#  $B_{c}$ meson exclusive decays to a $P$-wave charmonium and a pion at NLO
accuracy

Zi-Qiang Chen1,2222<EMAIL_ADDRESS>and Cong-Feng
<EMAIL_ADDRESS>corresponding author
1 School of Physics, University of Chinese Academy of Sciences, Beijing
100049, China
2 CAS Center for Excellence in Particle Physics, Beijing 10049, China

###### Abstract

In this paper, we calculate the next-to-leading order (NLO) quantum
chromodynamics (QCD) corrections to the exclusive processes
$B_{c}^{+}\to\chi_{cJ}(h_{c})\pi^{+}$ in the framework of the nonrelativistic
QCD (NRQCD) factorization formalism. The results show that NLO QCD corrections
markedly enhance the branching ratios with $K$ factors of about 2.5. In
combination with the study of $B_{c}^{+}\to J/\psi\pi^{+}$, we find that the
NLO NRQCD prediction for the ratio of branching fractions
$\frac{\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})}{\mathcal{B}(B_{c}^{+}\to
J/\psi\pi^{+})}$ is then compatible with the experimental measurement.

##  I Introduction

In the Standard Model (SM) of particle physics, the $B_{c}$ meson family is
unique as its states are composed of two different heavy quarks, beauty ($b$)
and charm ($c$). Unlike the charmonium and bottomonium states, the ground
state of $B_{c}$ meson family decays only via weak interactions, where the
main decay modes can be classified as: i) $\bar{b}$ decay with $c$ as a
spectator, ii) $c$ decay with $\bar{b}$ as a spectator, and iii) $\bar{b}c$
annihilate to a virtual $W$ boson. Study on these decay processes can deepen
our understanding of both strong and weak interactions, and provides
opportunities to search for new physics beyond the SM.

Experimentally, the $B_{c}$ meson decay to charmonium processes are essential
for the reconstruction of $B_{c}$ meson signals. After the discovery of ground
state $B_{c}$ by the CDF Collaboration in 1998 Abe:1998wi , there have been
continuous measurements on its lifetime Abulencia:2006zu ; Abazov:2008rba  and
mass Aaltonen:2007gv ; Abazov:2008kv ; Aaij:2012dd ; Aaij:2020jrx  via the
semileptonic decay $B_{c}^{+}\to J/\psi l^{+}\nu_{l}$ and the exclusive two-
body decay $B_{c}^{+}\to J/\psi\pi^{+}$. In 2014, the ATLAS Collaboration
reported an excited state with mass of $6842\pm 4\pm 5$ MeV Aad:2014laa ,
which is regarded as a candidate of $B_{c}(2S)$ state. This observation was
confirmed by the CMS and LHCb Collaborations in 2019, where the
$B_{c}^{+}(2S)$ and $B_{c}^{*+}(2S)$ states were reconstructed through the
decays $B_{c}^{+}(2S)\to B_{c}^{+}\pi^{+}\pi^{-}$ and $B_{c}^{*+}(2S)\to
B_{c}^{*+}(\to B_{c}^{+}\gamma)\pi^{+}\pi^{-}$ Sirunyan:2019osb ; Aaij:2019ldo
, respectively. Note, in these works, the intermediate $B_{c}^{+}$ was also
reconstructed through $B_{c}^{+}\to J/\psi\pi^{+}$.

Besides $B_{c}^{+}$ decay to $S$-wave charmonium, its transition to $P$-wave
charmonium is also interesting for three reasons:

  1. 1.

The branching ratio $\mathcal{B}(\chi_{c2}\to J/\psi\gamma)=19.0\pm 0.5\%$
Zyla:2020zbs  is sizable111Although $\mathcal{B}(\chi_{c1}\to
J/\psi\gamma)=34.3\pm 1.0\%$ is also sizable, theoretical prediction indicates
that $\mathcal{B}(B_{c}^{+}\to\chi_{c1}\pi^{+})$ is insignificant compared to
$\mathcal{B}(B_{c}^{+}\to J/\psi\pi^{+})$.. Hence the cascade decay
$B_{c}^{+}\to\chi_{c2}(\to J/\psi\gamma)\pi^{+}$ may contribute a substantial
background for the $B_{c}\to J/\psi\pi^{+}$ process.

  2. 2.

The $\bar{b}c$ annihilation decay of $B_{c}^{+}$ is an interesting topic in
$B$ physics. Any observation of significant enhancement over the SM prediction
could indicate the presence of new physics effects. The decay of $B_{c}^{+}$
to three light charged hardrons, like $B_{c}^{+}\to K^{+}K^{-}\pi^{+}$,
provides a good way to study this issue. The contributions from intermediate
states, like $B_{c}^{+}\to\chi_{c0}(\to K^{+}K^{-})\pi^{+}$ should be
subtracted to obtain the annihilation contribution Aaij:2016xas .

  3. 3.

Although the exact values of the branching ratios for $B_{c}^{+}\to
J/\psi\pi^{+}$ and $B_{c}^{+}\to\chi_{cJ}(h_{c})\pi^{+}$ have not been
measured yet, the ratio
$\frac{\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})}{\mathcal{B}(B_{c}^{+}\to
J/\psi\pi^{+})}$ is accessible by combining the results of Refs. Zyla:2020zbs
; Aaij:2016xas ; Aaij:2014ija . Phenomenological study on this issue provides
an opportunity to test the nonrelativistic QCD (NRQCD) effective theory.

Theoretically, the exclusive two-body decay of $B_{c}^{+}$ into $P$-wave
charmonium and a light meson has been studied in various approaches: the NRQCD
approach Kiselev:2001zb ; Zhu:2017lwi , the perturbative QCD approach
Rui:2017pre , the Bethe-Salpeter equation Chang:2001pm ; Wang:2011jt , the
nonrelativistic quark model Hernandez:2006gt , and the relativistic quark
model Ivanov:2006ni ; Ebert:2010zu . We notice that their predictions on the
branching ratios are generally incompatible with each other, and their
analyses are limited to the leading-order (LO) accuracy. Considering the fact
that the higher-order QCD corrections in quarkonium energy regime are normally
significant Qiao:2012hp , in this work we calculate the next-to-leading order
(NLO) QCD corrections to $B_{c}^{+}\to\chi_{cJ}(h_{c})\pi^{+}$ processes in
the framework of NRQCD factorization formalism Bodwin:1994jh .

The rest of the paper is organized as follows. In Sec. II, we present the
primary formulas employed in the calculation. In Sec. III, we elucidate some
technical details for the analytical calculation. In Sec. IV, the numerical
evaluation for concerned processes is performed at NLO QCD accuracy. The last
section is reserved for summary and conclusions.

##  II Formalism

###  II.1 Effective weak Hamiltonian

In the SM, $B^{+}_{c}\to\chi_{cJ}(h_{c})\pi^{+}$ occur through $W$-mediated
charge-current processes. However, since $m_{W}\gg m_{b,c}$, large logarithm
terms will arise in higher-order QCD corrections. Thus, the renormalization-
group (RG) improved effective weak Hamiltonian method is usually employed in
the calculation. The interaction term is

$\mathcal{H}_{\rm
eff}=\frac{G_{F}}{\sqrt{2}}V_{ud}V^{*}_{cb}\left(C_{1}(\mu)\mathcal{Q}_{1}(\mu)+C_{2}(\mu)\mathcal{Q}_{2}(\mu)\right),$
(1)

where $G_{F}$ is the Fermi constant, $V_{ud}$ and $V_{cb}$ are the Cabibbo-
Kobayashi-Maskawa (CKM) matrix elements; $C_{1,2}(\mu)$ are the perturbatively
calculable Wilson coefficients, $\mathcal{Q}_{1,2}(\mu)$ are the local four-
quark operators, which take the form

$\displaystyle\mathcal{Q}_{1}(\mu)=\bar{b}_{i}\gamma^{\mu}(1-\gamma_{5})c_{i}\otimes\bar{u}_{j}\gamma_{\mu}(1-\gamma_{5})d_{j},$
(2)
$\displaystyle\mathcal{Q}_{2}(\mu)=\bar{b}_{i}\gamma^{\mu}(1-\gamma_{5})c_{j}\otimes\bar{u}_{j}\gamma_{\mu}(1-\gamma_{5})d_{i}.$
(3)

Here $i$, $j$ are color indices and the summation convention for repeated
indices is understood. In contrast to conventional operators
$\mathcal{Q}_{1,2}$, we will adopt another basis

$\displaystyle\mathcal{Q}_{0}(\mu)=\bar{b}_{i}\gamma^{\mu}(1-\gamma_{5})c_{i}\otimes\bar{u}_{j}\gamma_{\mu}(1-\gamma_{5})d_{j},$
(4)
$\displaystyle\mathcal{Q}_{8}(\mu)=\bar{b}_{i}T^{a}_{ij}\gamma^{\mu}(1-\gamma_{5})c_{j}\otimes\bar{u}_{k}T^{a}_{kl}\gamma_{\mu}(1-\gamma_{5})d_{l},$
(5)

where $T^{a}$ is the generator of $SU(3)$ fundamental representation. By
applying the Fierz rearrangement relation

$T^{A}_{ij}T^{A}_{kl}=-\frac{1}{6}\delta_{ij}\delta_{kl}+\frac{1}{2}\delta_{il}\delta_{kj},$
(6)

we obtain

$\mathcal{Q}_{0}=\mathcal{Q}_{1},\quad\mathcal{Q}_{8}=-\frac{1}{6}\mathcal{Q}_{1}+\frac{1}{2}\mathcal{Q}_{2},$
(7)

and

$C_{0}=C_{1}+\frac{1}{3}C_{2},\quad C_{8}=2C_{2}.$ (8)

The Wilson coefficients $C_{0,8}$ can be obtained by solving the RG equation.
Under the leading logarithmic approximation,

$\displaystyle
C_{0}(\mu)=\frac{2}{3}\bigg{[}\frac{\alpha_{s}(m_{W})}{\alpha_{s}(\mu)}\bigg{]}^{\frac{\gamma_{+}}{2\beta_{0}}}+\frac{1}{3}\bigg{[}\frac{\alpha_{s}(m_{W})}{\alpha_{s}(\mu)}\bigg{]}^{\frac{\gamma_{-}}{2\beta_{0}}},$
(9) $\displaystyle
C_{8}(\mu)=\bigg{[}\frac{\alpha_{s}(m_{W})}{\alpha_{s}(\mu)}\bigg{]}^{\frac{\gamma_{+}}{2\beta_{0}}}-\bigg{[}\frac{\alpha_{s}(m_{W})}{\alpha_{s}(\mu)}\bigg{]}^{\frac{\gamma_{-}}{2\beta_{0}}},$
(10)

with

$\gamma_{\pm}=\pm 6\frac{N_{c}\mp 1}{N_{c}}.$ (11)

Here, $\alpha_{s}$ is the running coupling constant of QCD;
$\beta_{0}=(11/3)C_{A}-(4/3)T_{f}n_{f}$ is the one-loop coefficient of QCD
beta function, $n_{f}$ is the number of active quarks, and $C_{A}=N_{c}=3$,
$T_{f}=1/2$ are normal color factors.

###  II.2 Projection operators and decay amplitudes

According to the NRQCD factorization formalism Bodwin:1994jh  and the
factorization formalism for nonleptonic $B$ meson decay Beneke:2000ry , the
decay amplitude of $B_{c}^{+}\to\chi_{cJ}(h_{c})\pi^{+}$ is conjectured to be
factorized as

$\mathcal{M}\left(B_{c}^{+}\to\chi_{c}(h_{c})\pi^{+}\right)\simeq\frac{if_{\pi}}{8\pi}\frac{|R_{cc}^{\prime}(0)|}{\sqrt{8m_{c}^{3}}}\frac{|R_{bc}(0)|}{\sqrt{m_{b}+m_{c}}}\int_{0}^{1}dxT(x,\mu)\phi_{\pi}(x,\mu),$
(12)

where, $f_{\pi}$ is the decay constant of pion; $R_{bc}(0)$ is the radial wave
function at the origin for $B_{c}^{+}$, $R_{cc}^{\prime}(0)$ is the derivative
of radial wave function for $\chi_{cJ}(h_{c})$, $T(x,\mu)$ is the
perturbatively calculable hard kernel, and $\phi_{\pi}(x,\mu)$ is the leading-
twist light-cone distribution amplitude (LCDA) of pion.

With the decay amplitude $\mathcal{M}$, the branching ratio can be obtained
through

$\mathcal{B}=\frac{\tau_{B_{c}}}{16\pi}\frac{(m_{b}+m_{c})^{2}-4m_{c}^{2}}{(m_{b}+m_{c})^{3}}|\mathcal{M}|^{2},$
(13)

where $\tau_{B_{c}}$ is the mean life of $B_{c}^{+}$ meson. At the NLO QCD
accuracy, $|\mathcal{M}|^{2}\simeq|\mathcal{M}_{\rm born}|^{2}+2{\rm
Re(\mathcal{M}_{\rm loop}\mathcal{M}_{\rm born}^{*})}$, where
$\mathcal{M}_{\rm born}$ is the Born amplitude at $\mathcal{O}(\alpha_{s})$,
and $\mathcal{M}_{\rm loop}$ is the one-loop correction at
$\mathcal{O}(\alpha_{s}^{2})$.

The hard kernel $T(x,\mu)$ can be computed by using the covariant projection
operator method. The spin and color projection operators used in our
calculation are

$\displaystyle\Pi_{bc}^{0}=\frac{1}{2}(\not{k}_{1}+m_{b}+m_{c})\gamma_{5}\otimes\frac{1_{c}}{\sqrt{N_{c}}},$
(14)
$\displaystyle\Pi_{cc}^{\mu}=\Big{(}\frac{\not{k}_{2}}{2}-\not{q}-m_{c}\Big{)}\gamma^{\mu}\Big{(}\frac{\not{k}_{2}}{2}+\not{q}+m_{c}\Big{)}\otimes\frac{1_{c}}{\sqrt{N_{c}}},$
(15)
$\displaystyle\Pi_{cc}^{0}=\Big{(}\frac{\not{k}_{2}}{2}-\not{q}-m_{c}\Big{)}\gamma_{5}\Big{(}\frac{\not{k}_{2}}{2}+\not{q}+m_{c}\Big{)}\otimes\frac{1_{c}}{\sqrt{N_{c}}},$
(16)
$\displaystyle\Pi_{ud}^{0}=\frac{1}{2}\not{k}_{3}\gamma_{5}\otimes\frac{1_{c}}{\sqrt{N_{c}}},$
(17)

where $k_{1}$ denotes the momentum of $B_{c}^{+}$, $k_{2}$ the momentum of
$\chi_{cJ}(h_{c})$, $k_{3}$ the momentum of $\pi^{+}$, and $2q$ is the
relative momentum between the $c\bar{c}$ pair. Then the hard kernels can be
expressed as

$\displaystyle
T(\chi_{c0})=\frac{1}{\sqrt{3}}I_{\alpha\beta}\frac{d}{dq_{\beta}}{\rm
Tr}\big{(}\Pi_{cc}^{\alpha}\mathcal{A}\big{)}\bigg{|}_{q\to 0},$ (18)
$\displaystyle
T(\chi_{c1}(\lambda))=\frac{i}{2\sqrt{2}m_{c}}\varepsilon_{\alpha\beta\mu\nu}k_{1}^{\mu}\epsilon^{\nu}(\lambda)\frac{d}{dq_{\beta}}{\rm
Tr}\big{(}\Pi_{cc}^{\alpha}\mathcal{A}\big{)}\bigg{|}_{q\to 0},$ (19)
$\displaystyle
T(\chi_{c2}(\lambda))=\bigg{[}\frac{1}{2}(I_{\alpha\mu}I_{\beta\nu}+I_{\alpha\nu}I_{\beta\mu})-\frac{1}{3}I_{\alpha\beta}I_{\mu\nu}\bigg{]}\epsilon^{\mu\nu}(\lambda)\frac{d}{dq_{\beta}}{\rm
Tr}\big{(}\Pi_{cc}^{\alpha}\mathcal{A}\big{)}\bigg{|}_{q\to 0},$ (20)
$\displaystyle
T(h_{c}(\lambda))=\epsilon_{\beta}(\lambda)\frac{d}{dq_{\beta}}{\rm
Tr}\big{(}\Pi_{cc}^{0}\mathcal{A}\big{)}\bigg{|}_{q\to 0}.$ (21)

Here, $T(H(\lambda))$ denotes the hard kernel referring to $B_{c}^{+}\to
H(\lambda)\pi^{+}$ process, with $\lambda$ labeling the helicity state of $H$,
$\epsilon(\lambda)$ labeling the corresponding polarization vector;
$\mathcal{A}$ denotes the standard amplitude for partonic process, amputated
of quark spinors. Note, in Eqs. (18)$\sim$(21), the projectors $\Pi_{bc}^{0}$
and $\Pi_{ud}^{0}$ are suppressed for brevity. The tensor $I_{\alpha\beta}$ in
Eqs. (18) and (20) is

$I_{\alpha\beta}=-g_{\alpha\beta}+\frac{k_{2\alpha}k_{2\beta}}{4m_{c}^{2}}.$
(22)

Up to NLO, the hard kernel $T(x,\mu)$ can be written as

$\displaystyle T(x,\mu)=\frac{G_{F}}{\sqrt{2}}$ $\displaystyle
V_{ud}V^{*}_{cb}\frac{\alpha_{s}(\mu)}{\pi}\bigg{\\{}C_{0}(\mu)T_{0}^{(0)}(x)+C_{8}(\mu)T_{8}^{(0)}(x)+\frac{\alpha_{s}(\mu)}{\pi}\bigg{[}C_{0}(\mu)T_{0}^{(1)}(x)$
$\displaystyle+C_{8}(\mu)T_{8}^{(1)}(x)+\Big{(}C_{0}(\mu)\tilde{T}_{0}^{(1)}(x)+C_{8}(\mu)\tilde{T}_{8}^{(1)}(x)\Big{)}{\rm
ln}\frac{\mu^{2}}{(m_{b}+m_{c})^{2}}\bigg{]}\bigg{\\}},$ (23)

where $T^{(j)}_{i}$ and $\tilde{T}^{(j)}_{i}$ are dimensionless and can only
depend on $x$ and $m_{c}/m_{b}$. Note, here and throughout, the
renormalization scale is set equal to the factorization scale.

##  III Analytical calculation

###  III.1 Kinematics and LO calculation

Figure 1: Tree-level Feynman diagrams for
$B_{c}^{+}\to\chi_{c}(h_{c})\pi^{+}$. The vertex “$\otimes\otimes$” denotes
the insertion of effective weak interaction.

The tree-level Feynman diagrams for partonic processes are shown in Fig. 1.
Therein, (a) and (b) contribute to $T^{(0)}_{0}$, (c) and (d) contribute to
$T^{(0)}_{8}$. The momenta of incoming and outgoing particles are denoted as

$[\bar{b}c](k_{1})\to[c\bar{c}](k_{2},\epsilon)+[u\bar{d}](k_{3}).$ (24)

Here, the initial and final state particles are all on their mass shells:
$k_{1}^{2}=(m_{b}+m_{c})^{2}$, $k_{2}^{2}=4m_{c}^{2}$, $k_{3}^{2}=0$. By
introducing the orthonormal four-vector base: $n_{0}=(1,0,0,0)$,
$n_{1}=(0,1,0,0)$, $n_{2}=(0,0,1,0)$, $n_{3}=(0,0,0,1)$, the momenta of
external particles in the initial state rest frame can be assigned as

$\displaystyle k_{1}^{\mu}=(m_{b}+m_{c})n_{0}^{\mu},$ $\displaystyle
k_{2}^{\mu}=\frac{m_{b}+m_{c}}{2}[(4r^{2}-8r+5)n_{0}^{\mu}-(3-2r)(2r-1)n_{3}^{\mu}],$
$\displaystyle
k_{3}^{\mu}=\frac{m_{b}+m_{c}}{2}(3-2r)(2r-1)(n_{0}^{\mu}+n_{3}^{\mu}).$ (25)

Here we introduce $r=\frac{m_{b}}{m_{b}+m_{c}}$. For convenience, we also
introduce

$\displaystyle e_{1}^{\mu}=n_{1}^{\mu},\quad e_{2}^{\mu}=n_{2}^{\mu},$
$\displaystyle
e_{3}^{\mu}=\frac{1}{4(1-r)}[(3-2r)(2r-1)n_{0}^{\mu}+(4r^{2}-8r+5)n_{3}^{\mu}].$
(26)

Then the polarization vectors for $\chi_{c1}$ and $h_{c}$ can be chosen as

$\epsilon^{\mu}(1)=e_{1}^{\mu},\quad\epsilon^{\mu}(2)=e_{2}^{\mu},\quad\epsilon^{\mu}(3)=e_{3}^{\mu}.$
(27)

The polarization vectors for $\chi_{c2}$ can be constructed as

$\displaystyle\epsilon^{\mu\nu}(1)=\frac{1}{\sqrt{2}}(e_{1}^{\mu}e_{2}^{\nu}+e_{2}^{\mu}e_{1}^{\nu}),\quad\epsilon^{\mu\nu}(2)=\frac{1}{\sqrt{2}}(e_{1}^{\mu}e_{1}^{\nu}-e_{2}^{\mu}e_{2}^{\nu}),$
$\displaystyle\epsilon^{\mu\nu}(3)=\frac{1}{\sqrt{2}}(e_{1}^{\mu}e_{3}^{\nu}+e_{3}^{\mu}e_{1}^{\nu}),\quad\epsilon^{\mu\nu}(4)=\frac{1}{\sqrt{2}}(e_{2}^{\mu}e_{3}^{\nu}+e_{3}^{\mu}e_{2}^{\nu}),$
$\displaystyle\epsilon^{\mu\nu}(5)=\frac{1}{\sqrt{6}}(e_{1}^{\mu}e_{1}^{\nu}+e_{2}^{\mu}e_{2}^{\nu}-2e_{3}^{\mu}e_{3}^{\nu}).$
(28)

In fact, according to the helicity conservation rule Brodsky:1981kj , the only
nonzero helicity amplitudes for $\chi_{c1}$, $\chi_{c2}$, and $h_{c}$ channels
are $T(\chi_{c1}(3))$, $T(\chi_{c2}(5))$, and $T(h_{c}(3))$ respectively.

The tree-level calculation is straightforward, and the results are pretty
simple:

$\displaystyle
T^{(0)}_{0}(\chi_{c0})=-\frac{256\pi^{2}(2r-3)(4r^{2}-4r+3)}{(r-1)(2r-1)^{4}},$
(29) $\displaystyle
T^{(0)}_{8}(\chi_{c0})=\frac{64\pi^{2}}{9(2r-1)^{2}}\bigg{[}\frac{8r^{2}-14r+7}{(2r-1)(r-1)(x-x_{1}+i\varepsilon)}+\frac{4(2r^{2}+3r-3)}{(2r-1)(x-x_{2}-i\varepsilon)}$
$\displaystyle\quad\quad\quad\quad+\frac{2(r-1)}{(2r-3)(x-x_{1}+i\varepsilon)^{2}}+\frac{4(r-1)^{2}}{(2r-3)(x-x_{2}-i\varepsilon)^{2}}\bigg{]};$
(30) $\displaystyle
T^{(0)}_{0}(\chi_{c1})=-\frac{256\sqrt{6}\pi^{2}(2r-3)}{3(r-1)(2r-1)^{2}},$
(31) $\displaystyle
T^{(0)}_{8}(\chi_{c1})=\frac{64\sqrt{6}\pi^{2}}{9(2r-1)^{3}}\bigg{[}\frac{4r-3}{(r-1)(x-x_{1}+i\varepsilon)}-\frac{2(4r^{2}-6r+3)}{x-x_{2}-i\varepsilon}\bigg{]};$
(32) $\displaystyle
T^{(0)}_{0}(\chi_{c2})=-\frac{256\sqrt{2}\pi^{2}(2r-3)^{2}}{(r-1)(2r-1)^{4}},$
(33) $\displaystyle
T^{(0)}_{8}(\chi_{c2})=\frac{128\sqrt{2}\pi^{2}}{9(2r-1)^{2}}\bigg{[}\frac{4r^{2}-10r+5}{2(2r-1)(r-1)(x-x_{1}+i\varepsilon)}-\frac{2(2r^{2}-6r+3)}{(2r-1)(x-x_{2}-i\varepsilon)}$
$\displaystyle\quad\quad\quad\quad-\frac{r-1}{(2r-3)(x-x_{1}+i\varepsilon)^{2}}-\frac{2(r-1)^{2}}{(2r-3)(x-x_{2}-i\varepsilon)^{2}}\bigg{]};$
(34) $\displaystyle
T^{(0)}_{0}(h_{c})=-\frac{512\sqrt{3}\pi^{2}(2r-3)(4r^{3}-8r^{2}+7r-5)}{(r-1)(2r-1)^{4}},$
(35) $\displaystyle
T^{(0)}_{8}(h_{c})=\frac{128\sqrt{3}\pi^{2}}{9(2r-1)^{2}}\bigg{[}-\frac{1}{2(r-1)(x-x_{1}+i\varepsilon)}+\frac{2(r-1)}{x-x_{2}-i\varepsilon}$
$\displaystyle\quad\quad\quad\quad-\frac{r-1}{(2r-3)(x-x_{1}+i\varepsilon)^{2}}+\frac{2(r-1)^{2}}{(2r-3)(x-x_{2}-i\varepsilon)^{2}}\bigg{]}.$
(36)

Here, $x$ is the momentum fraction assigned to the $u$ quark, $x_{1}$ and
$x_{2}$ are defined as

$x_{1}=\frac{1-r}{3-2r},\quad\quad x_{2}=\frac{2-r}{3-2r}.$ (37)

For $m_{b},m_{c}>0$, we have $0<x_{1},x_{2}<1$.

The $\frac{1}{x-x_{1}+i\varepsilon}$ and $\frac{1}{x-x_{2}-i\varepsilon}$
terms in Eqs. (30), (32), (34), and (36) arise from the light quark
propagators of Fig. 1(c) and (d) (the red lines). The
$\frac{1}{(x-x_{1}+i\varepsilon)^{2}}$ and
$\frac{1}{(x+x_{2}-i\varepsilon)^{2}}$ terms arise due to the derivative
relative to $q$, and hence will absent in $B_{c}$ decay to $S$-wave charmonium
process. According to Eq. (12), the hard kernel $T(x,\mu)$ needs to be
convoluted with LCDA $\phi_{\pi}(x,\mu)$ to obtain the decay amplitude. The
integral interval $x\in[0,1]$ include the poles $x_{1}$ and $x_{2}$, while the
singularities are tamed by the $i\varepsilon$ prescription

$\displaystyle\int_{0}^{1}dx\frac{1}{x-x_{i}\pm
i\varepsilon}=\ln\frac{1-x_{i}}{x_{i}}\mp i\pi,\quad{\rm for}\ 0<x_{i}<1,$
(38) $\displaystyle\int_{0}^{1}dx\frac{1}{(x-x_{i}\pm
i\varepsilon)^{2}}=-\frac{1}{x_{i}(1-x_{i})},\quad{\rm for}\ 0<x_{i}<1.$ (39)

The above equations can be obtained by deforming the integral paths away from
poles. However, at the heavy quark limit $r\to 1$, the poles approach to the
end points. Considering the $x(1-x)$ factor in $\phi_{\pi}(x,\mu)$, we have

$\int_{0}^{1}dxT(x,\mu)\phi_{\pi}(x,\mu)\overset{r\to
1}{\sim}A\int_{0}^{1}dx\frac{1}{x}+B\int_{0}^{1}dx\frac{1}{1-x},$ (40)

which is divergent and indicates a potential violation of the factorization
formalism at the tree-level.

###  III.2 NLO corrections

The typical one-loop Feynman diagrams are shown in Fig. 2. Therein,
(a)$\sim$(c) contribute to $T^{(1)}_{0}$; (e)$\sim$(h) contribute to
$T^{(1)}_{8}$; (d) contributes to both $T^{(1)}_{0}$ and $T^{(1)}_{8}$. Note,
according to the Ward-Takahashi identity Ward:1950xp ; Takahashi:1957xn , the
contributions of type (c) diagrams are canceled by each other, which is
verified by our explicit calculation.

Figure 2: Typical one-loop Feynman diagrams for
$B_{c}^{+}\to\chi_{c}(h_{c})\pi^{+}$. The vertex “$\otimes\otimes$” denotes
the insertion of effective weak interaction.

In the computation of one-loop amplitudes, dimensional regularization with
$D=4-2\epsilon$ is employed to regularize the ultraviolet (UV) and infrared
(IR) singularities. Explicitly, in our calculation, the momenta and
polarization vectors of external particles are kept in four-dimensions, while
other quantities are continued to $D$-dimensions. The method proposed in Refs.
Kreimer:1989ke ; Korner:1991sx  is used to deal with the $D$-dimensional
$\gamma_{5}$ trace.

The UV singularities are removed by the renormalization procedure. The
standard renormalization constants include $Z_{2}$, $Z_{m}$, $Z_{l}$, and
$Z_{g}$, corresponding to heavy quark field, heavy quark mass, light quark
field, and strong coupling constant, respectively. We define $Z_{2}$, $Z_{m}$
and $Z_{l}$ in the on-shell (OS) scheme, $Z_{g}$ in the modified minimal-
subtraction ($\overline{\rm MS}$) scheme. The corresponding counterterms are

$\displaystyle\delta Z_{2}^{\rm
OS}=-C_{F}\frac{\alpha_{s}}{4\pi}\left[\frac{1}{\epsilon_{\rm
UV}}+\frac{2}{\epsilon_{\rm
IR}}-3\gamma_{E}+3\ln\frac{4\pi\mu^{2}}{m^{2}}+4\right],$ (41)
$\displaystyle\delta Z_{m}^{\rm
OS}=-3C_{F}\frac{\alpha_{s}}{4\pi}\left[\frac{1}{\epsilon_{\rm
UV}}-\gamma_{E}+\ln\frac{4\pi\mu^{2}}{m^{2}}+\frac{4}{3}\right],$ (42)
$\displaystyle\delta Z_{l}^{\rm
OS}=-C_{F}\frac{\alpha_{s}}{4\pi}\left[\frac{1}{\epsilon_{\rm
UV}}-\frac{1}{\epsilon_{\rm IR}}\right],$ (43) $\displaystyle\delta
Z_{g}^{\overline{\rm
MS}}=-\frac{\beta_{0}}{2}\frac{\alpha_{s}}{4\pi}\left[\frac{1}{\epsilon_{\rm
UV}}-\gamma_{E}+\ln(4\pi)\right].$ (44)

Here, $\gamma_{E}$ is the Euler’s constant; $m$ stands for $m_{c}$ and $m_{b}$
accordingly. Besides the standard QCD renormalization, the renormalization of
effective weak Hamiltonian leads to the counterterm

$\delta T_{i}^{(1)}=\delta Z^{\mathcal{Q}}_{ij}T^{(0)}_{j},$ (45)

where $i,j=0,8$. At the $\overline{\rm MS}$ scheme Gambino:2003zm ,

$\delta Z^{\mathcal{Q}}_{ij}=\frac{1}{4}\bigg{(}\begin{array}[]{cc}0&6\\\
4/3&-2\\\ \end{array}\bigg{)}\bigg{[}\frac{1}{\epsilon_{\rm
UV}}-\gamma_{E}+\ln(4\pi)\bigg{]}.$ (46)

Note, the overall constant factor of $\delta T_{i}^{(1)}$ is defined as the
same as that of $T_{i}^{(1)}$ (see Eq. (23)). Hence comparing to the
traditional form, the $\delta Z^{\mathcal{Q}}_{ij}$ here is divided by an
additional factor, $\alpha_{s}/\pi$.

While for the IR singularities, parts of them are canceled each other, the
remaining are canceled by the counterterm arising from the corrections of
LCDA. At the $\overline{\rm MS}$ scheme Braaten:1982yp ,

$\displaystyle\delta T^{(1)}_{i}=\frac{C_{F}}{2}\bigg{(}\frac{1}{\epsilon_{\rm
IR}}-\gamma_{E}$
$\displaystyle+\ln(4\pi)\bigg{)}\int_{0}^{1}dy\big{(}T^{(0)}_{i}(y)-T^{(0)}_{i}(x)\big{)}$
$\displaystyle\bigg{[}\frac{y}{x}\bigg{(}1+\frac{1}{x-y}\bigg{)}\theta(x-y)+\frac{1-y}{1-x}\bigg{(}1+\frac{1}{y-x}\bigg{)}\theta(y-x)\bigg{]}.$
(47)

Since $T^{(0)}_{0}$ is independent of $x$, the counterterm to $T^{(1)}_{0}$
vanishes.

At the regions near $x_{1}$ and $x_{2}$, the asymptotic behavior of
$T_{i}^{(1)}$ is
$T_{i}^{(1)}\sim\ln^{m}(x-x_{1}+i\varepsilon)/(x-x_{1}+i\varepsilon)^{n}$ and
$T_{i}^{(1)}\sim\ln^{m}(x-x_{2}-i\varepsilon)/(x-x_{2}-i\varepsilon)^{n}$
respectively. Since $x_{1}$ and $x_{2}$ are not pinched, the convolution of
Eq. (12) is free from singularities.

##  IV Numerical results

###  IV.1 Parameters and decay amplitudes

The input parameters taken in the numerical calculation go as follows
Zyla:2020zbs :

$\displaystyle G_{F}=1.166\times 10^{-5}\ {\rm
GeV}^{-2},\quad|V_{ud}|=0.9737,\quad|V_{cb}|=0.041,\quad\tau_{B_{c}}=0.510\
{\rm ps},$ $\displaystyle f_{\pi}=130.2\ {\rm MeV},\quad m_{W}=80.379\ {\rm
GeV},\quad\bar{m}_{c}=1.27^{+0.02}_{-0.02}\ {\rm GeV},$
$\displaystyle\bar{m}_{b}=4.18^{+0.03}_{-0.02}\ {\rm
GeV},\quad|R_{bc}(0)|^{2}=1.642\ {\rm
GeV}^{3},\quad|R_{cc}^{\prime}(0)|^{2}=0.075\ {\rm GeV}^{5}.$ (48)

Here, $\bar{m}_{Q}$ denotes the $\overline{\rm MS}$ mass of heavy quark $Q$,
$|R_{bc}(0)|$ and $|R_{cc}^{\prime}(0)|$ are evaluated in the QCD-motivated
Buchmöller-Tye potential Eichten:1995ch . At NLO, the pole mass $m_{Q}$ can be
obtained through

$m_{Q}=\bar{m}_{Q}\Big{(}1+C_{F}\frac{\alpha_{s}(\bar{m}_{Q})}{\pi}\Big{)},$
(49)

where $C_{F}=4/3$ is the color factor. Then we have
$m_{b}=4.60^{+0.03}_{-0.02}\ {\rm GeV}$ and $m_{c}=1.49^{+0.02}_{-0.02}\ {\rm
GeV}$.

The two-loop formula Deur:2016tte

$\frac{4\pi}{\alpha_{s}(\mu^{2})}-\frac{\beta_{1}}{\beta_{0}}\ln\left(\frac{4\pi}{\alpha_{s}(\mu^{2})}+\frac{\beta_{1}}{\beta_{0}}\right)=\frac{4\pi}{\alpha_{s}(\mu_{0}^{2})}-\frac{\beta_{1}}{\beta_{0}}\ln\left(\frac{4\pi}{\alpha_{s}(\mu_{0}^{2})}+\frac{\beta_{1}}{\beta_{0}}\right)+\beta_{0}\ln\frac{\mu^{2}}{\mu_{0}^{2}}$
(50)

of the running coupling constant is employed in our calculation, in which
$\beta_{1}=(34/3)C_{A}^{2}-4C_{F}T_{F}n_{f}-(20/3)C_{A}T_{F}n_{f}$. Here we
adopt $n_{f}=4$, $\alpha_{s}(\mu_{0}^{2})=0.184$ with $\mu_{0}=9.46$ GeV
Brambilla:2007cz  as the initial scale.

For the parametrization of LCDA, it is convenient and conventional to expand
it as:

$\phi_{\pi}(x,\mu)=6x(1-x)\bigg{(}1+\sum_{n=1}^{\infty}a_{n}(\mu)C_{n}^{3/2}(1-2x)\bigg{)},$
(51)

where $C_{n}^{3/2}$ are the Gegenbauer polynomials

$\displaystyle C_{0}^{3/2}(x)=1,\quad C_{1}^{3/2}(x)=3x,\quad
C_{2}^{3/2}(x)=\frac{3}{2}(5x^{2}-1),$ $\displaystyle
C_{3}^{3/2}(x)=\frac{5}{2}(7x^{2}-3),\quad
C_{4}^{3/2}(x)=\frac{15}{8}(21x^{4}-14x^{2}+1),\cdots$ (52)

Here we ignore the evolutionary effects, and take Braun:1989iv ; Ball:1998je ;
Lu:2002ny

$a_{2}=0.44,\quad a_{4}=0.25,\quad{\rm others}=0.$ (53)

For convenience, we also introduce the moments

$\displaystyle
M_{i}^{(j)}(n)=\int_{0}^{1}6x(1-x)C_{n}^{3/2}(1-2x)T_{i}^{(j)}(x),$ (54)
$\displaystyle\tilde{M}_{i}^{(j)}(n)=\int_{0}^{1}6x(1-x)C_{n}^{3/2}(1-2x)\tilde{T}_{i}^{(j)}(x).$
(55)

Figure 3: The integral path used in numerical calculation.

As has been discussed in Sec. III, the poles $x_{1}$ and $x_{2}$ will not
spoil the integral of Eq. (12). However, in numerical calculations, it is
inefficient to integrate along the real axis, as one has to approach the poles
at a distance of order $|i\varepsilon|$. Hence, in our calculations, we deform
the integration path from real axis to two conjoined semicircles, as shown in
Fig. 3. The corresponding analytic continuation of $T_{i}^{(1)}(x)$ is also
performed, which is a bit of a tedious task.

Table 1: Numerical results for $M_{i}^{(j)}$ and $\tilde{M}_{i}^{(j)}$. Here
$m_{b}=4.60$ GeV, $m_{c}=1.49$ GeV. $M_{i}^{(j)}(n)$ of
$B_{c}^{+}\to\chi_{c0}\pi^{+}$, in unit of $10^{4}$
---
|  $M_{0}^{(0)}(n)$ |  $M_{0}^{(1)}(n)$ |  $\tilde{M}_{0}^{(1)}(n)$ |  $M_{8}^{(0)}(n)$ |  $M_{8}^{(1)}(n)$ |  $\tilde{M}_{8}^{(1)}(n)$
$n=0$ |  $-51.1$ |  $-607+3.15i$ |  $-105-1.07i$ |  $-1.03+0.716i$ |  $51.9+29.6i$ |  $14.4+1.85i$
$n=1$ |  $0$ |  $14.4-1.44i$ |  $0.759-0.799i$ |  $-0.506+0.532i$ |  $4.00+56.6i$ |  $-1.76+1.85i$
$n=2$ |  $0$ |  $11.3-1.01i$ |  $-1.36-2.38i$ |  $0.908+1.58i$ |  $16.6+15.5i$ |  $3.61+6.29i$
$n=3$ |  $0$ |  $1.98-28.9i$ |  $-1.00-1.72i$ |  $0.670+1.14i$ |  $31.9+7.20i$ |  $2.90+4.95i$
$n=4$ |  $0$ |  $0.723-18.8i$ |  $-3.02+1.67i$ |  $2.01-1.11i$ |  $20.9-16.8i$ |  $9.27-5.12i$
$M_{i}^{(j)}(n)$ of $B_{c}^{+}\to\chi_{c1}\pi^{+}$, in unit of $10^{4}$
|  $M_{0}^{(0)}(n)$ |  $M_{0}^{(1)}(n)$ |  $\tilde{M}_{0}^{(1)}(n)$ |  $M_{8}^{(0)}(n)$ |  $M_{8}^{(1)}(n)$ |  $\tilde{M}_{8}^{(1)}(n)$
$n=0$ |  $-4.82$ |  $-54.2+3.01i$ |  $-10.9+0.708i$ |  $0.612-0.472i$ |  $12.1-6.23i$ |  $3.19-1.22i$
$n=1$ |  $0$ |  $-5.22-9.86i$ |  $0.233-1.60i$ |  $-0.155+1.07i$ |  $-0.518+18.6i$ |  $-0.538+3.71i$
$n=2$ |  $0$ |  $10.2+0.556i$ |  $1.03+1.33i$ |  $-0.686-0.888i$ |  $-6.57-18.6i$ |  $-2.73-3.53i$
$n=3$ |  $0$ |  $-15.0+13.3i$ |  $-2.12-0.208i$ |  $1.41+0.139i$ |  $23.2+4.98i$ |  $6.12+0.600i$
$n=4$ |  $0$ |  $-2.72-16.0i$ |  $1.57-1.39i$ |  $-1.04+0.923i$ |  $-24.9+3.93i$ |  $-4.81+4.25i$
$M_{i}^{(j)}(n)$ of $B_{c}^{+}\to\chi_{c2}\pi^{+}$, in unit of $10^{4}$
|  $M_{0}^{(0)}(n)$ |  $M_{0}^{(1)}(n)$ |  $\tilde{M}_{0}^{(1)}(n)$ |  $M_{8}^{(0)}(n)$ |  $M_{8}^{(1)}(n)$ |  $\tilde{M}_{8}^{(1)}(n)$
$n=0$ |  $47.6$ |  $554-3.59i$ |  $99.4-0.253i$ |  $-0.0790+0.169i$ |  $-54.3-16.7i$ |  $-16.1+0.436i$
$n=1$ |  $0$ |  $-4.06+5.58i$ |  $-1.09+1.28i$ |  $0.730-0.853i$ |  $-5.03-40.8i$ |  $2.53-2.96i$
$n=2$ |  $0$ |  $-12.6-10.4i$ |  $-0.649+0.0235i$ |  $0.433-0.0157i$ |  $-1.64+0.456i$ |  $1.72-0.0622i$
$n=3$ |  $0$ |  $4.30+7.99i$ |  $1.62+2.45i$ |  $-1.08-1.63i$ |  $-13.3-5.67i$ |  $-4.67-7.06i$
$n=4$ |  $0$ |  $-16.6+22.6i$ |  $0.349+1.11i$ |  $-0.232-0.741i$ |  $-2.51+2.35i$ |  $-1.07-3.41i$
$M_{i}^{(j)}(n)$ of $B_{c}^{+}\to h_{c}\pi^{+}$, in unit of $10^{4}$
|  $M_{0}^{(0)}(n)$ |  $M_{0}^{(1)}(n)$ |  $\tilde{M}_{0}^{(1)}(n)$ |  $M_{8}^{(0)}(n)$ |  $M_{8}^{(1)}(n)$ |  $\tilde{M}_{8}^{(1)}(n)$
$n=0$ |  $66.7$ |  $795-5.37i$ |  $138+0.484i$ |  $0.916-0.322i$ |  $-78.3-40.7i$ |  $-19.9-0.833i$
$n=1$ |  $0$ |  $-12.2+1.28i$ |  $-0.529+1.02i$ |  $0.353-0.677i$ |  $-12.2-68.3i$ |  $1.23-2.35i$
$n=2$ |  $0$ |  $-12.7-0.513i$ |  $-0.304+2.69i$ |  $0.202-1.80i$ |  $-8.48-5.48i$ |  $0.804-7.13i$
$n=3$ |  $0$ |  $-9.27+20.7i$ |  $1.31+1.09i$ |  $-0.876-0.727i$ |  $-37.0-2.78i$ |  $-3.79-3.15i$
$n=4$ |  $0$ |  $2.04+21.1i$ |  $4.31+1.25i$ |  $-2.88-0.836i$ |  $-7.85-0.762i$ |  $-13.2-3.85i$

The numerical results for $M_{i}^{(j)}$ and $\tilde{M}_{i}^{(j)}$ are shown in
Table 1. It can be seen that the values of $M_{0}^{(0)}(0)$, $M_{0}^{(1)}(0)$,
and $\tilde{M}_{0}^{(1)}(0)$ are more significant than others. These
significant moments corresponding to the first term of Eq. (51), which depicts
the asymptotic behavior of LCDA: $\phi_{\pi}(x,\mu\to\infty)=6x(1-x)$. Hence
the uncertainty, caused either by ignoring the evolutionary effect of $a_{n}$
or by taking another parametrization scheme, is unimportant to the final decay
width.

###  IV.2 Branching ratios

In the calculation of branching ratios, one meets a considerable freedom in
the choice of renormalization scale $\mu$, because the meaning of $\mu$ is not
quite evident. By convention, there are three reasonable schemes:

  1. 1.

Set $\mu$ to be some typical virtuality of internal particles. For example,
the virtuality of gluons of Fig. 1(a)$\sim$(b) lead to $\mu=1.1$ GeV; the
largest virtuality of internal particles of Fig. 1(a)$\sim$(b) leads to
$\mu=4.0$ GeV.

  2. 2.

Relate $\mu$ to kinematic variables of final state particles. For example, the
three-momentum of final state charmonium or pion in the $B_{c}^{+}$ rest frame
lead to $\mu=|\vec{k}_{2}|=2.3$ GeV.

  3. 3.

Determine $\mu$ through scale-setting procedure, like the Brodsky-Lepage-
Mackenzie (BLM) method Brodsky:1982gc ; Brodsky:2011ta , where the quark
vacuum-polarization corrections are re-summed into the running coupling. In
our case, the BLM scale can be acquired by setting $\mu$ such it that kills
the $n_{f}$ terms. We obtain $\mu_{\rm BLM}=0.38$ GeV for $\chi_{c0}$ channel,
$\mu_{\rm BLM}=0.47$ GeV for $\chi_{c1}$ channel, $\mu_{\rm BLM}=0.34$ GeV for
$\chi_{c2}$ and $h_{c}$ channels.

Since the BLM scales here are close to the perturbative QCD scale
$\Lambda_{\rm QCD}$, we will not use them in following evaluation. Instead, we
will vary $\mu$ in a range that covers most reasonable choices of scale.

The branching ratios can be obtained by employing Eq. (13). The NLO branching
ratios versus the running renormalization scale $\mu$ are exhibited in Fig. 4,
where the contributions from the Born terms and the uncertainties induced by
heavy quark masses are also shown. It can be seen that, after including the
NLO corrections, the branching ratios are enhanced by a factor (defined as the
$K$ factor) of about $2.5$, and the theoretical uncertainties induced by the
renormalization scale remain large, which may indicate that the contributions
from missing higher orders (beyond NLO) are significant.

Figure 4: The LO and NLO branching ratio versus running renormalization scale
$\mu$. (a) for $B_{c}^{+}\to\chi_{c0}\pi^{+}$, (b) for
$B_{c}^{+}\to\chi_{c1}\pi^{+}$, (c) for $B_{c}^{+}\to\chi_{c2}\pi^{+}$, (d)
for $B_{c}^{+}\to h_{c}\pi^{+}$. The plots in the left column corresponding to
$1.0<\mu<3.0$, and in the right column corresponding to $3.0<\mu<6.5$.

In Table 2, we compare our branching ratios with those from other literatures.
Among them, the predictions of Ref. Kiselev:2001zb  are based on LO NRQCD
calculation, with some technique details different from ours; the predictions
of Ref. Zhu:2017lwi  are also based on LO NRQCD, but with relativistic
corrections. It can be seen that our LO predictions are compatible with those
of Refs. Kiselev:2001zb ; Zhu:2017lwi , but generally larger than those of
other approaches. Our NLO branching ratios, however, are lager than almost all
other predictions.

Table 2: Comparison of the branching ratios of $B_{c}^{+}\to\chi_{cJ}(h_{c})\pi^{+}$. Here we vary the renormalization scale $\mu$ from $|\vec{k}_{2}|/2$ to $2|\vec{k}_{2}|$. The central values are obtained at $\mu=|\vec{k}_{2}|$. The uncertainties are induced by $\mu$ and heavy quark masses. Branching ratios ($10^{-3}$) |  NLO |  LO |  Kiselev:2001zb  |  Zhu:2017lwi  |  Rui:2017pre  |  Wang:2011jt  |  Hernandez:2006gt  |  Ivanov:2006ni  |  Ebert:2010zu
---|---|---|---|---|---|---|---|---|---
$\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})$ |  $28^{+38}_{-13}$ |  $11^{+16}_{-5.0}$ |  $9.7$ |  $6.5$ |  $1.6$ |  0.3 |  0.26 |  $0.55$ |  0.21
$10^{2}\times\mathcal{B}(B_{c}^{+}\to\chi_{c1}\pi^{+})$ |  $23^{+28}_{-10}$ |  $10^{+16}_{-4.7}$ |  $9.7$ |  $6.4$ |  $51$ |  2.2 |  0.14 |  $6.8$ |  20
$\mathcal{B}(B_{c}^{+}\to\chi_{c2}\pi^{+})$ |  $24^{+31}_{-11}$ |  $9.7^{+14}_{-4.5}$ |  $8.5$ |  $4.4$ |  $4.0$ |  0.23 |  0.22 |  $0.46$ |  0.38
$\mathcal{B}(B_{c}^{+}\to h_{c}\pi^{+})$ |  $48^{+66}_{-22}$ |  $19^{+28}_{-8.8}$ |  $16$ |  $9.7$ |  $0.54$ |  1.0 |  0.53 |  $1.1$ |  0.46

In Ref. Aaij:2016xas , the LHCb Collaboration reported evidence for the decay
$B_{c}^{+}\to\chi_{c0}\pi^{+}$ with a significance of 4.0 standard deviations.
The measured product of the ratio of cross sections and branching ratio is
$\frac{\sigma(B_{c}^{+})}{\sigma(B^{+})}\times\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})=(9.8^{+3.4}_{-3.0}({\rm
stat})\pm 0.8({\rm syst}))\times 10^{-6}$. Combining this result with the
measurement on
$\frac{\sigma(B_{c}^{+})}{\sigma(B^{+})}\times\frac{\mathcal{B}(B_{c}^{+}\to
J/\psi\pi^{+})}{\mathcal{B}(B^{+}\to J/\psi K^{+})}$ Aaij:2014ija  and the
data for $\mathcal{B}(B^{+}\to J/\psi K^{+})$ Zyla:2020zbs , we obtain

$\frac{\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})}{\mathcal{B}(B_{c}^{+}\to
J/\psi\pi^{+})}=1.4_{-0.6}^{+0.8}.$ (56)

In Ref. Qiao:2012hp , the exclusive processes $B_{c}^{+}\to J/\psi\pi^{+}$ is
investigated at the NLO QCD accuracy within the framework of the NRQCD
factorization formalism. By taking the same parameters as Ref. Qiao:2012hp ,
we obtain

$\frac{\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})}{\mathcal{B}(B_{c}^{+}\to
J/\psi\pi^{+})}\bigg{|}_{\rm NLO}^{\rm NRQCD}=2.7_{-1.0}^{+1.7},$ (57)

which is compatible with Eq. (56) with the uncertainties. It should be
remarked that the NRQCD predictions on both
$\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})$ and $\mathcal{B}(B_{c}^{+}\to
J/\psi\pi^{+})$ are larger than those of other approaches. Further
experimental studies, especially the measurement on the exact values of
branching ratios, are needed to clarify the situation.

In the “naive factorization” approach, the exclusive two-body decay amplitude
reduces to the product of a form factor and a decay constant. The
contributions of “naive factorization” come from the so-called “factorizable”
diagrams, like Fig. 1(a)$\sim$(b) and Fig. 2(a)$\sim$(b). Our numerical
estimation shows that the “factorizable” diagrams can mimic over $93\%$
contribution of the full calculation, which is sufficient in phenomenological
study. On the other hand, since the “factorizable” hard kernels are
independent of $x$, the convolution in Eq. (12) reduce to the integration of
$\phi_{\pi}(x,\mu)$, which can be easily obtained. The “factorizable” hard
kernels are presented in the Appendix.

With the $M_{i}^{(j)}$ and $\tilde{M}_{i}^{(j)}$ listed in Table 1, the
branching ratios for $B_{c}^{+}\to\chi_{cJ}(h_{c})K^{+}$ are also at hand.
Considering the fact that the contributions from “factorizable” diagrams are
dominant, we have
$\frac{\mathcal{B}(B_{c}^{+}\to\chi_{cJ}(h_{c})K^{+})}{\mathcal{B}(B_{c}^{+}\to\chi_{cJ}(h_{c})\pi^{+})}\approx\big{(}\frac{|V_{us}|f_{K}}{|V_{ud}|f_{\pi}}\big{)}^{2}\approx
0.0762$.

##  V Summary and conclusions

In this work, we investigate the $B_{c}$ meson exclusive decays to a $P$-wave
charmonium and a pion at the NLO QCD accuracy in the framework of the NRQCD
factorization formalism. Numerical results show that after including the NLO
corrections, the branching ratios are enhanced by a factor of about $2.5$. The
renormalization scale dependence, however, is still large at NLO, which may
indicate significant contributions from higher-order terms beyond NLO.

Although the exact values of
$\mathcal{B}(B_{c}^{+}\to\chi_{cJ}(h_{c})\pi^{+})$ have not been measured yet,
the quantity
$\frac{\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})}{\mathcal{B}(B_{c}^{+}\to
J/\psi\pi^{+})}$ can be extracted from relevant measurements Zyla:2020zbs ;
Aaij:2016xas ; Aaij:2014ija . With the prediction of $\mathcal{B}(B_{c}^{+}\to
J/\psi\pi^{+})$ in Ref. Qiao:2012hp  and our prediction of
$\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})$, we find the predicted
$\frac{\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})}{\mathcal{B}(B_{c}^{+}\to
J/\psi\pi^{+})}$ is compatible with experimental data. However, since the
NRQCD predictions on both $\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})$ and
$\mathcal{B}(B_{c}^{+}\to J/\psi\pi^{+})$ are generally larger than those of
other approaches, further experimental study is still needed to clarify this
issue.

Last, we notice that the contributions from “factorizable” diagrams are
dominant, hence the “naive factorization” approach works well in at least our
phenomenological study. Therefore we have
$\frac{\mathcal{B}(B_{c}^{+}\to\chi_{cJ}(h_{c})K^{+})}{\mathcal{B}(B_{c}^{+}\to\chi_{cJ}(h_{c})\pi^{+})}\approx\big{(}\frac{|V_{us}|f_{K}}{|V_{ud}|f_{\pi}}\big{)}^{2}\approx
0.0762$.

Acknowledgments

This work was supported in part by the National Key Research and Development
Program of China under Contracts No. 2020YFA0406400, and the National Natural
Science Foundation of China (NSFC) under the Grants No. 11975236, No.
11635009, and No. 12047553.

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## Appendix: “factorizable” hard kernels

In this appendix, we present the “factorizable” hard kernels $T_{0,{\rm
f}}^{(0)}$, $\tilde{T}_{0,{\rm f}}^{(1)}$ and $T_{0,{\rm f}}^{(1)}$. The LO
“factorizable” hard kernels $T_{0,{\rm f}}^{(0)}=T_{0}^{(0)}$, which have been
presented in Eqs. (29)(31)(33)(35). The NLO “factorizable” hard kernels
$\tilde{T}_{0,{\rm f}}^{(1)}=\frac{\beta_{0}}{4}T_{0}^{(0)}$, and
$T^{(1)}_{0,{\rm f}}$ are as follows:

$\displaystyle T$ ${}_{0,{\rm
f}}^{(1)}(\chi_{c0})=\Big{\\{}\tfrac{4(11z-24)}{z^{2}-2z-1}+(-\tfrac{4(74z-219)}{z^{2}-2z-1}+\tfrac{839z^{2}-4048z+5525}{32(z^{3}-4z^{2}+3z+4)}-\tfrac{4}{9(z-3)}+\tfrac{458}{z-1}+\tfrac{8(z-6)}{(z^{2}-2z-1)^{2}}$
$\displaystyle-\tfrac{64z}{(z^{2}-2z-1)^{3}}-\tfrac{14}{z+1}-\tfrac{3(7z-17)}{2(z^{2}-3z+4)}-\tfrac{47023}{288z}-\tfrac{20497}{24z^{2}}-\tfrac{1019}{2z^{3}}-\tfrac{762}{z^{4}}-\tfrac{224}{z^{5}}+\tfrac{480}{z^{6}}-\tfrac{384}{z^{7}})f_{2}$
$\displaystyle+(\tfrac{3(z-7)}{2(z^{2}-3z+4)}+\tfrac{839z^{2}-4048z+5525}{16(z^{3}-4z^{2}+3z+4)}-\tfrac{8}{9(z-3)}-\tfrac{106}{9(z-2)}+\tfrac{70}{3(z-2)^{2}}-\tfrac{870}{z-1}-\tfrac{976}{3(2z-1)}+\tfrac{224}{3(2z-1)^{2}}$
$\displaystyle+\tfrac{310}{9(z+1)}-\tfrac{56}{3(z+1)^{2}}+\tfrac{145865}{144z}+\tfrac{12971}{12z^{2}}+\tfrac{1865}{3z^{3}}+\tfrac{512}{z^{4}}+\tfrac{664}{3z^{5}}+\tfrac{480}{z^{6}}-\tfrac{384}{z^{7}})f_{3}$
$\displaystyle+(\tfrac{-839z^{2}+4048z-5525}{32(z^{3}-4z^{2}+3z+4)}+\tfrac{4}{9(z-3)}-\tfrac{22}{z-2}+\tfrac{14}{(z-2)^{2}}+\tfrac{128}{z-1}+\tfrac{320}{3(z-1)^{2}}-\tfrac{8(74z-219)}{z^{2}-2z-1}+\tfrac{16(z-6)}{(z^{2}-2z-1)^{2}}$
$\displaystyle-\tfrac{128z}{(z^{2}-2z-1)^{3}}-\tfrac{28}{z+1}-\tfrac{12(2z-3)}{2z^{2}-5z+1}+\tfrac{48}{z^{2}-3z+4}+\tfrac{142783}{288z}-\tfrac{5197}{8z^{2}}-\tfrac{3763}{6z^{3}}-\tfrac{586}{z^{4}}-\tfrac{1336}{3z^{5}})f_{4}$
$\displaystyle+(\tfrac{8(74z-219)}{z^{2}-2z-1}+\tfrac{128z}{(z^{2}-2z-1)^{3}}-\tfrac{734}{z-1}-\tfrac{320}{3(z-1)^{2}}+\tfrac{1952}{3(2z-1)}-\tfrac{448}{3(2z-1)^{2}}-\tfrac{16(z-6)}{(z^{2}-2z-1)^{2}}+\tfrac{320}{3(z+1)}$
$\displaystyle+\tfrac{112}{3(z+1)^{2}}+\tfrac{6(2z-3)}{2z^{2}-5z+1}+\tfrac{12(z-3)}{z^{2}-3z+4}-\tfrac{308}{z}+\tfrac{4328}{3z^{2}}+\tfrac{888}{z^{3}}+\tfrac{2000}{z^{4}})f_{5}+(\tfrac{12(z-3)}{z^{2}-3z+4}-\tfrac{6}{z-1}$
$\displaystyle+\tfrac{6(2z-3)}{2z^{2}-5z+1}-\tfrac{12}{z})f_{6}+(\tfrac{2(29z^{2}-143z-210)}{11(z^{3}-5z^{2}-8z-4)}+\tfrac{3967z^{2}-18000z+23101}{176(z^{3}-4z^{2}+3z+4)}+\tfrac{14}{z+1}-\tfrac{669}{16z}+\tfrac{83}{4z^{2}}$
$\displaystyle-\tfrac{13}{z^{3}}-\tfrac{16}{z^{4}})f_{7}+(\tfrac{6(z-7)}{z^{2}-3z+4}+\tfrac{1052}{9(z-2)}-\tfrac{140}{3(z-2)^{2}}-\tfrac{160}{z-1}+\tfrac{352}{9(z+1)}-\tfrac{2}{z}+\tfrac{36}{z^{2}}-\tfrac{64}{z^{3}}+\tfrac{16}{z^{4}}$
$\displaystyle-\tfrac{64}{z^{5}})f_{8}+(-\tfrac{28}{9(z-2)^{2}}-\tfrac{12122}{27(z+1)}-\tfrac{880}{9(z+1)^{2}}-\tfrac{6}{z^{2}-3z+4}+\tfrac{442}{z}-\tfrac{356}{z^{2}}+\tfrac{256}{z^{3}}-\tfrac{144}{z^{4}}+\tfrac{64}{z^{5}}$
$\displaystyle+\tfrac{188}{27(z-2)})f_{9}+(\tfrac{3(5z-3)}{2(z^{2}-3z+4)}-\tfrac{80}{9(z-2)}+\tfrac{14}{3(z-2)^{2}}-\tfrac{352}{9(z+1)}+\tfrac{81}{2z}-\tfrac{44}{z^{2}}+\tfrac{32}{z^{3}}-\tfrac{24}{z^{4}})f_{10}$
$\displaystyle+(\tfrac{3(7z-17)}{2(z^{2}-3z+4)}-\tfrac{8}{z-2}+\tfrac{14}{(z-2)^{2}}-\tfrac{5}{2z}-\tfrac{4}{z^{2}}-\tfrac{8}{z^{4}})f_{11}+(-3z+\tfrac{2}{z-1}-\tfrac{14}{z+1}-\tfrac{8}{z^{2}}-\tfrac{12}{z^{3}}+10$
$\displaystyle-\tfrac{2}{z})f_{12}+(\tfrac{14}{z+1}-\tfrac{57}{z}-\tfrac{63}{z^{2}}-\tfrac{52}{z^{3}}-\tfrac{76}{z^{4}}-37+\tfrac{18}{z-1})f_{13}+(-\tfrac{3(5z-3)}{z^{2}-3z+4}-\tfrac{72}{5(z+1)}$
$\displaystyle+\tfrac{36(2z-1)}{5(z^{2}-2z+2)}+\tfrac{3}{z}+3)f_{14}+(25z+\tfrac{112}{z-1}-\tfrac{36}{z^{2}-2z+2}-\tfrac{93}{z^{2}}+\tfrac{24}{z^{3}}-\tfrac{36}{z^{4}}+103-\tfrac{6}{z})f_{15}+(\tfrac{2}{z}$
$\displaystyle+\tfrac{10}{z^{2}}-\tfrac{28}{z^{3}}+\tfrac{16}{z^{4}}+\tfrac{6}{z+1})f_{16}+6f_{17}+(\tfrac{12(z+1)}{z^{2}-3z+4}-3z+\tfrac{6}{z}-\tfrac{16}{z^{2}}+\tfrac{4}{z^{3}}+1-\tfrac{32}{z+1})f_{18}$
$\displaystyle+(-2z^{2}-\tfrac{6}{z+1}+\tfrac{30}{z^{2}}+\tfrac{38}{z^{3}}-\tfrac{24}{z^{4}}-11-\tfrac{17}{z})f_{19}+(-2z^{2}-\tfrac{5z}{2}+\tfrac{27}{2z^{2}}-\tfrac{10}{z^{3}}+\tfrac{8}{z^{4}}-31+\tfrac{2}{z})f_{20}$
$\displaystyle+(-\tfrac{32}{z}+\tfrac{32}{z^{2}}-\tfrac{32}{z^{3}}+\tfrac{16}{z^{4}}+6+\tfrac{32}{z+1})f_{21}+(-\tfrac{29z}{2}+\tfrac{16}{z-1}+\tfrac{89}{2z^{2}}+\tfrac{4}{z^{3}}+\tfrac{16}{z^{4}}-51+\tfrac{16}{z})f_{22}$
$\displaystyle+(\tfrac{10}{z}+\tfrac{22}{z^{2}}+\tfrac{36}{z^{3}}+\tfrac{16}{z^{4}}-\tfrac{2}{z-1})f_{23}+(4z^{2}-20z+\tfrac{40}{z^{2}}+68-\tfrac{40}{z})f_{24}+(\tfrac{72}{5(z+1)}$
$\displaystyle-\tfrac{36(2z-1)}{5(z^{2}-2z+2)})f_{25}+(\tfrac{36}{z^{2}-2z+2}-\tfrac{72}{z^{2}-3z+4})f_{26}+\tfrac{152}{3(z-2)}-\tfrac{56}{(z-2)^{2}}+\tfrac{814}{3(z-1)}+\tfrac{224}{2z-1}$
$\displaystyle-\tfrac{32(z-2)}{(z^{2}-2z-1)^{2}}+\tfrac{436}{3(z+1)}-\tfrac{12(5z-9)}{z^{2}-3z+4}-\tfrac{1690}{3z}-\tfrac{892}{3z^{2}}-\tfrac{116}{z^{3}}-\tfrac{1240}{3z^{4}}+\tfrac{288}{z^{5}}-\tfrac{384}{z^{6}}\Big{\\}}\tfrac{64\pi^{2}}{9},$
(58) $\displaystyle T$ ${}_{0,{\rm
f}}^{(1)}(\chi_{c1})=\Big{\\{}(-\tfrac{8(2z-5)}{z^{2}-2z-1}+\tfrac{32(2z+1)}{(z^{2}-2z-1)^{2}}+\tfrac{8}{z-3}+\tfrac{1370}{3(z-2)}-\tfrac{164}{z-1}-\tfrac{8}{3(z+1)}-\tfrac{118}{z}-\tfrac{360}{z^{2}}+\tfrac{8}{z^{3}}$
$\displaystyle+\tfrac{76}{z^{4}}-\tfrac{64}{z^{5}})f_{2}+(\tfrac{341}{3(z-2)}-\tfrac{172}{z-1}-\tfrac{304}{3(2z-1)}+\tfrac{32}{(2z-1)^{2}}+\tfrac{88}{z+1}-\tfrac{16}{(z+1)^{2}}+\tfrac{121}{z}+\tfrac{506}{3z^{2}}+\tfrac{508}{3z^{3}}$
$\displaystyle+\tfrac{68}{z^{4}}-\tfrac{64}{z^{5}}+\tfrac{16}{z-3})f_{3}+(-\tfrac{16(2z-5)}{z^{2}-2z-1}+\tfrac{64(2z+1)}{(z^{2}-2z-1)^{2}}-\tfrac{8}{z-3}+\tfrac{1345}{3(z-2)}-\tfrac{32}{z-1}+\tfrac{64}{3(z-1)^{2}}-\tfrac{16}{3(z+1)}$
$\displaystyle+\tfrac{24(2z-1)}{2z^{2}-5z+1}-\tfrac{123}{z}-\tfrac{890}{3z^{2}}-\tfrac{484}{3z^{3}}+\tfrac{8}{z^{4}})f_{4}+(\tfrac{16(2z-5)}{z^{2}-2z-1}-\tfrac{460}{z-2}-\tfrac{244}{z-1}-\tfrac{64}{3(z-1)^{2}}+\tfrac{608}{3(2z-1)}$
$\displaystyle-\tfrac{64}{(2z-1)^{2}}-\tfrac{64(2z+1)}{(z^{2}-2z-1)^{2}}-\tfrac{544}{3(z+1)}+\tfrac{32}{(z+1)^{2}}-\tfrac{12(2z-1)}{2z^{2}-5z+1}+\tfrac{380}{z}+\tfrac{696}{z^{2}})f_{5}+(-\tfrac{12(2z-1)}{2z^{2}-5z+1}+\tfrac{16}{z-2}$
$\displaystyle-\tfrac{12}{z-1}+\tfrac{8}{z+1})f_{6}+(\tfrac{8}{3(z+1)}-\tfrac{11}{2z}-\tfrac{7}{z^{2}}+\tfrac{2}{z^{3}}+\tfrac{41}{6(z-2)})f_{7}+(-2z-\tfrac{7}{3(z-2)}+\tfrac{4}{z-1}+\tfrac{28}{3(z+1)}$
$\displaystyle+2-\tfrac{11}{z})f_{12}+(-8z-\tfrac{211}{6(z-2)}+\tfrac{36}{z-1}-\tfrac{28}{3(z+1)}-\tfrac{15}{2z}-\tfrac{41}{z^{2}}-18)f_{13}+(-6z-\tfrac{12}{z-2})f_{14}$
$\displaystyle+(-6z+\tfrac{83}{2(z-2)}-\tfrac{36}{z-1}-\tfrac{31}{2z}-\tfrac{3}{z^{2}}+18)f_{15}+(\tfrac{28}{3(z+1)}-\tfrac{13}{z}+\tfrac{6}{z^{2}}-\tfrac{13}{3(z-2)})f_{16}+(-\tfrac{16}{z+1}$
$\displaystyle-\tfrac{8}{z-2})f_{17}+(12z+\tfrac{97}{3(z-2)}+\tfrac{32}{3(z+1)}+2+\tfrac{1}{z})f_{18}+(4z^{2}-31z-\tfrac{13}{6(z-2)}-\tfrac{28}{3(z+1)}+\tfrac{31}{2z}$
$\displaystyle-\tfrac{9}{z^{2}}+38)f_{19}+(-4z^{2}-18z-\tfrac{461}{4(z-2)}+\tfrac{1}{4z}-\tfrac{3}{2z^{2}}-57)f_{20}+(-\tfrac{32}{3(z+1)}-\tfrac{6}{z}+\tfrac{4}{z^{2}}$
$\displaystyle-\tfrac{22}{3(z-2)})f_{21}+(-8z^{2}-25z-\tfrac{753}{4(z-2)}+\tfrac{34}{z-1}+\tfrac{5}{4z}+\tfrac{21}{2z^{2}}-35)f_{22}+(-\tfrac{4}{z-1}+\tfrac{13}{z}+\tfrac{10}{z^{2}}$
$\displaystyle-\tfrac{17}{z-2})f_{23}+(8z^{2}+8z+\tfrac{60}{z-2}+32+\tfrac{4}{z})f_{24}-\tfrac{500}{3(z-1)}+\tfrac{96}{2z-1}+\tfrac{16}{z^{2}-2z-1}+\tfrac{24}{z+1}+\tfrac{476}{3z}$
$\displaystyle+\tfrac{568}{3z^{2}}+\tfrac{32}{z^{3}}-\tfrac{64}{z^{4}}\Big{\\}}\tfrac{32}{3}\sqrt{\tfrac{2}{3}}\pi^{2},$
(59) $\displaystyle T$ ${}_{0,{\rm
f}}^{(1)}(\chi_{c2})=\Big{\\{}\tfrac{18(3z-5)}{z^{2}-3z+4}+\tfrac{4z}{z^{2}-2z-1}+(-\tfrac{3(65z-231)}{32(z^{2}-3z+4)}-\tfrac{9}{2(z-2)}-\tfrac{12}{z-1}+\tfrac{723}{32z}+\tfrac{141}{8z^{2}}-\tfrac{9}{2z^{3}}$
$\displaystyle-\tfrac{18}{z^{4}})f_{1}+(\tfrac{-839z^{2}+4048z-5525}{32(z^{3}-4z^{2}+3z+4)}+\tfrac{8(11z+6)}{(z^{2}-2z-1)^{2}}+\tfrac{64z}{(z^{2}-2z-1)^{3}}+\tfrac{4}{9(z-3)}-\tfrac{1411}{2(z-2)}+\tfrac{1017}{(z-2)^{2}}+\tfrac{490}{z-1}$
$\displaystyle+\tfrac{4(80z-243)}{z^{2}-2z-1}+\tfrac{28}{z+1}+\tfrac{3(37z-163)}{16(z^{2}-3z+4)}-\tfrac{32735}{288z}+\tfrac{14899}{24z^{2}}-\tfrac{79}{2z^{3}}+\tfrac{870}{z^{4}}+\tfrac{188}{z^{5}}-\tfrac{576}{z^{6}}+\tfrac{384}{z^{7}})f_{2}$
$\displaystyle+(\tfrac{-839z^{2}+4048z-5525}{16(z^{3}-4z^{2}+3z+4)}+\tfrac{8}{9(z-3)}+\tfrac{3905}{18(z-2)}-\tfrac{481}{3(z-2)^{2}}+\tfrac{186}{z-1}-\tfrac{336}{2z-1}+\tfrac{160}{3(2z-1)^{2}}-\tfrac{2884}{9(z+1)}$
$\displaystyle-\tfrac{424}{3(z+1)^{2}}-\tfrac{21(17z-23)}{16(z^{2}-3z+4)}+\tfrac{6313}{36z}-\tfrac{1723}{6z^{2}}-\tfrac{140}{3z^{3}}-\tfrac{204}{z^{4}}-\tfrac{700}{3z^{5}}-\tfrac{576}{z^{6}}+\tfrac{384}{z^{7}})f_{3}$
$\displaystyle+(\tfrac{8(80z-243)}{z^{2}-2z-1}+\tfrac{16(11z+6)}{(z^{2}-2z-1)^{2}}+\tfrac{839z^{2}-4048z+5525}{32(z^{3}-4z^{2}+3z+4)}+\tfrac{128z}{(z^{2}-2z-1)^{3}}-\tfrac{4}{9(z-3)}+\tfrac{137}{z-2}+\tfrac{484}{(z-2)^{2}}$
$\displaystyle+\tfrac{424}{z-1}-\tfrac{128}{3(z-1)^{2}}+\tfrac{48}{z+1}+\tfrac{12(2z-3)}{2z^{2}-5z+1}+\tfrac{3(27z-157)}{16(z^{2}-3z+4)}-\tfrac{376657}{288z}+\tfrac{623}{8z^{2}}-\tfrac{1163}{6z^{3}}+\tfrac{350}{z^{4}}$
$\displaystyle+\tfrac{1264}{3z^{5}})f_{4}+(-\tfrac{8(80z-243)}{z^{2}-2z-1}-\tfrac{1988}{3(z-2)}-\tfrac{28}{(z-2)^{2}}+\tfrac{146}{z-1}+\tfrac{128}{3(z-1)^{2}}+\tfrac{672}{2z-1}-\tfrac{320}{3(2z-1)^{2}}$
$\displaystyle-\tfrac{16(11z+6)}{(z^{2}-2z-1)^{2}}-\tfrac{128z}{(z^{2}-2z-1)^{3}}+\tfrac{860}{3(z+1)}+\tfrac{272}{3(z+1)^{2}}-\tfrac{6(2z-3)}{2z^{2}-5z+1}-\tfrac{12(z-3)}{z^{2}-3z+4}+\tfrac{552}{z}-\tfrac{620}{3z^{2}}$
$\displaystyle+\tfrac{192}{z^{3}}-\tfrac{2000}{z^{4}})f_{5}+(-\tfrac{12(z-3)}{z^{2}-3z+4}+\tfrac{22}{3(z-2)}-\tfrac{14}{(z-2)^{2}}+\tfrac{6}{z-1}+\tfrac{248}{3(z+1)}+\tfrac{64}{(z+1)^{2}}-\tfrac{6(2z-3)}{2z^{2}-5z+1}-\tfrac{78}{z}$
$\displaystyle+\tfrac{18}{z^{2}})f_{6}+(\tfrac{-3967z^{2}+18000z-23101}{176(z^{3}-4z^{2}+3z+4)}-\tfrac{2(29z^{2}-143z-210)}{11(z^{3}-5z^{2}-8z-4)}-\tfrac{7}{4(z-2)}+\tfrac{6}{(z-2)^{2}}-\tfrac{4}{z+1}+\tfrac{537}{16z}-\tfrac{65}{4z^{2}}$
$\displaystyle+\tfrac{10}{z^{3}}+\tfrac{10}{z^{4}})f_{7}+(-\tfrac{6(z-7)}{z^{2}-3z+4}-\tfrac{1052}{9(z-2)}+\tfrac{140}{3(z-2)^{2}}+\tfrac{160}{z-1}-\tfrac{352}{9(z+1)}+\tfrac{2}{z}-\tfrac{36}{z^{2}}+\tfrac{64}{z^{3}}-\tfrac{16}{z^{4}}$
$\displaystyle+\tfrac{64}{z^{5}})f_{8}+(\tfrac{28}{9(z-2)^{2}}+\tfrac{12122}{27(z+1)}+\tfrac{880}{9(z+1)^{2}}+\tfrac{6}{z^{2}-3z+4}-\tfrac{442}{z}+\tfrac{356}{z^{2}}-\tfrac{256}{z^{3}}+\tfrac{144}{z^{4}}-\tfrac{64}{z^{5}}$
$\displaystyle-\tfrac{188}{27(z-2)})f_{9}+(-\tfrac{3(5z-3)}{2(z^{2}-3z+4)}+\tfrac{80}{9(z-2)}-\tfrac{14}{3(z-2)^{2}}+\tfrac{352}{9(z+1)}-\tfrac{81}{2z}+\tfrac{44}{z^{2}}-\tfrac{32}{z^{3}}+\tfrac{24}{z^{4}})f_{10}$
$\displaystyle+(-\tfrac{3(7z-17)}{2(z^{2}-3z+4)}+\tfrac{8}{z-2}-\tfrac{14}{(z-2)^{2}}+\tfrac{5}{2z}+\tfrac{4}{z^{2}}+\tfrac{8}{z^{4}})f_{11}+(-\tfrac{4}{(z-2)^{2}}-\tfrac{2}{z-1}+\tfrac{2}{3(z+1)}+\tfrac{43}{2z}-\tfrac{13}{z^{2}}$
$\displaystyle+\tfrac{12}{z^{3}}-4-\tfrac{19}{6(z-2)})f_{12}+(-\tfrac{2}{(z-2)^{2}}-\tfrac{18}{z-1}-\tfrac{2}{3(z+1)}-\tfrac{189}{4z}+\tfrac{153}{2z^{2}}-\tfrac{11}{z^{3}}+\tfrac{76}{z^{4}}-14-\tfrac{457}{12(z-2)})f_{13}$
$\displaystyle+(\tfrac{3(5z-3)}{z^{2}-3z+4}-\tfrac{9}{z-2}-\tfrac{9}{(z-2)^{2}}-\tfrac{9(z-1)}{z^{2}-2z+2}-\tfrac{3}{z}-3)f_{14}+(\tfrac{9(z+1)}{z^{2}-2z+2}-22z+\tfrac{225}{4(z-2)}+\tfrac{15}{(z-2)^{2}}$
$\displaystyle-\tfrac{16}{z-1}+\tfrac{183}{4z}+\tfrac{21}{2z^{2}}-\tfrac{33}{z^{3}}+\tfrac{36}{z^{4}}+29)f_{15}+(-\tfrac{3}{(z-2)^{2}}-\tfrac{38}{z+1}+\tfrac{37}{z}-\tfrac{43}{z^{2}}+\tfrac{40}{z^{3}}-\tfrac{16}{z^{4}}-\tfrac{1}{z-2})f_{16}$
$\displaystyle+(-\tfrac{8}{(z-2)^{2}}+\tfrac{8}{3(z+1)}-\tfrac{128}{(z+1)^{2}}+\tfrac{36}{z}-6-\tfrac{8}{3(z-2)})f_{17}+(-\tfrac{12(z+1)}{z^{2}-3z+4}+\tfrac{29}{6(z-2)}+\tfrac{8}{(z-2)^{2}}$
$\displaystyle+\tfrac{104}{3(z+1)}-\tfrac{33}{2z}+\tfrac{19}{z^{2}}-\tfrac{4}{z^{3}}+5)f_{18}+(2z^{2}+36z+\tfrac{769}{4(z-2)}+\tfrac{447}{4(z-2)^{2}}+\tfrac{38}{z+1}+\tfrac{83}{4z}+\tfrac{27}{4z^{2}}-\tfrac{56}{z^{3}}$
$\displaystyle+\tfrac{24}{z^{4}}+17)f_{19}+(2z^{2}+37z-\tfrac{819}{4(z-2)}-\tfrac{276}{(z-2)^{2}}+\tfrac{145}{4z}-\tfrac{24}{z^{2}}+\tfrac{16}{z^{3}}-\tfrac{8}{z^{4}}-14)f_{20}$
$\displaystyle+(-\tfrac{44}{(z-2)^{2}}-\tfrac{104}{3(z+1)}+\tfrac{41}{z}-\tfrac{50}{z^{2}}+\tfrac{44}{z^{3}}-\tfrac{16}{z^{4}}-6-\tfrac{127}{3(z-2)})f_{21}+(z-\tfrac{105}{z-2}-\tfrac{507}{4(z-2)^{2}}-\tfrac{16}{z-1}$
$\displaystyle-\tfrac{5}{2z}-\tfrac{187}{4z^{2}}+\tfrac{8}{z^{3}}-\tfrac{16}{z^{4}}-39)f_{22}+(-\tfrac{12}{(z-2)^{2}}+\tfrac{2}{z-1}-\tfrac{10}{z}-\tfrac{16}{z^{2}}-\tfrac{24}{z^{3}}-\tfrac{16}{z^{4}}+\tfrac{6}{z-2})f_{23}$
$\displaystyle+(-4z^{2}-52z-\tfrac{306}{z-2}-\tfrac{162}{(z-2)^{2}}-\tfrac{22}{z^{2}}-104+\tfrac{34}{z})f_{24}+(\tfrac{9(z-1)}{z^{2}-2z+2}-\tfrac{9}{4(z-2)}-\tfrac{9}{4(z-2)^{2}}-\tfrac{27}{4z}$
$\displaystyle-\tfrac{9}{4z^{2}}+\tfrac{18}{z^{3}}-\tfrac{9}{z^{4}})f_{25}+(-\tfrac{9(z+1)}{z^{2}-2z+2}+\tfrac{9(93z+85)}{64(z^{2}-3z+4)}-\tfrac{9}{z-2}-\tfrac{27}{4(z-2)^{2}}+\tfrac{315}{64z}+\tfrac{9}{16z^{2}}-\tfrac{9}{4z^{3}})f_{26}$
$\displaystyle-\tfrac{152}{3(z-2)}+\tfrac{56}{(z-2)^{2}}-\tfrac{550}{3(z-1)}+\tfrac{160}{2z-1}+\tfrac{32(z-2)}{(z^{2}-2z-1)^{2}}-\tfrac{292}{3(z+1)}+\tfrac{580}{3z}+\tfrac{202}{3z^{2}}+\tfrac{160}{z^{3}}+\tfrac{1096}{3z^{4}}$
$\displaystyle-\tfrac{384}{z^{5}}+\tfrac{384}{z^{6}}\Big{\\}}\tfrac{64\sqrt{2}\pi^{2}}{9},$
(60) $\displaystyle T$ ${}_{0,{\rm
f}}^{(1)}(h_{c})=\Big{\\{}-\tfrac{4(10z-19)}{z^{2}-2z-1}+(\tfrac{-683z^{2}+3216z-4289}{32(z^{3}-4z^{2}+3z+4)}+\tfrac{40}{27(z-3)}-\tfrac{794}{3(z-2)}-\tfrac{272}{z-1}+\tfrac{4(127z-206)}{z^{2}-2z-1}$
$\displaystyle-\tfrac{8(12z+7)}{(z^{2}-2z-1)^{2}}-\tfrac{64(2z+1)}{(z^{2}-2z-1)^{3}}+\tfrac{26}{3(z+1)}-\tfrac{37271}{864z}+\tfrac{52447}{72z^{2}}+\tfrac{1117}{6z^{3}}+\tfrac{634}{z^{4}}+\tfrac{296}{z^{5}}-\tfrac{544}{z^{6}}$
$\displaystyle+\tfrac{384}{z^{7}})f_{2}+(\tfrac{-683z^{2}+3216z-4289}{16(z^{3}-4z^{2}+3z+4)}+\tfrac{80}{27(z-3)}-\tfrac{1301}{9(z-2)}-\tfrac{10}{(z-2)^{2}}+\tfrac{418}{z-1}+\tfrac{448}{9(2z-1)}-\tfrac{16}{3(2z-1)^{2}}$
$\displaystyle+\tfrac{80}{3(z+1)}+\tfrac{8}{3(z+1)^{2}}-\tfrac{125399}{432z}-\tfrac{15257}{36z^{2}}-\tfrac{147}{z^{3}}-\tfrac{1496}{3z^{4}}-\tfrac{688}{3z^{5}}-\tfrac{544}{z^{6}}+\tfrac{384}{z^{7}})f_{3}+(\tfrac{8(127z-206)}{z^{2}-2z-1}$
$\displaystyle+\tfrac{683z^{2}-3216z+4289}{32(z^{3}-4z^{2}+3z+4)}-\tfrac{40}{27(z-3)}-\tfrac{1615}{3(z-2)}-\tfrac{6}{(z-2)^{2}}+\tfrac{1052}{3(z-1)}-\tfrac{32}{3(z-1)^{2}}-\tfrac{16(12z+7)}{(z^{2}-2z-1)^{2}}$
$\displaystyle-\tfrac{128(2z+1)}{(z^{2}-2z-1)^{3}}+\tfrac{52}{3(z+1)}-\tfrac{797929}{864z}+\tfrac{12305}{72z^{2}}+\tfrac{223}{6z^{3}}+\tfrac{1334}{3z^{4}}+\tfrac{1576}{3z^{5}})f_{4}+(-\tfrac{8(127z-206)}{z^{2}-2z-1}$
$\displaystyle+\tfrac{128(2z+1)}{(z^{2}-2z-1)^{3}}+\tfrac{16(12z+7)}{(z^{2}-2z-1)^{2}}+\tfrac{6208}{9(z-2)}-\tfrac{164}{3(z-1)}+\tfrac{32}{3(z-1)^{2}}-\tfrac{896}{9(2z-1)}+\tfrac{32}{3(2z-1)^{2}}-\tfrac{20}{z+1}$
$\displaystyle-\tfrac{16}{3(z+1)^{2}}+\tfrac{1928}{3z}-\tfrac{2000}{3z^{2}}-\tfrac{168}{z^{3}}-\tfrac{2000}{z^{4}})f_{5}+(\tfrac{-3075z^{2}+14096z-18121}{176(z^{3}-4z^{2}+3z+4)}-\tfrac{2(17z^{2}-99z-114)}{11(z^{3}-5z^{2}-8z-4)}$
$\displaystyle-\tfrac{13}{3(z-2)}-\tfrac{26}{3(z+1)}+\tfrac{457}{16z}-\tfrac{51}{4z^{2}}+\tfrac{11}{z^{3}}+\tfrac{12}{z^{4}})f_{7}+(\tfrac{20}{(z-2)^{2}}+\tfrac{96}{z-1}-\tfrac{128}{3(z+1)}-\tfrac{8}{z}-\tfrac{68}{z^{2}}+\tfrac{48}{z^{3}}$
$\displaystyle-\tfrac{16}{z^{4}}+\tfrac{64}{z^{5}}-\tfrac{136}{3(z-2)})f_{8}+(\tfrac{4}{3(z-2)^{2}}+\tfrac{1376}{3(z+1)}+\tfrac{320}{3(z+1)^{2}}-\tfrac{456}{z}+\tfrac{356}{z^{2}}-\tfrac{240}{z^{3}}+\tfrac{144}{z^{4}}-\tfrac{64}{z^{5}}$
$\displaystyle-\tfrac{8}{3(z-2)})f_{9}+(-\tfrac{2}{(z-2)^{2}}+\tfrac{128}{3(z+1)}-\tfrac{46}{z}+\tfrac{38}{z^{2}}-\tfrac{32}{z^{3}}+\tfrac{24}{z^{4}}+\tfrac{10}{3(z-2)})f_{10}+(-\tfrac{6}{(z-2)^{2}}-\tfrac{2}{z}+\tfrac{2}{z^{2}}$
$\displaystyle+\tfrac{8}{z^{4}}+\tfrac{2}{z-2})f_{11}+(-z^{2}+4z-\tfrac{2}{z-2}-\tfrac{2}{z-1}+\tfrac{2}{z+1}-\tfrac{6}{z^{2}}+\tfrac{12}{z^{3}}-4+\tfrac{12}{z})f_{12}+(-9z-\tfrac{35}{2(z-2)}$
$\displaystyle-\tfrac{18}{z-1}-\tfrac{2}{z+1}-\tfrac{9}{2z}+\tfrac{73}{z^{2}}+\tfrac{10}{z^{3}}+\tfrac{76}{z^{4}}-8)f_{13}+(z^{2}+27z+\tfrac{45}{2(z-2)}-\tfrac{32}{z-1}+\tfrac{75}{2z}+\tfrac{33}{z^{2}}-\tfrac{30}{z^{3}}$
$\displaystyle+\tfrac{36}{z^{4}}+8)f_{15}+(-\tfrac{22}{z+1}+\tfrac{22}{z}-\tfrac{32}{z^{2}}+\tfrac{36}{z^{3}}-\tfrac{16}{z^{4}}+2+\tfrac{4}{z-2})f_{16}+(-z^{2}+2z-\tfrac{10}{3(z-2)}+\tfrac{64}{3(z+1)}$
$\displaystyle+\tfrac{18}{z^{2}}-\tfrac{4}{z^{3}}-1-\tfrac{20}{z})f_{18}+(-6z^{2}-9z-\tfrac{51}{z-2}+\tfrac{22}{z+1}-\tfrac{6}{z^{2}}-\tfrac{50}{z^{3}}+\tfrac{24}{z^{4}}-65+\tfrac{27}{z})f_{19}$
$\displaystyle+(-\tfrac{5z^{2}}{2}-\tfrac{89z}{2}-\tfrac{115}{2(z-2)}+\tfrac{55}{2z}-\tfrac{20}{z^{2}}+\tfrac{14}{z^{3}}-\tfrac{8}{z^{4}})f_{20}+(-\tfrac{64}{3(z+1)}+\tfrac{22}{z}-\tfrac{36}{z^{2}}+\tfrac{40}{z^{3}}-\tfrac{16}{z^{4}}$
$\displaystyle-\tfrac{2}{3(z-2)})f_{21}+(-\tfrac{z^{2}}{2}-\tfrac{z}{2}+\tfrac{213}{2(z-2)}-\tfrac{16}{z-1}-\tfrac{13}{2z}-\tfrac{46}{z^{2}}+\tfrac{4}{z^{3}}-\tfrac{16}{z^{4}}+55)f_{22}+(\tfrac{2}{z-1}-\tfrac{10}{z}-\tfrac{16}{z^{2}}$
$\displaystyle-\tfrac{28}{z^{3}}-\tfrac{16}{z^{4}}+2+\tfrac{16}{z-2})f_{23}+(4z^{2}+20z+\tfrac{48}{z-2}-\tfrac{24}{z^{2}}+36+\tfrac{24}{z})f_{24}-\tfrac{16}{z-2}+\tfrac{24}{(z-2)^{2}}-\tfrac{278}{3(z-1)}$
$\displaystyle-\tfrac{16}{2z-1}-\tfrac{32}{(z^{2}-2z-1)^{2}}-\tfrac{132}{z+1}+\tfrac{598}{3z}+\tfrac{152}{z^{2}}+\tfrac{52}{3z^{3}}+\tfrac{1432}{3z^{4}}-\tfrac{352}{z^{5}}+\tfrac{384}{z^{6}}\Big{\\}}\tfrac{64\pi^{2}}{3\sqrt{3}}.$
(61)

Here $z\equiv 2r-1$, and $f_{i}$ are defined as

$\displaystyle f_{1}=i\pi,\quad f_{2}=\ln(2),\quad
f_{3}=\ln(\tfrac{1-z}{2}),\quad f_{4}=\ln(\tfrac{1+z}{2}),\quad f_{5}=\ln(z),$
$\displaystyle f_{6}=\ln(\tfrac{z^{2}-3z+4}{2}),\quad
f_{7}=\sqrt{\tfrac{z^{3}-5z^{2}-8z-4}{(z-1)z^{2}}}\ln\Big{(}\tfrac{\sqrt{\tfrac{z^{3}-5z^{2}-8z-4}{(z-1)z^{2}}}-1}{\sqrt{\tfrac{z^{3}-5z^{2}-8z-4}{(z-1)z^{2}}}+1}\Big{)},$
$\displaystyle
f_{8}=\tfrac{1}{2}(\tfrac{1-z}{2})^{2}((1-2\ln(\tfrac{1-z}{2}))^{2}+\tfrac{\pi^{2}}{3}+1),\quad
f_{9}=\tfrac{1}{2}(\tfrac{1+z}{2})^{2}((1-2\ln(\tfrac{z+1}{2}))^{2}+\tfrac{\pi^{2}}{3}+1),$
$\displaystyle
f_{10}=\tfrac{(1-z)^{2}}{2(3-z)z}\ln(\tfrac{(1-z)^{2}}{1+z})(\ln(\tfrac{16(3-z)^{2}}{(1+z)^{3}(1-z)^{2}(2-z)^{2}})+4)+2(\ln(\tfrac{z+1}{2})-2)\ln(\tfrac{z+1}{2})$
$\displaystyle\quad+\tfrac{1}{2z}(\ln(\tfrac{(1+z)(1-z)^{4}}{16})-4)\ln(1+z)-2\ln(\tfrac{(1-z)^{2}}{1+z})\ln(\tfrac{3-z}{2-z}))$
$\displaystyle\quad+\tfrac{(1+z)(2-z)}{(3-z)z}({\rm
Li}_{2}(-\tfrac{1}{2-z})-{\rm
Li}_{2}(-\tfrac{(1-z)^{2}}{(1+z)(2-z)}))+\tfrac{\pi^{2}}{6}+4,$ $\displaystyle
f_{11}=-\tfrac{(6-7z)}{2z}\ln^{2}(\tfrac{1-z}{2})+\tfrac{2(2-3z)}{z}\ln(\tfrac{1-z}{2})+\tfrac{(z-2)}{z}({\rm
Li}_{2}(\tfrac{1}{2-z})-{\rm Li}_{2}(\tfrac{1-z}{2-z}))$
$\displaystyle\quad-\tfrac{(z-2)}{2z}(\ln(2)(3\ln(2)+4)-2\ln(1-z)\ln(2(2-z)))+\tfrac{\pi^{2}}{6}+4,$
$\displaystyle
f_{12}=C_{0}(0,\tfrac{1}{4}(z-1)z^{2},-\tfrac{1}{4}(z-3)z^{2},\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(z+1)^{2},\tfrac{1}{4}(z+1)^{2}),$
$\displaystyle
f_{13}=C_{0}(0,\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(-z^{2}+2z+1),\tfrac{1}{4}(z+1)^{2},\tfrac{1}{4}(z-1)^{2},0),$
$\displaystyle
f_{14}=C_{0}(0,\tfrac{1}{4}(z-1)^{2},-\tfrac{1}{4}(z-1)(2z^{2}-5z+1),\tfrac{1}{4}(z+1)^{2},\tfrac{1}{4}(z-1)^{2},0),$
$\displaystyle
f_{15}=C_{0}(0,\tfrac{1}{4}(z+1)^{2},\tfrac{1}{4}(z-1)(z+1)(2z-1),\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(z+1)^{2},0),$
$\displaystyle
f_{16}=C_{0}(-\tfrac{1}{4}(z-3)z^{2},\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(-z^{2}+2z+1),\tfrac{1}{4}(z+1)^{2},\tfrac{1}{4}(z-1)^{2},0),$
$\displaystyle
f_{17}=C_{0}(-\tfrac{1}{4}(z-3)z^{2},\tfrac{1}{4}(z-1)^{2},-\tfrac{1}{4}(z-1)(2z^{2}-5z+1),\tfrac{1}{4}(z+1)^{2},\tfrac{1}{4}(z-1)^{2},0),$
$\displaystyle
f_{18}=C_{0}(\tfrac{1}{4}(z-1)z^{2},-\tfrac{1}{4}(z-3)z^{2},0,\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(z+1)^{2}),$
$\displaystyle
f_{19}=C_{0}(\tfrac{1}{4}(z-1)z^{2},\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(z-1)^{2},0,0,\tfrac{1}{4}(z-1)^{2}),$
$\displaystyle
f_{20}=C_{0}(\tfrac{1}{4}(z-1)z^{2},\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(z-1)(z+1)(2z-1),0,0,\tfrac{1}{4}(z-1)^{2}),$
$\displaystyle
f_{21}=C_{0}(\tfrac{1}{4}(z-1)z^{2},\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(z-1)(z+1)(2z-1),\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(z-1)^{2},0),$
$\displaystyle
f_{22}=C_{0}(\tfrac{1}{4}(z-1)z^{2},\tfrac{1}{4}(z+1)^{2},\tfrac{1}{4}(-z^{2}+2z+1),0,0,\tfrac{1}{4}(z+1)^{2}),$
$\displaystyle
f_{23}=C_{0}(\tfrac{1}{4}(z-1)z^{2},\tfrac{1}{4}(z+1)^{2},\tfrac{1}{4}(-z^{2}+2z+1),\tfrac{1}{4}(z+1)^{2},\tfrac{1}{4}(z+1)^{2},0),$
$\displaystyle
f_{24}=C_{0}(\tfrac{1}{4}(z-1)^{2},(z-1)^{2},\tfrac{1}{4}(z-1)(z+1)(2z-1),0,\tfrac{1}{4}(z-1)^{2},\tfrac{1}{4}(z-1)^{2}),$
$\displaystyle
f_{25}=C_{0}(\tfrac{1}{4}(z-1)^{2},(z-1)^{2},\tfrac{1}{4}(z-1)(z+1)(2z-1),\tfrac{1}{4}(z-1)^{2},0,0),$
$\displaystyle
f_{26}=C_{0}((z-1)^{2},\tfrac{1}{4}(z+1)^{2},-\tfrac{1}{4}(z-1)(2z^{2}-5z+1),0,0,\tfrac{1}{4}(z+1)^{2}),$
(62)

where $C_{0}$ is the standard Passarino-Veltman scalar three-point function.