of SM singlet superfields,R i andŜ i . The neutral parts of two Higgs doublet superfieldsĤ The tree-level trilinear RPV coupling λ ijkL iQjDk can be added forL i sharing the same SM quantum number withĤ d . It is needed to point out that the RPV superpotential terms like λ ijkL iQjDk , λ ijkLiLjÊk , λ ijkÛ iDjDk as well as µ iLiĤu are all in principle allowed for the SM gauge invariance if there are no extra symmetries. Here we only consider the term λ ijkL iQjDk connecting the quark sector with lepton sector, without the pure-quark term λ ijkÛ iDjDk or the purely lepton interaction λ ijkLiLjÊk , because of the attempt to avoid the probable disastrously rapid proton decay [172,173] when there are nonzero parameters λ and λ simultaneously and the strong collider constraints on the lightest sneutrino mass when the λ and λ both exist [174][175][176][177][178][179][180][181][182][183][184]. The bilinear term µ iLiĤu is also allowed but we exclude it in order to avoid the extra contributions to neutrino masses [185].
With the scalar components of Higgs doublet superfields denoted by H u and H d , and squarks and sleptons denoted by "˜", the soft supersymmetry breaking Lagrangian is given by where L soft MSSM corresponds to MSSM part [170,171]. It should be mentioned that MSSM and singlet neutrino sectors are all at low scale (around 1 TeV) in this work, so some terms in the most general superpotential and soft breaking Lagrangian are already or will be eliminated ad hoc for the phenomenological consideration.
As to the three terms following W MSSM in Eq. (2.1) which give the neutrino mass spectrum at tree level, the 9 × 9 mass matrix of neutrino in the (ν, R, S) basis is given by And the µ S parameter can be obtained by Then we turn to the sneutrino mass square matrix in the (ν I(R) L ,R I(R) ,S I(R) ) basis, which is expressed as where the "±" above expresses the CP -even and CP -odd, and also CP -odd is denoted by I and CP -even is denoted by R. The soft mass m 2 L = m 2 L + 1 2 m 2 Z cos 2β +m D m T D can be regarded as one whole input where m 2 L is the soft mass square ofL in L soft MSSM . The CP -even and CP -odd masses can be nearly the same for tiny µ S and relatively small B µ S [187]. Besides, the value of m 2 S is set to be zero here. Thus, the approximate form is provided as [188] In the following we adopt this particular structure and then the mixing matricesṼ I(R) , which diagonalize sneutrino mass square matrices byṼ I(R) M 2 diag , are also the same whether CP even or odd. All theṼ R and the physical mass mνR can be expressed asṼ I and mνI , respectively in the rest of the paper. With respect to charged sleptons, the LH sector element is given by m 2 L + m 2 l − m D m T D − m 2 W cos 2β. Afterwards we discuss the last term of the superpotential. For the superpotential term λ ijkL iQjDk , the corresponding Lagrangian in the flavour basis is obtained as where "c" indicates the charge conjugated fermions. Then in the context of mass eigenstates for the down quarks and charged leptons, the Lagrangian above with other fieldsν L , ν L and u L (aligned withũ L ) rotated to mass eigenstates by mixing matricesṼ I(R) , V and K respectively, is given by where all the fields are represented by the mass eigenstates. Concretely, ν v andν v are in the mass eigenstate with the index v = 1, 2, . . . 9 and the three neo-λ terms are deduced as In the following, we call the diagrams including these λ couplings by "λ diagrams", otherwise by "non-λ diagrams".
By the end of this section, we should mention the chargino and neutralino mass matrices equations from here on. This kind of assumption is also taken in recent works for the similar phenomenological consideration, such as Refs. [43][44][45][46][47]. Authors of Ref. [45] further assume that λ ij1 = λ ij2 = 0 which are adopted in Refs. [46,47] considering the bounds of τ → µρ 0 and τ → µφ decays, while these constraints can also be negligible by setting sufficiently heavy mũ Lj .
We scrutinize all the one-loop  [189] and C NP 9(10) (µ b ) vanishes due to the approximate conservation of (axial-)vector currents. Thus we can constrain the model parameters related to C 9(10) (µ b ) using the global fit results introduced in section 1.
From the appendix, one can see thatl i andd Lj in the box diagrams can be forbidden under the assumption of a single value k. In the following we further assume that md Rk is sufficiently heavy to focus on the contributions of sneutrinos as the bridge between the trilinear RPV term and the inverse seesaw mechanism. Therefore, contributions includingd Rk are negligible and can be removed. Besides, we also set mũ Lj adequately heavy 3 . Because the LFU violating contributions mainly from µ + µ − channel are expected to explain b → s + − anomalies in both scenario A and B, we will set that M 2 ν I(R) has no flavour mixing and the electron-flavour elements with both LH and RH chirality are sufficiently heavy. Then nearly all box contributions to b → se + e − transition and some box contributions to b → sµ + µ − transition can be eliminated and afterwards we show which contributions remain.
Firstly among these non-negligible chargino box diagrams, the non-λ diagram with RH sneutrinos previously discussed in Ref. [56] is recalculated by us. We find the Wilson coefficient from this diagram equals to −C NP 10 , which is different from the condition that C NP in Ref. [56]. The related C NP 9 , namely as C in this paper, is given by where the Yukawa couplings . This formula can be seen in the second term of Eq (A.1).
Then we show the λ within chargino box diagram containing the RPV interactions between singlet sneutrinos and quarks. The contribution is given by 3) which appears in the third term of Eq (A.1).
where the mixing matrix elements . It is obvious that these W/H ± box contributions include SM effects, which cannot be separated naively from NP effects because of the generation and the chiral mixing of massive neutrinos. In addition, these contributions still contain both the µ + µ − channel sector and e + e − channel sector.
We will further investigate these contributions in detail at section 5.
Next we show the penguin contributions. Indeed, the Wilson coefficients of Z-boson penguin diagrams are negligible. While the contributions of photon penguin diagrams can be non- 4 The (g − 2) and other constraints In this section, we introduce the NP contributions to (g − 2) and other related processes.

The muon (electron) anomalous magnetic moment
The amplitude of the → γ ( = e, µ) transition is given by in the zero limit of photon moment q. The second term in the bracket gives the loop corrections and a is called the anomalous magnetic moment for the related lepton.
The SM-like diagrams that only involve SM particles give the same contributions to a as the SM. Hence the SUSY part can contribute to the observed anomaly ∆a [141]. The oneloop chargino and neutralino contributions in RPV-MSSMIS are given here, with referring to Refs. [122,128,141,190], and functions has different features compared with ∆a µ , we consider the scheme of |δa χ ± µ | |δa χ ± e | ≈ 0 and |δa χ 0 e | |δa χ 0 µ | ≈ 0 [131]. And thus, the muon generation associated with RH sleptons is set sufficiently heavy as well as heavyL 1 which is already assumed in section 3. Afterwards we expect 1.92 (1.33) |δa χ ± µ + δa χ 0 µ | × 10 9 3.10(3.69) to be in accord with a µ data at 1(2)σ respectively.

Tree-level processes
In the following we investigate related transition bounds which exchanged Rk at the tree level.
On account of the assumption of heavy md Rk ∼ 10 TeV, the neutral current processes B → K ( * ) νν, B → πνν, D 0 → µ + µ − as well as the charged current processes B → τ ν, D s → τ ν and τ → Kν provide no effective constraints. Besides, there are some tree-level processes which may make some slight bounds to mention here.
The effective Lagrangian for K → πνν decay can be described by 193]. While the NP contribution is Then the bound is obtained as |λ N i 2k λ N * i1k | < 0.074 when md Rk around 10 TeV [47], and hence, we can set |λ i1k | 10 −2 to avoid this bound.

B → X s γ
(4.10) The contribution of charged Higgs is given by 7 (y t ), (4.11) where the form factors are F 4(yt−1) 4 log y t and F 6(yt−1) 3 log y t with y t = m 2 t /m 2 H ± . One can see the F 7 (y t ) part is not suppressed when tan β is large and this is unlike the H ± contributions to C 9(10) which are entirely suppressed by large tan β. The formulas of C χ ± 7 engaging the chargino together withũ Rj and the QCD corrections can be seen in Ref [195].

B s −B s mixing
Another process we should consider is B s −B s mixing, mastered by the Lagrangian where the non-negligible NP contribution is given by including the λ diagram with double sneutrinos and the non-λ diagram with double RH su-quarks, and the SM contribution C SM  [198] leads to the bound of 0.90 < |1 + C NP Bs /C SM Bs | < 1.11, (4.14) at 2σ level.

Lepton flavour violating decays
We discuss the lepton flavour violating decays including τ → γ, µ → eγ, τ → , µ → eee and τ → . Firstly, the λ -diagram contributions can be eliminated whenb R is sufficiently heavy [47]. As to the non-λ diagrams, all the neutralino-slepton diagrams contain flavour mixings of charged sleptons and all the chargino-sneutrino diagrams contain flavour mixings of sneutrinos (see Ref. [199] for concrete formulas). So the effects of these two kinds of diagrams vanish when there are no flavour mixing in the two mass matrices. For contributions of W/H ± -neutrino diagrams, they are always connected to these terms which are and their conjugate terms, where α, β = e, µ, τ and α = β [199]. In section 5.1, we will show all these terms contribute no effects under the particular structure of neutrino mass matrix. The same analyses can also be applied to the non-λ diagrams in For the λ diagrams of these two processes, we refer to the detailed discussions in Ref. [46], and no obvious constraints are found.

Anomalous t → cV (h) decays
The SM predicts the branching ratios of t → cV decays (V stands for the vector bosons including Z, γ and the gluon g) and t → ch decay (h stands for SM-like Higgs) below the scale of 10 −15 − 10 −12 [200] due to the Glashow-Iliopoulos-Maiani suppression. This scale is beyond the detection capabilities at the collider in the near future. The most recent experimental 95% CL upper limits on the branching ratios of these top quark decays at the Large Hadron Collider (LHC) show that B(t → cZ) < 2.4 × 10 −4 [201], B(t → cγ) < 1.8 × 10 −4 [202], B(t → cg) < 4.1 × 10 −4 [203] and B(t → ch) < 1.1 × 10 −3 [204]. Compared with the effects from pure MSSM, the one-loop RPV diagrams can make more contributions [205,206], and hence we will investigate these effects in our model.
For the t → cV decays, the effective tcV vertices are expressed as, where k ν is the momentum of the vector boson. The form factors A Z and B V are given by [205] A Z =λ * i2kλ i3k where p t and p c are the momentums of top and charm quarks and functions B 1 and c ij are the Passarino-Veltman integrals totally referring to Ref. [207]. The constants (4.17) As for the t → ch decay, the effective tch vertex is expressed as, After omitting masses of charm quarks and all down-type quarks, one can obtain [206] A h R =λ * i2kλ i3k where factor Yl Li ≈ m Z sin θ W cos θ W ( 1 2 − sin 2 θ W ) cos 2β when the masses of leptons are omitted and the mass of CP -odd Higgs is sufficiently heavy. Then the decay width of t → ch is (4.20) The NP contributes to related branching ratios are given by B(t → cV (h)) NP = Γ(t → cV (h)) NP /Γ(t → bW ) SM where the dominant decay t → bW has the SM prediction Γ(t → bW ) SM = 1.42 GeV [208].
For the constraints on LH sleptons as mentioned in section 2, when considering non-zero λ couplings, the lower bounds of mν L and ml reach TeV scale [182][183][184]. Because only nonzero λ couplings are restricted in this work, LH sleptons decaying to pure leptons directly is secondary and processes without λ interactions should been taken into consideration mainly.
We consider the searches which focus on LH sleptons decaying into leptons and the lightest neutralino χ 0 1 [216][217][218], and the recent ATLAS results [217] show that the LH sleptons which are heavier than χ 0 1 can avoid the exclusion for m χ 0 1 300 GeV. Besides, the compressed scenario that the lightest chargino mass m χ ± 1 is slightly heavier than m χ 0 1 [219], is adopted. Thus we will let inputs inducing m χ ± 1 m χ 0 1 300 GeV and ml L > 300 GeV.

Numerical results and discussions
In this section, we investigate b → s + − anomalies numerically as well as the a µ anomaly and the related constraints.

Choice of input parameters
First parts of input parameters used throughout the paper are collected in table 1, which includes the lepton oscillation data [74] under the normal ordering (NO) assumption of LH neutrino masses. In addition, we further keep δ CP = π to omit the CP violation in U PMNS .
The lightest neutrino mass is set zero to have the masses of three-flavour light neutrinos as  [192] [192]     approximate numerical form of the mixing matrix V T , terms in the contributions of W/H ± -neutrino diagrams to the lepton flavour violating decays within section 4.5. V T * αv V T βv can be decomposed into the following two parts, 9 v βv h related to the nearly degenerate heavy neutrinos and light neutrinos respectively [188]. It can also be found that V T * αv V T βv provides no effects from Eq. (5.1). In sum, the lepton flavour violating decays mentioned in section 4.5 contribute no effective bounds in our input sets.
In the following we discuss the feature of C Our parameter set leads that the contribution ∆C In summary, the LFU violating coefficient and the LFU coefficient can be represented by U , respectively. We find that the factor c µL mv = −g 2 V m1Ṽ . Therefore, the large chiral mixing of sneutrinos will make some enhancements to both C V and a NP µ simultaneously.

Explanations of
In this part we will search for the common areas of these five variables, mL 2 , λ 223 , λ 233 , λ 323 and λ 333 , to explain b → s + − anomalies as well as (g − 2) µ deviations considering related constraints, in two fit scenarios mentioned in section 1. In scenario A, we fix mL 3 = mL 2 − 50 GeV which is benefit for satisfying the constraint of B s −B s mixing. In scenario B, we fix For the bounds of ml L in LHC searches mentioned in section 4.7, we focus on mL 2 370 GeV which makes the mass of lightest charged slepton ml 1 (sneutrino mν 1 ) be above 318 (301) GeV in scenario A and mν 1 be heavier than 100 GeV while ml 1 be above 352 GeV in scenario B. In particular, we choose k = 3 for a benchmark of the numerical calculation.

Results in scenario A
Next we investigate b → s + − anomalies further with the parameter regions of mL 2 we have obtained above. In scenario A, C U should be 0 as the definition. To make C U cancel out,

Results in scenario B
In scenario B, the common scopes are shown at figure 5 Figure 6(a) shows the common scopes constrained by the rare B-meson decay fits at 1σ level denoted by painted areas and 2σ level denoted by hatched areas, combined with other process constraints at 2σ level for mL 3 = mL 2 = 550 (green), 750 (orange) and 950 GeV (red). Figure 6(b) shows the regions which satisfy the rare B-meson decay fits with being 1(2)σ favored under the assumption λ * 323 λ 333 = −λ * 223 λ 233 , denoted by red (blue) points, with other process constraints being considered. The area on the left of the red dashed line is allowed to be accordant with a µ data at 2σ.

Predictions of t → cg decay
As the numerical discussions above, we have the final parameter spaces of mL 2 as well as the coupling combinations λ * 223 λ 233 and λ * 323 λ 333 to explain related LFU violating anomalies. While these variables also provide NP effects on the top decays t → cV (h).
We have checked that our final parameter spaces can satisfy the most recent upper limits on the branching ratios of t → cV (h) decays at LHC easily. The NP contributions to the branching ratios of t → cV (h) depend on the termλ * i2kλ i3k fl Li where fl Li stands for the loop integral including LH charged sleptons. And this term can be given byλ * i2kλ i3k fl Li . Because of cancelling out in λ * i2k λ i3k fl Li , the hierarchical structure between λ a2k and λ a3k (a = 2, 3) is considered to make prominent contributions and we set the large λ a3k here. We keep restricting k as 3 and set λ 233 = λ 333 = 2 or 3.
In the following we show that the prediction values of B(t → cg) NP from the parameter spaces to explain b → s + − and a µ anomalies can reach the sensitivity at the FCC-hh in figure 7. One can see that in scenario A, when 370 GeV mL 2 440 GeV and λ a33 = 3, the prediction B(t → cg) NP is higher than the prospect upper limit 9.87×10 −8 , at 100 TeV FCC-hh for the integrated luminosity of L = 10 ab −1 of data through the triple-top signal [223] , and the prediction in scenario B for the same λ a33 can also reach this upper limit. When λ a33 is set to be 2, the branching ratio is much lower and can not even reach the sensitivity at FCC-hh for the estimated L = 39 ab −1 in both scenario A and B. We conclude that, at FCC-hh, this model signal on the t → cg transition has considerable possibilities to be found for sufficiently large λ a33 , but the model can escape easily from the bound of this transition when the structure between λ a23 and λ a33 is not hierarchical enough.

Conclusions
Recent measurements on the transition b → s + − reveal the deviations from SM predictions.
The most motivative R ( * ) K anomaly and anomalies from other observables like P 5 , called b → s + − anomalies collectively, suggest the NP of LFU violation may exist. Besides, this NP may also affect the enduring muon anomalous magnetic moment, (g − 2) µ problem.
In this work, we have studied the chiral mixing effects of sneutrinos in the R-parity violating MSSM with inverse seesaw mechanisms to explore the explanation of b → s + − anomalies with (g − 2) µ problem simultaneously. Here all the one-loop contributions to b → s + − processes are scrutinized under the assumption of a single value k. Among them, the contributions of chiral mixing between LH and singlet (s)neutrinos within superpotential term λ ijkL iQjDk are given for the first time to our knowledge. To explain b → s + − anomalies in this model, two kinds of model-independent global fits are adopted. One is the single-parameter scenario of C NP 9,µ = −C NP 10,µ and the other scenario is the double-parameter one that (±)C V contributes to the C NP 9(10),µ part in µ + µ − channel and C U contributes to C NP 9 part in both µ + µ − and e + e − channels. Then in the numerical analyses, we find that b → s + − and (g − 2) µ anomalies can be explained simultaneously in both scenario A and B. The main constraints among related processes are from B s −B s mixing covering B → X s γ decay mostly but the other tree-level and one-loop processes provide no effective bounds. At last we make a prospect that NP contributions to t → cg process can reach the sensitivity at FCC-hh in parts of the parameter spaces of this model.

Acknowledgements
We thank Yi-Lei Tang

A One-loop box contributions in RPV-MSSMIS
In this appendix, we list the whole Wilson coefficients from the one-loop box diagrams of b → s + − in RPV-MSSMIS without the extra assumption of a single value k.
The LH-quark-current contributions of chargino box diagrams to b → s + − process are given by where the Yukawa couplings y u i = √ 2m u i /v u and Y I v = (Y ν ) j Ṽ I * v(j+3) . While the corresponding RH-quark-current contributions are The contributions of W/H ± box diagrams to b → s + − process are given by The contributions of 4λ box diagrams to b → s + − process are given by The contributions of neutralino box diagrams only contain RH-quark-current parts, which are given by