Probing new physics in $B_s \to (K,K^*)\tau \nu$ and $B \to \pi \tau \nu$ decays

TL;DR

This paper investigates potential new physics effects in specific B meson decays involving tau leptons, using an effective field theory approach and assuming similar new physics patterns across related decay channels.

Contribution

It provides predictions for various observables in B decays with tau leptons under new physics scenarios, extending the standard model analysis with vector-type couplings.

Findings

Predicted branching ratios and asymmetries in B decays with tau leptons.

Identified potential signatures of new physics in decay observables.

Compared standard model and new physics predictions for key decay parameters.

Abstract

Motivated by the anomalies present in $b \to u$ and $b \to c$ semileptonic decays, we study the corresponding $B_s \to (K,K^*) \tau \nu$ and $B \to \pi \tau \nu$ decays within an effective field theory formalism. Our analysis is based on a strict model dependent assumption, i.e., we assume that $b \to u$ and $b \to c$ transition decays exhibit similar new physics pattern. We give prediction of various observables such as the branching fraction, ratio of branching ratio, lepton side forward-backward asymmetry, longitudinal polarization fraction of the charged lepton and convexity parameter in the standard model and in the presence of vector type new physics couplings.

Figures

Tables

Table 1. Table 1: The SM prediction and the world averages of the ratio of branching ratios for various decay modes

Ratio of branching ratio	SM prediction	Experimental prediction
$R_D$	$0.300\pm 0.008$ Lattice:2015rga ; Na:2015kha ; Aoki:2016frl ; Bigi:2016mdz	$0.407\pm 0.039\pm 0.024$ Lees:2013uzd ; Huschle:2015rga ; Sato:2016svk ; Hirose:2016wfn ; Aaij:2015yra
$R_{D^{\ast }}$	$0.258\pm 0.005$ Fajfer:2012vx ; Bernlochner:2017jka ; Bigi:2017jbd ; Jaiswal:2017rve	$0.304\pm 0.013\pm 0.007$ Lees:2013uzd ; Huschle:2015rga ; Sato:2016svk ; Hirose:2016wfn ; Aaij:2015yra
$R_{J/\Psi }$	$[0.20,\mathrm{\hspace{0.17em}0.39}]$ Cohen:2018dgz	$0.71\pm 0.17\pm 0.18$ Aaij:2017tyk
$ℬ\left(B\to \tau \nu \right)$	$\left(0.84\pm 0.11\right)\times {10}^{-4}$ Bona:2009cj	$\left(1.09\pm 2.4\right)\times {10}^{-4}$ Bernlochner:2015mya
$R_{\pi }^l$	$0.566$	$0.698\pm 0.155$ Patrignani:2016xqp
$R_{\pi }$	$0.641$ Patrignani:2016xqp	Bernlochner:2015mya

Table 2. Table 2: The central values and 1 ​ σ 1 𝜎 1\sigma ranges of each observable for both μ 𝜇 \mu and τ 𝜏 \tau modes in SM are reported for B s → K ​ l ​ ν → subscript 𝐵 𝑠 𝐾 𝑙 𝜈 B_{s}\to Kl\nu , B s → K ∗ ​ l ​ ν → subscript 𝐵 𝑠 superscript 𝐾 𝑙 𝜈 B_{s}\to K^{*}l\nu and B → π ​ l ​ ν → 𝐵 𝜋 𝑙 𝜈 B\to\pi l\nu decays.

$B_s\to Kl\nu$		$BR\times {10}^{-4}$	$⟨A_{FB}^l⟩$	$⟨P^l⟩$	$⟨C_F^l⟩$	$R_{B_{s}K}$
$\mu$ mode	Central value	1.520	$6.647\times {10}^{-3}$	0.982	-1.479
	$1\sigma$ range	[1.098, 2.053]	[0.006, 0.007]	[0.979, 0.984]	[-1.482, -1.478]	0.636
$\tau$ mode	Central value	0.966	0.284	0.105	-0.607
	$1\sigma$ range	[0.649, 1.392]	[0.262, 0.291]	[-0.035, 0.279]	[-0.711, -0.525]	[0.586, 0.688]
$B_s\to K^{\ast }l\nu$		$BR\times {10}^{-4}$	$⟨A_{FB}^l⟩$	$⟨P^l⟩$	$⟨C_F^l⟩$	$R_{B_s{K}^{\ast }}$
$\mu$ mode	Central value	3.259	-0.281	0.993	-0.417
	$1\sigma$ range	[2.501, 4.179]	[-0.342, -0.222]	[0.989, 0.995]	[-0.575, -0.247]	0.578
$\tau$ mode	Central value	1.884	-0.132	0.539	-0.105
	$1\sigma$ range	[1.449, 2.419]	[-0.203, -0.061]	[0.458, 0.603]	[-0.208, -0.007]	[0.539, 0.623]
$B\to \pi l\nu$		$BR\times {10}^{-4}$	$⟨A_{FB}^l⟩$	$⟨P^l⟩$	$⟨C_F^l⟩$	$R_{\pi }$
$\mu$ mode	Central value	1.369	$4.678\times {10}^{-3}$	0.988	-1.486
	$1\sigma$ range	[1.030, 1.786]	[0.004, 0.006]	[0.981, 0.991]	[-1.489, -1.481]	0.641
$\tau$ mode	Central value	0.878	0.246	0.298	-0.737
	$1\sigma$ range	[0.690, 1.092]	[0.227, 0.262]	[0.195, 0.385]	[-0.781, -0.682]	[0.576, 0.725]

Table 3. Table 3: Allowed ranges of each observable in the presence of V L subscript 𝑉 𝐿 V_{L} and V ~ L subscript ~ 𝑉 𝐿 \widetilde{V}_{L} NP coupling.

	$V_L$		${\tilde{V}}_L$
	$⟨R⟩$	$⟨BR⟩\times {10}^{-4}$	$⟨R⟩$	$⟨BR⟩\times {10}^{-4}$	$⟨P^{\tau }⟩$
$B_s\to K\tau \nu$	$[0.644,0.891]$	$[0.735,1.746]$	$[0.638,0.898]$	$[0.731,1.774]$	$[-0.026,0.217]$
$B_s\to K^{\ast }\tau \nu$	$[0.593,0.804]$	$[1.684,2.993]$	$[0.582,0.802]$	$[1.579,3.098]$	$[0.249,0.513]$
$B\to \pi \tau \nu$	$[0.630,0.915]$	$[0.793,1.368]$	$[0.631,0.926]$	$[0.765,1.391]$	$[0.117,0.315]$

Equations

$$R_{D^{(\ast )}}=\frac{\mathcal{B}(B\to D^{(\ast )}\tau \nu )}{\mathcal{B}(B\to D^{(\ast )}l\nu )},R_{J/\Psi }=\frac{\mathcal{B}(B_c\to J/\Psi \tau \nu )}{\mathcal{B}(B_c\to J/\Psi l\nu )}$$

$${\mathcal{L}}_{\mathrm{eff}}$$

$$R=\frac{\mathcal{B}(B_q\to M\tau \nu )}{\mathcal{B}(B_q\to Ml\nu )},$$

Full text

I Introduction

Study of lepton flavor non-universality in the $B$ meson systems have been the center of interest both theoretically and experimentally over the last decade. Disagreement between the SM expectations and the experimental measurements (BaBar, Belle and LHCb) in $B\to D^{(*)}l\nu$ and $B_{c}\to J/\Psi l\nu$ undergoing $b\to(c,u)l\nu$ quark level transitions are well reflected in the flavor ratios $R_{D}$, $R_{D^{\ast}}$ and $R_{J/\Psi}$ defined as,

[TABLE]

In Table 1, we report the precise SM predictions and the experimental measurements of the various decay modes. The combined deviation of $3.78\sigma$ in $R_{D}$ and $R_{D^{*}}$ and around $1.3\sigma$ in $R_{J/\Psi}$ from SM expectation is observed. Similarly, the average value of the branching ratio $\mathcal{B}(B\to\tau\nu)$ reported by BaBar and Belle experiments is not in good agreement with the SM expectations. Although, the $\mathcal{B}(B\to\pi l\nu)$ is consistent with the SM, the ratio $R_{\pi}^{l}=\left(\tau_{B^{0}}/\tau_{B^{-}}\right){\mathcal{B}(B\to\tau\nu)}/$ ${\mathcal{B}(B\to\pi l\nu)}$ shows mild deviation. Similar deviations are also observed in the ratio $R_{\pi}={\mathcal{B}(B\to\pi\tau\nu)}/{\mathcal{B}(B\to\pi\,l\,\nu)}$ as well. Motivated by these anomalies, we study the implications of $R_{D}$, $R_{D^{\ast}}$, $R_{J/\Psi}$, and $R_{\pi}^{l}$ anomalies on $B_{s}\to(K,\,K^{\ast})\tau\nu$ and $B\to\pi\tau\nu$ semileptonic decays in a model dependent way.

II Theory

II.1 Effective Lagrangian

The effective Lagrangian for $b\to u\,l\,\nu$ transition decays in the presence of vector type NP couplings is of the form Dutta:2013qaa

[TABLE]

where, $G_{F}$ is the Fermi coupling constant and $|V_{ub}|$ is the CKM matrix element. $V_{L}$, $V_{R}$ are the NP Wilson coefficients (WCs) involving left-handed neutrinos, and the WCs referring to tilde terms involve right-handed neutrinos.

Using the effective Lagrangian, we calculate the three body differential decay distribution for the $B\to(P,\,V)\,l\,\nu$ decays. The final expressions pertaining to the psudoscalar and vector differential decay rates can be found in Rajeev:2018txm .

In general, we define the ratio of branching ratio as

[TABLE]

where $M=K,\,K^{\ast},\,\pi$ and $l=\mu$. We also define various $q^{2}$ dependent observables such as differential branching ratio $DBR(q^{2})$, ratio of branching ratio $R(q^{2})$, forward backward asymmetry $A_{FB}^{l}(q^{2})$, polarization fraction of the charged lepton $P^{l}(q^{2})$ and convexity parameter $C_{F}^{l}(q^{2})$ for the decay modes. For details one can refer to Rajeev:2018txm .

III Results and discussion

III.1 Standard model predictions

The SM central values are reported in Table 2. We calculate the central values by considering the central values of the input parameters. For the $1\sigma$ ranges, we perform a random scan over the theoretical inputs such as CKM matrix elements and the form factor inputs within $1\sigma$ of their central values. The significant difference in the $\mu$ mode and the $\tau$ mode are observed. The branching ratio of the order of $10^{-4}$ is observed in all the decay modes. The results pertaining $\langle P^{l}\rangle$ and $\langle C_{F}^{l}\rangle$ are calculated for the first time for these decay modes. In Fig. 1, we show the $q^{2}$ dependency of all the observables for the $\mu$ mode and the $\tau$ mode.

III.2 Beyond the SM predictions

We discuss the NP contributions coming from $V_{L}$ and $\widetilde{V}_{L}$ NP couplings. To get the allowed NP parameter space, we impose $2\sigma$ constraint coming from the measured values of $R_{D}$, $R_{D^{\ast}}$, $R_{J/\Psi}$, and $R_{\pi}^{l}$. In the left panel of Fig. 2, we show the allowed range of $V_{L}$ and $\widetilde{V}_{L}$ NP couplings once the $2\sigma$ constraints are imposed. Similarly, in the right panel the corresponding ranges in $\mathcal{B}(B\to\pi\tau\nu)$ and $R_{\pi}$ using the allowed ranges of $V_{L}$ and $\widetilde{V}_{L}$ NP couplings are shown. In Table 3 we display the allowed ranges of each observable in the presence of $V_{L}$ and $\widetilde{V}_{L}$ NP couplings. Also, in Fig. 3 and  4, we display the $q^{2}$ dependency of the various observables in the presence of $V_{L}$ and $\widetilde{V}_{L}$ NP couplings for the $B_{s}\to K\tau\nu$, $B_{s}\to K^{\ast}\tau\nu$ and $B\to\pi\tau\nu$ decays. The detailed observations are as follows:

• •

For the $V_{L}$ NP coupling, we notice a significant deviation from the SM prediction in ${\rm DBR}(q^{2})$ and $R(q^{2})$ for all the decay modes. In addition, in the presence of $\widetilde{V}_{L}$ NP coupling the $\tau$ polarization fraction show deviation along with $R(q^{2})$ and ${\rm DBR}(q^{2})$. So the measurement of $P^{\tau}(q^{2})$ can easily differentiate $V_{L}$ and $\widetilde{V}_{L}$ NP contributions.

• •

The other observable such as $A_{FB}^{\tau}(q^{2})$, $P^{\tau}(q^{2})$ and $C_{F}^{\tau}(q^{2})$ are not affected by $V_{L}$ NP coupling. Similarly, $A_{FB}^{\tau}(q^{2})$ and $C_{F}^{\tau}(q^{2})$ are not affected by $\widetilde{V}_{L}$ NP coupling.

IV Conclusion

We study $B_{s}\to(K,\,K^{\ast})\tau\nu$ and $B\to\pi\tau\nu$ decay modes within the SM and within the various NP scenarios. Although, there are hints of NP in various $B$ meson decays, the NP is not yet established. Studying $B_{s}\to(K,\,K^{\ast})\tau\nu$ and $B\to\pi\tau\nu$ decay modes theoretically as well as experimentally are well motivated as these can provide complementary information regarding NP.