Resolving $R_D$ and $R_{D^*}$ anomalies

TL;DR

This paper proposes a method to distinguish different new physics explanations for the observed anomalies in B meson decays by measuring specific polarization and angular asymmetries.

Contribution

It introduces a set of observables in B decay that can uniquely identify the Lorentz structure of potential new physics.

Findings

Measurement of polarization fractions can discriminate between new physics models.

Angular asymmetries provide additional constraints on new physics operators.

The approach can resolve degeneracies among different new physics explanations.

Abstract

The current world averages of the ratios $R_{D^{(*)}}$ are about $4\sigma$ away from their Standard Model prediction. These measurements indicate towards the violation of lepton flavor universality in $b\rightarrow c\,l\,\bar{\nu}$ decay. The different new physics operators, which can explain the $R_{D^{(*)}}$ measurements, have been identified previously. We show that a simultaneous measurement of the polarization fractions of $\tau$ and $D^*$ and the angular asymmetries $A_{FB}$ and $A_{LT}$ in $B\rightarrow D^*\tau\bar{\nu}$ decay can distinguish all the new physics amplitudes and hence uniquely identify the Lorentz structure of new physics.

Tables

Table 1. Table 1: Best fit values of NP WCs at Λ = 1 Λ 1 \Lambda=1 TeV, taken from table IV of ref.  Alok:2017qsi . We provide the predictions of ⟨ P τ ​ ( D ∗ ) ⟩ delimited-⟨⟩ subscript 𝑃 𝜏 superscript 𝐷 \langle P_{\tau}(D^{*})\rangle , ⟨ f L ⟩ delimited-⟨⟩ subscript 𝑓 𝐿 \langle f_{L}\rangle , ⟨ A F ​ B ⟩ delimited-⟨⟩ subscript 𝐴 𝐹 𝐵 \langle A_{FB}\rangle and ⟨ A L ​ T ⟩ delimited-⟨⟩ subscript 𝐴 𝐿 𝑇 \langle A_{LT}\rangle in decay B → D ∗ ​ τ ​ ν ¯ → 𝐵 superscript 𝐷 𝜏 ¯ 𝜈 B\rightarrow D^{*}\tau\bar{\nu} with their uncertainties for each of the allowed solutions.

NP WCs	Fit values	$⟨P_{\tau }\left(D^{\ast }\right)⟩$	$⟨f_L⟩$	$⟨A_{FB}⟩$	$⟨A_{LT}⟩$
SM	$C_i=0$	$-0.499\pm 0.004$	$0.45\pm 0.04$	$-0.011\pm 0.007$	$-0.245\pm 0.003$
$C_{V_L}$	$0.149\pm 0.032$	$-0.499\pm 0.004$	$0.45\pm 0.04$	$-0.011\pm 0.007$	$-0.245\pm 0.003$
$C_T$	$0.516\pm 0.015$	$+0.115\pm 0.013$	$0.14\pm 0.03$	$-0.114\pm 0.009$	$+0.110\pm 0.009$
$C_{S_L}^{\prime \prime }$	$-0.526\pm 0.102$	$-0.485\pm 0.003$	$0.46\pm 0.04$	$-0.087\pm 0.011$	$-0.211\pm 0.008$
$(C_{V_L},C_{V_R})$	$(-1.286,1.512)$	$-0.499\pm 0.004$	$0.45\pm 0.04$	$-0.371\pm 0.004$	$+0.007\pm 0.004$
$(C_{V_L}^{\prime },C_{V_R}^{\prime })$	$(0.124,-0.058)$	$-0.484\pm 0.005$	$0.45\pm 0.04$	$-0.003\pm 0.007$	$-0.243\pm 0.003$
$(C_{S_L}^{\prime \prime },C_{S_R}^{\prime \prime })$	$(-0.643,-0.076)$	$-0.477\pm 0.003$	$0.46\pm 0.04$	$-0.104\pm 0.005$	$-0.202\pm 0.002$

Equations

$$R_D=\frac{\Gamma (B\to D\tau \bar{\nu })}{\Gamma (B\to D\{e/\mu \}\bar{\nu })},R_{D^{\ast }}=\frac{\Gamma (B\to D^{\ast }\tau \bar{\nu })}{\Gamma (B\to D^{\ast }\{e/\mu \}\bar{\nu })}.$$

$$H_{eff}=\frac{4G_F}{\sqrt{2}}{V}_{cb}\left[O_{V_L}+\frac{\sqrt{2}}{4G_F{V}_{cb}}\frac{1}{{\Lambda }^2}\left\{\sum _i\left(C_i{O}_i+C_i^{{}^{\prime }}{O}_i^{{}^{\prime }}+C_i^{{}^{\prime \prime }}{O}_i^{{}^{\prime \prime }}\right)\right\}\right],$$

Full text

1 Introduction

In recent years, the evidence for charged lepton universality violation is observed in the charge current process $b\rightarrow c\tau\bar{\nu}$. The experiments, BaBar, Belle and LHCb, made several measurements of the ratios

[TABLE]

The current world averges of these measurements are about $4\sigma$ away from the Standard Model (SM) predictions Amhis:2016xyh .

All the meson decays in eq. (1) are driven by quark level transitions $b\rightarrow cl\bar{\nu}$. These transitions occur at tree level in the SM. The discrepancy between the measured values of $R_{D}$ and $R_{D^{*}}$ and their respective SM predictions is an indication of presence of new physics (NP) in the $b\rightarrow c\tau\bar{\nu}$ transition. The possibility of NP in $b\rightarrow c\mu\bar{\nu}$ is excluded by other data Alok:2017qsi . All possible NP four-Fermi operators for $b\rightarrow c\tau\bar{\nu}$ transition are listed in ref. Freytsis:2015qca . In ref Alok:2017qsi , a fit was performed between all the $b\rightarrow c\tau\bar{\nu}$ data and each of the NP interaction term. The NP terms, which can account for the all $b\rightarrow c\tau\bar{\nu}$ data, are identified and their Wilson coefficients (WCs) are calculated. It was found that there are six allowed NP solutions. Among those six solutions, four solutions are distinct with a different Lorentz structure. In ref. Alok:2016qyh it was found that the tensor NP solution could be distinguished from other possibilities provided $\langle f_{L}\rangle$, the $D^{*}$ polarization fraction can be measured with an absolute uncertainty of $0.1$.

Here, we consider four angular observables, $P_{\tau}(D^{*})$ ($\tau$ polarization fraction), $f_{L}$ ($D^{*}$ polarization fraction), $A_{FB}$ (the forward-backward asymmetry), $A_{LT}$ (longitudinal-transverse asymmetry) in the decay $B\rightarrow D^{*}\tau\bar{\nu}$. Note that these asymmetries can only be measured if the momentum of the $\tau$ lepton is reconstructed. We show that a measurement of these four quantities can uniquely identify the Lorentz structure of the NP operator responsible for the present discrepancy in $R_{D}$ and $R_{D^{*}}$ Alok:2018uft .

2 Distinguishing different new physics solutions

The most general effective Hamiltonian for $b\rightarrow c\tau\bar{\nu}$ transition can be written as

[TABLE]

where $G_{F}$ is the Fermi coupling constant, $V_{cb}$ is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element and the NP scale $\Lambda$ is assumed to be 1 TeV. We also assume that neutrino is always left chiral. The effective Hamiltonian for the SM contains only the $O_{V_{L}}$ operator. The explicit forms of the four-fermion operators $O_{i}$, $O^{{}^{\prime}}_{i}$ and $O^{{}^{\prime\prime}}_{i}$ are given in ref Freytsis:2015qca . The NP effects are encoded in the NP WCs $C_{i},C^{{}^{\prime}}_{i}$ and $C^{{}^{\prime\prime}}_{i}$. Each primed and double primed operator can be expressed as a linear combination of unprimed operators through Feirz transformation.

The values of NP WCs which fit the data on the observables $R_{D}$, $R_{D^{*}}$, $R_{J/\psi}$, $P_{\tau}(D^{*})$ and $\mathcal{B}(B_{c}\rightarrow\tau\bar{\nu})$, have been calculated previously Alok:2017qsi . Here $R_{J/\psi}$ is the ratio of $\mathcal{B}(B_{c}\rightarrow J/\psi\tau\bar{\nu})$ to $\mathcal{B}(B_{c}\rightarrow J/\psi\mu\bar{\nu})$ Aaij:2017tyk . The results of these fits are listed in table 1. This table also lists, for each of the NP solutions, the predicted values of the polarization fractions and the angular asymmetries in $B\rightarrow D^{*}\tau\bar{\nu}$ decay.

Here we compute $A_{FB}(q^{2})$ and $A_{LT}(q^{2})$ in $B\rightarrow D^{*}\tau\bar{\nu}$ decay, as functions of $q^{2}=(p_{B}-p_{D^{*}})^{2}$, where $p_{B}$ and $p_{D^{*}}$ are the four momenta of $B$ and $D^{*}$ respectively. The predictions for $P_{\tau}(D^{*})$, $f_{L}$ and $A_{FB}$ are calculated using the framework provided in Sakaki:2013bfa and for $A_{LT}(q^{2})$ we follow ref Alok:2010zd ; Duraisamy:2014sna .

The $B\rightarrow D^{(*)}\,l\,\bar{\nu}$ decay distributions depend upon hadronic form-factors. The form factors for $B\rightarrow D$ decay are well known in lattice QCD Aoki:2016frl and we use them in our analyses. For $B\rightarrow D^{*}$ decay, the HQET parameters are extracted using data from Belle and BaBar experiments along with lattice inputs. In this work, the numerical values of these parameters are taken from refs. Bailey:2014tva and Amhis:2016xyh .

This table lists six different NP solutions but only the first four solutions are distinct Alok:2017qsi . Thus we have four different NP solutions with different Lorentz structures. We explore methods to distinguish between them.

3 Results and Discussions

The average values of $P_{\tau}(D^{*})$ and $f_{L}$ for all six NP solutions are given in table 1. Not surprisingly, there is a large difference between the predicted values for $O_{T}$ solution and those for other NP solutions. If either of these observables is measured with an absolute uncertainty of $0.1$, then the $O_{T}$ solution is either confirmed or ruled out at $3\sigma$ level.

We now show that the angular asymmetries $A_{FB}$ and $A_{LT}$ have a good discrimination capability between the three remaining NP WCs. The plots for $A_{FB}$ and $A_{LT}$ as a function of $q^{2}$ are shown in the bottom row of fig. 1 and their average values are listed in table 1. We see that the plots of both $A_{FB}(q^{2})$ and $A_{LT}(q^{2})$, for $(O_{V_{L}},O_{V_{R}})$ solution, differ significantly from the plots of all other NP solutions as do the average values. If either of these asymmetries is measured with an absolute uncertainty of $0.07$, then the $(O_{V_{L}},O_{V_{R}})$ solution is either confirmed or ruled out at $3\sigma$ level.

So far we have identified observables which can clearly identify the $O_{T}$ and the $(O_{V_{L}},O_{V_{R}})$ solutions. As we can see from table 1, one needs to measure $\langle A_{FB}\rangle$ with an absolute uncertainty of $0.03$ or better to obtain a $3\sigma$ distinction between $O_{V_{L}}$ and $O^{\prime\prime}_{S_{L}}$ solutions. However, this ability to make the distinction can be improved by observing $q^{2}$ dependence of $A_{FB}$ for these solutions. We note that $A_{FB}(q^{2})$ for $O_{V_{L}}$ solution has a zero crossing at $q^{2}=5.6$ GeV2 whereas this crossing point occurs at $q^{2}=7.5$ GeV2 for $O^{\prime\prime}_{S_{L}}$ solution. A calculation of $\langle A_{FB}\rangle$ in the limited range $6$ GeV${}^{2}<q^{2}<q^{2}_{max}$ gives the result $+0.1$ for $O_{V_{L}}$ and $+0.01$ for $O^{\prime\prime}_{S_{L}}$. Hence, determining the sign of $\langle A_{FB}\rangle$, for the full $q^{2}$ range and for the limited higher $q^{2}$ range, provides a very useful tool for discrimination between these two solutions.

Hence, we find that a clear distinction can be made between the four different NP solutions to the $R_{D}$/$R_{D^{*}}$ puzzle by means of polarization fractions and angular asymmetries. Note that only the observables ($P_{\tau}(D^{*})$ and $f_{L}$) isolating $O_{T}$ do not require the reconstruction of $\tau$ momentum. The reconstruction of $\tau$ momentum is crucial to measure the asymmetries which can distinguish between the other three NP solutions.