Semileptonic decays $B_c \to (\eta_c,J/\psi) l \bar{\nu}_l $ in the "PQCD + Lattice" approach

TL;DR

This paper employs a combined PQCD and lattice QCD approach to analyze semileptonic $B_c$ decays, providing refined predictions for branching ratios, ratios, and polarization observables, with results consistent with current experimental data.

Contribution

It introduces a novel combination of PQCD and lattice QCD methods with new distribution amplitudes and parametrizations for form factor extrapolation in $B_c$ semileptonic decays.

Findings

Branching ratios decrease by 5-16% when including lattice inputs.

Predicted ratios $R_{\eta_c}$ and $R_{J/\psi}$ are 0.34 and 0.28 respectively.

Predicted polarization $P_{\tau}$ values are 0.37 and -0.55 for $\eta_c$ and $J/\psi$.

Abstract

In this paper, we studied the semileptonic decays $B_c^- \to (\eta_c, J/\psi) l ^- \bar{\nu}_l$ by employing the PQCD factorization approach, using the newly defined distribution amplitudes of the $B_c$ meson and the new kind of parametrization for extrapolation of the form factors , and also taking into account the lattice QCD results about the relevant form factors at several points. We found the following main results: (a) the PQCD predictions for the branching ratios of $B_c \to (\eta_c,J/\psi) l \bar{\nu}$ decays will become a little smaller by about $(5-16)\%$ when the lattice input are taken into account in the extrapolation of the relevant form factors; (b) the PQCD predictions for the ratio $R_{\eta_c, J/\psi}$ and the longitudinal polarization $P_{\tau}$ are $R_{\eta_c}=0.34\pm 0.01 , R_{J/\psi}=0.28\pm 0.01$, $P_{\tau}(\eta_c) = 0.37\pm 0.01$ and $P_{\tau}(J/\psi) = -0.55 \pm 0.01$ ; and (c) after the inclusion of the lattice input the theoretical predictions changed slightly: $R_{\eta_c}=0.31\pm 0.01$, $ R_{ J/\psi}=0.27\pm 0.01$, $P_{\tau}( \eta_c) = 0.36 \pm 0.01$ and $P_{\tau}( J/\psi) = -0.53\pm 0.01$. The theoretical predictions for $R_{ J/\psi}$ agree with the measured one within errors, and other predictions could be tested in the near future LHCb experiments.

Tables

Table 1. Table 1: The theoretical predictions for the form factors f 0 , + , V subscript 𝑓 0 𝑉 f_{\rm 0,+},V and A 0 , 1 , 2 subscript 𝐴 0 1 2 A_{\rm 0,1,2} at q 2 = 0 superscript 𝑞 2 0 q^{\rm 2}=0 , obtained by employing the PQCD approach, by using some other different approaches wang09 ; BSW ; nrqcd ; epjc51-833 ; jhep1905-094 ; prd68-094020 ; prd97-054014 or in Lattice QCD lattice1 .

Form factors	PQCD	PQCD	LFQM	BSW	NRQCD	LCSR	LCSR	RQM	CCQM	Lattice
	This work	cpc37-093102	wang09	BSW	nrqcd	epjc51-833	jhep1905-094	prd68-094020	prd97-054014	lattice1
$f_{0,+}^{B_c\to {\eta }_c}\left(0\right)$	$0.56\left(7\right)$	$0.48\left(7\right)$	$0.61$	$0.58$	$1.67$	$0.87$	$0.62$	$0.47$	$0.75$	$0.59$
$V^{B_c\to J/\psi }\left(0\right)$	$0.75\left(9\right)$	$0.42\left(2\right)$	$0.74$	$0.91$	$2.24$	$1.69$	$0.73$	$0.49$	$0.78$	$0.70$
$A_0^{B_c\to J/\psi }\left(0\right)$	$0.40\left(5\right)$	$0.59\left(3\right)$	$0.53$	$0.58$	$1.43$	$0.27$	$0.54$	$0.40$	$0.56$	$-$
$A_1^{B_c\to J/\psi }\left(0\right)$	$0.47\left(5\right)$	$0.46\left(3\right)$	$0.50$	$0.63$	$1.57$	$0.75$	$0.55$	$0.73$	$0.55$	$0.48$
$A_2^{B_c\to J/\psi }\left(0\right)$	$0.62\left(6\right)$	$0.64\left(3\right)$	$0.44$	$0.74$	$1.73$	$1.69$	$0.35$	$0.50$	$0.56$	$-$

Table 2. Table 2: The form factors f i ​ ( 0 ) subscript 𝑓 𝑖 0 f_{i}(0) from the PQCD calculations, the J P superscript 𝐽 𝑃 J^{P} and masses (in unit of GeV) of the low-laying B c subscript 𝐵 𝑐 B_{c} resonances jhep1905-094 used in the BCL fit for B c → ( η c , J / ψ ) → subscript 𝐵 𝑐 subscript 𝜂 𝑐 𝐽 𝜓 B_{c}\to(\eta_{c},J/\psi) form factors. The parameters α 0 , 1 subscript 𝛼 0 1 \alpha_{0,1} are determined from the fitting.

FFs	$f_i\left(0\right)$ in PQCD	$J^P$	$m_R$	${\alpha }_0$	${\alpha }_1$
$f_0$	0.56(7)	$0^{+}$	6.71	0.691	-7.74
$f_{+}$	0.56(7)	$1^{-}$	6.34	0.763	-12.2
$V$	0.75(9)	$1^{-}$	6.34	1.06	-20.6
$A_0$	0.40(5)	$0^{-}$	6.28	0.551	-10.5
$A_1$	0.47(5)	$1^{+}$	6.75	0.586	-7.73
$A_2$	0.62(6)	$1^{+}$	6.75	1.01	-26.8

Table 3. Table 3: The theoretical predictions (in unit of 10 − 3 superscript 10 3 10^{-3} ) for the branching ratios ℬ ​ ( B c → ( η c , J / ψ ) ​ l ​ ν ¯ l ) ℬ → subscript 𝐵 𝑐 subscript 𝜂 𝑐 𝐽 𝜓 𝑙 subscript ¯ 𝜈 𝑙 {\cal B}(B_{c}\to(\eta_{c},J/\psi)l\bar{\nu}_{l}) in the PQCD and “PQCD+Lattice” approaches. As a comparison, we also list the predictions as given in a previous PQCD work cpc37-093102 , and other four approaches wang09 ; epjc51-833 ; z-series ; jhep1905-094 .

Mode	PQCD	PQCD+Lattice	PQCD	LFQM	Z-Series	LCSR	LCSR
			cpc37-093102	wang09	z-series	epjc51-833	jhep1905-094
$ℬ\left(B_c\to {\eta }_c\mu {\overline{\nu }}_{\mu }\right)$	${8.14}_{-1.77}^{+1.96}$	${7.76}_{-1.51}^{+1.95}$	${4.4}_{-1.1}^{+1.2}$	$6.7$	$6.6$	$16.7$	${8.2}_{-1.1}^{+1.2}$
$ℬ\left(B_c\to {\eta }_c\tau {\overline{\nu }}_{\tau }\right)$	${2.79}_{-0.63}^{+0.84}$	${2.41}_{-0.40}^{+0.49}$	${1.4}_{-0.3}^{+0.4}$	$1.9$	$2.0$	$4.9$	${2.6}_{-0.5}^{+0.6}$
$ℬ\left(B_c\to J/\psi \mu {\overline{\nu }}_{\mu }\right)$	${16.1}_{-3.4}^{+4.5}$	${14.1}_{-2.2}^{+2.7}$	${10.0}_{-1.2}^{+1.3}$	$14.9$	$14.5$	$23.7$	${22.4}_{-4.9}^{+5.7}$
$ℬ\left(B_c\to J/\psi \tau {\overline{\nu }}_{\tau }\right)$	${4.54}_{-1.01}^{+1.29}$	${3.83}_{-0.58}^{+0.63}$	${2.9}_{-0.3}^{+0.4}$	$3.7$	$3.6$	$6.5$	${5.3}_{-1.4}^{+1.6}$

Table 4. Table 4: The theoretical predictions for the ratios R η c subscript 𝑅 subscript 𝜂 𝑐 R_{\eta_{c}} and R J / ψ subscript 𝑅 𝐽 𝜓 R_{J/\psi} obtained by employing the PQCD and “PQCD+Lattice” approaches, or as given in previous works cpc37-093102 ; wang09 ; epjc51-833 ; z-series ; jhep1905-094 ; mi1 ; mi2 .

Mode	PQCD	PQCD+Lattice	PQCD	LFQM	Z-Series	LCSR	LCSR	M-Ind.
			cpc37-093102	wang09	z-series	epjc51-833	jhep1905-094	mi1 ; mi2
$R_{{\eta }_c}$	$0.34\pm 0.01$	$0.31\pm 0.01$	$0.31$	$0.28$	$0.31$	$0.30$	$0.32\pm 0.02$	$0.29\pm 0.05$
$R_{J/\psi }$	$0.28\pm 0.01$	$0.27\pm 0.01$	$0.29$	$0.25$	$0.25$	$0.27$	$0.23\pm 0.01$	$[0.20,0.39]$

Equations

$$R_X$$

$$R_{J/\psi }^{\mathrm{Exp}}=0.71\pm 0.17(stst.)\pm 0.18(syst.),$$

$$R_{J/\psi }\approx 0.29,R_{{\eta }_c}\approx 0.31,$$

$${\varphi }_{B_c}(x,b)=\frac{f_{B_c}}{2\sqrt{6}}{N}_{B_c}x(1-x)\cdot \exp \left[-\frac{(1-x)m_c^2+x{m}_b^2}{8{\beta }_{B_c}^{2}x(1-x)}\right]\cdot \exp \left[-2{\beta }_{B_c}^{2}x(1-x)b^2\right],$$

$${\varphi }_{B_c}(x)=\frac{f_{B_c}}{2\sqrt{6}}\delta \left(x-\frac{m_c}{m_{B_c}}\right).$$

$$P_1=\frac{m_{{\mathrm{B}}_{\mathrm{c}}}}{\sqrt{2}}(1,1,0_{\perp }),P_2=r\frac{m_{{\mathrm{B}}_{\mathrm{c}}}}{\sqrt{2}}({\eta }^{+},{\eta }^{-},0_{\perp }),$$

$${\eta }^{\pm }=\eta \pm \sqrt{{\eta }^2-1},\eta =\frac{1}{2r}\left[1+r^2-\frac{q^2}{m_{B_c}^2}\right],$$

$${ϵ}_{\mathrm{L}}=\frac{1}{\sqrt{2}}({\eta }^{+},-{\eta }^{-},0_{\perp }),{ϵ}_{\mathrm{T}}=(0,0,1),$$

$${\Phi }_{B_c}(x,b)=\frac{i}{\sqrt{6}}(p{/}_1+m_{B_c}){\gamma }_5{\varphi }_{B_c}(x,b).$$

$${\int }_0^1{\varphi }_{B_c}(x,b=0)dx\equiv {\int }_0^1{\varphi }_{B_c}(x)dx=\frac{f_{B_c}}{2\sqrt{6}}$$

$${\Phi }_{{\eta }_c}(x)$$

$${\Phi }_{J/\Psi }^L(x)$$

$${\Phi }_{J/\Psi }^T(x)$$

$${\mathcal{H}}_{eff}(b\to c{l}^{-}{\bar{\nu }}_l)=\frac{G_F}{\sqrt{2}}{V}_{cb}\bar{c}{\gamma }_{\mu }(1-{\gamma }_5)b\cdot \bar{l}{\gamma }^{\mu }(1-{\gamma }_5){\nu }_l.$$

$$\frac{d\Gamma (B_c\to {\eta }_{c}l{\bar{\nu }}_{\mathrm{l}})}{d{q}^2}$$

$$f_{+}(q^2)$$

$$f_0(q^2)$$

$$f_1(q^2)$$

$$f_2(q^2)$$

$$H_i(t_i)=h_i(x_1,x_2,b_1,b_2)\cdot {\alpha }_s(t_i)\exp \left[-S_{ab}(t_i)\right],\mathrm{for}i=(1,2),$$

$$\frac{d{\Gamma }_{\mathrm{L}}}{d{q}^2}$$

$$\frac{d{\Gamma }_{\pm }}{d{q}^2}$$

$$\frac{d\Gamma }{d{q}^2}=\frac{d{\Gamma }_{\mathrm{L}}}{d{q}^2}+\frac{d{\Gamma }_{+}}{d{q}^2}+\frac{d{\Gamma }_{-}}{d{q}^2}.$$

$$V(q^2)$$

$$A_0(q^2)$$

$$A_1(q^2)$$

$$A_2(q^2)$$

$$m_{{\mathrm{B}}_{\mathrm{c}}}$$

$${\tau }_{{\mathrm{B}}_{\mathrm{c}}}$$

$$|V_{\mathrm{cb}}|$$

$$f_i(t)$$

$$0\le t_0$$

$$t_{\pm }$$

Full text

The semileptonic $B_{c}$ decays; the PQCD approach; Lattice QCD results; the ratio of the branching ratios; the longitudinal polarization;

I Introduction

In the standard model (SM), all electroweak gauge bosons ( $Z,\gamma$ and $W^{\pm}$) have equivalent couplings to three generation leptons, and the only differences arise due to the mass hierarchy: $m_{e}<m_{\mu}\ll m_{\tau}$: this is the so-called Lepton flavor universality (LFU) in the SM . The $B_{c}$ meson can only decay through weak interactions because it is below the B-D threshold, it is therefore an ideal system to study the weak decays of heavy quarks. Since the rare semileptonic decays governed by the flavor-changing neutral currents (FCNC) are forbidden at the tree level in the SM, the precise measurements for such semileptonic $B_{c}$ decays can play an important role in testing the SM and in searching for the signal and/or evidence of the new physics (NP) beyond the SM. In recent years, the measured values of $R(D)$ and $R(D^{*})$ , defined as the ratios of the branching fractions ${\cal B}(B\to D^{(*)}\tau\nu_{\tau})$ and ${\cal B}(B\to D^{(*)}l\nu_{l})$) , are clearly larger than the SM predictions: the combined deviation is about $3.8\sigma$ for $R(D)-R(D^{*})$ in 2017   hfag2017 , or $3.1\sigma$ after the inclusion of the newest Belle measurement Belle1904 ; Caria1903 . The semileptonic decays $B\to D^{(*)}l\nu_{l}$ with $l=(e,\mu,\tau)$ are therefore studied intensively by many authors in the framework of the SM  fajfer-prl109 ; csb59-125 ; csb59-3787 ; prd95-115008 ; jhep-1711 , or in various new physics (NP) models beyond the SM for example in Refs. prd95-115008 ; li-16a ; adam2019 ; Fajfer-2018 .

If the above mentioned $R(D^{(*)})$ anomalies are indeed the first signal of the LFU violation ( i.e. an indication of new physics ) in $B_{u,d}$ sector, it must appear in the similar semileptonic decays of $B_{s}$ and $B_{c}$ mesons, and should be studied systematically. The $B_{c}$ ($\bar{b}c$) meson, as a bound state of two heavy bottom and charm quarks, was firstly observed by the CDF Collaboration CDF1998 and then by the Large Hadron Collider (LHC) experiments in recent years pdg2018 . The properties of $B_{c}$ meson and the dynamics involved in $B_{c}$ decays could be fully exploited through the precise measurements at the LHC experiments, especially the measurements carried on by the LHCb Collaboration. Very recently, some hadronic and semileptonic $B_{c}$ meson decays have been measured by LHCb experiments lhcb-17a ; lhcb-18a . Analogous to the cases for $B$ decays, the generalization of the $R(D^{(*)})$ for the semileptonic $B_{c}$ decays are the ratio $R_{\eta_{c}}$ and $R_{J/\psi}$:

[TABLE]

But only the ratio $R_{J/\psi}$ has been measured by the LHCb Collaboration very recently lhcb-18a . ,

[TABLE]

which is consistent with currently available SM predictions epjc45-711 ; cpc37-093102 ; prd73-094006 ; epjc51-833 ; wang09 ; qiao2013 ; prd62-014019 ; jpg26 ; prd68-094020 ; prd82-034032 ; prd48-5208 ; npb569-473 ; prd74a within $2\sigma$ errors .

During the past two decades, the semileptonic $B_{c}\to(\eta_{c},J/\psi)l\bar{\nu}_{l}$ decays have been studied by many authors in rather different theories or models, for example, in the QCD sum rule (QCD SR) and light-cone sum rules (LCSR) epjc51-833 ; prd48-5208 ; npb569-473 ; zpc57-43 ; jhep1905-094 , the relativistic quark model (RQM) or non-relativistic quark model (NRQM)  prd68-094020 ; prd71-094006 , the light-front quark model (LFQM) wang09 ; prd89-017501 , the covariant confining quark model (CCQM) prd97-054014 , the nonrelativistic QCD (NRQCD) Chang-1992 ; Qiao-2011 ; qiao2013 ; Shen-2014 ; nrqcd , the model-independent investigations (MII) mi1 ; mi2 ; mi3 ; z-series , the lattice QCD (LQCD)  lattice1 ; lattice2 ; lattice3 and the perturbative QCD (PQCD) factorization approach cpc37-093102 ; epjc76-564 ; prd96-076001 .

In a previous work  cpc37-093102 , we calculated the ratio $R_{J/\psi}$ and $R_{\eta_{c}}$ by employing the PQCD approach pqcd1 ; pqcd2 and found the PQCD predictions  cpc37-093102 :

[TABLE]

which also agree well with the ones from the QCDSR or other different approaches in the frame work of the SM . In this paper, we will present a new systematic evaluation for the ratio $R_{J/\psi}$ and $R_{\eta_{c}}$ by using the PQCD factorization approach but with the following further improvements:

1. (1)

We here will use a newly developed distribution amplitude (DA) $\phi_{B_{c}}(x,b)$ for $B_{c}$ meson as proposed very recently in Ref. Bc-am :

[TABLE]

instead of the simple $\delta$-function as being used in Ref. epjc45-711 ; cpc37-093102 :

[TABLE] 2. (2)

For the relevant form factors, the new preliminary lattice QCD results from the HFQCD Collaboration include (a) the lattice QCD results for $V(q^{2})$ and $A_{1}(q^{2})$ at several $q^{2}$ values for $B_{c}\to J/\psi$ transition, and (b) the lattice QCD results for $f_{0}(q^{2})$ at five $q^{2}$ values and $f_{+}(q^{2})$ at four $q^{2}$ values lattice1 ; lattice2 . We will use four Lattice QCD results $(f_{0,+}(8.72),V(5.44),A_{1}(10.07))$ as the new input in the extrapolation of the relevant form factors from the low $q^{2}$ region to the $q_{max}^{2}$. 3. (3)

For the extrapolation of the form factors, analogous to the authors of Ref. jhep1905-094 , we will also use the Bourrely-Caprini-Lellouch (BCL) parametrization to make the series expansion of the form factors bcl09 instead of the exponential expansion formulae as being used in Ref. cpc37-093102 . We will calculate the branching ratios of the considered decays and the ratios $R_{J/\psi}$ and $R_{\eta_{c}}$ by using the PQCD approach itself and the “PQCD+Lattice” method respectively, and compare the resultant predictions. 4. (4)

Besides the ratios $R_{\eta_{c}}$ and $R_{J/\psi}$, we here will calculate the longitudinal polarization $P_{\tau}(\eta_{c})$ and $P_{\tau}(J/\psi)$ of the final state tau lepton, which was absent in Ref. cpc37-093102 . Just like the polarization $P_{\tau}^{D^{*}}$ firstly measured at Belle prl118-801 , both $P_{\tau}(\eta_{c})$ and $P_{\tau}(J/\psi)$ could be measured in the future LHCb experiment.

The paper is organized as follows: after this introduction, we give the distribution amplitudes of the $B_{c}$ meson and the final state $\eta_{c}$ and $J/\psi$ mesons in Section 2. By employing the PQCD factorization approach we calculate and present the expressions for the $B_{c}\to(\eta_{c},J/\psi)$ transition form factors in the low $q^{2}$ region in Section 3. The extrapolation of the six form factors from the low $q^{2}$ region to the $q^{2}_{max}$, the PQCD and the ”PQCD+Lattice” predictions for the branching ratios ${\cal B}(B_{c}\to(\eta_{c},J/\psi))(\mu^{-}\bar{\nu}_{\mu},\tau^{-}\bar{\nu}_{\tau})$, the ratios $R_{\eta_{c}}$ and $R_{J/\psi}$ of the branching ratios, and the longitudinal polarization $P_{\tau}(\eta_{c})$ and $P_{\tau}(J/\psi)$ are given in Section 4. A short summary is given in the final section.

II Kinematics and the Wave Functions

The lowest order Feynman diagrams for $B_{c}\to Xl\nu$ are displayed in Fig. 1. The kinematics of these decays are discussed in the large-recoil (low $q^{2}$) region, where the PQCD factorization approach is applicable to the considered semileptonic decays involving $\eta_{c}$ or $J/\psi$ as the final state meson li1995 . In the $B_{c}$ meson rest frame, we define the $B_{c}$ meson momentum $P_{1}$, and the final state meson momentum $P_{2}$ in the light-cone coordinates asprd67-054028 ; cpc37-093102

[TABLE]

with

[TABLE]

where the mass ratio $r=m_{\eta_{c}}/m_{B_{c}}$ or $m_{J/\psi}/m_{B_{c}}$, and $q=p_{\rm 1}-p_{\rm 2}$ is the momentum of the lepton pair. The longitudinal polarization vector $\epsilon_{\rm L}$ and transverse polarization vector $\epsilon_{\rm T}$ of the vector meson are defined in the same way as in Ref. cpc37-093102 :

[TABLE]

The momentum $k_{1}$ and $k_{2}$ of the spectator quark in $B_{c}$ or in final state $(J/\psi,\eta_{c})$ are parameterized in the same way as in Ref. cpc37-093102 .

For the $B_{c}$ meson wave function, we make use of the same one as being used for example in Ref.epjc45-711 ; cpc37-093102 ,

[TABLE]

We here will use the new DA $\phi_{B_{c}}(x,b)$  Bc-am as given in Eq. (4) instead of the simple $\delta$-function as given in Eq. (5). As usual, the normalization constant $N_{B_{c}}$ in Eq. (4) is fixed by the relation

[TABLE]

where the decay constant $f_{B_{c}}=0.489\pm 0.005$ GeV has been obtained in lattice QCD by the TWQCD Collaboration TW . We will set $\beta_{B_{c}}=1.0\pm 0.1$ GeV in Eq. (4) in order to estimate the uncertainty  Bc-am .

For the pseudoscalar charmonium state $\eta_{c}$ and the vector one $J/\Psi$, we use the same wave function as those in Refs. prd73-094006 ; cpc37-093102 :

[TABLE]

where the twist-2 asymptotic DAs $(\phi^{v}(x),\phi^{L}(x),\phi^{T}(x))$ and the twist-3 ones $(\phi^{s}(x),\phi^{t}(x),\phi^{V}(x))$ are the same ones as those being used in Refs. prd73-094006 ; cpc37-093102 .

III The form factors and differential decay widths

For the considered charged current $B_{c}\to(\eta_{c},J/\psi)l^{-}\bar{\nu}_{l}$ decays, the quark-level transition is the $b\to cl^{-}\bar{\nu}_{l}$ decay with the effective Hamiltonian

[TABLE]

where $G_{F}=1.16637\times 10^{-5}GeV^{-2}$ is the Fermi-coupling constant and $V_{cb}$ is the CKM matrix element. The differential decay widths of the semi-leptonic decays $B^{-}_{c}\to\eta_{c}l^{-}\bar{\nu}_{\rm l}$ can be written wang09 ; cpc37-093102 in the following form:

[TABLE]

where $m_{\rm l}$ is the mass of the charged leptons, $0\leq q^{2}\leq(m_{\rm B_{c}}-m_{\rm\eta_{c}})^{2}$ and $\lambda(q^{2})=(m_{\rm B_{c}}^{2}+m_{\rm\eta_{c}}^{2}-q^{2})^{2}-4m_{\rm B_{c}}^{2}m_{\rm\eta_{c}}^{2}$ is the phase space factor. In the PQCD factorization approach, the form factor $f_{0}(q^{2})$ and $f_{+}(q^{2})$ in Eq. (15) defined through the matrix element $<\eta_{c}(p_{2})|\bar{c}(0)\gamma_{\mu}b(0)|B_{c}(p_{1})>$ wang09 ; cpc37-093102 can be calculated and written as a summation of the auxiliary form factor $f_{1,2}(q^{2})$:

[TABLE]

After making the analytical calculations in PQCD approach, one found the function $f_{1,2}(q^{2})$:

[TABLE]

[TABLE]

with the functions $H_{i}(t_{i})$ can be written in the following form

[TABLE]

where $C_{F}=4/3$ is a color factor, $r_{c}=m_{c}/m_{B_{c}}$, $r_{b}=m_{b}/m_{B_{c}}$, $r=m_{\eta_{c}}/m_{B_{c}}$ . The explicit expressions of the hard functions $h_{i}(x_{1},x_{2},b_{1},b_{2})$ and the Sudakov functions $\exp\left[-S_{ab}(t_{i})\right]$ will be given in Appendix.

For $B^{-}_{c}\to J/\psi l^{-}\bar{\nu}_{\rm l}$ decays, the differential decay widths can be written in the following form wang09 ; cpc37-093102 :

[TABLE]

[TABLE]

where $0\leq q^{2}\leq(m_{\rm B_{c}}-m_{\rm J/\psi})^{2}$ and $\lambda(q^{2})=(m_{\rm B_{c}}^{2}+m_{J/\psi}^{2}-q^{2})^{2}-4m_{\rm B_{c}}^{2}m_{J/\psi}^{2}$ . The total differential decay widths is defined as

[TABLE]

The form factors $V(q^{2})$ and $A_{0,1,2}(q^{2})$ can also be calculated in the framework of the PQCD factorization approach:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $r_{c}=m_{c}/m_{B_{c}}$, $r_{b}=m_{b}/m_{B_{c}}$ and $r=m_{J/\psi}/m_{B_{c}}$, the parameter $\eta$ is defined in Eq. (7), and the functions $H_{i}(t_{i})$ are the same ones as those defined in Eq. (19).

4     Numerical Results

In the numerical calculations we use the following input parameters (here masses and decay constants are in units of GeV)hfag2017 ; pdg2018 ; TW :

[TABLE]

For the considered semileptonic $B_{c}$ meson decays, it is easy to see that the theoretical predictions for the differential decay rates and other physical observables strongly depend on the form factors $f_{\rm 0,+}(q^{\rm 2})$ for $B_{c}\to\eta_{c}l\nu_{l}$ decays , and the form factors $V(q^{\rm 2})$ and $A_{\rm 0,1,2}(q^{\rm 2})$ for $B_{c}\to J/\psi l\nu_{l}$ decays wang09 ; cpc37-093102 . The value of these form factors at $q^{2}=0$ and their $q^{2}$ dependence in the whole range of $0\leq q^{2}\leq q^{2}_{max}$ contain a lot of information of the physical process. Up to now, these form factors have been calculated in many rather different methods, for example, in Refs. zpc57-43 ; prd48-5208 ; npb569-473 ; epjc51-833 ; jpg26 ; prd68-094020 ; prd71-094006 .

In Refs. csb59-125 ; csb59-3787 ; prd67-054028 ; prd78-014018 , the authors examined the applicability of the PQCD factorization approach to $(B\to D^{(*)})$ transitions, and have shown that the PQCD approach with the inclusion of the Sudakov effects is applicable to study the semileptonic decays $B\to D^{(*)}l\bar{\nu}_{\rm l}$ csb59-125 ; csb59-3787 . Since the PQCD predictions for the considered form factors are reliable only at the low $q^{\rm 2}$ region, we first calculate explicitly the values of the relevant form factors at the sixteen points in the lower region $0\leq q^{\rm 2}\leq m_{\rm\tau}^{\rm 2}$ by using the expressions as given in Eqs. (17,18,23-26) and the definitions in Eq. (16). In the second column of Table 1, we show the PQCD predictions for six relevant form factors at $q^{2}=0$. The errors of the PQCD predictions are the combination of the major errors from the uncertainty of $\beta_{\rm B_{c}}=1.0\pm 0.1$ GeV, $m_{\rm c}=1.27\pm 0.03$ GeV and $|V_{cb}|=(42.2\pm 0.8)\times 10^{-3}$. In the third column of Table 1, we show the previous PQCD predictions presented in Ref. cpc37-093102 . As a comparison, we also list the central values of the form factors $f_{i}(0)$ obtained in some other different approaches: such as the BSW BSW , the NRQCD nrqcd , the LCSR epjc51-833 ; jhep1905-094 , RQM and CCQM msthods prd68-094020 ; prd97-054014 or from the lattice QCD lattice1 .

It is easy to see from the numerical values as given in Table 1 that (a) the PQCD predictions for $f_{0,+}(0)$, $V(0)$ and $A_{1}(0)$ agree very well with the corresponding lattice QCD results, and (b) the theoretical predictions from different approaches can also be rather different in values, by a factor of three for $f_{0,+}(0)$ for instance. Since the PQCD calculations for form factor are not reliable in the large $q^{\rm 2}$ region , we have to make an extrapolation for all relevant form factors from the lower $q^{\rm 2}$ region to the larger $q^{\rm 2}$ region. In this work we will make the extrapolation by using the following two different methods.

In the first method, we use our PQCD predictions for all relevant form factors $f_{i}(q^{2})$ at the sixteen points of $0\leq q^{2}\leq m^{2}_{\tau}$ as input, and then make the extrapolation from low $q^{2}$ region to the $q_{\rm max}^{2}$ by using the Bourrely-Caprini-Lellouch (BCL) parametrization bcl09 . Analogous to Ref. jhep1905-094 , we here also consider only the first two terms of the series in the parameter $z$:

[TABLE]

where $t=q^{2}$, $m_{R}$ are the masses of the low-laying $B_{c}$ resonance as listed in Table 2, and the parameters $t_{\pm}$ and $t_{0}$ 111We here use the same optimized value of $t_{0}$ as the one in Refs. bfw10 ; jhep1905-094 without further discussion. are as follows:

[TABLE]

where $m_{x}=m_{\eta_{c}}$ or $m_{J/\psi}$ for $B_{c}\to\eta_{c}$ and $J/\psi$ transition, respectively. In Table 2, we list the PQCD input $f_{i}(0)$, the masses $m_{R}$, the parameters $\alpha_{0}$ and $\alpha_{1}$ determined from the BCL fitting procedure for $B_{c}\to\eta_{c}$ and $B_{c}\to J/\psi$ form factors. The values of $m_{R}$ are taken from Ref. jhep1905-094 directly.

The second method is the “PQCD+Lattice” method, similar with what we did in Ref. sb60-2009 for the studies of $R(D^{*})$. As mentioned in the introduction, the authors in HPQCD Collaboration lattice1 ; lattice2 calculated the form factors $f_{0,+}(q^{2})$ for $B_{c}\to\eta_{c}$ transition, and $V(q^{2})$ and $A_{1}(q^{2})$ for $B_{c}\to J/\psi$ transition by using the lattice QCD method ( working directly at $m_{b}$ with an improved non-relativiatic QCD (NRQCD) effective theory formulism ) at $q^{2}=0$ and several other points of $q^{2}$. In order to improve the reliability of the extrapolation of $f_{i}(q^{2})$ to the larger $q^{2}$ region, we use currently available “Lattice” results at points $q^{2}=(5.44,8.72,10.07)$ GeV2 as given in Refs. lattice1 ; lattice2 ,

[TABLE]

as the lattice QCD input in the fitting process for the form factors $(f_{0,+}(q^{2}),V(q^{2}),A_{1}(q^{2}))$. In order to estimate the effects of possible uncertainties of the lattice QCD inputs, we here set a roughly five percent error ( $\pm 0.05$ ) to the four form factors in Eq. (30). For other two form factors $A_{0}(q^{2})$ and $A_{2}(q^{2})$, unfortunately, no lattice QCD results are available at present.

In Figs. 2 and 3 , we show the theoretical predictions for the $q^{2}$-dependence of the six relevant form factors for $B_{c}\to(\eta_{c},J/\psi)$ transitions, obtained by using the PQCD approach and the “PQCD+ Lattice” approach, respectively. In these two figures, the blue solid curves show the theoretical predictions for the $q^{2}$-dependence of $f_{0,+}(q^{2})$, $V(q^{2})$ and $A_{0,1,2}(q^{2})$ in the PQCD approach, while the red dashed curves show the four form factors $(f_{0,+}(q^{2}),V(q^{2}),A_{1}(q^{2}))$ obtained by using the ”PQCD+Lattice” approach. The band in these figures show the uncertainties of the corresponding theoretical predictions for the form factors. The four black dots symbols in Fig. 2 and 3 show the lattice QCD input in Eq. 30 used in the fitting procedure. One can see from the theoretical predictions as illustrated in Figs. 2 and 3 that the values and their $q^{2}$-dependence of the form factors obtained by using the two different methods agree very well with each other in the whole range of $q^{2}$.

In Fig. 4, we show the $q^{2}$-dependence of the theoretical predictions for the differential decay rates $d\Gamma/dq^{2}$ for the semileptonic decays $B_{c}\to(\eta_{c},J/\psi)l\bar{\nu}_{l}$ with $l=(\mu,\tau)$, where the blue solid curve and the red dashed ones show the $d\Gamma/dq^{2}$ in the PQCD approach and “PQCD+Lattice” method respectively. For the four considered $B_{c}\to(\eta_{c},J/\psi)(\mu^{-}\bar{\nu}_{\mu},\tau^{-}\bar{\nu}_{\tau})$ decays, the theoretical predictions for the differential decay rates from the PQCD and the “PQCD+Lattice” approach agree well within errors in the whole $q^{2}$ region. For $B_{c}\to J/\psi\mu^{-}\bar{\nu}_{\mu}$ decay, on the other hand, the difference between the central values is a little evident in the large $q^{2}$ region but remains small in size. We do wish the lattice results for the form factors $A_{0,2}(q^{2})$ become available soon and which will help us to improve our work.

From the formulae of the differential decay rates as given in Eqs. (15,22), it is straightforward to make the integration over the range of $m^{2}_{l}\leq q^{2}\leq(m^{2}_{B_{c}}-m_{x}^{2})$ with $x=(\eta_{c},J/\psi)$. The theoretical predictions (in unit of $10^{-3}$) for the branching ratios of the considered semileptonic decays are the following:

[TABLE]

[TABLE]

where the major theoretical errors come from the uncertainties of the input parameters $\beta_{\rm B_{c}}=1.0\pm 0.1$ GeV, $|V_{\rm cb}|=(42.2\pm 0.8)\times 10^{-3}$ and $m_{\rm c}=1.27\pm 0.03$ GeV.

In Table 3, we list the theoretical predictions (in unit of $10^{-3}$) for the branching ratios of the considered decays $B_{c}\to(\eta_{c},J/\psi)l^{-}\bar{\nu}_{l}$ with $l=(\mu,\tau)$, obtained in this paper by employing the PQCD and the “PQCD+Lattice” approaches. And as a comparison, we also show the results from the previous PQCD work cpc37-093102 , and from several different models or approaches wang09 ; epjc51-833 ; z-series ; jhep1905-094 . One can see that the difference between different theoretical predictions can be as large as a factor of two for the same decay mode. In Table 4, we show the theoretical predictions for the ratios $R_{\eta_{c}}$ and $R_{J/\psi}$ of the branching ratios for the considered semileptonic $B_{c}$ decays, as defined in Eq. (1) and evaluated in this paper. Some previous results as given in Refs. cpc37-093102 ; wang09 ; epjc51-833 ; z-series ; jhep1905-094 ; mi1 ; mi2 are also listed as comparison..

From the theoretical predictions for the branching ratios and the ratios $R_{\eta_{c},J/\psi}$ as listed in Eqs. (33-42) and Table 3 and 4, we find the following points:

• (1)

The theoretical predictions for the branching ratios of all considered $B_{c}\to(\eta_{c},J/\psi)l^{-}\bar{\nu}_{l}$ decays in both the PQCD and “PQCD+Lattice” approach agree well within errors ( around $30\%$ in magnitude). Numerically, the theoretical predictions for a fixed decay mode will become a little smaller by a degree of $(5-16)\%$ when the Lattice QCD results for the form factors $(f_{0,+},V,A_{1})$ are taken into account in the extrapolation of the relevant form factors to higher $q^{2}$ region.

• (2)

The theoretical predictions for the ratios $R_{\eta_{c}}$ and $R_{J/\psi}$ in both the PQCD and “PQCD+Lattice” approach agree very well, and have very small errors ( less than $5\%$ in magnitude) due to the strong cancellation between the errors of the theoretical predictions for branching ratios. Although the theoretical predictions for $R_{J/\psi}$ as listed in Table 4 in both the PQCD and “PQCD+Lattice” approaches are still smaller than the measured value $0.71\pm 0.24$ as reported by LHCb Collaboration lhcb-18a , but still agree with it because of the still large errors of the experimental measurements. We believe that the ratios $R_{\eta_{c}}$ and $R_{J/\psi}$ could be measured in high precision at the future LHCb experiment and can help us to test the theoretical models or approaches.

• (3)

Although the theoretical predictions for the decay rates from different methods or approaches can be rather different, even reaches a factor of two or three, the theoretical predictions for the ratios $R_{\eta_{c}}$ and $R_{J/\psi}$ from different works  cpc37-093102 ; wang09 ; nrqcd ; epjc51-833 ; z-series ; jhep1905-094 , however, agree very well with each other within $30\%$ of the central value.

For both kinds of the semileptonic decays $B\to D^{(*)}l^{-}\bar{\nu}_{l}$ and $B_{c}^{-}\to(\eta_{c},J/\psi)l^{-}\bar{\nu}_{l}$, their quark level weak decays are indeed the same charged current tree transitions: $b\to cl^{-}\bar{\nu}_{l}$ with $l=(e,\mu,\tau)$. The only difference between them is the spectator quark: one is the heavy charm quark, another is the light up or down quark. Consequently, it is reasonable to assume that the dynamics for these two kinds of semileptonic decays are similar in nature, we therefore can use similar method to study these two kinds of semileptonic decays.

For $B\to D^{(*)}\tau\bar{\nu}_{\tau}$ decays, besides the decay rates and the ratios $R(D^{(*)})$, the longitudinal polarization $P_{\tau}(D^{(*)})$ of the tau lepton and the fraction of $D^{*}$ longitudinal polarization $F_{L}^{D^{*}}$ are also the additional physical observables and sensitive to some kinds of new physics ptau1 ; ptau2 ; ptau3 ; ptau4 . The first measurement of $P_{\tau}(D^{*})$ and $F_{L}^{D^{*}}$ have been reported very recently by Belle Collaboration prl118-801 ; prd97-012004 ; 1903tau :

[TABLE]

They are compatible with the SM predictions : $P_{\tau}(D^{*})=-0.497\pm 0.013$ for $\bar{B}\to D^{*}\tau^{-}\bar{\nu}_{\tau}$ ptau2 ; ptau4 , and $F_{L}(D^{*})=0.441\pm 0.006$ huang18 or $0.457\pm 0.010$ bhatt18 .

For $B_{c}\to(\eta_{c},J/\psi)\tau\bar{\nu}_{\tau}$ decays, we consider the relevant longitudinal polarizations $P_{\tau}(\eta_{c})$ and $P_{\tau}(J/\psi)$, and define them in the same way as the one for $P_{\tau}(D^{(*)})$ in Refs. ptau1 ; ptau2 ; ptau3 ; ptau4 :

[TABLE]

where $\Gamma^{\pm}(X)$ denotes the decay rates of $B_{c}\to X\tau\bar{\nu}_{\tau}$ with a $\tau$ lepton helicity $\pm 1/2$. Following Ref. ptau3 , the explicit expressions of $d\Gamma^{\pm}/dq^{2}$ for the considered semileptonic $B_{c}$ decays here can be written in the following form:

[TABLE]

with the functions $H_{i}(q^{2})$

[TABLE]

where $0\leq q^{2}\leq\left(m_{\rm B_{c}}-m_{X}\right)^{2}$ and $\lambda(q^{2})=\left(m_{\rm B_{c}}^{2}+m_{X}^{2}-q^{2}\right)^{2}-4m_{\rm B_{c}}^{2}m_{X}^{2}$ with $X=(\eta_{c},J/\psi)$, and the explicit expressions of the form factors $f_{+,0}(q^{2})$, $V(q^{2})$ and $A_{0,1,2}(q^{2})$ in PQCD approach have been given in Eqs. (16,23-26).

After making the proper integrations over $q^{2}$, we found the theoretical predictions for the longitudinal polarization $P_{\tau}$ for the considered semileptonic $B_{c}\to(\eta_{c},J/\psi)l^{-}\bar{\nu}_{l}$ decays :

[TABLE]

in the PQCD approach, and

[TABLE]

in the “ PQCD + Lattice” approach. The dominant errors come from the uncertainty of $\beta_{B_{c}}$ and $m_{c}$. Following the new measurement of the longitudinal polarization $P_{\tau}(D^{*})$ for $B\to D^{*}\tau\nu_{\tau}$ at Belle prl118-801 , we believe that the similar longitudinal polarization $P_{\tau}(\eta_{c})$ and $P_{\tau}(J/\psi)$ could be measured in the near future LHCb experiment when enough amount of $B_{c}$ decay events are collected.

5. Summary

In this paper, we studied the semileptonic decays $B_{c}\to(\eta_{c},J/\psi)l\bar{\nu}$ by employing the pQCD factorization approach with the usage of some new inputs: (a) we used the newly defined DAs of the $B_{c}$ meson instead of the old delta-function; (b) we used the new BCL parametrization for the extrapolation of the form factors from low $q^{2}$ region to $q^{2}_{max}$; and (c) we take into account currently known lattice QCD results of form factors at several points as the new input in our fitting procedure. We calculated the form factors $f_{\rm 0,+}(q^{\rm 2})$, $V(q^{\rm 2})$ and $A_{\rm 0,1,2}(q^{\rm 2})$ of the $B_{c}\to(\eta_{c},J/\psi)$ transitions, presented the predictions for the branching ratios ${\cal B}(B_{c}\to(\eta_{c},J/\psi)l\bar{\nu}_{l})$ , the ratios $R_{\eta_{c}}$ and $R_{J/\psi}$ of the branching ratios, and the longitudinal polarizations $P_{\tau}(\eta_{c})$ and $P_{\tau}(J/\psi)$ of the final $\tau$ lepton.

From the numerical calculations and phenomenological analysis we found the following points:

1. (1)

The theoretical predictions for the branching ratios of $B_{c}\to(\eta_{c},J/\psi)l\bar{\nu}$ decays from both PQCD and “PQCD+Lattice” approach agree very well with each other, a small decrease by about $(5-16)\%$ will be produced when the lattice QCD input for the form factors $(f_{0,+}(8.72),V(5.44),A_{1}(10.07))$ are taken into account in the extrapolation of the relevant form factors to higher $q^{2}$ region. 2. (2)

The theoretical predictions for the ratios $R_{\eta_{c}}$ and $R_{J/\psi}$ are the following:

[TABLE]

The central values of above theoretical predictions for $R_{J/\psi}$ are smaller than the measured one as shown in Eq. (2), but still agree with it within the errors. 3. (3)

The theoretical predictions for the longitudinal polarization $P(\tau)$ of the tau lepton are the following:

[TABLE]

These predictions could be tested in the near future LHCb experiments.

Acknowledgements.

We wish to thank Wen-Fei Wang and Ying-Ying Fan for valuable discussions. This work was supported by the National Natural Science Foundation of China under Grant No. 11775117 and 11235005.

Appendix A Relevant functions

In this appendix, we present the explicit expressions for some functions appeared in the previous sections. The hard functions $h_{1}$ and $h_{2}$ appeared in Eq. (19) can be written as

[TABLE]

with

[TABLE]

where $r_{q}=m_{q}/m_{B_{c}}$ with $q=(c,b)$, $r=m_{\eta_{c}}/m_{B_{c}}$ ( $r=m_{J/\psi}/m_{B_{c}}$) when it appears in the form factors $f_{+,0}(q^{2})$ ( $V(q^{2})$ and $A_{0,1,2}(q^{2})$). $\eta$ and $\eta^{+}$ are defined in Eq. (7). The functions $K_{0}$ and $I_{0}$ in Eq. (61) are the modified Bessel functions. The term inside the square-root symbol of $\alpha_{(1,2)}$ and $\beta_{(1,2)}$ may be positive or negative. When such term is negative, the argument of the functions $K_{0}$ and $I_{0}$ becomes imaginary, and the associated Bessel functions $K_{0}$ and $I_{0}$ will consequently transform in the following way

[TABLE]

where the functions $J_{0}(x)$ and $Y_{0}(x)$ can be written in the following formsf8e 222One can find the expression of $J_{0}(x)$ and $Y_{0}(x)$ in Sec.8.411 and 8.415 of Ref. sf8e .

[TABLE]

The factor $\exp[-S_{ab}(t)]$ in Eq. (19) contains the Sudakov logarithmic corrections and the renormalization group evolution effects of both the wave functions and the hard scattering amplitude with $S_{ab}(t)=S_{B_{c}}(t)+S_{X}(t)$ as given in Ref. Bc-am

[TABLE]

where $\eta^{+}$ is defined in Eq. (7), while the hard scale $t$ and the quark anomalous dimension $\gamma_{q}=-\alpha_{s}/\pi$, which governs the aforementioned renormalization group evolution. The Sudakov exponent $s_{c}(Q,b)$ for an energetic charm quark is expressed Bc-am as the difference

[TABLE]

The hard scales $t_{i}$ are chosen as the largest scale of the virtuality of the internal particles in the hard $b$-quark decay diagram,

[TABLE]