---
abstract: 'As part of a study BSM corrections to leptonic decays of the $B_c$ meson, Tran et al. [@Tran:2018kuv] use the covariant confining quark model (CCQM) to estimate the matrix element of the pseudo-scalar curent between the vacuum and the $B_c$ meson. We note that this matrix element can be determined using existing lattice QCD results.'
author:
- 'C. T. H. Davies'
- 'C. McNeile'
title: 'Comment on Implications of new physics in the decays $B_c \to (J/\psi,\eta_c)\tau\nu$'
---

Introduction
============

The paper by Tran et al.  [@Tran:2018kuv] discusses Beyond the Standard Model (BSM) contributions to leptonic and semi-leptonic decays of the $B_c$ meson. This is a very topical calculation because of the tantalizing hints that there are lepton symmetry violations in various B and $B_c$ meson decays found by the LHCb collaboration [@Aaij:2017tyk]. To quantify the constraints from these analyses it is important to have reliable values for the operator matrix elements involved, with quantified uncertainties.

The pseudo-scalar matrix element {#sec:psme}
================================

[@Tran:2018kuv] considers a Hamiltonian of corrections to the standard model: $${\cal H}_{eff} =
\frac{4 G_F V_{cb}}{\sqrt{2}}
(
{\cal O}_{V_L}
+ \sum_{X=S_i, V_i, T_L}  \delta_{l \tau}   X {\cal O_X}
)
\label{eq:Hnew}$$ and works out the phenomenology for the leptonic and semi-leptonic decays of the $B_c$ meson. The operators considered are: $$\begin{aligned}
{\cal O}_{V_i} & = & (\overline{c} \gamma^\mu P_i b  )
 (\overline{l} \gamma_{\mu} P_L \nu_l ) ,  \\
{\cal O}_{S_i} & = & (\overline{c}  P_i b  )
 (\overline{l}  P_L \nu_l ), \\
{\cal O}_{T_L} & = & (\overline{c} \sigma^{\mu\nu} P_L b  )
 (\overline{l} \sigma_{\mu\nu} P_L \nu_l ) ,
\label{eq:BSMcontribution}\end{aligned}$$ where $\sigma_{\mu\nu} = i [\gamma_\mu , \gamma_\nu] / 2 $, $P_L = (1 - \gamma_5 ) / 2 $, and $P_R = (1 + \gamma_5 ) / 2 $.

The delta function in the Hamiltonian in equation \[eq:Hnew\] takes into account lepton flavor violation in this model. The complex $X$ are the Wilson coefficients from the Beyond the Standard Model (BSM) theory. We note that there is no suppression of the operators by the scale of the BSM physics, because the three additional operators all have the same dimension as the operators in the standard model:

The leptonic decay constant of the $B_c$ meson, $f_{B_c}$: $$\label{eq:fbc}
\langle 0 \mid \overline{c} \gamma_5 \gamma_{\mu} b \mid B_c \rangle =
f_{Bc}   p_{\mu} ,$$ is used in the standard model calculation of the annihilation rate of the $B_c$ meson to leptons via a $W$ boson. The additional operators in equation \[eq:BSMcontribution\] require the introduction of the pseudo-scalar matrix element of the $B_c$ meson defined via $$\langle 0 \mid \overline{c} \gamma_5 b \mid B_c \rangle = f_{Bc}^{P}(\mu)
  M_{bc}.$$ The matrix element $f_{Bc}^{P}$ depends on the renormalization scale $\mu$ in QCD. A physical result is obtained when it is combined with the Wilson coefficient, which also depends on $\mu$, from the BSM theory.

The leptonic branching fraction of the $B_c$ meson is $$\begin{gathered}
{\cal B} (B_c \rightarrow \tau \nu) = 
\frac{G_F^2}{8 \pi} \mid V_{cb} \mid^2 \tau_{B_c}  m_{B_c} m_{\tau}^2  \\
\left( 1 - \frac{m_\tau^2}{m_{B_c}^2}  \right)^2 
f_{B_c}^2  
A_{BSM},\end{gathered}$$ where $A_{BSM}$ is

$$A_{BSM}  = \mid 1 - (V_R - V_L) + 
\frac{m_{B_c}}{m_\tau} \frac{f_{B_c}^{P} }{f_{B_c}} (S_R - S_L) 
\mid^2 .$$

In the standard model $A_{BSM}$ = 1. If there are experimental deviations of the leptonic decay of the $B_c$ meson from the value in the standard model, then the values of $f_{B_c}$ and $f_{B_c}^{P}$ are required to constrain values of the Wilson coefficients $V_R$, $V_L$, $S_R$, and $S_L$, of the BSM theory. The Wilson coefficients also contribute to semi-leptonic decays of heavy light mesons, so additional constraints on them can be obtained. This is an modern update of the experimental origins of the V-A theory in the standard model, where experimental data was used to constrain the interactions between quarks (see [@Das:2009zzd] for example). Although the leptonic decay of the $B_c$ meson has not been observed experimentally, the constraints from a LEP1 measurement allowed, Tran et al. [@Tran:2018kuv] to put bounds on the $S_L$ and $S_R$ couplings.  [@Tran:2018kuv; @Ivanov:2016qtw] use a CCQM to estimate $f_{Bc}^{P}(\mu)$, although without giving a scale, $\mu$, at which it is determined.

Lattice QCD results {#sec:lattice}
===================

The decay constant of the $B_c$ has been calculated in lattice QCD using two different approaches which give results in good agreement [@McNeile:2012qf; @Colquhoun:2015oha]. The most accurate results comes from using the Highly Improved Staggered Quark (HISQ) formalism [@Follana:2006rc]. In this formalism there is an exact partially conserved axial current (PCAC) [@Kilcup:1986dg] relation $$\partial_{\mu} A_{\mu} =  (m_1  +  m_2)  P  \;\;.
\label{eq:PCACdefn}$$ From the pseudoscalar matrix element times quark mass we can then obtain the matrix element of the temporal axial current (at zero spatial momentum) needed for eq. (\[eq:fbc\]) with absolute normalisation. This is done in [@McNeile:2012qf] for heavy-charm pseudoscalar mesons for a range of heavy quark masses and values of the lattice spacing, $a$. This enables the heavy quark mass dependence of the heavy-charm decay constant to be mapped out in the continuum ($a \rightarrow 0$) limit and a result for $f_{B_c}$ to be obtained when the heavy quark mass corresponds to that of the $b$. The value obtained is $$\label{eq:fbcresult}
f_{B_c} = 0.427(6)(2) \,\mathrm{GeV},$$ and a complete error budget is given in [@McNeile:2012qf].

A completely different approach for $f_{B_c}$ based on the lattice discretisation of nonrelativistic QCD (NRQCD) [@Lepage:1992tx] is given in [@Colquhoun:2015oha]. There the matrix element of the temporal axial current is calculated directly but, since there is no PCAC relation on the lattice in this case, the current is matched to that of continuum QCD using lattice QCD perturbation theory through $\mathcal{O}(\alpha_s)$ [@Monahan:2012dq]. A result for $f_{B_c}$ of 0.434(15) GeV is obtained, where the uncertainty is dominated by that from lattice discretisation effects and systematic uncertainties in matching the current. Although the uncertainty is larger here than in the HISQ case, the agreement between the two results is confirmation of our understanding of the errors from the two approaches.

Since, in the HISQ case [@McNeile:2012qf], the lattice PCAC relation was used to determine $f_{B_c}$, it is clear that we could also have determined $f_{Bc}^{P}(\mu)$. Since $f_{Bc}^P(\mu)$ runs with $\mu$ it is much more convenient to determine it in combination with quark masses. The PCAC relation, eq. (\[eq:PCACdefn\]) on the lattice yields the following relationship between $f_{B_c}$ and $f_{Bc}^P$: $$(m_b+m_c)f_{Bc}^{P} = M_{B_c} f_{Bc} .
\label{eq:pcaclatt}$$ Here $m_b$ and $m_c$ are the bare lattice quark masses. Since both sides of this equation are scheme- and scale-invariant, we can instead apply this relationship in the continuum using the continuum results for $f_{B_c}$ obtained from lattice QCD calculations.

Then $$f_{Bc}^{P}(\mu) = \frac{M_{B_c} f_{Bc} }{m_b(\mu) + m_c(\mu)}, 
\label{eq:pcacdecay}$$ where $m_b(\mu)$ and $m_c(\mu)$ are the bottom and charm quark masses at the scale $\mu$ in a standard continuum scheme, such as $\overline{\mathrm{MS}}$. The quark masses are also most conveniently and accurately obtained from lattice QCD calculations, see for example [@McNeile:2010ji].

We use results from [@McNeile:2010ji] for the quark masses in the $\overline{\mathrm{MS}}$ scheme at a standard scale of 3 GeV, $\overline{m}_c$(3 GeV, $n_f$=4) = 0.986(6) GeV, $m_b/m_c$ = 4.51(4), $f_{B_c}$ from [@McNeile:2012qf] (0.427(6) GeV) and $M_{B_c}$ = 6.274(1) GeV from experiment [@Aaij:2016qlz]. This gives, in the $\overline{\mathrm{MS}}$ scheme $$\label{eq:fbcpresult}
\overline{f}_{Bc}^P (\mbox{3 GeV}) = 0.493(9)  \,\,\mathrm{GeV}$$ where the uncertainty is dominated by that from the lattice QCD result for $f_{B_c}$. The result for $\overline{f}_{Bc}^{P}$ can be run to different values of $\mu$ using the inverse of the running of the $\overline{\mathrm{MS}}$ quark mass [@Chetyrkin:1997dh; @Vermaseren:1997fq].

The result for $f_{B_c}$ computed using the CCQM [@Tran:2018kuv] of 0.489 GeV is 15% larger than that obtained from the lattice QCD results discussed above. The systematic uncertainty from using the CCQM is estimated in [@Tran:2018kuv] as 10%. The result given in [@Tran:2018kuv] for $f_{B_c}^{P}$ of 0.645 GeV is hard to interpret or compare to the lattice QCD values since no scheme or scale for it is given. Lattice QCD results for the form factors of $B_c$ semileptonic decay to charmonium states are as yet preliminary [@Colquhoun:2016osw] but will provide a further point of comparison in future.

Conclusions
===========

Weak decays of the $B_c$ meson provide exciting opportunities for constraining new physics as growing datasets from LHC, along with new analyses, become available [@Tran:2018kuv]. The theoretical input to this of hadronic parameters such as decay constants and form factors for the $B_c$ need to be firmly based on ‘first-principles’ approaches to QCD, such as lattice QCD. This allows not only the result to be given but also a well-motivated uncertainty on its value. To this end, we collect here existing lattice QCD results, with their associated uncertainty, for the $B_c$ decay constant and we derive from them a value for the pseudoscalar current matrix element.

Our calculations were done on the Darwin Supercomputer as part of STFC’s DiRAC facility jointly funded by STFC, BIS and the Universities of Cambridge and Glasgow. This work was funded by STFC.

[10]{}

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