Explaining the $B_{d(s)}\rightarrow K^{(\ast)}\bar{K}^{(\ast)}$ puzzle via chiral-flip in $R$-parity violating MSSM with seesaw mechanism

First let us briefly review the RPV-MSSMIS [14]. Here are given the superpotential and the soft supersymmetric (SUSY) breaking Lagrangian,

	$\displaystyle{\cal W}=$	$\displaystyle{\cal W}_{\rm MSSM}+Y_{\nu}^{ij}\hat{R}_{i}\hat{L}_{j}\hat{H}_{u}+M_{R}^{ij}\hat{R}_{i}\hat{S}_{j}+\frac{1}{2}\mu_{S}^{ij}\hat{S}_{i}\hat{S}_{j}+\lambda^{\prime}_{ijk}\hat{L}_{i}\hat{Q}_{j}\hat{D}_{k},$
	$\displaystyle{\cal L}^{\rm soft}=$	$\displaystyle{\cal L}^{\rm soft}_{\rm MSSM}-(m^{2}_{\tilde{R}})_{ij}\tilde{R}^{\ast}_{i}\tilde{R}_{j}-(m^{2}_{\tilde{S}})_{ij}\tilde{S}^{\ast}_{i}\tilde{S}_{j}$
		$\displaystyle-(A_{\nu}Y_{\nu})_{ij}\tilde{R}^{\ast}_{i}\tilde{L}_{j}H_{u}-B_{M_{R}}^{ij}\tilde{R}^{\ast}_{i}\tilde{S}_{j}-\frac{1}{2}B_{\mu_{S}}^{ij}\tilde{S}_{i}\tilde{S}_{j},$		(2.1)

where the generation indices $i,j,k=1,2,3$ while the colour ones are omitted, and squarks (sleptons) are denoted by the symbol “$\tilde{\ }$”, and as for the MSSM parts, ${\cal W}_{\rm MSSM}$ and ${\cal L}^{\rm soft}_{\rm MSSM}$, the reader can refer to Refs [18, 19]. All repeated indices are assumed to be summed over throughout this paper unless otherwise stated. The neutral scalar fields of the two Higgs doublet superfields, $\hat{H}_{u}=(\hat{H}^{+}_{u},\hat{H}^{0}_{u})^{T}$ and $\hat{H}_{d}=(\hat{H}^{0}_{d},\hat{H}^{-}_{d})^{T}$, acquire the non-zero vacuum expectation value, i.e. $\langle H^{0}_{u}\rangle=v_{u}$ and $\langle H^{0}_{d}\rangle=v_{d}$, respectively, and their mixing is expressed by $\tan\beta=v_{u}/v_{d}$.

The neutrino sector in the superpotential ${\cal W}$ provides the neutrino mass spectrum at the tree level, and in the $(\nu,R,S)$ basis, the $9\times 9$ mass matrix ${\cal M}_{\nu}$ is given by

$\displaystyle{\cal M}_{\nu}=\left(\begin{array}[]{ccc}0&m_{D}^{T}&0\\ m_{D}&0&M_{R}\\ 0&M_{R}^{T}&\mu_{S}\end{array}\right),$

(2.5)

where the Dirac mass matrix $m_{D}=\frac{1}{\sqrt{2}}v_{u}Y_{\nu}^{T}$. Then ones can diagonalize ${\cal M}_{\nu}$ through ${\cal M}^{\text{diag}}_{\nu}={\cal V}{\cal M}_{\nu}{\cal V}^{T}$. As to the sneutrino mass square matrix ${\cal M}_{\tilde{\nu}^{\cal I(R)}}^{2}$ in the $(\tilde{\nu}^{\cal I(R)}_{L},\tilde{R}^{\cal I(R)},\tilde{S}^{\cal I(R)})$ basis, it is expressed as

	$\displaystyle{\cal M}_{\tilde{\nu}^{\cal I(R)}}^{2}=$	$\displaystyle\left(\begin{array}[]{ccc}m^{2}_{\tilde{L}^{\prime}}&(A_{\nu}-\mu\cot\beta)m_{D}^{T}&m_{D}^{T}M_{R}\\ (A_{\nu}-\mu\cot\beta)m_{D}&m^{2}_{\tilde{R}}+M_{R}M_{R}^{T}+m_{D}m_{D}^{T}&\pm M_{R}\mu_{S}+B_{M_{R}}\\ M_{R}^{T}m_{D}&\pm\mu_{S}M_{R}^{T}+B_{M_{R}}^{T}&m^{2}_{\tilde{S}}+\mu_{S}^{2}+M_{R}^{T}M_{R}\pm B_{\mu_{S}}\end{array}\right)$		(2.9)
	$\displaystyle\approx$	$\displaystyle\left(\begin{array}[]{ccc}m^{2}_{\tilde{L}^{\prime}}&(A_{\nu}-\mu\cot\beta)m_{D}^{T}&m_{D}^{T}M_{R}\\ (A_{\nu}-\mu\cot\beta)m_{D}&m^{2}_{\tilde{R}}+M_{R}M_{R}^{T}+m_{D}m_{D}^{T}&B_{M_{R}}\\ M_{R}^{T}m_{D}&B_{M_{R}}^{T}&m^{2}_{\tilde{S}}+M_{R}^{T}M_{R}\pm B_{\mu_{S}}\end{array}\right),$		(2.13)

where the “$\pm$”, as well as “${\cal R(I)}$”, denotes the even (odd) CP, and the mass square $m^{2}_{\tilde{L}^{\prime}}\equiv m^{2}_{\tilde{L}}+\frac{1}{4}g^{2}_{2}v^{2}\cos 2\beta+m_{D}m_{D}^{T}$ is regarded as the model input, with $m^{2}_{\tilde{L}}$ being the soft mass square of $\tilde{L}$. We set $\mu_{S}\approx 0$ while $B_{\mu_{S}}$ is non-negligible [20] in Eq. (2.9), which induces the mass splitting between the CP-even and CP-odd sneutrinos for the same flavor. This is different from the degenerate-mass approximation adopted in our recent researches [14, 15]⁴⁴4Although the quasi-degenerate-mass scenario is favored by the direct dark matter (DM) detection [21], we focus on the field of $B$-meson processes, and given RPV is involved, DM is out of the scope of this work.. In the following sections, ones will see that this splitting-mass scenario can provide the chiral-flip contributions in some processes.

Afterwards we introduce the trilinear RPV interaction in this model. The superpotential term $\lambda^{\prime}_{ijk}\hat{L}_{i}\hat{Q}_{j}\hat{D}_{k}$ induces the relevant Lagrangian in the context of mass eigenstates for the down-type quarks and charged leptons, which is given by

	$\displaystyle{\cal L}_{\text{LQD}}=$	$\displaystyle\lambda^{\prime\cal I(R)}_{vjk}\tilde{\nu}_{v}^{\cal I(R)}\bar{d}_{Rk}d_{Lj}+\lambda^{\prime\cal N}_{vjk}\big{(}\tilde{d}_{Lj}\bar{d}_{Rk}\nu_{v}+\tilde{d}_{Rk}^{\ast}\bar{\nu}_{v}^{c}d_{Lj}\big{)}$
		$\displaystyle-\tilde{\lambda}^{\prime}_{ilk}\big{(}\tilde{l}_{Li}\bar{d}_{Rk}u_{Ll}+\tilde{u}_{Ll}\bar{d}_{Rk}l_{Li}+\tilde{d}_{Rk}^{\ast}\bar{l}_{Li}^{c}u_{Ll}\big{)}+{\rm h.c.},$		(2.14)

where “$c$” indicates the charge conjugated fermions, and the fields $\tilde{\nu}_{L}^{\cal I(R)}$, $\nu_{L}$, and $u_{L}$ (aligned with $\tilde{u}_{L}$) in the flavor basis have been rotated into mass eigenstates by the mixing matrices $\tilde{\cal V}^{\cal I(R)}$, ${\cal V}$, and $K$, respectively. Besides, the index $v=1,2,\dots 9$ denotes the generation of the physical (s)neutrinos, and the three $\lambda^{\prime}$ couplings are deduced as $\lambda^{\prime\cal I(R)}_{vjk}\equiv\lambda^{\prime}_{ijk}\tilde{\cal V}^{\cal I(R)\ast}_{vi}$, $\lambda^{\prime\cal N}_{vjk}\equiv\lambda^{\prime}_{ijk}{\cal V}_{vi}$, and $\tilde{\lambda}^{\prime}_{ilk}\equiv\lambda^{\prime}_{ijk}K^{\ast}_{lj}$. In the following, we adopt the “single-value-$k$” assumption, i.e. both $\lambda^{\prime}_{ij1}$ and $\lambda^{\prime}_{ij2}$ are set negligible, and the NP Wilson coefficients are given at the scale $\mu_{\rm NP}=1$ TeV.

In RPV-MSSMIS, the most favored operator to explain the $B_{d(s)}\rightarrow K^{(\ast)}\bar{K}^{(\ast)}$ puzzle is the magnetic one, ${\cal O}_{8gs}$, extracted from the gluon-penguins shown in Fig. 1. Given the recent LHC constraints which will be discussed in Sec. 3.1, we set all colored SUSY particles with masses above $10$ TeV, so the penguins engaged by squarks and gluinos contribute negligibly.

Next, we will show that, the NP Wilson coefficients ${C}_{7\gamma p}^{\rm NP}$ and ${C}_{8gp}^{\rm NP}$ in RPV-MSSMIS can include the chiral-flip contributions, in the mass-splitting scenario mentioned in Sec.2. The coefficient ${C}_{8gp}^{\rm NP}$, extracted from the diagrams containing sneutrinos (other suppressed contributions omitted) is given by,

	$\displaystyle{C}_{8gp}^{\rm NP}=-\frac{1}{48\sqrt{2}G_{F}\eta_{t}}\biggl{\{}$	$\displaystyle\frac{\lambda^{\prime{\cal I}\ast}_{vp3}\lambda^{\prime{\cal I}\ast}_{v33}}{m_{\tilde{\nu}^{\cal I}_{v}}^{2}}\left[8+6\log\left(\frac{m_{b}^{2}}{m_{\tilde{\nu}^{\cal I}_{v}}^{2}}\right)\right]-\frac{\lambda^{\prime{\cal R}\ast}_{vp3}\lambda^{\prime{\cal R}\ast}_{v33}}{m_{\tilde{\nu}^{\cal R}_{v}}^{2}}\left[8+6\log\left(\frac{m_{b}^{2}}{m_{\tilde{\nu}^{\cal R}_{v}}^{2}}\right)\right]$
		$\displaystyle+\frac{\lambda^{\prime{\cal I}\ast}_{vp3}\lambda^{\prime{\cal I}}_{v33}}{m^{2}_{\tilde{\nu}^{\cal I}_{v}}}+\frac{\lambda^{\prime{\cal R}\ast}_{vp3}\lambda^{\prime{\cal R}}_{v33}}{m^{2}_{\tilde{\nu}^{\cal R}_{v}}}\biggr{\}},$		(2.15)

and ${C}_{7\gamma p}^{\rm NP}$ is calculated as $-{C}_{8gp}/3$ because that in the setup of this model, the difference between $bp\gamma$ diagram and $bpg$ one is merely $-1/3$. In Eq. (2.2), ones can find that the chiral-flip is contained in the first two terms containing double-$\lambda^{\prime\ast}$ couplings. If we utilize the degenerate-mass scenario, these two terms totally cancel with each other and then only the non-flip terms, ${\lambda^{\prime{\cal I}\ast}_{v23}\lambda^{\prime{\cal I}}_{v33}}/{m^{2}_{\tilde{\nu}^{\cal I}_{v}}}+{\lambda^{\prime{\cal R}\ast}_{v23}\lambda^{\prime{\cal R}}_{v33}}/{m^{2}_{\tilde{\nu}^{\cal R}_{v}}}=2{\lambda^{\prime{\cal I}\ast}_{v23}\lambda^{\prime{\cal I}}_{v33}}/{m^{2}_{\tilde{\nu}^{\cal I}_{v}}}$ are remained, which agrees with the result in Ref. [22], with the formula-sign checked. Instead, if there exists a sufficient split between the masses $m_{\tilde{\nu}^{\cal I}_{v}}$ and $m_{\tilde{\nu}^{\cal R}_{v}}$ for the same $v$, the unique chiral-flip part is dominating, enhanced by logarithm terms. Here we analyse Fig. 1a qualitatively in the flavor basis to illustrate how double-$\lambda^{\prime\ast}$ terms are related to the chiral-flip. First the leading order term only contains normal $\lambda^{\prime}\lambda^{\prime\ast}$ couplings. When we consider the next order with the mixing of chirality for one single virtual quark, the chirality of sneutrino should also be flipped, inducing that double-$\lambda^{\prime\ast}$ couplings emerge. This situation is unique since (s)neutrino chiral-flip is forbidden in original RPV-MSSM, with Dirac neutrinos instead of Majorana ones.

Then it is worth mentioning that we also calculate the Wilson coefficient $C^{\rm NP}_{9\rm U}$ related to the operator, ${\cal O}_{9}=\frac{e^{2}}{16\pi^{2}}(\bar{s}_{L}\gamma^{\mu}b_{L})(\bar{\ell}\gamma_{\mu}\ell)$, which providing the lepton flavor universal contributions to $b\rightarrow s\ell^{+}\ell^{-}$, also dominated by the $\tilde{\nu}dd$ loop. The result is given that,

	$\displaystyle C^{\rm NP}_{9\rm U}=-\frac{\sqrt{2}}{144G_{F}\eta_{t}}\biggl{\{}$	$\displaystyle\frac{\lambda^{\prime{\cal I}\ast}_{v23}\lambda^{\prime{\cal I}}_{v33}}{m_{\tilde{\nu}^{\cal I}_{v}}^{2}}\left[\frac{8}{3}+2\log\left(\frac{m_{b}^{2}}{m_{\tilde{\nu}^{\cal I}_{v}}^{2}}\right)\right]+\frac{\lambda^{\prime{\cal R}\ast}_{v23}\lambda^{\prime{\cal R}}_{v33}}{m_{\tilde{\nu}^{\cal R}_{v}}^{2}}\left[\frac{8}{3}+2\log\left(\frac{m_{b}^{2}}{m_{\tilde{\nu}^{\cal R}_{v}}^{2}}\right)\right]$
		$\displaystyle+\frac{\lambda^{\prime{\cal I}\ast}_{v23}\lambda^{\prime{\cal I}\ast}_{v33}}{m^{2}_{\tilde{\nu}^{\cal I}_{v}}}-\frac{\lambda^{\prime{\cal R}\ast}_{v23}\lambda^{\prime{\cal R}\ast}_{v33}}{m^{2}_{\tilde{\nu}^{\cal R}_{v}}}\biggr{\}},$		(2.16)

and ones can find that, the chiral-flip part in Eq. (2.2), expressed by the double-$\lambda^{\prime\ast}$ terms, is not enhanced by logarithm terms. In the degenerate-mass scenario, the result returns to the one shown in Ref. [14]. We also examine the $Z$-penguin contribution to $C^{\rm NP}_{9\rm U}$, and find that it is negligible in the setup of this work.

As mentioned in Sec. 1, there are divergences between the experiment data and SM results, corresponding to the non-leptonic ratios $L_{K^{(\ast)}\bar{K}^{(\ast)}}$. If we consider NP is only within $B_{s}\rightarrow K^{(\ast)}\bar{K}^{(\ast)}$ decays without the $B_{d}$ decays, the predictions of the ratios, $L_{K^{(\ast)}\bar{K}^{(\ast)}}$, can be given by [2, 13],

	$\displaystyle L_{K\bar{K}}/L_{K\bar{K}}^{\rm SM}\approx$	$\displaystyle 1+1.13C_{8gs}^{\rm NP}(\mu_{\rm EW})+0.34C_{8gs}^{\rm NP}(\mu_{\rm EW})^{2},$		(2.17)
	$\displaystyle L_{K^{\ast}\bar{K}^{\ast}}/L_{K^{\ast}\bar{K}^{\ast}}^{\rm SM}\approx$	$\displaystyle 1+2.41C_{8gs}^{\rm NP}(\mu_{\rm EW})+1.74C_{8gs}^{\rm NP}(\mu_{\rm EW})^{2},$		(2.18)

where the electroweak (EW) broken scale $\mu_{\rm EW}=160$ GeV. Then at $2\sigma$ level, we need $C_{8gs}^{\rm NP}\lesssim-0.08$ to explain the non-leptonic tension. The NP in $bd$ sector is constrained very strictly which will be shown in Sec. 3.2.

In this section, we revisit the $B\rightarrow K^{(\ast)}\nu\bar{\nu}$ processes related to the quark transition $d_{j}\rightarrow d_{m}\nu_{i}\bar{\nu}_{i^{\prime}}$, and the corresponding effective Lagrangian is,

	$\displaystyle{\cal L}_{\rm eff}^{dd\nu\bar{\nu}}=$	$\displaystyle(C^{\rm SM}_{mj}\delta_{ii^{\prime}}+C^{\rm NP}_{mj})(\bar{d}_{m}\gamma_{\mu}P_{L}d_{j})(\bar{\nu}_{i}\gamma^{\mu}P_{L}\nu_{i^{\prime}})+C_{mj}^{\rm 1SRR}(\bar{d}_{m}P_{R}d_{j})(\bar{\nu}_{i}P_{R}\nu_{i^{\prime}})$
		$\displaystyle+C_{mj}^{\rm 2SRR}(\bar{d}_{m}\sigma_{\mu\nu}P_{R}d_{j})(\bar{\nu}_{i}\sigma^{\mu\nu}P_{R}\nu_{i^{\prime}})+{\rm h.c.},$		(2.19)

where the SM contribution is $C^{\rm SM}_{mj}=-\frac{\sqrt{2}G_{F}e^{2}K_{tj}K^{\ast}_{tm}}{4\pi^{2}\sin^{2}\theta_{W}}X(x_{t})$ and the loop function $X(x_{t})\equiv\frac{x_{t}(x_{t}+2)}{8(x_{t}-1)}+\frac{3x_{t}(x_{t}-2)}{8(x_{t}-1)^{2}}\log(x_{t})$ with $x_{t}\equiv m^{2}_{t}/m^{2}_{W}$ [23]. The NP contribution of vector current is [15]

$\displaystyle C^{\rm NP}_{mj}=\frac{\lambda^{\prime\cal N}_{i^{\prime}j3}\lambda^{\prime\cal N\ast}_{im3}}{2m^{2}_{\tilde{b}_{R}}}=\frac{{\cal V}_{i^{\prime}\alpha^{\prime}}{\cal V}^{\ast}_{i\alpha}\lambda^{\prime}_{\alpha^{\prime}j3}\lambda^{\prime\ast}_{\alpha m3}}{2m^{2}_{\tilde{b}_{R}}}.$

(2.20)

Besides, the NP coefficients $C_{mj}^{\rm 1SRR}$ and $C_{mj}^{\rm 2SRR}$ express the chiral-flip contributions of neutrino with sbottoms. However, the global fit shows that, these scalar and tensor contributions are negligible, relative to the vector one [24]. For simplicity, we consider negligible mixing for the $\tilde{b}$ sector to omit these two coefficients.

To study the NP effects on $B^{+}\rightarrow K^{+}\nu\bar{\nu}$ as well as $B\rightarrow K^{\ast}\nu\bar{\nu}$, ones can define the ratio, $R^{\nu\bar{\nu}}_{K^{(\ast)}}\equiv{{\cal B}(B\rightarrow K^{(\ast)}\nu\bar{\nu})}/{{\cal B}(B\rightarrow K^{(\ast)}\nu\bar{\nu})_{\rm SM}}$. In RPV-MSSMIS, we get that,

$\displaystyle R^{\nu\bar{\nu}}_{K}=R^{\nu\bar{\nu}}_{K^{\ast}}=\frac{\sum\limits_{i=1}^{3}\left|C_{23}^{\rm SM}+C_{23}^{\rm NP}\right|^{2}+\sum\limits_{i\neq i^{\prime}}^{3}\left|C_{23}^{\rm NP}\right|^{2}}{3\left|C_{23}^{\rm SM}\right|^{2}}.$

(2.21)

The recent Belle-II data [16] of $B\rightarrow K^{+}\nu\bar{\nu}$, and the updated SM prediction [17], induce $R^{\nu\bar{\nu}}_{K}=5.3\pm 1.7$ [25]. Recent research [25] shows that, the case of $R^{\nu\bar{\nu}}_{K}=R^{\nu\bar{\nu}}_{K^{\ast}}$ cannot simultaneously fulfill the Belle-II data at $1\sigma$ level as well as the upper limit ${\cal B}(B\rightarrow K^{\ast}\nu\bar{\nu})_{\rm exp}<1.8\times 10^{-5}$ [26], i.e. $R^{\nu\bar{\nu}}_{K^{\ast}}<1.9$, at $90\%$ confidence level (CL). Besides, with the theoretical result ${\cal B}(B^{+}\rightarrow K^{+}\nu\bar{\nu})/{\cal B}(B\rightarrow K^{\ast}\nu\bar{\nu})\approx 0.4$ for the single left-handed vector operator, it is also found that this case cannot explain the Belle-II data, without staying below the upper limit of ${\cal B}(B\rightarrow K^{\ast}\nu\bar{\nu})$ at $90\%$ CL [27, 28]. In this work, we will investigate the degree of $R^{\nu\bar{\nu}}_{K}$ approaching to its upper limit $1.9$, in the parameter space of RPV-MSSMIS.

Firstly, direct searches for SUSY particles should be considered. Since there are no signs of NP particles until the end of the LHC run II which reaching around $140$ fb^-1 at the center energy $13$ TeV, which providing stringent bounds on SUSY models. The allowed masses of colored sparticles, such as gluinos, the first-two generation squarks, stops and sbottoms have been excluded up to $1-2$ TeV scale [29, 30, 31, 32, 33, 34, 35]. In this work, the masses of all the colored SUSY particles are set above $10$ TeV, whereas the masses of sleptons as well as the heavy neutrinos are all around $10^{2}-10^{3}$ GeV. Some recent experiments have pushed the upper limit of slepton masses over TeV scale [36, 37, 38], however, these searches consider nonzero $\lambda$ related to the superpotential $\lambda_{ijk}\hat{L}_{i}\hat{L}_{j}\hat{E}_{k}$. Given that we only consider nonzero $\lambda^{\prime}$ in the model, this bound can be relaxed . It is worth mentioning that, ATLAS has recently made searches for the NP signs of this type of model, only containing $\lambda^{\prime}$ couplings [39]. Using the first collider limits for this model type, we keep $m_{\tilde{\mu}}\gtrsim 470$ GeV.

Next, we check the tree-level processes exchanging sbottoms, including $K^{+}\to\pi^{+}\nu\bar{\nu}$, $B\to\pi\nu\bar{\nu}$, $D^{0}\to\ell^{+}\ell^{-}$, $\tau\rightarrow\ell\rho^{0}$ as well as $B\to\tau\nu$, $D_{s}\to\tau\nu$, $\tau\to K(\pi)\nu$ and $\pi\to\ell\nu(\gamma)$.

As ones know that the experimental measurement ${\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})_{\rm exp}=(1.14^{+0.40}_{-0.33})\times 10^{-10}$ [40] and the SM prediction ${\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})_{\rm SM}=(9.24\pm 0.83)\times 10^{-11}$ [41] induce the strong constraint, $|\lambda^{\prime\cal N}_{i^{\prime}2k}\lambda^{\prime\cal N\ast}_{i1k}|\lesssim 10^{-4}(m_{\tilde{b}_{R}}/1{\rm TeV})^{2}$. Even for ${\tilde{b}_{R}}$ with $10$ TeV mass, there still exists the bound of $|\lambda^{\prime\cal N}_{i^{\prime}2k}\lambda^{\prime\cal N\ast}_{i1k}|\lesssim 0.01$. Thus, we assume $\lambda^{\prime}_{i1k}$ negligible to avoid this bound from this process, as well as the $B\to\pi\nu\bar{\nu}$ decay.

In table 1, we collect the experimental results and SM predictions of $D^{0}\to\ell^{+}\ell^{-}$, $\tau\rightarrow\ell\rho^{0}$ decays with the charged current processes, $B\to\tau\nu$, $D_{s}\to\tau\nu$ and $\tau\to K\nu$, as well as the processes discussed above. Following the same/analogical numerical calculations in the ordinary RPV-MSSM (see Refs. [42, 43]), we update the constraint from ${\cal B}(D^{0}\to\mu^{+}\mu^{-})$, as $|\lambda^{\prime}_{223}|^{2}<0.22(m_{\tilde{b}_{R}}/1{\rm TeV})^{2}$, and the bound from ${\cal B}(D^{0}\to e^{+}e^{-})$ is negligible due to the small $m_{e}$. We also update the calculation of the process ${\cal B}(\tau\rightarrow\ell\rho^{0})$, which provides the bound $|\lambda^{\prime}_{323}\lambda^{\prime\ast}_{223}|<0.45(m_{\tilde{b}_{R}}/1{\rm TeV})^{2}$ as well as $|\lambda^{\prime}_{323}\lambda^{\prime\ast}_{123}|<0.51(m_{\tilde{b}_{R}}/1{\rm TeV})^{2}$. The functions $R_{133}$, $R_{223}$, and $R_{123}$ (see concrete definitions in Ref. [43]), are utilized to express the ratios of the measurement values versus the SM predictions for ${\cal B}(B\to\tau\nu)$, ${\cal B}(D_{s}\to\tau\nu)$, and ${\cal B}(\tau\to K\nu)$, respectively, and we also consider these constraints.

As for the $\pi\to\ell\nu(\gamma)$ decay, similar to the formula in Ref. [44], the bound (here also including $\lambda^{\prime}$-loop corrections) can be shown with

$\displaystyle\frac{1+\eta_{\mu\mu}+h^{\prime}_{\mu\mu}}{1+\eta_{ee}+h^{\prime}_{ee}}=1.0010(9),$

(3.1)

where the function $\eta$ and $h^{\prime}$ express the non-unitary part of neutrino and $\lambda^{\prime}$-loop corrections to $Wl\nu$-vertex, respectively, and they are given by  [15]

	$\displaystyle\eta_{ij}\equiv$	$\displaystyle\left({\cal V}^{T}_{3\times 3}\right)_{ik}{\cal U}^{-1}_{kj}-\delta_{ij},$
	$\displaystyle h^{\prime}_{li}=$	$\displaystyle-\frac{3}{64\pi^{2}}x_{\tilde{b}_{R}}f_{W}(x_{\tilde{b}_{R}})\tilde{\lambda}^{\prime\ast}_{l33}\tilde{\lambda}^{\prime}_{i33},$		(3.2)

where ${\cal U}$ is unitary PMNS-like, and the loop function $f_{W}(x)\equiv\frac{1}{x-1}+\frac{(x-2)\log x}{(x-1)^{2}}$ with $x_{\tilde{b}_{R}}\equiv m^{2}_{t}/m^{2}_{\tilde{b}_{R}}$, from the dominant $u_{i}d_{i}\tilde{b}_{R}$-loop diagram. In the inverse seesaw framework, the Hermitian $\eta$ can be figured out, i.e. $\eta\approx-\frac{1}{2}m_{D}^{\dagger}(M_{R}^{\ast})^{-1}(M_{R}^{T})^{-1}m_{D}$. We can translate the bound Eq. (3.1) into $|\eta_{ee}+h^{\prime}_{ee}|\lesssim 0.0028$ at the $2\sigma$ level, with the negligible $\eta(h^{\prime})_{\mu\mu}$. In this work, we can set sufficiently small $\lambda^{\prime}_{2jk}$ to keep $h^{\prime}_{\mu\mu}$ (negative as well as $\eta_{\mu\mu}$) negligible to avoid enlarging the Cabbibo anomaly [45, 46].

In RPV-MSSMIS, the neutrino mixing matrix, ${\cal V}$, is also bounded by the $\tau(\mu)$ decaying to charged leptons and neutrinos at the tree level. However, at one-loop level, both ${\cal V}$ and $\lambda^{\prime}$ couplings are constrained by these decays as well as the charged lepton flavor violating (cLFV) decays. We will address $\tau(\mu)$ decays totally in the following subsection 3.3, and before that, we can make a summary that couplings $\lambda^{\prime}_{i13}$ and $\lambda^{\prime}_{2j3}$ are already set negligible (at $\mu_{\rm NP}$ scale), considering the constraints investigated above, and that is, NP is mainly not contained in the $d$ and $\mu$ sectors.

In this section illustrating loop-level bounds, we firstly investigate the $B_{s}-\bar{B}_{s}$ mixing, which is mastered by

$\displaystyle{\cal L}_{\rm eff}^{b\bar{s}b\bar{s}}=(C^{\rm VLL}_{\rm SM}+C^{\rm VLL}_{\rm NP})(\bar{s}\gamma_{\mu}P_{L}b)(\bar{s}\gamma^{\mu}P_{L}b)+C^{\rm 1SRR}_{\rm NP}(\bar{s}P_{R}b)(\bar{s}P_{R}b)+{\rm h.c.},$

(3.3)

where the SM contribution is $C_{B_{s}}^{\rm SM}=-\frac{1}{4\pi^{2}}G_{F}^{2}m_{W}^{2}\eta_{t}^{2}S(x_{t})$ with the defined function $S(x_{t})\equiv\frac{x_{t}(4-11x_{t}+x_{t}^{2})}{4(x_{t}-1)^{2}}+\frac{3x_{t}^{3}\log(x_{t})}{2(x_{t}-1)^{3}}$, and the non-negligible NP contributions are,

	$\displaystyle C_{\rm NP}^{\rm VLL}$	$\displaystyle=\frac{1}{8i}\bigl{(}\frac{1}{4}\Lambda^{\prime 1\cal XY}_{vv^{\prime}}D_{2}[m_{\tilde{\nu}^{\cal X}_{v}},m_{\tilde{\nu}^{\cal Y}_{v^{\prime}}},m_{b},m_{b}]+\Lambda^{\prime\cal N}_{vv^{\prime}}D_{2}[m_{\nu_{v}},m_{\nu_{v^{\prime}}},m_{\tilde{b}_{R}},m_{\tilde{b}_{R}}]\bigr{)},$
	$\displaystyle C_{\rm NP}^{\rm 1SRR}$	$\displaystyle=\frac{1}{8i}\bigl{(}\Lambda^{\prime 2\cal XY}_{vv^{\prime}}(-1)^{\delta_{\cal XY}+1}m_{b}^{2}D_{0}[m_{\tilde{\nu}^{\cal X}_{v}},m_{\tilde{\nu}^{\cal Y}_{v^{\prime}}},m_{b},m_{b}]$
		$\displaystyle+\Lambda^{\prime 3\cal XY}_{vv^{\prime}}(\delta_{\cal XR}-\delta_{\cal XI})m_{b}^{2}D_{0}[m_{\tilde{\nu}^{\cal X}_{v}},m_{\tilde{\nu}^{\cal Y}_{v^{\prime}}},m_{b},m_{b}]-\Lambda^{\prime 1\cal XY}_{vv^{\prime}}m_{b}^{2}D_{1}[m_{\tilde{\nu}^{\cal X}_{v}},m_{\tilde{\nu}^{\cal Y}_{v^{\prime}}},m_{b},m_{b}]$
		$\displaystyle+\Lambda^{\prime 4\cal XY}_{vv^{\prime}}(\delta_{\cal YI}-\delta_{\cal YR})m_{b}^{2}D_{1}[m_{\tilde{\nu}^{\cal X}_{v}},m_{\tilde{\nu}^{\cal Y}_{v^{\prime}}},m_{b},m_{b}]-\frac{\lambda^{\prime{\cal I}\ast 2}_{v23}}{2m^{2}_{\tilde{\nu}^{\cal I}_{v}}}+\frac{\lambda^{\prime{\cal R}\ast 2}_{v23}}{2m^{2}_{\tilde{\nu}^{\cal R}_{v}}}\bigr{)},$		(3.4)

where $\Lambda^{\prime 1\cal XY}_{vv^{\prime}}\equiv\lambda^{\prime\cal X}_{v33}\lambda^{\prime\cal X\ast}_{v23}\lambda^{\prime\cal Y}_{v^{\prime}33}\lambda^{\prime\cal Y\ast}_{v^{\prime}23}$, $\Lambda^{\prime 2\cal XY}_{vv^{\prime}}\equiv\lambda^{\prime\cal X\ast}_{v33}\lambda^{\prime\cal X\ast}_{v23}\lambda^{\prime\cal Y\ast}_{v^{\prime}33}\lambda^{\prime\cal Y\ast}_{v^{\prime}23}$, $\Lambda^{\prime 3\cal XY}_{vv^{\prime}}\equiv\lambda^{\prime\cal X\ast}_{v33}\lambda^{\prime\cal X\ast}_{v23}\lambda^{\prime\cal Y}_{v^{\prime}33}\lambda^{\prime\cal Y\ast}_{v^{\prime}23}$, and $\Lambda^{\prime 4\cal XY}_{vv^{\prime}}\equiv\lambda^{\prime\cal X}_{v33}\lambda^{\prime\cal X\ast}_{v23}\lambda^{\prime\cal Y\ast}_{v^{\prime}33}\lambda^{\prime\cal Y\ast}_{v^{\prime}23}$ with ${\cal X}$, ${\cal Y}$ being ${\cal I}$ or ${\cal R}$, and $\Lambda^{\prime\cal N}_{vv^{\prime}}\equiv\lambda^{\prime\cal N}_{v33}\lambda^{\prime\cal N\ast}_{v23}\lambda^{\prime\cal N}_{v^{\prime}33}\lambda^{\prime\cal N\ast}_{v^{\prime}23}$. The formulas of Passarino-Veltman functions [52], $D_{0}$ and $D_{2}$, are defined as

	$\displaystyle D_{0}[m_{1},m_{2},m_{3},m_{4}]\equiv$	$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}-m_{1}^{2})(k^{2}-m_{2}^{2})(k^{2}-m_{3}^{2})(k^{2}-m_{4}^{2})},$
	$\displaystyle D_{2}[m_{1},m_{2},m_{3},m_{4}]\equiv$	$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{2}}{(k^{2}-m_{1}^{2})(k^{2}-m_{2}^{2})(k^{2}-m_{3}^{2})(k^{2}-m_{4}^{2})},$		(3.5)

and $D_{1}$ is given by $D_{\mu}=p_{i\mu}D_{i}$ [53], which is defined as

$\displaystyle D_{\mu}\equiv\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\mu}}{(k^{2}-m_{1}^{2})((k+p_{1})^{2}-m_{2}^{2})((k+p_{2})^{2}-m_{3}^{2})((k+p_{3})^{2}-m_{4}^{2})},$

(3.6)

with the limit $p_{i}\cdot p_{j}\rightarrow 0$ applied. The chiral-flip contributions are all contained in the coefficient $C_{\rm NP}^{\rm 1SRR}$, where the last two terms are extracted from the tree-level diagram. Combined with recent results of bag parameters, $B_{s}^{(i)}(m_{b})$, including the new value of $B_{s}^{(1)}(m_{b})$ [54, 55], we get the ratio,

$\displaystyle{\cal R}_{B_{s}}\equiv\frac{\Delta M_{s}}{\Delta M_{s}^{\rm SM}}=\left|1+\frac{C^{\rm VLL}_{\rm NP}}{C^{\rm VLL}_{\rm SM}}-2.38\frac{C_{\rm NP}^{\rm 1SRR}}{C^{\rm VLL}_{\rm SM}}\right|.$

(3.7)

The recent result averaged by HFLAV, $\Delta M_{s}^{\rm exp}=(17.765\pm 0.006)$ ${\rm ps}^{-1}$ [56], along with the SM prediction $\Delta M_{s}^{\rm SM}=(18.23\pm{0.63})~{}{\rm ps}^{-1}$ [57], leads to the strong constraint $0.90<{\cal R}_{B_{s}}<1.04$ at $2\sigma$ level. Given that the mass-splitting of sneutrinos is considered in this work, the tree-level contributions to the ratio ${\cal R}_{B_{s}}$ need the cancellation to fulfill the bound.