Probing signatures of beyond standard model physics through $B_s^* \rightarrow \mu^+ \mu^-$ decay

TL;DR

This paper investigates potential new physics effects on the rare decay $B_s^* ightarrow \mu^+ \mu^-$, finding that current models fitting other anomalies do not increase its predicted rate beyond the Standard Model expectation.

Contribution

It provides a model-independent analysis showing that existing new physics explanations for $b ightarrow s \mu^+ \mu^-$ anomalies do not enhance the $B_s^* ightarrow \mu^+ \mu^-$ decay rate.

Findings

No new physics operators improve the decay rate beyond the SM prediction.

Current explanations for $b ightarrow s \\mu^+ \\mu^-$ anomalies are consistent with the SM prediction for this decay.

The decay remains a clean probe for future new physics searches.

Abstract

We perform a model independent analysis to identify new physics operators which can enhance the branching ratio of $B_s^* \rightarrow \mu^+ \mu^-$ above its Standard Model (SM) prediction. We find that none of the new physics operators which provide a good fit to $b \rightarrow s \mu^+ \mu^-$ data can enhance the Br($B_s^* \rightarrow \mu^+ \mu^-$) above its SM value.

Tables

Table 1. Table 1: Calculation of the branching ratios of B s ∗ → μ + ​ μ − → superscript subscript 𝐵 𝑠 superscript 𝜇 superscript 𝜇 B_{s}^{*}\rightarrow\mu^{+}\mu^{-} for various new physics scenarios. Here Δ ​ χ 2 = χ S ​ M 2 − χ b ​ f 2 Δ superscript 𝜒 2 subscript superscript 𝜒 2 𝑆 𝑀 subscript superscript 𝜒 2 𝑏 𝑓 \Delta\chi^{2}=\chi^{2}_{SM}-\chi^{2}_{bf} and χ b ​ f 2 subscript superscript 𝜒 2 𝑏 𝑓 \chi^{2}_{bf} is the χ 2 superscript 𝜒 2 \chi^{2} at the best fit points.

Scenario	New physics couplings	$\Delta {\chi }^2$	Branching Ratio
\svhline $C_i=0\left(SM\right)$	-	0	$\left(1.23\pm 0.48\right)\times {10}^{-11}$
$C_9^{NP}$	-1.24 $\pm$ 0.18	43.27	$\left(0.95\pm 0.48\right)\times {10}^{-11}$
$C_{10}^{NP}$	$0.91\pm 0.19$	29.47	$\left(1.01\pm 0.51\right)\times {10}^{-11}$
$C_9^{{}^{\prime }}$	0.13 $\pm$ 0.16	0.66	$\left(1.30\pm 0.65\right)\times {10}^{-11}$
$C_{10}^{{}^{\prime }}$	-0.11 $\pm$ 0.13	0.68	$\left(1.29\pm 0.65\right)\times {10}^{-11}$
$C_9^{NP}=C_{10}^{NP}$	0.01 $\pm$ 0.18	0.001	$\left(1.26\pm 0.64\right)\times {10}^{-11}$
$C_9^{NP}=-C_{10}^{NP}$	-0.65 $\pm$ 0.11	43.04	$\left(0.89\pm 0.45\right)\times {10}^{-11}$
$C_9^{{}^{\prime }}=C_{10}^{{}^{\prime }}$	-0.04 $\pm$ 0.17	0.06	$\left(1.26\pm 0.64\right)\times {10}^{-11}$
$C_9^{{}^{\prime }}=-C_{10}^{{}^{\prime }}$	0.07 $\pm$ 0.08	0.81	$\left(1.30\pm 0.65\right)\times {10}^{-11}$
$[C_9^{NP},C_{10}^{NP}]$	[-1.10,0.33]	47.33	$\left(0.88\pm 0.44\right)\times {10}^{-11}$
$[C_9^{{}^{\prime }},C_{10}^{{}^{\prime }}]$	[0.08,-0.07]	0.81	$\left(1.31\pm 0.66\right)\times {10}^{-11}$
$[C_9^{NP}=C_{10}^{NP}$, $C_9^{{}^{\prime }}=C_{10}^{{}^{\prime }}]$	[-0.02,-0.02]	0.07	$\left(0.97\pm 0.49\right)\times {10}^{-11}$
$[C_9^{NP}=-C_{10}^{NP}$, $C_9^{{}^{\prime }}=-C_{10}^{{}^{\prime }}]$	[-0.67,0.16]	46.27	$\left(1.00\pm 0.52\right)\times {10}^{-11}$
$[C_9^{NP},C_{10}^{NP}$, $C_9^{{}^{\prime }},C_{10}^{{}^{\prime }}]$	[-1.31,0.26,0.34,-0.25]	56.04	$\left(1.00\pm 0.52\right)\times {10}^{-11}$

Equations

$${\mathcal{H}}_{SM}$$

$$\displaystyle+C_{9}\frac{\alpha_{em}}{4\pi}(\overline{s}\gamma^{\mu}P_{L}b)(\overline{\mu}\gamma_{\mu}\mu)+C_{10}\frac{\alpha_{em}}{4\pi}(\overline{s}\gamma^{\mu}P_{L}b)(\overline{\mu}\gamma_{\mu}\gamma_{5}\mu)\bigg{]}.$$

$${\mathcal{H}}_{eff}(b\to s{\mu }^{+}{\mu }^{-})$$

$${\mathcal{H}}^{VA}$$

$$\displaystyle+C_{9}^{{}^{\prime}NP}(\overline{s}\gamma^{\mu}P_{R}b)(\overline{\mu}\gamma_{\mu}\mu)+C_{10}^{{}^{\prime}NP}(\overline{s}\gamma^{\mu}P_{R}b)(\overline{\mu}\gamma_{\mu}\gamma_{5}\mu)\bigg{]}$$

$${\mathcal{H}}^{SP}$$

$$\displaystyle+R_{S}^{{}^{\prime}}(\overline{s}P_{L}b)(\overline{\mu}\mu)+R_{P}^{{}^{\prime}}(\overline{s}P_{L}b)(\overline{\mu}\gamma_{5}\mu)\bigg{]}$$

$$⟨0|\overline{s}b|B_s^{\ast }⟩=0,⟨0|\overline{s}{\gamma }^{5}b|B_s^{\ast }⟩=0.$$

$$\Gamma (B_s^{\ast }\to {\mu }^{+}{\mu }^{-})$$

$$\displaystyle 2\frac{m_{b}f_{B_{s}^{*}}^{T}}{m_{B^{*}_{s}}f_{B_{s}^{8}}}C^{eff}_{7}(m_{B_{s}^{*}}^{2})+C^{NP}_{9}+C^{{}^{\prime}NP}_{9}\bigg{|}^{2}+\bigg{|}C_{10}+C^{NP}_{10}+C^{{}^{\prime}NP}_{10}\bigg{|}^{2}~{}\bigg{]}.$$

$${\chi }^2(C_i)$$

Full text

1 Introduction

Several observables related to the decays of B meson do not agree with their Standard Model (SM) predictions. For e.g., the measurement of the ratios $R_{K^{(*)}}$, angular observable $P^{{}^{\prime}}_{5}$ in $B\rightarrow K^{*}\mu^{+}\mu^{-}$ decay in the 4.3-8.68 $q^{2}$-bin, branching ratio of $B_{s}\rightarrow\phi\mu^{+}\mu^{-}$ do not agree with their SM value. All of these discrepancies are related to the $b\rightarrow s\mu^{+}\mu^{-}$ sector. This can be attributed to the presence of new physics (NP) in $b\rightarrow s\mu^{+}\mu^{-}$ transition.

In AKA121 ; AKA122 , new physics in $b\rightarrow s\mu^{+}\mu^{-}$ decays were analysed by making use of an effective Hamiltonian with all possible Lorentz structures. It was shown that any large effects in $b\rightarrow s\mu^{+}\mu^{-}$ sector, in particular decays like $B\rightarrow K^{*}\mu^{+}\mu^{-}$ and $B_{s}\rightarrow\phi\mu^{+}\mu^{-}$, can only be due to new physics vector (V ) and axial-vector operators (A). After the advent of $R_{K^{*}}$ data, several groups performed global fits to identify the Lorentz structure of NP Capdevila ; Descotes2013 ; Altmannshofer382 ; Altmannshofer1503 ; Hurth909 ; Capdevila1610 ; Altmannshofer ; AKA1703 ; AKA1704 . Many NP solutions, all in the form of vector V and A operators, were obtained. In order to discriminate between these NP solutions, one needs new observables. It would be desirable to have an access to observables which are theoretically clean.

The purely leptonic decay of $B_{s}^{*}$ meson is such a decay channel Grinstein1509 . In this work we perform a model independent analysis of $B^{*}_{s}\rightarrow\mu^{+}\mu^{-}$ decay to see whether Br($B^{*}_{s}\rightarrow\mu^{+}\mu^{-}$) can discriminate between various NP solutions which provide a good fit to the $b\rightarrow s\mu^{+}\mu^{-}$ data js .

2 $B_{s}^{*}\rightarrow\mu^{+}\mu^{-}$ decay

The effective Hamiltonian for the quark level transition $b\rightarrow s\mu^{+}\mu^{-}$ within the SM is given by

[TABLE]

Here $G_{F}$ is the Fermi constant, $V_{ij}$ are elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and $P_{L,R}=(1\mp\gamma^{5})/2$. The effect of the operators $O_{i}$, i = 1 − 6, 8 can be included in the effective Wilson Coefficients by redefining $C_{7}(\mu)\rightarrow C_{eff}^{7}(\mu,q^{2})$ and $C_{9}(\mu)\rightarrow C_{eff}^{9}(\mu,q^{2})$.

To study NP effects in $B^{*}_{s}\rightarrow\mu^{+}\mu^{-}$ decay, we consider the addition of V , A, S and P operators to the SM effective Hamiltonian of $b\rightarrow s\mu^{+}\mu^{-}$

[TABLE]

where $\mathcal{H}^{VA}$ and $\mathcal{H}^{SP}$ are as

[TABLE]

where $C_{9}^{NP},C_{10}^{NP},C_{9}^{{}^{\prime}NP},C_{10}^{{}^{\prime}NP},R_{S},R_{P},R_{S}^{{}^{\prime}},R_{P}^{{}^{\prime}}$ are NP couplings.

We find that,

[TABLE]

Hence the Br($B^{*}_{s}\rightarrow\mu^{+}\mu^{-}$) is not affected by NP in the form of S and P operators.

The decay rate including NP V and A contribution is obtained to be,

[TABLE]

As the total decay width of $B^{*}_{s}$ meson,$\Gamma(B^{*tot}_{s})$ is not yet known precisely , it is assumed that $\Gamma(B^{*tot}_{s})$ is comparable to the dominant decay process $B^{*}_{s}\rightarrow B_{s}\gamma$ which is found to be $\Gamma(B^{*}_{s}\rightarrow B_{s}\gamma)=0.10\pm 0.05$ KeV Grinstein1509 .

3 Methodology

We perform a $\chi^{2}$ fit to all CP conserving data $b\rightarrow s\mu^{+}\mu^{-}$ sector for all possible combinations of NP couplings. The observables used in the fit are listed in js .

The $\chi^{2}$ function is constructed as

[TABLE]

The total covariance matrix $\mathcal{C}$ is obtained by adding the individual theoretical and experimental covariance matrices.