BARI-TH/22-734

Semileptonic $B_{c}$ decays to $P-$wave charmonium
and the nature of $\chi_{c1}(3872)$

P. Colangeloa, F. De Fazioa, F. Loparcoa , N. Losaccoa,b, M. Novoa-Bruneta

aIstituto Nazionale di Fisica Nucleare, Sezione di Bari, Via Orabona 4,
I-70126 Bari, Italy
bDipartimento Interateneo di Fisica “M. Merlin”, Università e Politecnico di
Bari,
via Orabona 4, 70126 Bari, Italy

###### Abstract

The hadronic form factors of $B_{c}$ semileptonic decays to the $P-$wave
charmonium 4-plet can be expressed near the zero-recoil point in terms of
universal functions, performing a systematic expansion in QCD in the relative
velocity of the heavy quarks and in $1/m_{Q}$. Such functions are independent
of the member of the multiplet involved in the transitions. We present the
results of an NLO calculation up to $O(1/m_{Q}^{2})$ classifying the universal
functions at this order. We work out a set of relations among the form factors
of the same mode and of different modes, which should be reproduced by
explicit calculations, reducing the hadronic uncertainty affecting such
channels. The approach is also helpful to investigate the debated nature of
$\chi_{c1}(3872)$, studying the production in $B_{c}$ semileptonic decays and
comparing it to the modes involving the other $2P$ charmonia.

##  1 Introduction

There is a manifold interest in the study of semileptonic decays of heavy
hadrons, namely those induced by the $b\to c\ell\bar{\nu}_{\ell}$ transition
at the quark level. The first one is the possibility to precisely measure
fundamental parameters of the Standard Model (SM), in this case the element
$|V_{cb}|$ of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix. The analyses
of processes involving different hadrons, modes and final states
(exclusive/inclusive) should provide compatible results: such a compatibility
has not been achieved, yet, considering the debated $|V_{cb}|$ determinations
from inclusive and exclusive $B$ decays [1] which fuel discussions on possible
explanations of the tension within the SM [2] or beyond [3]. Other processes,
e.g. $B_{c}$ to charmonium, give access to such a fundamental parameter which
has tight correlations with flavour observables [4].

Another important interest relies on the possibility of testing fundamental
features of the Standard Model, namely Lepton Flavour Universality (LFU).
Signals of LFU violation have been detected in $B$ decays induced by this
transition [5]. They hint to physics beyond SM [6], the structure of which can
be constrained by studying sets of decay observables [7, 8, 9, 10, 11, 12, 13,
14]. Related LFU violating effects should be observed in processes involving
different hadrons: the first measurements are available for $B_{c}\to
J/\psi\ell\bar{\nu}_{\ell}$ [15], other $B_{c}$ modes can be considered for
such investigations.

In addition to the above motivations concerning the structure of the theory of
electroweak interaction and of its extensions, there is interest related to
strong interaction effects. For decays involving hadrons comprising a single
heavy quark $Q$, a double expansion in QCD in powers of $1/m_{Q}$ and of
$\alpha_{s}$ provides a powerful method to classify the hadronic matrix
elements, both in the exclusive and inclusive transitions. This is the basis
for an efficient control of the theoretical uncertainty in the above mentioned
measurements. For mesons comprising two heavy quarks such as $B_{c}$, the
expansion parameter is the relative three-velocity of the heavy quarks, with
counting rules given within Nonrelativistic QCD (NRQCD). At the various orders
in the expansion the form factors governing $B_{c}\to
J/\psi(\eta_{c})\ell\bar{\nu}_{\ell}$ can be given in terms of universal
functions, the same for the two final states, in selected kinematical ranges
[16, 17, 18]. Relations can be established among the form factors, with a
reduction of the hadronic uncertainties affecting the description of the
semileptonic channels. It is worth considering such an expansion also for the
form factors governing the $B_{c}$ transitions to the $P$-wave charmonia
involving a spin 4-plet of final states, for which little information is
available at present.111The peculiar role of $B_{c}$, a weakly decaying meson
comprising only heavy quarks, has been recently discussed in [19, 20].

There is a further reason of interest. Semileptonic decays provide us with a
probe on the structure of the hadrons involved in the transitions. This is
important, for example, if one considers the meson $\chi_{c1}(3872)$ (usually
denoted as $X(3872)$). After the observation by the Belle Collaboration [21]
and the confirmation and measurements by other collaborations [22, 23, 24, 25,
26, 27, 28, 29, 30], such a meson, whose quantum numbers are $J^{PC}=1^{++}$
[31, 32], is under intense scrutiny due to features hinting a non conventional
charmonium structure. Puzzling properties are the closeness of the mass of
$X(3872)$ to the $D^{*0}\bar{D}^{0}$ threshold and the large decay rate to
$J/\psi\,\pi\pi\,(J/\psi\,\rho)$ compared to
$J/\psi\,\pi\pi\pi\,(J/\psi\,\omega)$ [33], which can be explained invoking a
multiquark structure of $X(3872)$, compact or molecular (see the discussions
in Refs.[34, 35] and in references therein). On the other hand, the large
$\Gamma(X(3872)\to\gamma\,\psi(2S))/\Gamma(X(3872)\to\gamma\,J/\psi)$ ratio
[36] and data on the production in $\gamma\gamma^{*}$ interactions [37]
support the identification of $\chi_{c1}(3872)$ with the state $\chi_{c1}(2P)$
sitting on the $D^{*0}\bar{D}^{0}$ threshold [38, 39]. Experimental tests have
been proposed to help decipher the structure of the meson [40], in the vast
literature on the subject.

Semileptonic $B_{c}$ decays to $\chi_{c1}(3872)$ can provide us with
information on the nature of this state. The way is inspired by our analysis
in [18] and is based on the systematic comparison of the $B_{c}$ transitions
to the $2P$ charmonia. In a selected kinematical range the $B_{c}$ and the
P-wave charmonium matrix elements can be expressed as an expansion in the
heavy quark relative three-velocity in the heavy hadrons, together with an
expansion in the inverse heavy quark mass. In this range the $B_{c}$ to the
P-wave charmonia form factors (for lowest lying or radial excitations) are
related. The violation or confirmation of such relations in
$B_{c}\to\chi_{c1}(3872)\ell\bar{\nu}$ with respect to the decays to the other
$2P$ charmonia would support the interpretation of $X(3872)$ as an exotic or a
conventional state.222 The comparison between $B_{c}$ semileptonic and
nonleptonic $B_{c}\to X(3872)\rho(a_{1})$ decays, fixing the $X(3872)$
polarization, has been proposed as a way to investigate the structure of this
meson [41]. Semileptonic $B_{c}$ decays to $X(3872)$ have been considered in
[42].

In this paper we focus on the $B_{c}$ to the $P$-wave charmonia form factors,
and compute their expressions in terms of universal functions at NLO. We
classify the universal functions and work out a set of relations useful both
to test calculations based on nonperturbative methods and in phenomenological
analyses. In Sec. 2 we parametrize the $B_{c}\to\chi_{c0,1,2}$ and $B_{c}\to
h_{c}$ matrix elements of the quark currents appearing in a generalized low-
energy Hamiltonian governing the $b\to c\ell\bar{\nu}_{\ell}$ transition,
providing the definition of the set of form factors considered in our study.
For the sake of completeness, in Appendix A we also report another
parameterization often employed in the literature, with straightforward
relations between the two. We give the decay distributions of the four
$B_{c}\to\chi_{c0,1,2}(h_{c})\ell\bar{\nu}_{\ell}$ modes, which can also be
used for $B_{c}$ decays to the $2P$ charmonium resonances. In Sec. 3 we
briefly describe the heavy quark expansion in QCD, we introduce the
$(B_{c},B_{c}^{*})$ spin doublet and the $(\chi_{c0,1,2},h_{c})$ spin 4-plet
and the trace formalism to express the relevant hadronic matrix elements. In
Sec. 4 we write the form factors in terms of universal functions, with the
resulting formulae collected in Appendix B. In Sec. 5 we present relations
among several form factors for single modes and for pairs of modes, other
relations being collected in Appendix C. Applications to phenomenology are
discussed in Sec. 6, for the lowest lying $1P$ charmonia and for the $2P$
excitations in connection with the $\chi_{c1}(3872)$ issue. Then we conclude.

##  2 $B_{c}\to\chi_{c0,1,2}$ and $B_{c}\to h_{c}$ form factors

The semileptonic $B_{c}$ decays to the charmonium states, including the
positive parity $\chi_{c0,1,2}$ and $h_{c}$, are governed by a low energy
Hamiltonian with general form

$\displaystyle H_{\rm eff}^{b\to c\ell\bar{\nu}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}\Big{[}(1+\epsilon_{V}^{\ell})\left({\bar{c}}\gamma_{\mu}(1-\gamma_{5})b\right)\left(\bar{\ell}\gamma^{\mu}(1-\gamma_{5})\nu_{\ell}\right)+\epsilon_{R}^{\ell}\left({\bar{c}}\gamma_{\mu}(1+\gamma_{5})b\right)\left(\bar{\ell}\gamma^{\mu}(1-\gamma_{5})\nu_{\ell}\right)$
$\displaystyle+$
$\displaystyle\epsilon_{S}^{\ell}\,({\bar{c}}b)\left({\bar{\ell}}(1-\gamma_{5})\nu_{\ell}\right)+\epsilon_{P}^{\ell}\,\left({\bar{c}}\gamma_{5}b\right)\left({\bar{\ell}}(1-\gamma_{5})\nu_{\ell}\right)$
$\displaystyle+$
$\displaystyle\epsilon_{T}^{\ell}\,\left({\bar{c}}\sigma_{\mu\nu}(1-\gamma_{5})b\right)\,\left({\bar{\ell}}\sigma^{\mu\nu}(1-\gamma_{5})\nu_{\ell}\right)\Big{]}$

considering the full set of $D=6$ semileptonic $b\to c$ operators with left-
handed neutrinos. The general Hamiltonian (2) comprises the Fermi constant
$G_{F}$, the element $V_{cb}$ of the CKM matrix and the Standard Model
operator ${\cal
O}_{SM}=4(\bar{c}_{L}\gamma^{\mu}b_{L})\left({\bar{\ell}_{L}}\gamma_{\mu}\nu_{\ell
L}\right)$. It also includes the operators ${\cal
O}_{R}=4(\bar{c}_{R}\gamma^{\mu}b_{R})\left({\bar{\ell_{L}}}\gamma_{\mu}\nu_{\ell
L}\right)$, ${\cal
O}_{S}=\left({\bar{c}}b\right)\left({\bar{\ell}}(1-\gamma_{5})\nu_{\ell}\right)$,
${\cal
O}_{P}=\left({\bar{c}}\gamma_{5}b\right)\left({\bar{\ell}}(1-\gamma_{5})\nu_{\ell}\right)$
and ${\cal
O}_{T}=\left({\bar{c}}\sigma_{\mu\nu}(1-\gamma_{5})b\right)\left({\bar{\ell}}\sigma^{\mu\nu}(1-\gamma_{5})\nu_{\ell}\right)$
arising in extensions of SM, with Wilson coefficients
$\epsilon^{\ell}_{V,R,S,P,T}$ in general complex and lepton-flavour dependent.
The SM Hamiltonian corresponds to $\epsilon^{\ell}_{i}=0$. Eq. (2) has been
considered in connection with the $B\to D^{(*)}\tau\nu_{\tau}$, $B\to
D^{(*)}\ell\nu_{\ell}$ LFU anomaly [12, 8, 3, 10, 9, 11, 13].

It is common to parametrize the $B_{c}\to\chi_{ci}$ ($i=0,1,2$) and $B_{c}\to
h_{c}$ matrix elements of the quark currents in Eq. (2) in terms of form
factors as written in Appendix A [43, 44, 45, 46]. $p$ and $p^{\prime}$ are
the four momenta of the initial and final meson, $\epsilon$ ($\eta$) the
polarization vector (tensor) of the spin $1$ (spin $2$) charmonium,
$q=p-p^{\prime}$ the lepton pair momentum. We use $\epsilon^{0123}=-1$, hence
$\sigma^{\mu\nu}\gamma_{5}=-\frac{i}{2}\varepsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta}$.
In our analysis it is more convenient to use a parametrization of the matrix
elements in terms of the four-velocities of $B_{c}$ and of the charmonium
state $C=\chi_{c0,1,2},h_{c}$: $v=p/m_{B_{c}}$ and
$v^{\prime}=p^{\prime}/m_{C}$, with $w=v\cdot v^{\prime}$. They are defined as
follows.

$B_{c}\to\chi_{c0}$:

$\displaystyle\langle\chi_{c0}(v^{\prime})|\bar{c}\gamma_{\mu}\gamma_{5}b|B_{c}(v)\rangle$
$\displaystyle=$
$\displaystyle\sqrt{m_{\chi_{c0}}\,m_{B_{c}}}\,\big{[}g_{+}(w)(v+v^{\prime})_{\mu}+g_{-}(w)(v-v^{\prime})_{\mu}\big{]}$
$\displaystyle\langle\chi_{c0}(v^{\prime})|\bar{c}\gamma_{5}b|B_{c}(v)\rangle$
$\displaystyle=$ $\displaystyle\sqrt{m_{\chi_{c0}}\,m_{B_{c}}}\,\,g_{P}(w)$
(2)
$\displaystyle\langle\chi_{c0}(v^{\prime})|\bar{c}\sigma_{\mu\nu}b|B_{c}(v)\rangle$
$\displaystyle=$
$\displaystyle\sqrt{m_{\chi_{c0}}\,m_{B_{c}}}\,\,g_{T}(w)\,\epsilon_{\mu\nu\alpha\beta}v^{\alpha}v^{\prime\beta}$

$B_{c}\to\chi_{c1}$:

$\displaystyle\langle\chi_{c1}(v^{\prime},\epsilon)|\bar{c}\gamma_{\mu}b|B_{c}(v)\rangle$
$\displaystyle=$ $\displaystyle
i\,\sqrt{m_{\chi_{c1}}\,m_{B_{c}}}\,\Big{[}g_{V_{1}}(w)\epsilon_{\mu}^{*}$
$\displaystyle+$ $\displaystyle(\epsilon^{*}\cdot
v)[g_{V_{2}}(w)(v+v^{\prime})_{\mu}+g_{V_{3}}(w)(v-v^{\prime})_{\mu}]\Big{]}$
$\displaystyle\langle\chi_{c1}(v^{\prime},\epsilon)|\bar{c}\gamma_{\mu}\gamma_{5}b|B_{c}(v)\rangle$
$\displaystyle=$
$\displaystyle\sqrt{m_{\chi_{c1}}\,m_{B_{c}}}\,g_{A}(w)\,\epsilon_{\mu\alpha\beta\sigma}\epsilon^{*\alpha}v^{\beta}v^{\prime\sigma}$
$\displaystyle\langle\chi_{c1}(v^{\prime},\epsilon)|\bar{c}b|B_{c}(v)\rangle$
$\displaystyle=$ $\displaystyle
i\,\sqrt{m_{\chi_{c1}}\,m_{B_{c}}}\,g_{S}(w)\,(\epsilon^{*}\cdot v)$ (3)
$\displaystyle\langle\chi_{c1}(v^{\prime},\epsilon)|\bar{c}\sigma_{\mu\nu}b|B_{c}(v)\rangle$
$\displaystyle=$
$\displaystyle\sqrt{m_{\chi_{c1}}\,m_{B_{c}}}\Big{[}g_{T_{1}}(w)\big{(}\epsilon_{\mu}^{*}(v+v^{\prime})_{\nu}-\epsilon_{\nu}^{*}(v+v^{\prime})_{\mu}\big{)}$
$\displaystyle+$ $\displaystyle
g_{T_{2}}(w)\big{(}\epsilon_{\mu}^{*}(v-v^{\prime})_{\nu}-\epsilon_{\nu}^{*}(v-v^{\prime})_{\mu}\big{)}$
$\displaystyle+$ $\displaystyle g_{T_{3}}(w)(\epsilon^{*}\cdot
v)(v_{\mu}v^{\prime}_{\nu}-v_{\nu}v_{\mu}^{\prime})\Big{]}$

$B_{c}\to\chi_{c2}$:

$\displaystyle\langle\chi_{c2}(v^{\prime},\eta)|\bar{c}\gamma_{\mu}b|B_{c}(v)\rangle$
$\displaystyle=$
$\displaystyle\sqrt{m_{\chi_{c2}}\,m_{B_{c}}}\,i\,k_{V}(w)\,\epsilon_{\mu\alpha\beta\sigma}\eta^{*\alpha\tau}v_{\tau}v^{\beta}v^{\prime\sigma}$
$\displaystyle\langle\chi_{c2}(v^{\prime},\eta)|\bar{c}\gamma_{\mu}\gamma_{5}b|B_{c}(v)\rangle$
$\displaystyle=$
$\displaystyle\sqrt{m_{\chi_{c2}}\,m_{B_{c}}}\,\big{[}k_{A_{1}}(w)\,\eta_{\mu\alpha}^{*}v^{\alpha}+\eta_{\alpha\beta}^{*}v^{\alpha}v^{\beta}\left(k_{A_{2}}(w)v_{\mu}+k_{A_{3}}(w)v^{\prime}_{\mu}\right)\big{]}$
$\displaystyle\langle\chi_{c2}(v^{\prime},\eta)|\bar{c}\gamma_{5}b|B_{c}(v)\rangle$
$\displaystyle=$
$\displaystyle\sqrt{m_{\chi_{c2}}\,m_{B_{c}}}k_{P}(w)\,\eta_{\alpha\beta}^{*}v^{\alpha}v^{\beta}$
(4)
$\displaystyle\langle\chi_{c2}(v^{\prime},\eta)|\bar{c}\sigma_{\mu\nu}\gamma_{5}b|B_{c}(v)\rangle$
$\displaystyle=$ $\displaystyle
i\sqrt{m_{\chi_{c2}}\,m_{B_{c}}}\,\big{[}k_{T_{1}}(w)({\eta^{*}}_{\mu}^{\alpha}v_{\alpha}v_{\nu}-{\eta^{*}}_{\nu}^{\alpha}v_{\alpha}v_{\mu})+$
$\displaystyle+$ $\displaystyle
k_{T_{2}}(w)({\eta^{*}}_{\mu}^{\alpha}v_{\alpha}v^{\prime}_{\nu}-{\eta^{*}}_{\nu}^{\alpha}v_{\alpha}v^{\prime}_{\mu})+k_{T_{3}}(w)\eta_{\alpha\beta}^{*}v^{\alpha}v^{\beta}(v_{\mu}v^{\prime}_{\nu}-v_{\nu}v^{\prime}_{\mu})\big{]}$

$B_{c}\to h_{c}$:

$\displaystyle\langle
h_{c}(v^{\prime},\epsilon)|\bar{c}\gamma_{\mu}b|B_{c}(v)\rangle$
$\displaystyle=$
$\displaystyle\sqrt{m_{h_{c}}\,m_{B_{c}}}\,\Big{[}f_{V_{1}}(w)\epsilon_{\mu}^{*}$
$\displaystyle+$ $\displaystyle(\epsilon^{*}\cdot
v)\big{(}f_{V_{2}}(w)(v+v^{\prime})_{\mu}+f_{V_{3}}(w)(v-v^{\prime})_{\mu}\big{)}\Big{]}$
$\displaystyle\langle
h_{c}(v^{\prime},\epsilon)|\bar{c}\gamma_{\mu}\gamma_{5}b|B_{c}(v)\rangle$
$\displaystyle=$ $\displaystyle
i\sqrt{m_{h_{c}}\,m_{B_{c}}}\,f_{A}(w)\,\epsilon_{\mu\alpha\beta\sigma}\epsilon^{*\alpha}v^{\beta}v^{\prime\sigma}$
$\displaystyle\langle h_{c}(v^{\prime},\epsilon)|\bar{c}b|B_{c}(v)\rangle$
$\displaystyle=$ $\displaystyle\sqrt{m_{h_{c}}\,m_{B_{c}}}\,(\epsilon^{*}\cdot
v)f_{S}(w)$ (5) $\displaystyle\langle
h_{c}(v^{\prime},\epsilon)|\bar{c}\sigma_{\mu\nu}b|B_{c}(v)\rangle$
$\displaystyle=$ $\displaystyle
i\,\sqrt{m_{h_{c}}\,m_{B_{c}}}\Big{[}f_{T_{1}}(w)\big{(}\epsilon_{\mu}^{*}(v+v^{\prime})_{\nu}-\epsilon_{\nu}^{*}(v+v^{\prime})_{\mu}\big{)}$
$\displaystyle+$ $\displaystyle
f_{T_{2}}(w)\big{(}\epsilon_{\mu}^{*}(v-v^{\prime})_{\nu}-\epsilon_{\nu}^{*}(v-v^{\prime})_{\mu}\big{)}$
$\displaystyle+$ $\displaystyle\,f_{T3}(w)\,(\epsilon^{*}\cdot
v)\left(v_{\mu}v^{\prime}_{\nu}-v_{\nu}v^{\prime}_{\mu}\right)\Big{]}.$

The decay distributions governed by the Hamiltonian (2) can be written in the
form:

$\displaystyle\frac{d\Gamma(B_{c}\to C\ell\bar{\nu})}{dw}$ $\displaystyle=$
$\displaystyle{\tilde{\Gamma}}\Big{\\{}|1+\epsilon_{V}|^{2}\,\frac{d\Gamma}{dw}^{SM}+|\epsilon_{R}|^{2}\frac{d\Gamma}{dw}^{R}+|\epsilon_{X}|^{2}\frac{d\Gamma}{dw}^{X}+|\epsilon_{T}|^{2}\frac{d\Gamma}{dw}^{T}$
$\displaystyle+$ $\displaystyle 2\,{\rm
Re}\left[\epsilon_{R}(1+\epsilon_{V}^{*})\right]\frac{d\Gamma}{dw}^{SMR}+2\,{\rm
Re}\left[\epsilon_{X}(1+\epsilon_{V}^{*})\right]\frac{d\Gamma}{dw}^{SMX}$
$\displaystyle+$ $\displaystyle 2\,{\rm
Re}\left[\epsilon_{T}(1+\epsilon_{V}^{*})\right]\frac{d\Gamma}{dw}^{SMT}+2\,{\rm
Re}\left[\epsilon_{R}\epsilon_{T}^{*}\right]\frac{d\Gamma}{dw}^{RT}$
$\displaystyle+$ $\displaystyle 2\,{\rm
Re}\left[\epsilon_{X}\epsilon_{R}^{*}\right]\frac{d\Gamma}{dw}^{XR}+2\,{\rm
Re}\left[\epsilon_{X}\epsilon_{T}^{*}\right]\frac{d\Gamma}{dw}^{XT}\Big{\\}},\,\,\,\,$

with $X=P$ in case of $\chi_{c0},\,\chi_{c2}$, and $X=S$ in case of
$\chi_{c1},\,h_{c}$. In Eq. (2) we define

${\tilde{\Gamma}}=\displaystyle\frac{G_{F}^{2}|V_{cb}|^{2}m_{B_{c}}^{5}\,r^{3}}{48\pi^{3}}\sqrt{w^{2}-1}\left(1-\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\right)^{2}$
(7)

with $r=m_{C}/m_{B_{c}}$ ($C=\chi_{c0,c1,c2},h_{c}$) and
${\hat{m}}_{\ell}=m_{\ell}/m_{B_{c}}$. For the four decay modes the
expressions of the functions in (2) in terms of the form factors in (2)-(5)
are as follows.

  * •

$C=\chi_{c0}$:

$\displaystyle\frac{d\Gamma}{dw}^{SM}=\frac{d\Gamma}{dw}^{R}=-\frac{d\Gamma}{dw}^{SMR}$
(8) $\displaystyle=$
$\displaystyle[g_{+}(w)]^{2}\,(w+1)\Big{[}(w-1)(1+r)^{2}+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\Big{(}(2w+1)(1+r^{2})-2r(w+2)\Big{)}\Big{]}$
$\displaystyle+$
$\displaystyle[g_{-}(w)]^{2}\,(w-1)\Big{[}(w+1)(1-r)^{2}+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\Big{(}(2w-1)(1+r^{2})+2r(w-2)\Big{)}\Big{]}\quad$
$\displaystyle-$ $\displaystyle
2g_{+}(w)g_{-}(w)\,(w^{2}-1)(1-r^{2})\left(1+\frac{2{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\right)$
$\displaystyle\frac{d\Gamma}{dw}^{P}$ $\displaystyle=$
$\displaystyle\frac{3}{2}[g_{P}(w)]^{2}(1+r^{2}-2rw)$ (9)
$\displaystyle\frac{d\Gamma}{dw}^{T}$ $\displaystyle=$ $\displaystyle
8[g_{T}(w)]^{2}\left(1+r^{2}-2rw+2{\hat{m}}_{\ell}^{2}\right)(w^{2}-1)$ (10)
$\displaystyle\frac{d\Gamma}{dw}^{SMP}$ $\displaystyle=$
$\displaystyle-\frac{d\Gamma}{dw}^{PR}=\frac{3}{2}g_{P}(w){\hat{m}}_{\ell}\,\Big{[}(w-1)(1+r)g_{-}(w)-(w+1)(1-r)g_{+}(w)\Big{]}$
(11) $\displaystyle\frac{d\Gamma}{dw}^{SMT}$ $\displaystyle=$
$\displaystyle-\frac{d\Gamma}{dw}^{RT}=-6g_{T}(w)\,{\hat{m}}_{\ell}(w^{2}-1)\,\Big{[}(1+r)g_{+}(w)-(1-r)g_{-}(w)\Big{]}\,\,.$
(12)

  * •

$C=\chi_{c1}$:

$\displaystyle\frac{d\Gamma}{dw}^{SM}=\frac{d\Gamma}{dw}^{R}$
$\displaystyle=[g_{V_{1}}(w)]^{2}\,\Big{[}3(w-r)^{2}-2(w^{2}-1)+\frac{{\hat{m}}_{\ell}^{2}}{2(1+r^{2}-2rw)}\Big{(}3(w-r)^{2}+w^{2}-1\Big{)}\Big{]}$
$\displaystyle+[g_{V_{2}}(w)]^{2}\,(w^{2}-1)(w+1)\Big{[}(1+r)^{2}(w-1)+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\Big{(}(2w+1)(1+r^{2})-2r(w+2)\Big{)}\Big{]}$
$\displaystyle+[g_{V_{3}}(w)]^{2}\,(w^{2}-1)(w-1)\Big{[}(1-r)^{2}(w+1)+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\Big{(}(2w-1)(1+r^{2})+2r(w-2)\Big{)}\Big{]}$
$\displaystyle+[g_{A}(w)]^{2}\,(w^{2}-1)\Big{(}2(1+r^{2}-2rw)+{\hat{m}}_{\ell}^{2}\Big{)}$
(13)
$\displaystyle+g_{V_{1}}(w)\,g_{V_{2}}(w)\,(w^{2}-1)\Big{[}2(1+r)(w-r)-\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\Big{(}-3+r^{2}-4w+2r(w+2)\Big{)}\Big{]}$
$\displaystyle-
g_{V_{1}}(w)\,g_{V_{3}}(w)\,(w^{2}-1)\Big{[}2(1-r)(w-r)+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\Big{(}-3+r^{2}+4w+2r(w-2)\Big{)}\Big{]}$
$\displaystyle-2g_{V_{2}}(w)\,g_{V_{3}}(w)\,(w^{2}-1)^{2}(1-r^{2})\left(1+\frac{2{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\right)$
$\displaystyle\frac{d\Gamma}{dw}^{SMR}$ $\displaystyle=$
$\displaystyle\frac{d\Gamma}{dw}^{SM}-2[g_{A}(w)]^{2}\,(w^{2}-1)\Big{(}2(1+r^{2}-2rw)+{\hat{m}}_{\ell}^{2}\Big{)}$
(14) $\displaystyle\frac{d\Gamma}{dw}^{S}$ $\displaystyle=$
$\displaystyle\frac{3}{2}[g_{S}(w)]^{2}(1+r^{2}-2rw)(w^{2}-1)$ (15)
$\displaystyle\frac{d\Gamma}{dw}^{T}$ $\displaystyle=$ $\displaystyle
8\left(1+\frac{2{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\right)\,$
$\displaystyle\times$
$\displaystyle\Big{\\{}[g_{T_{1}}(w)]^{2}\,(w+1)\Big{[}(5w+1)(1+r^{2})-2r(w^{2}+w+4)\Big{]}$
$\displaystyle+$
$\displaystyle\,[g_{T_{2}}(w)]^{2}\,(w-1)\Big{[}(5w-1)(1+r^{2})-2r(w^{2}-w+4)\Big{]}$
$\displaystyle+$
$\displaystyle\,[g_{T_{3}}(w)]^{2}\,(w^{2}-1)^{2}(1+r^{2}-2rw)$
$\displaystyle+$ $\displaystyle
2\,g_{T_{1}}(w)\,g_{T_{2}}(w)\,(w^{2}-1)(5r^{2}-2rw-3)$ $\displaystyle-$
$\displaystyle 2\,g_{T_{1}}(w)\,g_{T_{3}}(w)\,(w^{2}-1)(w+1)(1+r^{2}-2rw)$
$\displaystyle-$ $\displaystyle
2\,g_{T_{2}}(w)\,g_{T_{3}}(w)\,(w^{2}-1)(w-1)(1+r^{2}-2rw)\Big{\\}}$
$\displaystyle\frac{d\Gamma}{dw}^{SMS}=\frac{d\Gamma}{dw}^{SR}$
$\displaystyle=$ $\displaystyle\frac{3}{2}g_{S}(w){\hat{m}}_{\ell}(w^{2}-1)$
(17) $\displaystyle\times$
$\displaystyle\Big{[}g_{V_{1}}(w)+(w+1)(1-r)g_{V_{2}}(w)-(w-1)(1+r)g_{V_{3}}(w)\Big{]}$
$\displaystyle\frac{d\Gamma}{dw}^{SMT}=$ $\displaystyle
6\,{\hat{m}}_{\ell}\Big{\\{}-g_{T_{1}}(w)(w+1)\Big{[}(2-3r+w)\,g_{V_{1}}(w)+(w^{2}-1)\big{[}(1+r)g_{V_{2}}(w)-(1-r)g_{V_{3}}(w)\big{]}\Big{]}$
$\displaystyle-
g_{T_{2}}(w)(w-1)\Big{[}(-2-3r+w)\,g_{V_{1}}(w)+(w^{2}-1)\big{[}(1+r)g_{V_{2}}(w)-(1-r)g_{V_{3}}(w)\big{]}\Big{]}$
$\displaystyle+g_{T_{3}}(w)(w^{2}-1)\Big{[}(w-r)\,g_{V_{1}}(w)+(w^{2}-1)\big{[}(1+r)g_{V_{2}}(w)-(1-r)g_{V_{3}}(w)\big{]}\Big{]}$
(18)
$\displaystyle+2(w^{2}-1)g_{A}(w)\Big{[}(1+r)g_{T_{1}}(w)-(1-r)g_{T_{2}}(w)\Big{]}\Big{\\}}$
$\frac{d\Gamma}{dw}^{RT}=\frac{d\Gamma}{dw}^{SMT}-24\,{\hat{m}}_{\ell}(w^{2}-1)g_{A}(w)\Big{[}(1+r)g_{T_{1}}(w)-(1-r)g_{T_{2}}(w)\Big{]}\,\,.$
(19)

  * •

$C=\chi_{c2}$:

$\displaystyle\frac{d\Gamma}{dw}^{SM}$ $\displaystyle=$
$\displaystyle\frac{d\Gamma}{dw}^{R}=\frac{w^{2}-1}{6}\Big{\\{}[k_{V}(w)]^{2}\,3\left(2+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\right)(w^{2}-1)(1+r^{2}-2rw)$
$\displaystyle+$
$\displaystyle[k_{A_{1}}(w)]^{2}\Big{[}2(3+5r^{2}-10rw+2w^{2})+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\,(-3+5r^{2}-10rw+8w^{2})\Big{]}$
$\displaystyle+$
$\displaystyle[k_{A_{2}}(w)]^{2}2(w^{2}-1)\Big{[}2r^{2}(w^{2}-1)+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\,\Big{(}3-6rw+r^{2}(4w^{2}-1)\Big{)}\Big{]}$
$\displaystyle+$
$\displaystyle[k_{A_{3}}(w)]^{2}2(w^{2}-1)\Big{[}2(w^{2}-1)+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\,\Big{(}-1+3r^{2}-6rw+4w^{2})\Big{)}\Big{]}$
$\displaystyle+$ $\displaystyle
4(w^{2}-1)\Big{[}k_{A_{1}}(w)\,k_{A_{2}}(w)\,\big{[}2r(w-r)+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\,(3-r^{2}-2rw)\big{]}$
$\displaystyle-$ $\displaystyle
k_{A_{2}}(w)\,k_{A_{3}}(w)\,\big{[}-2r(w^{2}-1)+\frac{{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\,\big{[}2r(w^{2}+2)-3w(1+r^{2})\big{]}$
$\displaystyle+$ $\displaystyle
2k_{A_{1}}(w)\,k_{A_{3}}(w)\,\left(1+\frac{2{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\right)(w-r)\Big{]}\Big{\\}}$
$\displaystyle\frac{d\Gamma}{dw}^{SMR}$ $\displaystyle=$
$\displaystyle-\frac{d\Gamma}{dw}^{SM}+[k_{V}(w)]^{2}\,\Big{(}2(1+r^{2}-2rw)+{\hat{m}}_{\ell}^{2}\Big{)}(w^{2}-1)^{2}$
(21) $\displaystyle\frac{d\Gamma}{dw}^{P}$ $\displaystyle=$
$\displaystyle[k_{P}(w)]^{2}\,(w^{2}-1)^{2}(1+r^{2}-2rw)$ (22)
$\displaystyle\frac{d\Gamma}{dw}^{T}$ $\displaystyle=$
$\displaystyle\frac{8}{3}\left(1+\frac{2{\hat{m}}_{\ell}^{2}}{1+r^{2}-2rw}\right)(w^{2}-1)\Big{\\{}[k_{T_{1}}(w)]^{2}\Big{[}(3+2w^{2})(1-2rw)+r^{2}(8w^{2}-3)\Big{]}$
$\displaystyle+$
$\displaystyle[k_{T_{2}}(w)]^{2}\,(-1+5r^{2}-10rw+6w^{2})+[k_{T_{3}}(w)]^{2}\,2(w^{2}-1)^{2}(1+r^{2}-2rw)\quad$
$\displaystyle-$ $\displaystyle
2k_{T_{1}}(w)\,k_{T_{2}}(w)\,\big{[}6r-5w(1+r^{2})+4rw^{2}\big{]}$
$\displaystyle-$ $\displaystyle
4(w^{2}-1)(1+r^{2}-2rw)k_{T_{3}}(w)\big{[}w\,k_{T_{1}}(w)+k_{T_{2}}(w)\big{]}\Big{\\}}$
$\displaystyle\frac{d\Gamma}{dw}^{SMP}$ $\displaystyle=$
$\displaystyle-\frac{d\Gamma}{dw}^{PR}$ (24) $\displaystyle=$ $\displaystyle-
k_{P}(w)\,{\hat{m}}_{\ell}(w^{2}-1)^{2}\Big{[}k_{A_{1}}(w)+(1-rw)k_{A_{2}}(w)+(w-r)k_{A_{3}}(w)\Big{]}$
$\displaystyle\frac{d\Gamma}{dw}^{SMT}=2{\hat{m}}_{\ell}(w^{2}-1)$
$\displaystyle\times\Big{\\{}k_{T_{1}}(w)\Big{[}(3-5rw+2w^{2})\,k_{A_{1}}(w)+(w^{2}-1)\Big{(}2rw\,k_{A_{2}}(w)+2w\,k_{A_{3}}(w)+3rk_{V}(w)\Big{)}\Big{]}$
$\displaystyle+k_{T_{2}}(w)\Big{[}5(w-r)\,k_{A_{1}}(w)+(w^{2}-1)\Big{(}2r\,k_{A_{2}}(w)+2\,k_{A_{3}}(w)+3k_{V}(w)\Big{)}\Big{]}$
(25)
$\displaystyle-2k_{T_{3}}(w)\,(w^{2}-1)\Big{[}(w-r)k_{A_{1}}(w)+(w^{2}-1)\Big{(}r\,k_{A_{2}}(w)+k_{A_{3}}(w)\Big{)}\Big{]}\Big{\\}}$
$\displaystyle\frac{d\Gamma}{dw}^{RT}=-\frac{d\Gamma}{dw}^{SMT}+12{\hat{m}}_{\ell}(w^{2}-1)^{2}k_{V}(w)\Big{[}r\,k_{T_{1}}(w)+k_{T_{2}}(w)\Big{]}\,\,.$
(26)

  * •

$C=h_{c}$:

Looking at the matrix elements for $\chi_{c1}$ and $h_{c}$ in (3) and (5), the
distributions for $h_{c}$ are obtained from those of $\chi_{c1}$ replacing

$\displaystyle g_{V_{i}}g_{V_{j}}\to f_{V_{i}}f_{V_{j}},\quad g_{A}^{2}\to
f_{A}^{2},\quad g_{S}^{2}\to f_{S}^{2},\quad g_{T_{i}}g_{T_{j}}\to
f_{T_{i}}f_{T_{j}},$ $\displaystyle g_{S}g_{V_{i}}\to f_{S}f_{V_{i}},\quad
g_{V_{i}}g_{T_{j}}\to-f_{V_{i}}f_{T_{j}},\quad g_{A}g_{T_{i}}\to
f_{A}f_{T_{i}},$ (27)

with $i,j=1,2,3$.

The above expressions also hold for $B_{c}$ decays to the $2P$ (and higher)
charmonium resonances. They are the subject of our analysis together with the
form factors in Eqs. (2)-(5).

##  3 Form factors in the effective theory

As discussed in [18], we aim at expressing the form factors in the effective
theory of QCD resulting by an expansion in the inverse heavy quark mass. The
heavy quark QCD field $Q(x)$ with mass $m_{Q}$ is written, with the notations
in [47], as

$Q(x)=e^{-i\,m_{Q}v\cdot x}\psi(x)=e^{-i\,m_{Q}v\cdot
x}\Big{(}\psi_{+}(x)+\psi_{-}(x)\Big{)}$ (28)

with $\psi_{\pm}=P_{\pm}\psi(x)$ and $\displaystyle
P_{\pm}=\frac{1\pm\not{v}}{2}$. $\psi_{+}$ is the positive energy component of
the field, $v$ is the heavy meson (quarkonium) 4-velocity with $v^{2}=1$.
Hence, $Q(x)$ is expressed as

$Q(x)=e^{-i\,m_{Q}v\cdot
x}\left(1+\frac{i\not{D}_{\perp}}{2m_{Q}}+\frac{(-iv\cdot
D)}{2m_{Q}}\frac{i\not{D}_{\perp}}{2m_{Q}}+\dots\right)\psi_{+}(x),$ (29)

with $D_{\perp\mu}=D_{\mu}-(v\cdot D)v_{\mu}$. In the rest frame
$v=(1,\vec{0})$ we have $v\cdot D=D_{t}$ and $D_{\perp\mu}=(0,D_{i})$. The QCD
Lagrangian is expressed in terms of $\psi_{+}$ as an expansion,

$\displaystyle{\cal L}_{QCD}$ $\displaystyle=$
$\displaystyle{\bar{\psi}}_{+}(x)\left(iv\cdot
D+\frac{(iD_{\perp})^{2}}{2m_{Q}}+\frac{g}{4m_{Q}}\sigma\cdot
G_{\perp}+\frac{i\not{D}_{\perp}}{2m_{Q}}\frac{(-iv\cdot
D)}{2m_{Q}}(i\not{D}_{\perp})+\dots\right)\psi_{+}(x)$ (30) $\displaystyle=$
$\displaystyle{\cal L}_{0}+{\cal L}_{1}+\dots\,\,\,,$

with
$G_{\perp\mu\nu}=(g_{\mu\alpha}-v_{\mu}v_{\alpha})(g_{\nu\beta}-v_{\nu}v_{\beta})G^{\alpha\beta}$.

Nonrelativistic QCD provides us with power counting of the operators in terms
of the relative heavy quark 3-velocity in the hadron rest frame
$\tilde{v}=|\vec{\tilde{v}}|\ll 1$, starting from $D_{t}\sim{\tilde{v}}^{2}$,
$D_{\perp}\sim{\tilde{v}}$ and $\psi_{+}\sim\tilde{v}^{3/2}$ [48]. Hence, the
second term in Eq. (29) is ${\cal O}(\tilde{v}\times\tilde{v}^{3/2})$, the
third one is ${\cal O}(\tilde{v}^{3}\times\tilde{v}^{3/2})$, and so on. In the
following we omit the power $\tilde{v}^{3/2}$ for each quark field in the
power counting of the various operators. Then, the chromoelectric field
components $E_{i}=G_{0i}$ and the chromomagnetic ones
$B_{i}=\displaystyle\frac{1}{2}\epsilon_{ijk}G^{jk}$ are ${\cal
O}(\tilde{v}^{3})$ and ${\cal O}(\tilde{v}^{4})$, respectively [49].

The first two terms in Eq. (30) are ${\cal O}(\tilde{v}^{2})$. They give the
leading order effective Lagrangian

$\displaystyle{\cal L}_{0}$ $\displaystyle=$
$\displaystyle{\bar{\psi}}_{+}(x)\Big{(}iv\cdot
D+\frac{(iD_{\perp})^{2}}{2m_{Q}}\Big{)}\psi_{+}(x).$ (31)

The third and fourth term in Eq. (30) are ${\cal O}(\tilde{v}^{4})$. They
correspond to the NLO term

$\displaystyle{\cal L}_{1}$ $\displaystyle=$ $\displaystyle{\cal
L}_{1,1}+{\cal L}_{1,2}\,\,,$ (32)

where

$\displaystyle{\cal L}_{1,1}$ $\displaystyle=$
$\displaystyle{\bar{\psi}}_{+}(x)\frac{g\sigma\cdot
G_{\perp}}{4m_{Q}}\psi_{+}(x)$ $\displaystyle{\cal L}_{1,2}$ $\displaystyle=$
$\displaystyle\bar{\psi}_{+}(x)\frac{i\not{D}_{\perp}}{2m_{Q}}\frac{(-iv\cdot
D)}{2m_{Q}}(i\not{D}_{\perp})\psi_{+}(x).$ (33)

${\cal L}_{1,2}$ can be written as

$\displaystyle{\cal L}_{1,2}$ $\displaystyle=$
$\displaystyle-\frac{1}{4m_{Q}^{2}}\Bigg{(}{\bar{\psi}}_{+}(x)\left(-\frac{(iD_{\perp})^{4}}{2m_{Q}}\right)\psi_{+}(x)+{\bar{\psi}}_{+}(x)\frac{g}{2}\sigma\cdot
G_{\perp}\left(-\frac{(iD_{\perp})^{2}}{2m_{Q}}\right)\psi_{+}(x)$
$\displaystyle+$ $\displaystyle
igv^{\alpha}{\bar{\psi}}_{+}(x)iD_{\perp}^{\sigma}G_{\alpha\sigma}\psi_{+}(x)+gv^{\alpha}{\bar{\psi}}_{+}(x)iD_{\perp\tau}\sigma^{\tau\sigma}G_{\alpha\sigma}\psi_{+}(x)\Bigg{)}$
$\displaystyle=$ $\displaystyle{\cal L}_{1,2}^{(1)}+{\cal L}_{1,2}^{(2)}+{\cal
L}_{1,2}^{(3)}+{\cal L}_{1,2}^{(4)},$

with ${\cal L}_{1,2}^{(2)}$ of higher order in $\tilde{v}$. ${\cal L}_{1}$ can
be arranged as

${\cal L}_{1}={\cal L}_{1}^{A}+{\cal L}_{1}^{B}\,\,\\\ $ (35)

with

$\displaystyle{\cal L}_{1}^{A}$ $\displaystyle=$ $\displaystyle{\cal
L}_{1,1}+{\cal
L}_{1,2}^{(4)}=\frac{1}{4m_{Q}}{\bar{\psi}}_{+}(x)A_{\tau\sigma}\sigma^{\tau\sigma}\psi_{+}(x)$
(36) $\displaystyle{\cal L}_{1}^{B}$ $\displaystyle=$ $\displaystyle{\cal
L}_{1,2}^{(1)}+{\cal
L}_{1,2}^{(3)}=\frac{1}{4m_{Q}^{2}}{\bar{\psi}}_{+}(x)B\psi_{+}(x),$ (37)

factorizing in (36) the leading $1/m_{Q}$ power.

An analogous expansion describes the antiquark: $Q(x)$ is written as

$Q(x)=e^{i\,m_{Q}v\cdot x}X(x)=e^{i\,m_{Q}v\cdot
x}\Big{(}X_{+}(x)+X_{-}(x)\Big{)}$ (38)

giving

${\cal L}_{QCD}={\overline{X}}_{-}(x)\left(-iv\cdot
D+\frac{(iD_{\perp})^{2}}{2m_{Q}}+\dots\right)X_{-}(x).$ (39)

To express the heavy meson form factors in the effective QCD theory, we expand
the weak current involving two heavy quarks $\bar{Q}^{\prime}\Gamma Q$, with
$\Gamma$ a generic Dirac matrix:

$\displaystyle{\bar{Q}^{\prime}(x)}\Gamma Q(x)$ $\displaystyle=$
$\displaystyle{\bar{\psi}}^{\prime}_{+}(x)\Big{(}1-\frac{i{\overleftarrow{\not{D}}^{\prime}}_{\perp}}{2m_{Q^{\prime}}}-\frac{1}{4m_{Q^{\prime}}^{2}}(i{\overleftarrow{\not{D}}^{\prime}}_{\perp})(iv^{\prime}\cdot{\overleftarrow{D}})+\dots\Big{)}$
(40)
$\displaystyle\Gamma\Big{(}1+\frac{i{\overrightarrow{\not{D}}}_{\perp}}{2m_{Q}}+\frac{1}{4m_{Q}^{2}}(-iv\cdot{\overrightarrow{D}})i{\overrightarrow{\not{D}}}_{\perp}+\dots\Big{)}\psi_{+}(x)\,\,,$

where $D_{\perp\mu}^{\prime}=D_{\mu}-(v^{\prime}\cdot D)v_{\mu}^{\prime}$. Up
to ${\cal O}(1/m_{Q}^{2})$ the expansion reads:

${\bar{Q}^{\prime}}(x)\Gamma
Q(x)=J_{0}+\Big{(}\frac{J_{1,0}}{2m_{Q}}+\frac{J_{0,1}}{2m_{Q^{\prime}}}\Big{)}+\Big{(}-\frac{J_{2,0}}{4m_{Q}^{2}}-\frac{J_{0,2}}{4m_{Q^{\prime}}^{2}}+\frac{J_{1,1}}{4m_{Q}m_{Q^{\prime}}}\Big{)},$
(41)

with

$\displaystyle J_{0}$ $\displaystyle=$
$\displaystyle{\bar{\psi}}_{+}^{\prime}\Gamma\psi_{+}$ $\displaystyle J_{1,0}$
$\displaystyle=$ $\displaystyle{\bar{\psi}}_{+}^{\prime}\Gamma
i{\overrightarrow{\not{D}}}_{\perp}\psi_{+}$ $\displaystyle J_{0,1}$
$\displaystyle=$
$\displaystyle{\bar{\psi}}_{+}^{\prime}\left(-i{\overleftarrow{\not{D}}^{\prime}}_{\perp}\right)\Gamma\psi_{+}$
$\displaystyle J_{2,0}$ $\displaystyle=$
$\displaystyle{\bar{\psi}}_{+}^{\prime}\Gamma\left(iv\cdot\overrightarrow{D}\right)i{\overrightarrow{\not{D}}}_{\perp}\psi_{+}$
(42) $\displaystyle J_{0,2}$ $\displaystyle=$
$\displaystyle{\bar{\psi}}_{+}^{\prime}i{\overleftarrow{\not{D}}^{\prime}}_{\perp}\left(iv^{\prime}\cdot\overleftarrow{D}\right)\Gamma\psi_{+}$
$\displaystyle J_{1,1}$ $\displaystyle=$
$\displaystyle{\bar{\psi}}_{+}^{\prime}\left(-i{\overleftarrow{\not{D}}^{\prime}}_{\perp}\right)\Gamma\left(i{\overrightarrow{\not{D}}}_{\perp}\right)\psi_{+}.$

Eq. (41) comprises terms up to ${\cal O}(\tilde{v}^{3})$. ${\cal
O}(1/m_{Q}^{3})$ terms with three derivatives have not been included, even
though they can be of the same order in $\tilde{v}$ of some terms in the
corrections discussed in the following: we assume that their contribution is
small.

The hadronic matrix elements of the operators in Eq. (41) can be written using
the trace formalism [50]. For states comprising a heavy quark and a heavy
antiquark, spin symmetry is expected to hold in the preasymptotic mass range
which includes beauty and charm, outside the QCD Coulombic regime [16, 51].
Hence, the two states $(B_{c},B_{c}^{*})$ and the four states
$(\chi_{c0,1,2},h_{c})$ can be organized in a negative parity spin doublet and
in a positive parity spin 4-plet. They are described by $4\times 4$ matrices,
for the spin doublet [16]

$\mathcal{M}(v)=P_{+}(v)\,\left[B_{c}^{*\mu}\,\gamma_{\mu}-B_{c}\,\gamma_{5}\right]\,P_{-}(v)\;,$
(43)

for the spin 4-plet [52]

$\mathcal{M}^{\prime\mu}(v^{\prime})=P_{+}(v^{\prime})\,\bigg{[}\chi_{c2}^{\mu\nu}\,\gamma_{\nu}+\frac{1}{\sqrt{2}}\,\chi_{c1,\gamma}\,\epsilon^{\mu\alpha\beta\gamma}\,v^{\prime}_{\alpha}\,\gamma_{\beta}\,+\frac{1}{\sqrt{3}}\,\chi_{c0}\,(\gamma^{\mu}-v^{\prime\mu})+h_{c}^{\mu}\,\gamma_{5}\bigg{]}\,P_{-}(v^{\prime})$
(44)

with the condition

$v^{\prime}_{\mu}\,\mathcal{M}^{\prime\mu}=0\;.$ (45)

The normalization is $\sqrt{m_{M}}$, with $M$ a meson in the spin multiplet.
Radial excitations belong to multiplets analogous to (43) and (44). The trace
formalism has been used to obtain the effective Lagrangians governing strong
and radiative heavy quarkonium transitions in the soft-exchange approximation
[51, 52, 53]. We apply it to express the various form factors in terms of
(universal) functions independent of the particular decay mode.

##  4 Form factors in terms of universal functions

Applying the trace formalism, the matrix elements of the leading order term of
the expansion of the quark current (41) are written as

$\braket{M^{\prime}(v^{\prime})}{J_{0}}{M(v)}=-\Xi(w)\,v_{\mu}\,\text{Tr}\big{[}\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}\;$
(46)

involving the single universal function $\Xi(w)$.

At $1/m_{Q}$ the relevant matrix elements are parametrized as

$\displaystyle\braket{M^{\prime}(v^{\prime})}{\bar{\psi}^{\prime}_{+}\,\Gamma\,i\,\overrightarrow{D}_{\alpha}\,\psi_{+}}{M(v)}=-\text{Tr}\big{[}\Sigma_{\mu\alpha}^{(b)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}\;,$
(47a)
$\displaystyle\braket{M^{\prime}(v^{\prime})}{\bar{\psi}^{\prime}_{+}\,(-i\,\overleftarrow{D}_{\alpha})\,\Gamma\,\psi_{+}}{M(v)}=-\text{Tr}\big{[}\Sigma_{\mu\alpha}^{(c)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}\;.$
(47b)

The general expression of the functions $\Sigma_{\mu\alpha}^{(Q)}$ is

$\Sigma_{\mu\alpha}^{(Q)}=\Sigma_{1}^{(Q)}\,g_{\mu\alpha}+\Sigma_{2}^{(Q)}\,v_{\mu}\,v_{\alpha}+\Sigma_{3}^{(Q)}\,v_{\mu}\,v^{\prime}_{\alpha}\,+\Sigma_{4}^{(Q)}\,v_{\mu}\,\gamma_{\alpha}+\Sigma_{5}^{(Q)}\,\gamma_{\mu}\,v_{\alpha}+\Sigma_{6}^{(Q)}\,\gamma_{\mu}\,v^{\prime}_{\alpha}+\Sigma_{7}^{(Q)}\,i\,\sigma_{\mu\alpha}.$
(48)

The terms proportional to $v^{\prime\mu}$ vanish due to (45).
$\Sigma_{i}^{(Q)}(w)$ $(i=1,\dots 7)$ are the universal functions at this
order. Upon integration by parts, relations can be worked out. Using

$i\,\partial_{\alpha}\,(\bar{\psi}^{\prime}_{+}\,\Gamma\,\psi_{+})=\bar{\psi}^{\prime}_{+}\,i\,\overleftarrow{D}_{\alpha}\,\Gamma\,\psi_{+}+\bar{\psi}^{\prime}_{+}\,\Gamma\,i\,\overrightarrow{D}_{\alpha}\,\psi_{+}\;,$
(49)

we obtain:

$-\big{(}\tilde{\Lambda}\,v_{\alpha}-\tilde{\Lambda}^{\prime}\,v^{\prime}_{\alpha}\big{)}\,\Xi\,v_{\mu}\,\text{Tr}\big{[}\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}=\text{Tr}\big{[}\Sigma^{(c)}_{\mu\alpha}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}-\text{Tr}\big{[}\Sigma_{\mu\alpha}^{(b)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}\;.$
(50)

For the $Q{\bar{Q}}^{\prime}$ mesons in the initial and final state the
parameters $\tilde{\Lambda}$, $\tilde{\Lambda}^{\prime}$ are given by

$\tilde{\Lambda}=m_{H}-m_{Q}-m_{{\bar{Q}}^{\prime}}$ (51)

and analogously for $\tilde{\Lambda}^{\prime}$. The relations follow:

$\displaystyle\Sigma_{i}^{(b)}(w)-\Sigma_{i}^{(c)}(w)$
$\displaystyle=0\qquad\qquad i=1,4,5,6,7\;,$ (52a)
$\displaystyle\Sigma_{2}^{(b)}(w)-\Sigma_{2}^{(c)}(w)$
$\displaystyle=\tilde{\Lambda}\,\Xi\;,$ (52b)
$\displaystyle\Sigma_{3}^{(b)}(w)-\Sigma_{3}^{(c)}(w)$
$\displaystyle=-\tilde{\Lambda}^{\prime}\,\Xi(w)\;.$ (52c)

The same procedure allows us to parametrize the matrix elements relevant for
the $1/m_{Q}^{2}$ terms in (41):

$\displaystyle\braket{M^{\prime}(v^{\prime})}{\bar{\psi}^{\prime}_{+}\,\Gamma\,i\,\overrightarrow{D}_{\alpha}\,i\,\overrightarrow{D}_{\beta}\,\psi_{+}}{M(v)}$
$\displaystyle=-\text{Tr}\big{[}\Omega_{\mu\alpha\beta}^{(b)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}\;$
(53a)
$\displaystyle\braket{M^{\prime}(v^{\prime})}{\bar{\psi}^{\prime}_{+}\,i\,\overleftarrow{D}_{\alpha}\,i\,\overleftarrow{D}_{\beta}\,\Gamma\,\psi_{+}}{M(v)}$
$\displaystyle=-\text{Tr}\big{[}\Omega_{\mu\alpha\beta}^{(c)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}\;.$
(53b)

$\Omega_{\mu\alpha\beta}^{(Q)}$ have the general expression

$\displaystyle\Omega_{\mu\alpha\beta}^{(Q)}$ $\displaystyle=$
$\displaystyle\Omega_{1}^{(Q)}\,g_{\mu\alpha}\,v_{\beta}+\Omega_{2}^{(Q)}\,g_{\alpha\beta}\,v_{\mu}+\Omega_{3}^{(Q)}\,g_{\beta\mu}\,v_{\alpha}+\Omega_{4}^{(Q)}\,g_{\mu\alpha}\,v^{\prime}_{\beta}+\Omega_{5}^{(Q)}\,g_{\beta\mu}\,v^{\prime}_{\alpha}+\Omega_{6}^{(Q)}\,g_{\mu\alpha}\,\gamma_{\beta}$
$\displaystyle+$
$\displaystyle\Omega_{7}^{(Q)}\,g_{\alpha\beta}\,\gamma_{\mu}+\Omega_{8}^{(Q)}\,g_{\beta\mu}\,\gamma_{\alpha}+\Omega_{9}^{(Q)}\,v_{\mu}\,v_{\alpha}\,v_{\beta}+\Omega_{10}^{(Q)}\,v_{\mu}\,v_{\alpha}\,v^{\prime}_{\beta}+\Omega_{11}^{(Q)}\,v_{\mu}\,v^{\prime}_{\alpha}\,v_{\beta}$
$\displaystyle+$
$\displaystyle\Omega_{12}^{(Q)}\,v_{\mu}\,v^{\prime}_{\alpha}\,v^{\prime}_{\beta}+\Omega_{13}^{(Q)}\,v_{\mu}\,v_{\alpha}\,\gamma_{\beta}+\Omega_{14}^{(Q)}\,v_{\mu}\,\gamma_{\alpha}\,v_{\beta}+\Omega_{15}^{(Q)}\,\gamma_{\mu}\,v_{\alpha}\,v_{\beta}+\Omega_{16}^{(Q)}\,v_{\mu}\,v^{\prime}_{\alpha}\,\gamma_{\beta}$
$\displaystyle+$
$\displaystyle\Omega_{17}^{(Q)}\,\gamma_{\mu}\,v_{\alpha}\,v^{\prime}_{\beta}+\Omega_{18}^{(Q)}\,v_{\mu}\,\gamma_{\alpha}\,v^{\prime}_{\beta}+\Omega_{19}^{(Q)}\,\gamma_{\mu}\,v^{\prime}_{\alpha}\,v_{\beta}+\Omega_{20}^{(Q)}\,\gamma_{\mu}\,v^{\prime}_{\alpha}\,v^{\prime}_{\beta}+\Omega_{21}^{(Q)}\,i\,\sigma_{\mu\alpha}\,v_{\beta}$
$\displaystyle+$
$\displaystyle\Omega_{22}^{(Q)}\,i\,\sigma_{\alpha\beta}\,v_{\mu}+\Omega_{23}^{(Q)}\,i\,\sigma_{\beta\mu}\,v_{\alpha}+\Omega_{24}^{(Q)}\,i\,\sigma_{\mu\alpha}\,v^{\prime}_{\beta}+\Omega_{25}^{(Q)}\,i\,\sigma_{\beta\mu}\,v^{\prime}_{\alpha}\;.$

The terms proportional to $v^{\prime\mu}$ vanish. The matrix element with two
derivatives, one acting on $\psi_{+}$, the other one on
$\bar{\psi}^{\prime}_{+}$, can be parametrized integrating by parts either Eq.
(53a),

$\begin{split}\braket{M^{\prime}(v^{\prime})}{\bar{\psi}^{\prime}_{+}\,(-i\,\overleftarrow{D}_{\alpha})\,\Gamma\,i\,\overrightarrow{D}_{\beta}\,\psi_{+}}{M(v)}&=\big{(}\tilde{\Lambda}\,v_{\alpha}-\tilde{\Lambda}^{\prime}\,v^{\prime}_{\alpha}\big{)}\,\text{Tr}\big{[}\Sigma_{\mu\beta}^{(b)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}-\text{Tr}\big{[}\Omega_{\mu\alpha\beta}^{(b)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]},\end{split}$
(55)

or Eq. (53b),

$\begin{split}\braket{M^{\prime}(v^{\prime})}{\bar{\psi}^{\prime}_{+}\,(-i\,\overleftarrow{D}_{\alpha})\,\Gamma\,i\,\overrightarrow{D}_{\beta}\,\psi_{+}}{M(v)}&=-\big{(}\tilde{\Lambda}\,v_{\beta}-\tilde{\Lambda}^{\prime}\,v^{\prime}_{\beta}\big{)}\,\text{Tr}\big{[}\Sigma_{\mu\alpha}^{(c)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}-\text{Tr}\big{[}\Omega_{\mu\alpha\beta}^{(c)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}.\end{split}$
(56)

This gives

$\Omega_{\mu\alpha\beta}^{(b)}-\Omega_{\mu\alpha\beta}^{(c)}=\big{(}\tilde{\Lambda}\,v_{\alpha}-\tilde{\Lambda}^{\prime}\,v^{\prime}_{\alpha}\big{)}\,\Sigma_{\mu\beta}^{(b)}+\big{(}\tilde{\Lambda}\,v_{\beta}-\tilde{\Lambda}^{\prime}\,v^{\prime}_{\beta}\big{)}\,\Sigma_{\mu\alpha}^{(c)}.$
(57)

To express the form factors in the effective theory we have also to take into
account the corrections from the expansion of the states. They can be
parametrized as

$\displaystyle\braket{M^{\prime}(v^{\prime})}{i\,\int\text{d}^{4}x\,\text{T}\left[J_{0}(0),\mathcal{L}_{1}(x)\right]}{M(v)}=\hskip
56.9055pt$
$\displaystyle-\frac{1}{4\,m_{b}}\,\underbrace{\left(-\frac{i}{2}\right)\,\text{Tr}\big{[}\Upsilon_{2\mu\alpha\beta}^{(b)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,P_{+}\,\sigma^{\alpha\beta}\,\mathcal{M}\big{]}}_{G^{(b)}}-\frac{1}{2\,m_{b}^{2}}\,\underbrace{\text{Tr}\big{[}\Upsilon_{1\mu}^{(b)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}}_{K^{(b)}}\;,$
(58)
$\displaystyle\braket{M^{\prime}(v^{\prime})}{i\,\int\text{d}^{4}x\,\text{T}\left[J_{0}(0),\mathcal{L}^{\prime}_{1}(x)\right]}{M(v)}=$
$\displaystyle-\frac{1}{4\,m_{c}}\,\underbrace{\left(-\frac{i}{2}\right)\,\text{Tr}\big{[}\Upsilon_{2\mu\alpha\beta}^{(c)}\,\overline{\mathcal{M}}^{\prime\mu}\,\sigma^{\alpha\beta}\,P^{\prime}_{+}\,\Gamma\,\mathcal{M}\big{]}}_{G^{(c)}}-\frac{1}{2\,m_{c}^{2}}\,\underbrace{\text{Tr}\big{[}\Upsilon_{1\mu}^{(c)}\,\overline{\mathcal{M}}^{\prime\mu}\,\Gamma\,\mathcal{M}\big{]}}_{K^{(c)}}\;,$
(59)

where $\mathcal{L}_{1},\mathcal{L}^{\prime}_{1}$ are given in (32) with
$m_{Q}\to m_{b},m_{c}$, respectively. The functions $\Upsilon^{(Q)}$ have the
general expression

$\displaystyle\Upsilon_{1\mu}^{(Q)}$ $\displaystyle=$
$\displaystyle\Upsilon_{1A}^{(Q)}\,v_{\mu}+\Upsilon_{1B}^{(Q)}\,\gamma_{\mu}\;,$
(60) $\displaystyle\Upsilon_{2\mu\alpha\beta}^{(Q)}$ $\displaystyle=$
$\displaystyle\Upsilon_{2A}^{(Q)}\,(g_{\mu\alpha}\,v_{\beta}-g_{\mu\beta}\,v_{\alpha})+\Upsilon_{2B}^{(Q)}\,(g_{\mu\alpha}\,v^{\prime}_{\beta}-g_{\mu\beta}\,v^{\prime}_{\alpha})+\Upsilon_{2C}^{(Q)}\,(g_{\mu\alpha}\,\gamma_{\beta}-g_{\mu\beta}\,\gamma_{\alpha})$
$\displaystyle+$
$\displaystyle\Upsilon_{2D}^{(Q)}\,v_{\mu}\,(v_{\alpha}\,v^{\prime}_{\beta}-v^{\prime}_{\alpha}\,v_{\beta})+\Upsilon_{2E}^{(Q)}\,v_{\mu}\,(v_{\alpha}\,\gamma_{\beta}-\gamma_{\alpha}\,v_{\beta})+\Upsilon_{2F}^{(Q)}\,v_{\mu}\,(v^{\prime}_{\alpha}\,\gamma_{\beta}-\gamma_{\alpha}\,v^{\prime}_{\beta})$
$\displaystyle+$
$\displaystyle\Upsilon_{2G}^{(Q)}\,\gamma_{\mu}\,(v_{\alpha}\,v^{\prime}_{\beta}-v^{\prime}_{\alpha}\,v_{\beta})+\Upsilon_{2H}^{(Q)}\,i\,(\sigma_{\mu\alpha}\,v_{\beta}-\sigma_{\mu\beta}\,v_{\alpha})+\Upsilon_{2I}^{(Q)}\,i\,(\sigma_{\mu\alpha}\,v^{\prime}_{\beta}-\sigma_{\mu\beta}\,v^{\prime}_{\alpha})$
$\displaystyle+$
$\displaystyle\Upsilon_{2J}^{(Q)}\,i\,v_{\mu}\,\sigma_{\alpha\beta}\;.$

Since the terms proportional to $v_{\alpha}$ or $v_{\beta}$ do not contribute
to $G^{(b)}$, and those proportional to $v^{\prime}_{\alpha}$ or
$v^{\prime}_{\beta}$ do not contribute to $G^{(c)}$, we have:
$\Upsilon_{2D}^{(Q)}=\Upsilon_{2G}^{(Q)}=0$ (for $Q=c,b$),
$\Upsilon_{2B}^{(c)}=\Upsilon_{2F}^{(c)}=\Upsilon_{2I}^{(c)}=0$ and
$\Upsilon_{2A}^{(b)}=\Upsilon_{2E}^{(b)}=\Upsilon_{2H}^{(b)}=0$.

With this set of relations, the form factors in Eqs. (2)-(5) are expressed in
terms of universal functions up to ${\cal O}(1/m^{2}_{Q})$. The formulae are
collected in Appendix B.

##  5 Relations among form factors

At the leading order all form factors in $B_{c}\to\chi_{c0,1,2},h_{c}$ matrix
elements are expressed near zero recoil in terms of the single function
$\Xi(w)$, see Appendix B. Increasing the order of the expansion, relations
among the form factors can be worked out exploiting the results in Appendix C.

The expansion in the relative quark velocity $\tilde{v}$ involves a large
number of universal functions, which renders difficult the derivations of the
relations among form factors. It is interesting to consider the various orders
in $1/m_{Q}$. In particular, at ${\cal O}(1/m_{Q})$ there are relations among
form factors in the same decay mode, in pairs of decay modes and in more than
two modes, near the zero-recoil point.

i) Relations among the form factors in the same channel

For the $B_{c}\to\chi_{ci}$ (i=0,1,2) and $B_{c}\to h_{c}$ form factors we
have:

  * •

$B_{c}\to\chi_{c0}$:

$g_{T}(w)=-\frac{1}{w+1}\big{[}2g_{-}(w)+g_{P}(w)\big{]}\,.$ (62)

  * •

$B_{c}\to\chi_{c1}$:

$\displaystyle g_{T_{2}}(w)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\big{[}g_{V_{1}}(w)-(1+w)g_{A}(w)\big{]}$ (63)
$\displaystyle g_{T_{3}}(w)$ $\displaystyle=$
$\displaystyle-\frac{1}{2(w-1)}\big{[}g_{V_{1}}(w)+4g_{V_{2}}(w)\big{]}+\frac{1}{2}g_{A}(w)+\frac{1}{w-1}\big{[}g_{S}(w)+g_{T_{1}}(w)\big{]}\quad$
(64)

with the condition

$-\frac{1}{2}\big{[}g_{V_{1}}(1)+4g_{V_{2}}(1)\big{]}+g_{S}(1)+g_{T_{1}}(1)=0\,.$
(65)

  * •

$B_{c}\to\chi_{c2}$:

$\displaystyle k_{T_{1}}(w)$ $\displaystyle=$ $\displaystyle-
wk_{V}(w)+k_{A_{2}}(w)+wk_{A_{3}}(w)+k_{P}(w)$ (66) $\displaystyle
k_{T_{2}}(w)$ $\displaystyle=$ $\displaystyle
k_{V}(w)-k_{A_{1}}(w)-k_{A_{2}}(w)-wk_{A_{3}}(w)-k_{P}(w)$ (67) $\displaystyle
k_{T_{3}}(w)$ $\displaystyle=$ $\displaystyle-k_{V}(w)+k_{A_{3}}(w)\,.$ (68)

  * •

$B_{c}\to h_{c}$:

$\displaystyle f_{T_{2}}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}f_{V_{1}}(w)+(1+w)f_{A}(w)\big{]}$ (69)
$\displaystyle f_{T_{3}}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{2(w-1)}\big{[}f_{V_{1}}(w)+4f_{V_{2}}(w)\big{]}+\frac{1}{2}f_{A}(w)-\frac{1}{w-1}\big{[}f_{S}(w)-f_{T_{1}}(w)\big{]}\quad$
(70)

with the condition

$\frac{1}{2}\big{[}f_{V_{1}}(1)+4f_{V_{2}}(1)\big{]}-\big{[}f_{S}(1)-f_{T_{1}}(1)\big{]}=0.$
(71)

ii) Relations among form factors of pairs of decay channels

We have:

  * •

$B_{c}\to\chi_{c0}$ and $B_{c}\to\chi_{c1}$:

$\displaystyle(w+1)g_{+}(w)-(w-1)g_{-}(w)+g_{P}(w)=$
$\displaystyle\frac{w+1}{\sqrt{6}}\Big{\\{}2g_{V_{1}}(w)+(w+1)g_{V_{2}}(w)-(w-1)\big{[}g_{V_{3}}(w)+g_{A}(w)\big{]}-g_{S}(w)+2g_{T_{1}}(w)\Big{\\}}.\quad$

  * •

$B_{c}\to h_{c}$ and $B_{c}\to\chi_{c1}$:

$f_{V_{1}}(w)+(w-1)f_{A}(w)-2f_{T_{1}}(w)=\sqrt{2}\Big{\\{}g_{V_{1}}(w)+(w+1)g_{V_{2}}(w)-(w-1)g_{V_{3}}(w)-g_{S}(w)\Big{\\}}$
(73) $\displaystyle
3f_{V_{1}}(w)+2(w+1)f_{V_{2}}(w)-(w-1)\big{[}2f_{V_{3}}(w)-f_{A}(w)\big{]}-2\big{[}f_{S}(w)+f_{T_{1}}(w)\big{]}=\qquad$
$\displaystyle\sqrt{2}\Big{\\{}g_{V_{1}}(w)-(w-1)g_{A}(w)+2g_{T_{1}}(w)\Big{\\}}\,\,.$
(74)

##  6 Phenomenology

We now discuss some consequences of the relations found in the previous
Sections. We use $m_{B_{c}}=6274.47\pm 0.28\pm 0.17$ MeV,
$\tau_{B_{c}}=0.510\pm 0.009$ ps, $m_{\chi_{c0}(1P)}=3414.71\pm 0.30$ MeV,
$m_{\chi_{c1}(1P)}=3510.67\pm 0.05$ MeV, $m_{\chi_{c2}(1P)}=3556.17\pm 0.07$
MeV and $m_{h_{c}(1P)}=3525.38\pm 0.11$ MeV [33]. We first focus on the LO
relations, then on NLO, mainly considering the Standard Model. The form
factors are expressed in terms of universal functions in a selected kinematic
range, close to the zero recoil point $w=1$. In the numerical analyses we
extrapolate the relations to the full kinematic range in the various channels.
The range is not wide ($w_{max}\sim 1.16-1.09$ for $B_{c}\to\chi_{c0}$,
$B_{c}\to\chi_{c2}$ and light leptons or $\tau$, $w_{max}\sim 1.11-1.05$ for
$2P$ charmonia). This allows to get information on the results in the full
kinematical range. In general, the extrapolation can be constrained by making
use of dispersive matrices, which allow to reconstruct the form factors
knowing their values in few kinematical points [54]: the application of such
methods is beyond the purposes of the present study.

The LO relations obtained from the formulae in appendix B connect all form
factors of $B_{c}$ transitions to the $P-$wave charmonium 4-plet to the single
function $\Xi(w)$, which is different for the $B_{c}\to 1P$ and the $B_{c}\to
2P$ modes. In ratios of decay distributions, for the same value of $w$ the
form factor dependence cancels out. The ratios are depicted in fig. 1 both for
$1P$ and the $2P$ channels, in the range of $w$ common to all modes. For the
$2P$ mesons we use the masses $m_{\chi_{c0}}(2P)=3860$ MeV and
$m_{\chi_{c2}}(2P)=3930$ MeV [33], even though another $J^{PC}=0^{++}$ state
is also reported by the Particle Data Group, $\chi_{c0}(3915)$ [33].

Figure 1: Ratios of decay distributions
$\frac{d\Gamma(B_{c}\to\chi_{c1}\ell\bar{\nu})/dw}{d\Gamma(B_{c}\to\chi_{c0}\ell\bar{\nu})/dw}$
(continuous blue line) and
$\frac{d\Gamma(B_{c}\to\chi_{c2}\ell\bar{\nu})/dw}{d\Gamma(B_{c}\to\chi_{c1}\ell\bar{\nu})/dw}$
(dashed red line) in the Standard Model, in the case $\ell=\mu$ (left) and
$\ell=\tau$ (right) for the $1P$ (top row) and $2P$ final charmonia (bottom
row), with the meson masses quoted in the text. The LO relations among form
factors are extrapolated to the full kinematical range.

The results in the figure can be understood considering the connection among
the decay distributions, holding at this order if the $\chi_{ci}$ mass
differences are neglected:

$2\frac{d\Gamma}{dw}(B_{c}\to\chi_{c0}\ell{\bar{\nu}}_{\ell})+\frac{d\Gamma}{dw}(B_{c}\to\chi_{c1}\ell{\bar{\nu}}_{\ell})-\frac{d\Gamma}{dw}(B_{c}\to\chi_{c2}\ell{\bar{\nu}}_{\ell})=0.$
(75)

Notice that this relation holds in the SM and also including the full set of
NP operators in the generalized Hamiltonian (2) regardless of the Wilson
coefficients $\epsilon_{i}$.

A simple parametrization of the function $\Xi(w)$ involves the intercept,
slope and curvature at the zero recoil point,

$\Xi(w)=\Xi_{0}+\Xi_{1}(w-1)+\Xi_{2}(w-1)^{2}\,\,.$ (76)

The three parameters $\Xi_{0},\,\Xi_{1},\,\Xi_{2}$ can be constrained, namely
$\Xi_{0}\in[0.05,1]$, $\Xi_{1}\in[-1,1]$ and $\Xi_{2}\in[-1,1]$, requiring
that the measurement ${\cal
B}(B_{c}^{-}\to\chi_{c0}\pi^{-})=(2.4\pm^{0.9}_{0.8})\times 10^{-5}$ [33] is
reproduced at 1$\sigma$ by naive factorization. Correlations between ratios of
branching fractions are found varying $\Xi_{1}/\Xi_{0}$ and $\Xi_{2}/\Xi_{0}$
in the selected regions. We choose the same ranges also for $2P$ excitations
in the final state. For the $B_{c}$ transitions to the $1P$ charmonium a
negative (positive) correlation is found between the ratios
$\displaystyle\frac{\Gamma(B_{c}\to\chi_{c2}\ell\bar{\nu}_{\ell})}{\Gamma(B_{c}\to\chi_{c1}\ell\bar{\nu}_{\ell})}$
and
$\displaystyle\frac{\Gamma(B_{c}\to\chi_{c1}\ell\bar{\nu}_{\ell})}{\Gamma(B_{c}\to\chi_{c0}\ell\bar{\nu}_{\ell})}$
for $\ell=\mu\,\,(\tau$), as in fig. 2. There are positive correlations also
between the Lepton Flavour Universality ratios $R(C)=\Gamma(B_{c}\to
C\tau\bar{\nu}_{\tau})/\Gamma(B_{c}\to C\ell\bar{\nu}_{\ell})$ which compare
the decay rates to $\tau$ and to light leptons, see fig. 3.

Figure 2: Correlations between the ratios of branching fractions $\frac{{\cal
B}(B_{c}\to\chi_{c2}\ell\bar{\nu})}{{\cal B}(B_{c}\to\chi_{c1}\ell\bar{\nu})}$
and $\frac{{\cal B}(B_{c}\to\chi_{c1}\ell\bar{\nu})}{{\cal
B}(B_{c}\to\chi_{c0}\ell\bar{\nu})}$ for $\ell=\mu$ (left) and $\ell=\tau$
(right). The LO expression of the form factors is considered, with $\Xi(w)$
parametrized as in Eq. (76).  Figure 3: Correlations between Lepton Flavour
Universality ratios $R(\chi_{c0,2})$ and $R(h_{c})$ with $R(\chi_{c1})$ in the
SM. The LO expression of the form factors is considered, with $\Xi(w)$
parametrized in Eq. (76).

At the same LO, the results for $B_{c}$ decays to the $2P$ charmonium are
shown in figs. 1, 4 and 5, the behaviour expected for $\chi_{c1}(3872)$ if it
corresponds to $\chi_{c1}(2P)$.

Figure 4: Correlation between ratios of branching fractions of $B_{c}$ decays
to $2P$ charmonia and $\ell=\mu$ (left), $\ell=\tau$ (right), with the LO
expression of the form factors and $\Xi(w)$ parametrized in Eq. (76).  Figure
5: Correlations between Lepton Flavour Universality ratios
$R(\chi_{c0,2}(2P))$ with $R(\chi_{c1}(2P))$ in the SM. The LO expression of
the form factors is considered, with $\Xi(w)$ parametrized in Eq. (76).

The above results are independent of the function $\Xi$, a benefit of the
expansion of the form factors dominated by the leading order. Including the
NLO terms the phenomenology is more involved: nevertheless, the expansion
allows us to perform systematic analyses and improvements. An important
feature is that the obtained relations connect the form factors to the matrix
elements of operators in the effective theory: they should be verified in
explicit calculations, and represent a testing ground for the various
computations based on QCD methods. This is the case of the relations among the
form factors in the same channel, such as (62) for $J^{PC}=0^{++}$, and (64),
(67) for the $J^{PC}=1^{++},2^{++}$ modes.

At ${\cal O}(1/m_{Q})$, once the relations in Appendix C are taken into
account, the number of independent structures is $13$ in terms of universal
functions. For $\chi_{c0},\,\chi_{c1},\,h_{c}$ the decay distributions in SM
are related for $\frac{1}{\tilde{\Gamma}}\frac{d\Gamma}{dw}\Big{|}_{w\to 1}$,
with $\tilde{\Gamma}$ in (7). The expressions only involve
$\Sigma_{\chi_{c1},1}^{(b)}$ and $\Sigma_{\chi_{c1},1}^{(c)}$ defined in
Appendix B:

$\displaystyle\lim_{w\to
1}\frac{1}{\tilde{\Gamma}}\frac{d\Gamma}{dw}(B_{c}\to\chi_{c0}\ell{\bar{\nu}}_{\ell})$
$\displaystyle=$ $\displaystyle
18\,{\hat{m}}_{\ell}^{2}(\epsilon_{b}+\epsilon_{c})^{2}\Big{[}\Sigma_{\chi_{c1},1}^{(b)}(1)\Big{]}^{2}$
(77) $\displaystyle\lim_{w\to
1}\frac{1}{\tilde{\Gamma}}\frac{d\Gamma}{dw}(B_{c}\to\chi_{c1}\ell{\bar{\nu}}_{\ell})$
$\displaystyle=$ $\displaystyle
12\Big{[}2(1-r_{1})^{2}+{\hat{m}}_{\ell}^{2}\Big{]}\Big{[}\epsilon_{b}\Sigma_{\chi_{c1},1}^{(b)}(1)-\epsilon_{c}\Sigma_{\chi_{c1},1}^{(c)}(1)\Big{]}^{2}$
(78) $\displaystyle\lim_{w\to
1}\frac{1}{\tilde{\Gamma}}\frac{d\Gamma}{dw}(B_{c}\to
h_{c}\ell{\bar{\nu}}_{\ell})$ $\displaystyle=$ $\displaystyle
6\Big{[}2(1-r_{h})^{2}+{\hat{m}}_{\ell}^{2}\Big{]}\Big{[}(\epsilon_{b}-\epsilon_{c})\Sigma_{\chi_{c1},1}^{(b)}(1)+2\epsilon_{c}\Sigma_{\chi_{c1},1}^{(c)}(1)\Big{]}^{2}\,\,.$
(79)

Such relations also hold for the decays into the $2P$ resonances, and can be
used to compare the mode involving $\chi_{c1}(3872)$ to other modes in the
$2P$ 4-plet. Admittedly, this is a difficult measurement.

As for the NP extension in Eq. (2) with the pseudoscalar and tensor operators
included, the form factors parametrizing the scalar and tensor matrix elements
give contributions related to the SM ones as in Eq. (62), a useful connection
for phenomenological analyses.

Other improvements are possible if reliable results are available even for a
single form factor. As shown in [18] using input from lattice NRQCD, details
on a form factor can be employed to get information on universal functions,
predicting other form factors and establishing connections among observables.

##  7 Conclusions

We have derived the expressions of the form factors of the semileptonic
$B_{c}$ decays to the $P$-wave charmonium 4-plet as an expansion in the
relative velocity of the heavy quarks in the charmonium and of $1/m_{Q}$. The
expressions involve universal functions, independent of the specific channel,
and allow to connect different modes. They are a testing ground for explicit
calculations of the form factors and can be used in studying LFU ratios and
the effects of SM extensions. They are useful in analyzing the semileptonic
$B_{c}$ decays to the $2P$ charmonia: the comparison of measurements of
$B_{c}$ transitions to $\chi_{c1}(3872)$ with other states in the $2P$
charmonium 4-plet is a tool to gain new information on the nature of
$X(3872)$.

Acknowledgements. This study has been carried out within the INFN project
(Iniziativa Specifica) QFT-HEP.

##  Appendix A Another parametrization of the $B_{c}\to\chi_{c\,0,1,2}$ and
$B_{c}\to h_{c}$ matrix elements

For the sake of completeness we report another parametrization of the
$B_{c}\to\chi_{c\,0,1,2}$ and $B_{c}\to h_{c}$ matrix elements in terms of
form factors, often used in the literature:

$\displaystyle\langle
S(p^{\prime})|\bar{c}\gamma^{\mu}\gamma_{5}b|{B_{c}}(p)\rangle$
$\displaystyle=$ $\displaystyle
f_{0}^{S}(q^{2})\frac{m_{B_{c}}^{2}-m_{S}^{2}}{q^{2}}q^{\mu}+f_{+}^{S}(q^{2})\left(p^{\mu}+p^{\prime\mu}-\frac{m_{B_{c}}^{2}-m_{S}^{2}}{q^{2}}q^{\mu}\right),$
$\displaystyle\langle S(p^{\prime})|\bar{c}\sigma^{\mu\nu}b|{B_{c}}(p)\rangle$
$\displaystyle=$
$\displaystyle-\frac{2f_{T}^{S}(q^{2})}{m_{B_{c}}+m_{S}}\varepsilon^{\mu\nu\rho\sigma}p_{\rho}p^{\prime}_{\sigma},$
(A.1) $\displaystyle\langle
S(p^{\prime})|\bar{c}\sigma^{\mu\nu}\gamma_{5}b|{B_{c}}(p)\rangle$
$\displaystyle=$
$\displaystyle-i\frac{2f_{T}^{S}(q^{2})}{m_{B_{c}}+m_{S}}\left(p^{\mu}p^{\prime\nu}-p^{\nu}p^{\prime\mu}\right),$
$\displaystyle\langle
A(p^{\prime},\epsilon)|\bar{c}\gamma^{\mu}b|{B_{c}}(p)\rangle$
$\displaystyle=$
$\displaystyle-i\Big{[}2m_{A}A_{0}^{A}(q^{2})\frac{\epsilon^{*}\cdot
q}{q^{2}}q^{\mu}+(m_{B_{c}}+m_{A})A_{1}^{A}(q^{2})\left(\epsilon^{*\mu}-\frac{\epsilon^{*}\cdot
q}{q^{2}}q^{\mu}\right)$ $\displaystyle-$ $\displaystyle
A_{2}^{A}(q^{2})\frac{\epsilon^{*}\cdot
q}{m_{B_{c}}+m_{A}}\left(p^{\mu}+p^{\prime\mu}-\frac{m_{B_{c}}^{2}-m_{A}^{2}}{q^{2}}q^{\mu}\right)\Big{]},$
$\displaystyle\langle
A(p^{\prime},\epsilon)|\bar{c}\gamma^{\mu}\gamma_{5}b|{B_{c}}(p)\rangle$
$\displaystyle=$
$\displaystyle\frac{2V^{A}(q^{2})}{m_{B_{c}}+m_{A}}\varepsilon^{\mu\nu\rho\sigma}\epsilon_{\nu}^{*}p_{\rho}^{\prime}p_{\sigma},$
$\displaystyle\langle
A(p^{\prime},\epsilon)|\bar{c}\sigma^{\mu\nu}b|{B_{c}}(p)\rangle$
$\displaystyle=$ $\displaystyle i\,T_{0}^{A}(q^{2})\frac{\epsilon^{*}\cdot
q}{(m_{B_{c}}+m_{A})^{2}}(p^{\mu}p^{\prime\nu}-p^{\nu}p^{\prime\mu})$
$\displaystyle+$ $\displaystyle
i\,T_{1}^{A}(q^{2})(p^{\mu}\epsilon^{*\nu}-\epsilon^{*\mu}p^{\nu})+i\,T_{2}^{A}(q^{2})(p^{\prime\mu}\epsilon^{*\nu}-\epsilon^{*\mu}p^{\prime\nu}),$
$\displaystyle\langle
A(p^{\prime},\epsilon)|\bar{c}\sigma^{\mu\nu}\gamma_{5}b|{B_{c}}(p)\rangle$
$\displaystyle=$ $\displaystyle T_{0}^{A}(q^{2})\frac{\epsilon^{*}\cdot
q}{(m_{B_{c}}+m_{A})^{2}}\varepsilon^{\mu\nu\alpha\beta}p_{\alpha}p^{\prime}_{\beta}+T_{1}^{A}(q^{2})\varepsilon^{\mu\nu\alpha\beta}p_{\alpha}\epsilon^{*}_{\beta}+T_{2}^{A}(q^{2})\varepsilon^{\mu\nu\alpha\beta}p^{\prime}_{\alpha}\epsilon^{*}_{\beta},$
$\displaystyle\langle
T(p^{\prime},\eta)|\bar{c}\gamma^{\mu}b|{B_{c}}(p)\rangle$ $\displaystyle=$
$\displaystyle\frac{2V^{T}(q^{2})}{m_{B_{c}}\left(m_{B_{c}}+m_{T}\right)}\varepsilon^{\mu\nu\rho\sigma}\eta_{\nu\alpha}^{*}p_{\rho}^{\prime}p_{\sigma}q^{\alpha},$
$\displaystyle\langle
T(p^{\prime},\eta)|\bar{c}\gamma^{\mu}\gamma_{5}b|{B_{c}}(p)\rangle$
$\displaystyle=$
$\displaystyle\Big{[}2m_{T}A_{0}^{T}(q^{2})\frac{\eta^{*\alpha\beta}q_{\beta}}{q^{2}}q^{\mu}+(m_{B_{c}}+m_{T})A_{1}^{T}(q^{2})\left(\eta^{*\mu\alpha}-\frac{\eta^{*\alpha\beta}q_{\beta}}{q^{2}}q^{\mu}\right)$
$\displaystyle-$ $\displaystyle
A_{2}^{T}(q^{2})\frac{\eta^{*\alpha\beta}q_{\beta}}{m_{B_{c}}+m_{T}}\left(p^{\mu}+p^{\prime\mu}-\frac{m_{B_{c}}^{2}-m_{T}^{2}}{q^{2}}q^{\mu}\right)\Big{]}\frac{-iq_{\alpha}}{m_{B_{c}}},$
$\displaystyle\langle
T(p^{\prime},\eta)|\bar{c}\sigma^{\mu\nu}b|{B}(p)\rangle$ $\displaystyle=$
$\displaystyle\Big{[}T_{0}^{T}(q^{2})\frac{\eta^{*}_{\rho\tau}q^{\rho}}{(m_{B_{c}}+m_{T})^{2}}\varepsilon^{\mu\nu\alpha\beta}p^{\prime}_{\alpha}p_{\beta}+T_{1}^{T}(q^{2})\varepsilon^{\mu\nu\alpha\beta}\eta^{*}_{\alpha\tau}p_{\beta}$
$\displaystyle+$ $\displaystyle
T_{2}^{T}(q^{2})\varepsilon^{\mu\nu\alpha\beta}\eta^{*}_{\alpha\tau}p^{\prime}_{\beta}\Big{]}\frac{q^{\tau}}{m_{B_{c}}},$
$\displaystyle\langle
T(p^{\prime},\eta)|\bar{c}\sigma^{\mu\nu}\gamma_{5}b|{B}(p)\rangle$
$\displaystyle=$
$\displaystyle\Big{[}T_{0}^{T}(q^{2})\frac{\eta^{*\alpha\beta}q_{\alpha}}{(m_{B_{c}}+m_{T})^{2}}(p^{\mu}p^{\prime\nu}-p^{\nu}p^{\prime\mu})$
$\displaystyle+$ $\displaystyle
T_{1}^{T}(q^{2})(p^{\mu}\eta^{*\nu\beta}-\eta^{*\mu\beta}p^{\nu})+T_{2}^{T}(q^{2})(p^{\prime\mu}\eta^{*\nu\beta}-\eta^{*\mu\beta}p^{\prime\nu})\Big{]}\frac{-iq_{\beta}}{m_{B_{c}}}.$

In Eqs. (A.1), (A) and (A) we denote
$S={\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{c_{0}}$,
$A={\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{c_{1}}$,$h_{c}$,
and
$T={\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{c_{2}}$.
The conditions

$\displaystyle f_{0}^{S}(0)$ $\displaystyle=$ $\displaystyle f_{+}^{S}(0)$
$\displaystyle A_{0}^{A(T)}(0)$ $\displaystyle=$
$\displaystyle\frac{m_{B}+m_{A(T)}}{2m_{A(T)}}A_{1}^{A(T)}(0)-\frac{m_{B}-m_{A(T)}}{2m_{A(T)}}A_{2}^{A(T)}(0)$
(A.4)

remove the singularity at $q^{2}=0$.

##  Appendix B Form factors in terms of universal functions at
$\mathcal{O}(1/m^{2}_{Q})$

In this Appendix we collect the relations between the form factors in Eqs.
(2)-(5) and those expressed in the effective theory at
$\mathcal{O}(1/m^{2}_{Q})$. We define
$\epsilon_{b(c)}=\displaystyle\frac{1}{2m_{b(c)}}$ and
$\Omega_{i}^{(Q),\text{mix}}=\Omega_{i}^{(b),\text{mix}}+\Omega_{i}^{(c),\text{mix}}$
(for $i=0,1,2$), with $\Omega_{i}^{(b,c),\text{mix}}$ given below.

###  B.1 $B_{c}\to\chi_{c0}$

We define:

$\displaystyle\Sigma_{\chi_{c0}}^{(b)}$
$\displaystyle=(2+w)\Sigma_{1}^{(b)}+(w^{2}-1)\Sigma_{3}^{(b)}-3(w-1)[\Sigma_{4}^{(b)}+\Sigma_{6}^{(b)}]-(w-7)\Sigma_{7}^{(b)}\;,$
(B.1.1) $\displaystyle\Sigma_{\chi_{c0}}^{(c)}$
$\displaystyle=3\Sigma_{1}^{(c)}-(w^{2}-1)\Sigma_{2}^{(c)}-(w-1)[\Sigma_{4}^{(c)}-3\Sigma_{5}^{(c)}]+6\Sigma_{7}^{(c)}\;,$
(B.1.2) $\displaystyle\Omega_{\chi_{c0}}^{(b)}$
$\displaystyle=(w+2)[\Omega_{3}^{(b)}+w\Omega_{5}^{(b)}-\Omega_{8}^{(b)}]+(w^{2}-1)[\Omega_{4}^{(b)}+\Omega_{10}^{(b)}+w\Omega_{12}^{(b)}-\Omega_{18}^{(b)}]$
$\displaystyle-3(w-1)[\Omega_{6}^{(b)}+\Omega_{13}^{(b)}+w\Omega_{16}^{(b)}+\Omega_{17}^{(b)}+w\Omega_{20}^{(b)}+\Omega_{22}^{(b)}]+(w-7)[\Omega_{23}^{(b)}+w\Omega_{25}^{(b)}]$
$\displaystyle-(w-1)(w-2)\Omega_{24}^{(b)}\;,$ (B.1.3)
$\displaystyle\Omega_{\chi_{c0}}^{(c)}$
$\displaystyle=-3[w\Omega_{1}^{(c)}+\Omega_{4}^{(c)}-\Omega_{6}^{(c)}+2w\Omega_{21}^{(c)}+2\Omega_{24}^{(c)}]+(w^{2}-1)[w\Omega_{9}^{(c)}+\Omega_{10}^{c}-\Omega_{13}^{(c)}]$
$\displaystyle+(w-1)[w\Omega_{14}^{(c)}-3w\Omega_{15}^{(c)}-3\Omega_{17}^{(c)}+\Omega_{18}^{(c)}+\Omega_{22}^{(c)}+3\Omega_{23}^{(c)}]\;,$
(B.1.4) $\displaystyle\Omega_{\chi_{c0}}^{(Q),\text{mix}}$
$\displaystyle=(w+1)(w-2)\Omega_{2}^{(Q)}+(w+1)(w+2)\Omega_{3}^{(Q)}-3(w+1)[\Omega_{4}^{(Q)}+2\Omega_{24}^{Q}]+9\Omega_{6}^{(Q)}$
$\displaystyle-(w-2)[3\Omega_{7}^{(Q)}-\Omega_{8}^{(Q)}]+(w-1)(w+1)^{2}\Omega_{10}^{(Q)}-(w^{2}-1)[3\Omega_{13}^{(Q)}+3\Omega_{17}^{(Q)}-\Omega_{18}^{(Q)}]$
$\displaystyle-(w+1)^{2}\Omega_{22}^{(Q)}+(w+1)(w-7)\Omega_{23}^{(Q)}\;,$
(B.1.5) $\displaystyle\Upsilon_{2,\chi_{c0}}^{(b)}$
$\displaystyle=(w+1)[2\Upsilon_{2B}^{(b)}+2(w-1)\Upsilon_{2F}^{(b)}+4\Upsilon_{2I}^{(b)}+3\Upsilon_{2J}^{(b)}]+2(w-2)\Upsilon_{2C}^{(b)}\;,$
(B.1.6) $\displaystyle\Upsilon_{2,\chi_{c0}}^{(c)}$
$\displaystyle=(w+1)[2\Upsilon_{2A}^{(c)}+2(w-1)\Upsilon_{2E}^{(c)}+4\Upsilon_{2H}^{(c)}-\Upsilon_{2J}^{(c)}]-6\Upsilon_{2C}^{(c)}\;.$
(B.1.7)

Using these definitions we obtain the expressions of the form factors in terms
of universal functions:

$\displaystyle g_{+}$
$\displaystyle=-\frac{1}{\sqrt{3}}\big{[}\epsilon_{b}\Sigma_{\chi_{c0}}^{(b)}+\epsilon_{c}\Sigma_{\chi_{c0}}^{(c)}\big{]}+\frac{1}{\sqrt{3}}\big{[}\epsilon_{b}^{2}\Omega_{\chi_{c0}}^{(b)}-\epsilon_{c}^{2}\Omega_{\chi_{c0}}^{(c)}\big{]}\;,$
(B.1.8) $\displaystyle g_{-}$
$\displaystyle=\frac{w+1}{\sqrt{3}}\Xi+\frac{1}{2\sqrt{3}}\big{[}\epsilon_{b}\Upsilon_{2,\chi_{c0}}^{(b)}+\epsilon_{c}\Upsilon_{2,\chi_{c0}}^{(c)}\big{]}$
$\displaystyle+\frac{2}{\sqrt{3}}\Big{[}\epsilon_{b}^{2}[(w+1)\Upsilon_{1A}^{(b)}-3\Upsilon_{1B}^{(b)}]+\epsilon_{c}^{2}[(w+1)\Upsilon_{1A}^{(c)}-3\Upsilon_{1B}^{(c)}]\Big{]}$
$\displaystyle+\frac{1}{2\sqrt{3}}\epsilon_{b}\epsilon_{c}\big{[}(w+1)(\tilde{\Lambda}\Sigma_{\chi_{c0}}^{(b)}-\tilde{\Lambda}^{\prime}\Sigma_{\chi_{c0}}^{(c)})-\,\Omega_{\chi_{c0}}^{(Q),\text{mix}}\big{]}\;,$
(B.1.9) $\displaystyle g_{P}$
$\displaystyle=\frac{w^{2}-1}{\sqrt{3}}\Xi-\frac{w+1}{\sqrt{3}}\big{[}\epsilon_{b}\Sigma_{\chi_{c0}}^{(b)}-\epsilon_{c}\Sigma_{\chi_{c0}}^{(c)}\big{]}+\frac{w-1}{2\sqrt{3}}\big{[}\epsilon_{b}\Upsilon_{2,\chi_{c0}}^{(b)}+\epsilon_{c}\Upsilon_{2,\chi_{c0}}^{(c)}\big{]}$
$\displaystyle+\frac{2(w-1)}{\sqrt{3}}\Big{[}\epsilon_{b}^{2}[(w+1)\Upsilon_{1A}^{(b)}-3\Upsilon_{1B}^{(b)}]+\epsilon_{c}^{2}[(w+1)\Upsilon_{1A}^{(c)}-3\Upsilon_{1B}^{(c)}]\Big{]}$
$\displaystyle+\frac{w+1}{\sqrt{3}}\big{[}\epsilon_{b}^{2}\Omega_{\chi_{c0}}^{(b)}+\epsilon_{c}^{2}\Omega_{\chi_{c0}}^{(c)}\big{]}-\frac{w-1}{2\sqrt{3}}\epsilon_{b}\epsilon_{c}\big{[}(w+1)(\tilde{\Lambda}\Sigma_{\chi_{c0}}^{(b)}-\tilde{\Lambda}^{\prime}\Sigma_{\chi_{c0}}^{(c)})-\,\Omega_{\chi_{c0}}^{(Q),\text{mix}}\big{]}\;,$
(B.1.10) $\displaystyle g_{T}$
$\displaystyle=-\frac{w+1}{\sqrt{3}}\Xi+\frac{1}{\sqrt{3}}\big{[}\epsilon_{b}\Sigma_{\chi_{c0}}^{(b)}-\epsilon_{c}\Sigma_{\chi_{c0}}^{(c)}\big{]}-\frac{1}{2\sqrt{3}}\big{[}\epsilon_{b}\Upsilon_{2,\chi_{c0}}^{(b)}+\epsilon_{c}\Upsilon_{2,\chi_{c0}}^{(c)}\big{]}$
$\displaystyle-\frac{2}{\sqrt{3}}\Big{[}\epsilon_{b}^{2}[(w+1)\Upsilon_{1A}^{(b)}-3\Upsilon_{1B}^{(b)}]+\epsilon_{c}^{2}[(w+1)\Upsilon_{1A}^{(c)}-3\Upsilon_{1B}^{(c)}]\Big{]}$
$\displaystyle-\frac{1}{\sqrt{3}}\big{[}\epsilon_{b}^{2}\Omega_{\chi_{c0}}^{(b)}+\epsilon_{c}^{2}\Omega_{\chi_{c0}}^{(c)}\big{]}+\frac{1}{2\sqrt{3}}\epsilon_{b}\epsilon_{c}\big{[}(w+1)(\tilde{\Lambda}\Sigma_{\chi_{c0}}^{(b)}-\tilde{\Lambda}^{\prime}\Sigma_{\chi_{c0}}^{(c)})-\,\Omega_{\chi_{c0}}^{(Q),\text{mix}}\big{]}\;.$
(B.1.11)

###  B.2 $B_{c}\to\chi_{c1}$

We define:

$\displaystyle\Sigma_{\chi_{c1},1}^{(b)}$
$\displaystyle=\Sigma_{1}^{(b)}-(w-1)\Sigma_{6}^{(b)}+2\Sigma_{7}^{(b)}\;,$
(B.2.1) $\displaystyle\Sigma_{\chi_{c1},2}^{(b)}$
$\displaystyle=\Sigma_{1}^{(b)}+(w+1)\Sigma_{3}^{(b)}-3\Sigma_{4}^{(b)}-\Sigma_{7}^{(b)}\;,$
(B.2.2) $\displaystyle\Sigma_{\chi_{c1},1}^{(c)}$
$\displaystyle=\Sigma_{1}^{(c)}-(w-1)\Sigma_{5}^{(c)}\;,$ (B.2.3)
$\displaystyle\Sigma_{\chi_{c1},2}^{(c)}$
$\displaystyle=(w+1)\Sigma_{2}^{(c)}-\Sigma_{4}^{(c)}-4\Sigma_{5}^{(c)}\;,$
(B.2.4) $\displaystyle\Omega_{\chi_{c1},1}^{(b)}$
$\displaystyle=-\Omega_{3}^{(b)}-w\Omega_{5}^{(b)}+\Omega_{8}^{(b)}+(w-1)[\Omega_{17}^{(b)}+w\Omega_{20}^{(b)}-\Omega_{24}^{(b)}]+2\,\Omega_{23}^{(b)}+2w\Omega_{25}^{(b)}\;,$
(B.2.5) $\displaystyle\Omega_{\chi_{c1},2}^{(b)}$
$\displaystyle=\Omega_{3}^{(b)}+(w+1)[\Omega_{4}^{(b)}+\Omega_{10}^{(b)}+w\Omega_{12}^{(b)}-\Omega_{18}^{(b)}-\Omega_{24}^{(b)}]$
$\displaystyle+w\Omega_{5}^{(b)}-3\Omega_{6}^{(b)}-\Omega_{8}^{(b)}-3\Omega_{13}^{(b)}-3w\Omega_{16}^{(b)}-3\Omega_{22}^{(b)}+\Omega_{23}^{(b)}+w\Omega_{25}^{(b)}\;,$
(B.2.6) $\displaystyle\Omega_{\chi_{c1},1}^{(c)}$
$\displaystyle=w\Omega_{1}^{(c)}+\Omega_{4}^{(c)}-\Omega_{6}^{(c)}-(w-1)[w\Omega_{15}^{(c)}+\Omega_{17}^{(c)}-\Omega_{23}^{(c)}]\;,$
(B.2.7) $\displaystyle\Omega_{\chi_{c1},2}^{(c)}$
$\displaystyle=-(w+1)[w\Omega_{9}^{(c)}+\Omega_{10}^{(c)}-\Omega_{13}^{(c)}]+w[\Omega_{14}^{(c)}+4\Omega_{15}^{(c)}]$
$\displaystyle+4\Omega_{17}^{(c)}+\Omega_{18}^{(c)}+\Omega_{22}^{(c)}-4\Omega_{23}^{(c)}\;,$
(B.2.8) $\displaystyle\Omega_{\chi_{c1},1}^{(Q),\text{mix}}$
$\displaystyle=(w+1)[\Omega_{3}^{(Q)}+\Omega_{4}^{(Q)}-2\Omega_{23}^{(Q)}]-3\Omega_{6}^{(Q)}-(w+2)\Omega_{7}^{(Q)}-\Omega_{8}^{(Q)}-(w^{2}-1)\Omega_{17}^{Q}\;,$
(B.2.9) $\displaystyle\Omega_{\chi_{c1},2}^{(Q),\text{mix}}$
$\displaystyle=w\Omega_{2}^{(Q)}+(w+3)\Omega_{3}^{(Q)}-4\Omega_{7}^{(Q)}-\Omega_{8}^{(Q)}+(w^{2}-1)\Omega_{10}^{(Q)}$
$\displaystyle-(w-1)[3\Omega_{13}^{(Q)}+4\Omega_{17}^{(Q)}+\Omega_{18}^{(Q)}]-(w-3)\Omega_{22}^{(Q)}+(w-9)\Omega_{23}^{(Q)}\;,$
(B.2.10) $\displaystyle\Upsilon_{2,\chi_{c1},1}^{(b)}$
$\displaystyle=(1+w)[\Upsilon_{2B}^{(b)}+\Upsilon_{2I}^{(b)}]-2\Upsilon_{2C}^{(b)}\;,$
(B.2.11) $\displaystyle\Upsilon_{2,\chi_{c1},2}^{(b)}$
$\displaystyle=\Upsilon_{2B}^{(b)}-2\Upsilon_{2C}^{(b)}-2(w-1)\Upsilon_{2F}^{(b)}-\Upsilon_{2I}^{(b)}-3\Upsilon_{2J}^{(b)}\;,$
(B.2.12) $\displaystyle\Upsilon_{2,\chi_{c1},1}^{(c)}$
$\displaystyle=\Upsilon_{2A}^{(c)}+\Upsilon_{2H}^{(c)}\;,$ (B.2.13)
$\displaystyle\Upsilon_{2,\chi_{c1},2}^{(c)}$
$\displaystyle=3\Upsilon_{2A}^{(c)}+\Upsilon_{2H}^{(c)}-\Upsilon_{2J}^{(c)}\;.$
(B.2.14)

With these definitions, we express the form factors in terms of universal
functions:

$\displaystyle g_{V_{1}}$
$\displaystyle=\frac{w^{2}-1}{\sqrt{2}}\Xi-\frac{w+1}{\sqrt{2}}\Big{[}\epsilon_{b}[2\Sigma_{\chi_{c1},1}^{(b)}+(w-1)\Sigma_{\chi_{c1},2}^{(b)}]-\epsilon_{c}[2\Sigma_{\chi_{c1},1}^{(c)}-(w-1)\Sigma_{\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle+\frac{w-1}{2\sqrt{2}}\Big{[}\epsilon_{b}[2\Upsilon_{2,\chi_{c1},1}^{(b)}-(w+1)\Upsilon_{2,\chi_{c1},2}^{(b)}]-\epsilon_{c}(w+1)[2\Upsilon_{2,\chi_{c1},1}^{(c)}-\Upsilon_{2,\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle+\sqrt{2}(w-1)\Big{[}\epsilon_{b}^{2}[(w+1)\Upsilon_{1A}^{(b)}-2\Upsilon_{1B}^{(b)})]+\epsilon_{c}^{2}[(w+1)\Upsilon_{1A}^{(c)}-2\Upsilon_{1B}^{(c)}]\Big{]}$
$\displaystyle-\frac{w+1}{\sqrt{2}}\Big{[}\epsilon_{b}^{2}[2\Omega_{\chi_{c1},1}^{(b)}-(w-1)\Omega_{\chi_{c1},2}^{(b)}]+\epsilon_{c}^{2}[2\Omega_{\chi_{c1},1}^{(c)}+(w-1)\Omega_{\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle-\frac{w-1}{2\sqrt{2}}\epsilon_{b}\epsilon_{c}\Big{[}(w+1)\Big{(}\tilde{\Lambda}[2\Sigma_{\chi_{c1},1}^{(b)}+(w-1)\Sigma_{\chi_{c1},2}^{(b)}]-\tilde{\Lambda}^{\prime}[2\Sigma_{\chi_{c1},1}^{(c)}-(w-1)\Sigma_{\chi_{c1},2}^{(c)}]\Big{)}$
$\displaystyle+[2\Omega_{\chi_{c1},1}^{(Q),\text{mix}}-(w+1)\Omega_{\chi_{c1},2}^{(Q),\text{mix}}]\Big{]}\;,$
(B.2.15) $\displaystyle g_{V_{2}}$
$\displaystyle=-\frac{w-1}{2\sqrt{2}}\Xi+\frac{1}{2\sqrt{2}}\Big{[}\epsilon_{b}[2\Sigma_{\chi_{c1},1}^{(b)}+(w-1)\Sigma_{\chi_{c1},2}^{(b)}]-\epsilon_{c}[2\Sigma_{\chi_{c1},1}^{(c)}-(w-1)\Sigma_{\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle-\frac{1}{4\sqrt{2}}\Big{[}\epsilon_{b}[2\Upsilon_{2,\chi_{c1},1}^{(b)}-(w-1)\Upsilon_{2,\chi_{c1},2}^{(b)}]-\epsilon_{c}[2(w+1)\Upsilon_{2,\chi_{c1},1}^{(c)}-(w-1)\Upsilon_{2,\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle-\frac{1}{\sqrt{2}}\Big{[}\epsilon_{b}^{2}[(w-1)\Upsilon_{1A}^{(b)}-2\Upsilon_{1B}^{(b)}]+\epsilon_{c}^{2}[(w-1)\Upsilon_{1A}^{(c)}-2\Upsilon_{1B}^{(c)}]\Big{]}$
$\displaystyle+\frac{1}{2\sqrt{2}}\Big{[}\epsilon_{b}^{2}[2\Omega_{\chi_{c1},1}^{(b)}-(w-1)\Omega_{\chi_{c1},2}^{(b)}]+\epsilon_{c}^{2}[2\Omega_{\chi_{c1},1}^{(c)}+(w-1)\Omega_{\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle+\frac{1}{4\sqrt{2}}\epsilon_{b}\epsilon_{c}\Big{[}\tilde{\Lambda}[2(w-3)\Sigma_{\chi_{c1},1}^{(b)}+(w-1)^{2}\Sigma_{\chi_{c1},2}^{(b)}]-\tilde{\Lambda}^{\prime}[2(w+1)\Sigma_{\chi_{c1},1}^{(c)}-(w-1)^{2}\Sigma_{\chi_{c1},2}^{(c)}]$
$\displaystyle+[2\Omega_{\chi_{c1},1}^{(Q),\text{mix}}-(w-1)\Omega_{\chi_{c1},2}^{(Q),\text{mix}}]\Big{]}\,,$
(B.2.16) $\displaystyle g_{V_{3}}$
$\displaystyle=\frac{w+1}{2\sqrt{2}}\Xi-\frac{1}{2\sqrt{2}}\Big{[}\epsilon_{b}[2\Sigma_{\chi_{c1},1}^{(b)}+(w+1)\Sigma_{\chi_{c1},2}^{(b)}]-\epsilon_{c}[2\Sigma_{\chi_{c1},1}^{(c)}-(w+1)\Sigma_{\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle+\frac{1}{4\sqrt{2}}\Big{[}\epsilon_{b}[2\Upsilon_{2,\chi_{c1},1}^{(b)}-(w+1)\Upsilon_{2,\chi_{c1},2}^{(b)}]-\epsilon_{c}(w+1)[2\Upsilon_{2,\chi_{c1},1}^{(c)}-\Upsilon_{2,\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle+\frac{1}{\sqrt{2}}\Big{[}\epsilon_{b}^{2}[(w+1)\Upsilon_{1A}^{(b)}-2\Upsilon_{1B}^{(b)}]+\epsilon_{c}^{2}[(w+1)\Upsilon_{1A}^{(c)}-2\Upsilon_{1B}^{(c)}]\Big{]}$
$\displaystyle-\frac{1}{2\sqrt{2}}\Big{[}\epsilon_{b}^{2}[2\Omega_{\chi_{c1},1}^{(b)}-(w+1)\Omega_{\chi_{c1},2}^{(b)}]+\epsilon_{c}^{2}[2\Omega_{\chi_{c1},1}^{(c)}+(w+1)\Omega_{\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle-\frac{1}{4\sqrt{2}}\epsilon_{b}\epsilon_{c}\Bigg{[}(w+1)\Big{(}\tilde{\Lambda}[2\Sigma_{\chi_{c1},1}^{(b)}+(w-1)\Sigma_{\chi_{c1},2}^{(b)}]-\tilde{\Lambda}^{\prime}[2\Sigma_{\chi_{c1},1}^{(c)}-(w-1)\Sigma_{\chi_{c1},2}^{(c)}]\Big{)}$
$\displaystyle+[2\Omega_{\chi_{c1},1}^{(Q),\text{mix}}-(w+1)\Omega_{\chi_{c1},2}^{(Q),\text{mix}}]\Bigg{]}\;,$
(B.2.17) $\displaystyle g_{A}$
$\displaystyle=\frac{w+1}{\sqrt{2}}\Xi-\frac{1}{\sqrt{2}}\Big{[}\epsilon_{b}[2\Sigma_{\chi_{c1},1}^{(b)}+(w-1)\Sigma_{\chi_{c1},2}^{(b)}]-\epsilon_{c}[2\Sigma_{\chi_{c1},1}^{(c)}-(w-1)\Sigma_{\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle+\frac{1}{2\sqrt{2}}\Big{[}\epsilon_{b}[2\Upsilon_{2,\chi_{c1},1}^{(b)}-(w+1)\Upsilon_{2,\chi_{c1},2}^{(b)}]-\epsilon_{c}(w+1)[2\Upsilon_{2,\chi_{c1},1}^{(c)}-\Upsilon_{2,\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle+\sqrt{2}\Big{[}\epsilon_{b}^{2}[(w+1)\Upsilon_{1A}^{(b)}-2\Upsilon_{1B}^{(b)}]+\epsilon_{c}^{2}[(w+1)\Upsilon_{1A}^{(c)}-2\Upsilon_{1B}^{(c)}]\Big{]}$
$\displaystyle-\frac{1}{\sqrt{2}}\Big{[}\epsilon_{b}^{2}[2\Omega_{\chi_{c1},1}^{(b)}-(w-1)\Omega_{\chi_{c1},2}^{(b)}]+\epsilon_{c}^{2}[2\Omega_{\chi_{c1},1}^{(c)}+(w-1)\Omega_{\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle-\frac{1}{2\sqrt{2}}\epsilon_{b}\epsilon_{c}\Big{[}(w+1)\Big{(}\tilde{\Lambda}[2\Sigma_{\chi_{c1},1}^{(b)}+(w-1)\Sigma_{\chi_{c1},2}^{(b)}]-\tilde{\Lambda}^{\prime}[2\Sigma_{\chi_{c1},1}^{(c)}-(w-1)\Sigma_{\chi_{c1},2}^{(c)}]\Big{)}\Big{]}$
$\displaystyle+(2\Omega_{\chi_{c1},1}^{(Q),\text{mix}}-(w+1)\Omega_{\chi_{c1},2}^{(Q),\text{mix}})\Big{]}\;,$
(B.2.18) $\displaystyle g_{S}$
$\displaystyle=\sqrt{2}\big{[}\epsilon_{b}\Sigma_{\chi_{c1},1}^{(b)}+\epsilon_{c}\Sigma_{\chi_{c1},1}^{(c)}\big{]}-\frac{1}{\sqrt{2}}\big{[}\epsilon_{b}\Upsilon_{2,\chi_{c1},1}^{(b)}-\epsilon_{c}(w+1)\Upsilon_{2,\chi_{c1},1}^{(c)}\big{]}$
$\displaystyle+2\sqrt{2}\big{[}\epsilon_{b}^{2}\Upsilon_{1B}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1B}^{(c)}\big{]}+\sqrt{2}\big{[}\epsilon_{b}^{2}\Omega_{\chi_{c1},1}^{(b)}-\epsilon_{c}^{2}\Omega_{\chi_{c1},1}^{(c)}\big{]}$
$\displaystyle+\frac{1}{\sqrt{2}}\epsilon_{b}\epsilon_{c}\big{[}(w+1)(\tilde{\Lambda}\Sigma_{\chi_{c1},1}^{(b)}+\tilde{\Lambda}^{\prime}\Sigma_{\chi_{c1},1}^{(c)})-\Omega_{\chi_{c1},1}^{(Q),\text{mix}}\big{]}\;,$
(B.2.19) $\displaystyle g_{T_{1}}$
$\displaystyle=-\frac{1}{\sqrt{2}}\Big{[}\epsilon_{b}[2\Sigma_{\chi_{c1},1}^{(b)}+(w-1)\Sigma_{\chi_{c1},2}^{(b)}]+\epsilon_{c}[2\Sigma_{\chi_{c1},1}^{(c)}-(w-1)\Sigma_{\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle-\frac{1}{\sqrt{2}}\Big{[}\epsilon_{b}^{2}[2\Omega_{\chi_{c1},1}^{(b)}-(w-1)\Omega_{\chi_{c1},2}^{(b)}]-\epsilon_{c}^{2}[2\Omega_{\chi_{c1},1}^{(c)}+(w-1)\Omega_{\chi_{c1},2}^{(c)}]\Big{]}\;,$
(B.2.20) $\displaystyle g_{T_{2}}$
$\displaystyle=\frac{w+1}{\sqrt{2}}\Xi+\frac{1}{2\sqrt{2}}\Big{[}\epsilon_{b}[2\Upsilon_{2,\chi_{c1},1}^{(b)}-(w+1)\Upsilon_{2,\chi_{c1},2}^{(b)}]-\epsilon_{c}(w+1)[2\Upsilon_{2,\chi_{c1},1}^{(c)}-\Upsilon_{2,\chi_{c1},2}^{(c)}]\Big{]}$
$\displaystyle+\sqrt{2}\Big{[}\epsilon_{b}^{2}[(w+1)\Upsilon_{1A}^{(b)}-2\Upsilon_{1B}^{(b)}]+\epsilon_{c}^{2}[(w+1)\Upsilon_{1A}^{(c)}-2\Upsilon_{1B}^{(c)}]\Big{]}$
$\displaystyle+\frac{1}{2\sqrt{2}}\epsilon_{b}\epsilon_{c}\Big{[}(w+1)\Big{(}\tilde{\Lambda}[2\Sigma_{\chi_{c1},1}^{(b)}+(w-1)\Sigma_{\chi_{c1},2}^{(b)}]-\tilde{\Lambda}^{\prime}[2\Sigma_{\chi_{c1},1}^{(c)}-(w-1)\Sigma_{\chi_{c1},2}^{(c)}]\Big{)}$
$\displaystyle+[2\Omega_{\chi_{c1},1}^{(Q),\text{mix}}-(w+1)\Omega_{\chi_{c1},2}^{(Q),\text{mix}}]\Big{]}\;,$
(B.2.21) $\displaystyle g_{T_{3}}$
$\displaystyle=\frac{1}{\sqrt{2}}\Xi-\frac{1}{\sqrt{2}}\big{[}\epsilon_{b}\Sigma_{\chi_{c1},2}^{(b)}-\epsilon_{c}\Sigma_{\chi_{c1},2}^{(c)}\big{]}-\frac{1}{2\sqrt{2}}\big{[}\epsilon_{b}\Upsilon_{2,\chi_{c1},2}^{(b)}-\epsilon_{c}\Upsilon_{2,\chi_{c1},2}^{(c)}\big{]}$
$\displaystyle+\sqrt{2}\big{[}\epsilon_{b}^{2}\Upsilon_{1A}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1A}^{(c)}\big{]}+\frac{1}{\sqrt{2}}\big{[}\epsilon_{b}^{2}\Omega_{\chi_{c1},2}^{(b)}+\epsilon_{c}^{2}\Omega_{\chi_{c1},2}^{(c)}\big{]}$
$\displaystyle+\frac{1}{2\sqrt{2}}\epsilon_{b}\epsilon_{c}\Big{[}\tilde{\Lambda}[4\Sigma_{\chi_{c1},1}^{(b)}+(w-1)\Sigma_{\chi_{c1},2}^{(b)}]+\tilde{\Lambda}^{\prime}(w-1)\Sigma_{\chi_{c1},2}^{(c)}-\Omega_{\chi_{c1},2}^{(Q),\text{mix}}\Big{]}\;.$
(B.2.22)

###  B.3 $B_{c}\to\chi_{c2}$

We define:

$\displaystyle\Sigma_{\chi_{c2}}^{(b)}$
$\displaystyle=\Sigma_{1}^{(b)}+(w+1)\Sigma_{3}^{(b)}-3\Sigma_{4}^{(b)}-\Sigma_{7}^{(b)}\;,$
(B.3.1) $\displaystyle\Sigma_{\chi_{c2},1}^{(c)}$
$\displaystyle=(w+1)\Sigma_{2}^{(c)}-\Sigma_{4}^{(c)}\;,$ (B.3.2)
$\displaystyle\Sigma_{\chi_{c2},2}^{(c)}$
$\displaystyle=(w+1)\Sigma_{2}^{(c)}+\Sigma_{4}^{(c)}\;,$ (B.3.3)
$\displaystyle\Omega_{\chi_{c2}}^{(b)}$
$\displaystyle=\Omega_{3}^{(b)}+(w+1)[\Omega_{4}^{(b)}+\Omega_{10}^{(b)}+w\Omega_{12}^{(b)}-\Omega_{18}^{(b)}-\Omega_{24}^{(b)}]+w\Omega_{5}^{(b)}$
$\displaystyle-3[\Omega_{6}^{(b)}+\Omega_{13}^{(b)}+w\Omega_{16}^{(b)}+\Omega_{22}^{(b)}]-\Omega_{8}^{(b)}+\Omega_{23}^{(b)}+w\Omega_{25}^{(b)}\;,$
(B.3.4) $\displaystyle\Omega_{\chi_{c2},1}^{(c)}$
$\displaystyle=-(w+1)[w\Omega_{9}^{(c)}+\Omega_{10}^{(c)}-\Omega_{13}^{(c)}]+w\Omega_{14}^{(c)}+\Omega_{18}^{(c)}+\Omega_{22}^{(c)}\;,$
(B.3.5) $\displaystyle\Omega_{\chi_{c2},2}^{(c)}$
$\displaystyle=(w+1)[w\Omega_{9}^{(c)}+\Omega_{10}^{(c)}-\Omega_{13}^{(c)}]+w\Omega_{14}^{(c)}+\Omega_{18}^{(c)}+\Omega_{22}^{(c)}\;,$
(B.3.6) $\displaystyle\Omega_{\chi_{c2},1}^{(Q),\text{mix}}$
$\displaystyle=\Omega_{8}^{(Q)}-w\Omega_{2}^{(Q)}+(1-w)[\Omega_{3}^{(Q)}+(1+w)\Omega_{10}^{(Q)}$
$\displaystyle-3\Omega_{13}^{(Q)}-\Omega_{18}^{(Q)}+\Omega_{23}^{(Q)}]+(w-3)\Omega_{22}^{(Q)}\;,$
(B.3.7) $\displaystyle\Omega_{\chi_{c2},2}^{(Q),\text{mix}}$
$\displaystyle=-\Omega_{8}^{(Q)}+(2-w)\Omega_{2}^{(Q)}+(1-w)[\Omega_{3}^{(Q)}+(1+w)\Omega_{10}^{(Q)}$
$\displaystyle-3\Omega_{13}^{(Q)}+\Omega_{18}^{(Q)}+\Omega_{23}^{(Q)}]+(1+w)\Omega_{22}^{(Q)}\;,$
(B.3.8) $\displaystyle\Upsilon_{2,\chi_{c2}}^{(b)}$
$\displaystyle=\Upsilon_{2B}^{(b)}-2\Upsilon_{2C}^{(b)}-2(w-1)\Upsilon_{2F}^{(b)}-\Upsilon_{2I}^{(b)}-3\Upsilon_{2J}^{(b)}\;,$
(B.3.9) $\displaystyle\Upsilon_{2,\chi_{c2},1}^{(c)}$
$\displaystyle=\Upsilon_{2A}^{(c)}-\Upsilon_{2H}^{(c)}+\Upsilon_{2J}^{(c)}\;,$
(B.3.10) $\displaystyle\Upsilon_{2,\chi_{c2},2}^{(c)}$
$\displaystyle=\Upsilon_{2A}^{(c)}-2(w-1)\Upsilon_{2E}^{(c)}-\Upsilon_{2H}^{(c)}+\Upsilon_{2J}^{(c)}\;.$
(B.3.11)

With these definitions the form factors are expressed in terms of universal
functions:

$\displaystyle k_{V}$
$\displaystyle=-\Xi+\big{[}\epsilon_{b}\Sigma_{\chi_{c_{2}}}^{(b)}+\epsilon_{c}\Sigma_{\chi_{c_{2}},1}^{(c)}\big{]}+\frac{1}{2}\big{[}\epsilon_{b}\Upsilon_{2,\chi_{c_{2}}}^{(b)}+\epsilon_{c}\Upsilon_{2,\chi_{c_{2}},1}^{(c)}\big{]}-2\big{[}\epsilon_{b}^{2}\Upsilon_{1A}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1A}^{(c)}\big{]}$
$\displaystyle-\big{[}\epsilon_{b}^{2}\Omega_{\chi_{c_{2}}}^{(b)}-\epsilon_{c}^{2}\Omega_{\chi_{c_{2}},1}^{(c)}\big{]}+\frac{1}{2}\epsilon_{b}\epsilon_{c}\Big{[}(w-1)\Big{(}\tilde{\Lambda}\Sigma_{\chi_{c_{2}}}^{(b)}+\tilde{\Lambda}^{\prime}\Sigma_{\chi_{c_{2}},1}^{(c)}\Big{)}+\Omega_{\chi_{c_{2}},1}^{(Q),\text{mix}}\Big{]}\;,$
(B.3.12) $\displaystyle k_{A_{1}}$
$\displaystyle=(w+1)\Xi-(w-1)\big{[}\epsilon_{b}\Sigma_{\chi_{c_{2}}}^{(b)}+\epsilon_{c}\Sigma_{\chi_{c_{2}},1}^{(c)}\big{]}$
$\displaystyle-\frac{w+1}{2}\big{[}\epsilon_{b}\Upsilon_{2,\chi_{c_{2}}}^{(b)}+\epsilon_{c}\Upsilon_{2,\chi_{c_{2}},1}^{(c)}\big{]}+2(w+1)\big{[}\epsilon_{b}^{2}\Upsilon_{1A}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1A}^{(c)}\big{]}$
$\displaystyle+(w-1)\big{[}\epsilon_{b}^{2}\Omega_{\chi_{c_{2}}}^{(b)}-\epsilon_{c}^{2}\Omega_{\chi_{c_{2}},1}^{(c)}\big{]}-\frac{w+1}{2}\epsilon_{b}\epsilon_{c}\big{[}(w-1)\big{(}\tilde{\Lambda}\Sigma_{\chi_{c_{2}}}^{(b)}+\tilde{\Lambda}^{\prime}\Sigma_{\chi_{c_{2}},1}^{(c)}\big{)}+\Omega_{\chi_{c_{2}},1}^{(Q),\text{mix}}\big{]}\;,$
(B.3.13) $\displaystyle k_{A_{2}}$
$\displaystyle=-\frac{\epsilon_{c}}{w+1}\big{[}\Sigma_{\chi_{c_{2}},1}^{(c)}+\Sigma_{\chi_{c_{2}},2}^{(c)}\big{]}-\frac{\epsilon_{c}}{2(w-1)}\big{[}\Upsilon_{2,\chi_{c_{2}},1}^{(c)}-\Upsilon_{2,\chi_{c_{2}},2}^{(c)}\big{]}-\frac{\epsilon_{c}^{2}}{w+1}\big{[}\Omega_{\chi_{c_{2}},1}^{(c)}-\Omega_{\chi_{c_{2}},2}^{(c)}\big{]}$
$\displaystyle-\frac{\epsilon_{b}\epsilon_{c}}{2(w-1)}\Big{[}(w-1)\Big{(}2\tilde{\Lambda}\Sigma_{\chi_{c_{2}}}^{(b)}+\tilde{\Lambda}^{\prime}(\Sigma_{\chi_{c_{2}},1}^{(c)}+\Sigma_{\chi_{c_{2}},2}^{(c)})\Big{)}+\Omega_{\chi_{c_{2}},1}^{(Q),\text{mix}}+\Omega_{\chi_{c_{2}},2}^{(Q),\text{mix}}\Big{]}\;,$
(B.3.14) $\displaystyle k_{A_{3}}$
$\displaystyle=-\Xi+\Big{[}\epsilon_{b}\Sigma_{\chi_{c_{2}}}^{(b)}+\frac{\epsilon_{c}}{w+1}\Big{(}w\Sigma_{\chi_{c_{2}},1}^{(c)}-\Sigma_{\chi_{c_{2}},2}^{(c)}\Big{)}\Big{]}+\frac{1}{2}\Big{[}\epsilon_{b}\Upsilon_{2,\chi_{c_{2}}}^{(b)}+\frac{\epsilon_{c}}{w-1}\Big{(}w\Upsilon_{2,\chi_{c_{2}},1}^{(c)}-\Upsilon_{2,\chi_{c_{2}},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle-2\big{[}\epsilon_{b}^{2}\Upsilon_{1A}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1A}^{(c)}\big{]}-\Big{[}\epsilon_{b}^{2}\Omega_{\chi_{c_{2}}}^{(b)}-\frac{\epsilon_{c}^{2}}{w+1}\Big{(}w\Omega_{\chi_{c_{2}},1}^{(c)}+\Omega_{\chi_{c_{2}},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle+\frac{\epsilon_{b}\epsilon_{c}}{2(w-1)}\Big{[}(w-1)\Big{(}(w+1)\tilde{\Lambda}\Sigma_{\chi_{c_{2}}}^{(b)}+\tilde{\Lambda}^{\prime}(w\Sigma_{\chi_{c_{2}},1}^{(c)}+\Sigma_{\chi_{c_{2}},2}^{(c)})\Big{)}+w\Omega_{\chi_{c_{2}},1}^{(Q),\text{mix}}+\Omega_{\chi_{c_{2}},2}^{(Q),\text{mix}}\Big{]}\;,$
(B.3.15) $\displaystyle k_{P}$
$\displaystyle=-\Xi+\big{[}\epsilon_{b}\Sigma_{\chi_{c_{2}}}^{(b)}+\epsilon_{c}\Sigma_{\chi_{c_{2}},2}^{(c)}\big{]}+\frac{1}{2}\big{[}\epsilon_{b}\Upsilon_{2,\chi_{c_{2}}}^{(b)}+\epsilon_{c}\Upsilon_{2,\chi_{c_{2}},2}^{(c)}\big{]}-2\big{[}\epsilon_{b}^{2}\Upsilon_{1A}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1A}^{(c)}\big{]}$
$\displaystyle-\big{[}\epsilon_{b}^{2}\Omega_{\chi_{c_{2}}}^{(b)}+\epsilon_{c}^{2}\Omega_{\chi_{c_{2}},2}^{(c)}\big{]}+\frac{1}{2}\epsilon_{b}\epsilon_{c}\big{[}(w-1)\big{(}\tilde{\Lambda}\Sigma_{\chi_{c_{2}}}^{(b)}+\tilde{\Lambda}^{\prime}\Sigma_{\chi_{c_{2}},2}^{(c)}\big{)}+\Omega_{\chi_{c_{2}},2}^{(Q),\text{mix}}\big{]}\;,$
(B.3.16) $\displaystyle k_{T_{1}}$
$\displaystyle=-\Xi+[\epsilon_{b}\Sigma_{\chi_{c_{2}}}^{(b)}-\epsilon_{c}\Sigma_{\chi_{c_{2}},1}^{(c)}]+\frac{1}{2}[\epsilon_{b}\Upsilon_{2,\chi_{c_{2}}}^{(b)}+\epsilon_{c}\Upsilon_{2,\chi_{c_{2}},1}^{(c)}]-2[\epsilon_{b}^{2}\Upsilon_{1A}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1A}^{(c)}]$
$\displaystyle-[\epsilon_{b}^{2}\Omega_{\chi_{c_{2}}}^{(b)}+\epsilon_{c}^{2}\Omega_{\chi_{c_{2}},1}^{(c)}]-\frac{1}{2}\epsilon_{b}\epsilon_{c}\Big{[}(w-1)\Big{(}\tilde{\Lambda}\Sigma_{\chi_{c_{2}}}^{(b)}+\tilde{\Lambda}^{\prime}\Sigma_{\chi_{c_{2}},1}^{(c)}\Big{)}+\Omega_{\chi_{c_{2}},1}^{(Q),\text{mix}}\Big{]}\;,$
(B.3.17) $\displaystyle k_{T_{2}}$
$\displaystyle=-\Xi-[\epsilon_{b}\Sigma_{\chi_{c_{2}}}^{(b)}-\epsilon_{c}\Sigma_{\chi_{c_{2}},1}^{(c)}]+\frac{1}{2}[\epsilon_{b}\Upsilon_{2,\chi_{c_{2}}}^{(b)}+\epsilon_{c}\Upsilon_{2,\chi_{c_{2}},1}^{(c)}]-2[\epsilon_{b}^{2}\Upsilon_{1A}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1A}^{(c)}]$
$\displaystyle+[\epsilon_{b}^{2}\Omega_{\chi_{c_{2}}}^{(b)}+\epsilon_{c}^{2}\Omega_{\chi_{c_{2}},1}^{(c)}]-\frac{1}{2}\epsilon_{b}\epsilon_{c}\Big{[}(w-1)\Big{(}\tilde{\Lambda}\Sigma_{\chi_{c_{2}}}^{(b)}+\tilde{\Lambda}^{\prime}\Sigma_{\chi_{c_{2}},1}^{(c)}\Big{)}+\Omega_{\chi_{c_{2}},1}^{(Q),\text{mix}}\Big{]}\;,$
(B.3.18) $\displaystyle k_{T_{3}}$
$\displaystyle=-\frac{\epsilon_{c}}{w+1}[\Sigma_{\chi_{c_{2}},1}^{(c)}+\Sigma_{\chi_{c_{2}},2}^{(c)}]+\frac{\epsilon_{c}}{2(w-1)}[\Upsilon_{2,\chi_{c_{2}},1}^{(c)}-\Upsilon_{2,\chi_{c_{2}},2}^{(c)}]-\frac{\epsilon_{c}^{2}}{w+1}[\Omega_{\chi_{c_{2}},1}^{(c)}-\Omega_{\chi_{c_{2}},2}^{(c)}]$
$\displaystyle-\frac{\epsilon_{b}\epsilon_{c}}{2(w-1)}\Big{[}(w-1)\Big{(}2\tilde{\Lambda}\Sigma_{\chi_{c_{2}}}^{(b)}+\tilde{\Lambda}^{\prime}(\Sigma_{\chi_{c_{2}},1}^{(c)}+\Sigma_{\chi_{c_{2}},2}^{(c)})\Big{)}+\Omega_{\chi_{c_{2}},1}^{(Q),\text{mix}}+\Omega_{\chi_{c_{2}},2}^{(Q),\text{mix}}\Big{]}\;.$
(B.3.19)

###  B.4 $B_{c}\to h_{c}$

We define:

$\displaystyle\Sigma_{h_{c},1}^{(b)}$
$\displaystyle=w\Sigma_{1}^{(b)}+(w^{2}-1)\Sigma_{3}^{(b)}-(w-1)[3\Sigma_{4}^{(b)}+\Sigma_{6}^{(b)}]-(w-3)\Sigma_{7}^{(b)}\;,$
(B.4.1) $\displaystyle\Sigma_{h_{c},2}^{(b)}$
$\displaystyle=\Sigma_{1}^{(b)}+(w+1)\Sigma_{3}^{(b)}-3\Sigma_{4}^{(b)}-\Sigma_{7}^{(b)}\;,$
(B.4.2) $\displaystyle\Sigma_{h_{c},1}^{(c)}$
$\displaystyle=\Sigma_{1}^{(c)}-(w^{2}-1)\Sigma_{2}^{(c)}+(w-1)[3\Sigma_{4}^{(c)}+\Sigma_{5}^{(c)}]-2\Sigma_{7}^{(c)}\;,$
(B.4.3) $\displaystyle\Sigma_{h_{c},2}^{(c)}$
$\displaystyle=(w+1)\Sigma_{2}^{(c)}-3\Sigma_{4}^{(c)}-2\Sigma_{5}^{(c)}\;,$
(B.4.4) $\displaystyle\Omega_{h_{c},1}^{(b)}$
$\displaystyle=w[\Omega_{3}^{(b)}+w\Omega_{5}^{(b)}-\Omega_{8}^{(b)}]+(w^{2}-1)[\Omega_{4}^{(b)}+\Omega_{10}^{(b)}+w\Omega_{12}^{(b)}-\Omega_{18}^{(b)}]$
$\displaystyle-(w-1)[3\Omega_{6}^{(b)}+3\Omega_{13}^{(b)}+3w\Omega_{16}^{(b)}+\Omega_{17}^{(b)}+w\Omega_{20}^{(b)}+3\Omega_{22}^{(b)}+w\Omega_{24}^{(b)}]$
$\displaystyle+(w-3)\,[\Omega_{23}^{(b)}+w\Omega_{25}^{(b)}]\;,$ (B.4.5)
$\displaystyle\Omega_{h_{c},2}^{(b)}$
$\displaystyle=\Omega_{3}^{(b)}+(w+1)[\Omega_{4}^{(b)}+\Omega_{10}^{(b)}+w\Omega_{12}^{(b)}-\Omega_{18}^{(b)}-\Omega_{24}^{(b)}]+w\Omega_{5}^{(b)}-3\Omega_{6}^{(b)}-\Omega_{8}^{(b)}-3\Omega_{13}^{(b)}$
$\displaystyle-3w\Omega_{16}^{(b)}-3\Omega_{22}^{(b)}+\Omega_{23}^{(b)}+w\Omega_{25}^{(b)}\;,$
(B.4.6) $\displaystyle\Omega_{h_{c},1}^{(c)}$
$\displaystyle=w\Omega_{1}^{(c)}+\Omega_{4}^{(c)}-\Omega_{6}^{(c)}-(w^{2}-1)[w\Omega_{9}^{(c)}+\Omega_{10}^{(c)}-\Omega_{13}^{(c)}]$
$\displaystyle+(w-1)[3w\Omega_{14}^{(c)}+w\Omega_{15}^{(c)}+\Omega_{17}^{(c)}+3\Omega_{18}^{(c)}+3\Omega_{22}^{(c)}-\Omega_{23}^{(c)}]-2w\Omega_{21}^{(c)}-2\Omega_{24}^{(c)}\;,$
(B.4.7) $\displaystyle\Omega_{h_{c},2}^{(c)}$
$\displaystyle=(w+1)[w\Omega_{9}^{(c)}+\Omega_{10}^{(c)}-\Omega_{13}^{(c)}]-3w\Omega_{14}^{(c)}-2w\Omega_{15}^{(c)}-2\Omega_{17}^{(c)}-3\Omega_{18}^{(c)}-3\Omega_{22}^{(c)}+2\Omega_{23}^{(c)}\;,$
(B.4.8) $\displaystyle\Omega_{h_{c},1}^{(Q),\text{mix}}$
$\displaystyle=(w+1)(w+2)\Omega_{2}^{(Q)}+(w+1)[w\Omega_{3}^{(Q)}-\Omega_{4}^{(Q)}+2\Omega_{24}^{(Q)}]+3\Omega_{6}^{(Q)}-(w+2)\Omega_{7}^{(Q)}$
$\displaystyle-3w\Omega_{8}^{(Q)}+(w-1)(w+1)^{2}\Omega_{10}^{(Q)}-(w^{2}-1)[3\Omega_{13}^{(Q)}+\Omega_{17}^{Q}+3\Omega_{18}^{(Q)}]$
$\displaystyle-(w-7)(w+1)\Omega_{22}^{(Q)}+(w-3)(w+1)\Omega_{23}^{(Q)}\;,$
(B.4.9) $\displaystyle\Omega_{h_{c},2}^{(Q),\text{mix}}$
$\displaystyle=(w+2)\Omega_{2}^{(Q)}+(w+1)\Omega_{3}^{(Q)}-2\Omega_{7}^{(Q)}-3\Omega_{8}^{(Q)}+(w^{2}-1)\Omega_{10}^{(Q)}$
$\displaystyle-(w-1)[3\Omega_{13}^{(Q)}+2\Omega_{17}^{(Q)}+3\Omega_{18}^{(Q)}]-(w-7)\Omega_{22}^{(Q)}+(w-5)\Omega_{23}^{(Q)}\;,$
(B.4.10) $\displaystyle\Upsilon_{2,h_{c},1}^{(b)}$
$\displaystyle=2w\Upsilon_{2C}^{(b)}+(w+1)[2(w-1)\Upsilon_{2F}^{(b)}+2\Upsilon_{2I}^{(b)}+3\Upsilon_{2J}^{(b)}]\;,$
(B.4.11) $\displaystyle\Upsilon_{2,h_{c},2}^{(b)}$
$\displaystyle=\Upsilon_{2B}^{(b)}-2\Upsilon_{2C}^{(b)}-2(w-1)\Upsilon_{2F}^{(b)}-\Upsilon_{2I}^{(b)}-3\Upsilon_{2J}^{(b)}\;,$
(B.4.12) $\displaystyle\Upsilon_{2,h_{c},1}^{(c)}$
$\displaystyle=2\Upsilon_{2C}^{(c)}-(w+1)[2(w-1)\Upsilon_{2E}^{(c)}+2\Upsilon_{2H}^{(c)}-3\Upsilon_{2J}^{(c)}]\;,$
(B.4.13) $\displaystyle\Upsilon_{2,h_{c},2}^{(c)}$
$\displaystyle=\Upsilon_{2A}^{(c)}-2(w-1)\Upsilon_{2E}^{(c)}-3\Upsilon_{2H}^{(c)}+3\Upsilon_{2J}^{(c)}\;.$
(B.4.14)

With these definitions we obtain:

$\displaystyle f_{V_{1}}$
$\displaystyle=-(w+1)\Big{[}\epsilon_{b}\Big{(}\Sigma_{h_{c},1}^{(b)}-(w-1)\Sigma_{h_{c},2}^{(b)}\Big{)}+\epsilon_{c}\Big{(}\Sigma_{h_{c},1}^{(c)}+(w-1)\Sigma_{h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle+\frac{w-1}{2}\Big{[}\epsilon_{b}\Big{(}\Upsilon_{2,h_{c},1}^{(b)}+(w+1)\Upsilon_{2,h_{c},2}^{(b)}\Big{)}+\epsilon_{c}\Big{(}\Upsilon_{2,h_{c},1}^{(c)}-(w+1)\Upsilon_{2,h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle-2(w-1)\big{[}\epsilon_{b}^{2}\Upsilon_{1B}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1B}^{(c)}\big{]}$
$\displaystyle+(w+1)\Big{[}\epsilon_{b}^{2}\Big{(}\Omega_{h_{c},1}^{(b)}-(w-1)\Omega_{h_{c},2}^{(b)}\Big{)}+\epsilon_{c}^{2}\Big{(}\Omega_{h_{c},1}^{(c)}+(w-1)\Omega_{h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle-\frac{w-1}{2}\epsilon_{b}\epsilon_{c}\Big{[}(w+1)\Big{(}\tilde{\Lambda}(\Sigma_{h_{c},1}^{(b)}-(w-1)\Sigma_{h_{c},2}^{(b)})+\tilde{\Lambda}^{\prime}(\Sigma_{h_{c},1}^{(c)}+(w-1)\Sigma_{h_{c},2}^{(c)})\Big{)}$
$\displaystyle+\Omega_{h_{c},1}^{(Q),\text{mix}}-(w+1)\Omega_{h_{c},2}^{(Q),\text{mix}}\Big{]}\;,$
(B.4.15) $\displaystyle f_{V_{2}}$
$\displaystyle=-\Xi+\frac{1}{2}\Big{[}\epsilon_{b}\Big{(}\Sigma_{h_{c},1}^{(b)}-(w-1)\Sigma_{h_{c},2}^{(b)}\Big{)}+\epsilon_{c}\Big{(}\Sigma_{h_{c},1}^{(c)}+(w-1)\Sigma_{h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle-\frac{1}{4}\Big{[}\epsilon_{b}\Big{(}\Upsilon_{2,h_{c},1}^{(b)}+(w-1)\Upsilon_{2,h_{c},2}^{(b)}\Big{)}+\epsilon_{c}\Big{(}\Upsilon_{2,h_{c},1}^{(c)}-(w-1)\Upsilon_{2,h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle-\Big{[}\epsilon_{b}^{2}\Big{(}2\Upsilon_{1A}^{(b)}-\Upsilon_{B}^{(b)}\Big{)}+\epsilon_{c}^{2}\Big{(}2\Upsilon_{1A}^{(c)}-\Upsilon_{1B}^{(c)}\Big{)}\Big{]}$
$\displaystyle-\frac{1}{2}\Big{[}\epsilon_{b}^{2}\Big{(}\Omega_{h_{c},1}^{(b)}-(w-1)\Omega_{h_{c},2}^{(b)}\Big{)}+\epsilon_{c}^{2}\Big{(}\Omega_{h_{c},1}^{(c)}+(w-1)\Omega_{h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle+\frac{1}{4}\epsilon_{b}\epsilon_{c}\Big{[}\tilde{\Lambda}\Big{(}(w-3)\Sigma_{h_{c},1}^{(b)}-(w-1)^{2}\Sigma_{h_{c},2}^{(b)}\Big{)}+\tilde{\Lambda}^{\prime}\Big{(}(w+1)\Sigma_{h_{c},1}^{(c)}+(w-1)^{2}\Sigma_{h_{c},2}^{(c)}\Big{)}$
$\displaystyle+\Omega_{h_{c},1}^{(Q),\text{mix}}-(w-1)\Omega_{h_{c},2}^{(Q),\text{mix}}\Big{]}\;,$
(B.4.16) $\displaystyle f_{V_{3}}$
$\displaystyle=-\frac{1}{2}\Big{[}\epsilon_{b}\Big{(}\Sigma_{h_{c},1}^{(b)}-(w+1)\Sigma_{h_{c},2}^{(b)}\Big{)}+\epsilon_{c}\Big{(}\Sigma_{h_{c},1}^{(c)}+(w+1)\Sigma_{h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle+\frac{1}{4}\Big{[}\epsilon_{b}\Big{(}\Upsilon_{2,h_{c},1}^{(b)}+(w+1)\Upsilon_{2,h_{c},2}^{(b)}\Big{)}+\epsilon_{c}\Big{(}\Upsilon_{2,h_{c},1}^{(c)}-(w+1)\Upsilon_{2,h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle-\big{[}\epsilon_{b}^{2}\Upsilon_{1B}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1B}^{(c)}\big{]}$
$\displaystyle+\frac{1}{2}\Big{[}\epsilon_{b}^{2}\Big{(}\Omega_{h_{c},1}^{(b)}-(w+1)\Omega_{h_{c},2}^{(b)}\Big{)}+\epsilon_{c}^{2}\Big{(}\Omega_{h_{c},1}^{(c)}+(w+1)\Omega_{h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle-\frac{1}{4}\epsilon_{b}\epsilon_{c}\Big{[}(w+1)\Big{(}\tilde{\Lambda}(\Sigma_{h_{c},1}^{(b)}-(w-1)\Sigma_{h_{c},2}^{(b)})+\tilde{\Lambda}^{\prime}(\Sigma_{h_{c},1}^{(c)}+(w-1)\Sigma_{h_{c},2}^{(c)})\Big{)}$
$\displaystyle+\Omega_{h_{c},1}^{(Q),\text{mix}}-(w+1)\Omega_{h_{c},2}^{(Q),\text{mix}}\Big{]}\;,$
(B.4.17) $\displaystyle f_{A}$
$\displaystyle=\Big{[}\epsilon_{b}\Big{(}\Sigma_{h_{c},1}^{(b)}-(w-1)\Sigma_{h_{c},2}^{(b)}\Big{)}+\epsilon_{c}\Big{(}\Sigma_{h_{c},1}^{(c)}+(w-1)\Sigma_{h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle-\frac{1}{2}\Big{[}\epsilon_{b}\Big{(}\Upsilon_{2,h_{c},1}^{(b)}+(w+1)\Upsilon_{2,h_{c},2}^{(b)}\Big{)}+\epsilon_{c}\Big{(}\Upsilon_{2,h_{c},1}^{(c)}-(w+1)\Upsilon_{2,h_{c},2}^{(c)}\Big{)}\Big{]}+2\big{[}\epsilon_{b}^{2}\Upsilon_{1B}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1B}^{(c)}\big{]}$
$\displaystyle-\Big{[}\epsilon_{b}^{2}\Big{(}\Omega_{h_{c},1}^{(b)}-(w-1)\Omega_{h_{c},2}^{(b)}\Big{)}+\epsilon_{c}^{2}\Big{(}\Omega_{h_{c},1}^{(c)}+(w-1)\Omega_{h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle+\frac{1}{2}\epsilon_{b}\epsilon_{c}\Big{[}(w+1)\Big{(}\tilde{\Lambda}(\Sigma_{h_{c},1}^{(b)}-(w-1)\Sigma_{h_{c},2}^{(b)})+\tilde{\Lambda}^{\prime}(\Sigma_{h_{c},1}^{(c)}+(w-1)\Sigma_{h_{c},2}^{(c)})\Big{)}$
$\displaystyle+\Omega_{h_{c},1}^{(Q),\text{mix}}-(w+1)\Omega_{h_{c},2}^{(Q),\text{mix}}\Big{]}\;,$
(B.4.18) $\displaystyle f_{S}$
$\displaystyle=-(w+1)\Xi+\big{[}\epsilon_{b}\Sigma_{h_{c},1}^{(b)}-\epsilon_{c}\Sigma_{h_{c},1}^{(c)}\big{]}-\frac{1}{2}\big{[}\epsilon_{b}\Upsilon_{2,h_{c},1}^{(b)}+\epsilon_{c}\Upsilon_{2,h_{c},1}^{(c)}\big{]}$
$\displaystyle-2\Big{[}\epsilon_{b}^{2}\Big{(}w+1)\Upsilon_{1A}^{(b)}-\Upsilon_{1B}^{(b)}\Big{)}+\epsilon_{c}^{2}\Big{(}(w+1)\Upsilon_{1A}^{(c)}-\Upsilon_{1B}^{(c)}\Big{)}\Big{]}-\big{[}\epsilon_{b}^{2}\Omega_{h_{c},1}^{(b)}-\epsilon_{c}^{2}\Omega_{h_{c},1}^{(c)}\big{]}$
$\displaystyle+\frac{1}{2}\epsilon_{b}\epsilon_{c}\Big{[}(w+1)\Big{(}\tilde{\Lambda}\Sigma_{h_{c},1}^{(b)}-\tilde{\Lambda}^{\prime}\Sigma_{h_{c},1}^{(c)}\Big{)}-\Omega_{h_{c},1}^{(Q),\text{mix}}\Big{]}\;,$
(B.4.19) $\displaystyle f_{T_{1}}$
$\displaystyle=\Big{[}\epsilon_{b}\Big{(}\Sigma_{h_{c},1}^{(b)}-(w-1)\Sigma_{h_{c},2}^{(b)}\Big{)}-\epsilon_{c}\Big{(}\Sigma_{h_{c},1}^{(c)}+(w-1)\Sigma_{h_{c},2}^{(c)}\Big{)}\Big{]}$
$\displaystyle-\Big{[}\epsilon_{b}^{2}\Big{(}\Omega_{h_{c},1}^{(b)}-(w-1)\Omega_{h_{c},2}^{(b)}\Big{)}-\epsilon_{c}^{2}\Big{(}\Omega_{h_{c},1}^{(c)}+(w-1)\Omega_{h_{c},2}^{(c)}\Big{)}\Big{]}\;,$
(B.4.20) $\displaystyle f_{T_{2}}$
$\displaystyle=-\frac{1}{2}\Big{[}\epsilon_{b}\Big{(}\Upsilon_{2,h_{c},1}^{(b)}+(w+1)\Upsilon_{2,h_{c},2}^{(b)}\Big{)}+\epsilon_{c}\Big{(}\Upsilon_{2,h_{c},1}^{(c)}-(w+1)\Upsilon_{2,h_{c},2}^{(c)}\Big{)}\Big{]}+2\big{[}\epsilon_{b}^{2}\Upsilon_{1B}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1B}^{(c)}\big{]}$
$\displaystyle\,\,\,\,-\frac{1}{2}\epsilon_{b}\epsilon_{c}\Big{[}(w+1)\Big{(}\tilde{\Lambda}(\Sigma_{h_{c},1}^{(b)}-(w-1)\Sigma_{h_{c},2}^{(b)})+\tilde{\Lambda}^{\prime}(\Sigma_{h_{c},1}^{(c)}+(w-1)\Sigma_{h_{c},2}^{(c)})\Big{)}$
$\displaystyle+\Omega_{h_{c},1}^{(Q),\text{mix}}-(w+1)\Omega_{h_{c},2}^{(Q),\text{mix}}\Big{]}\;,$
(B.4.21) $\displaystyle f_{T_{3}}$
$\displaystyle=\Xi-\big{[}\epsilon_{b}\Sigma_{h_{c},2}^{(b)}+\epsilon_{c}\Sigma_{h_{c},2}^{(c)}\big{]}-\frac{1}{2}\big{[}\epsilon_{b}\Upsilon_{2,h_{c},2}^{(b)}-\epsilon_{c}\Upsilon_{2,h_{c},2}^{(c)}\big{]}+2[\epsilon_{b}^{2}\Upsilon_{1A}^{(b)}+\epsilon_{c}^{2}\Upsilon_{1A}^{(c)}]$
$\displaystyle+\big{[}\epsilon_{b}^{2}\Omega_{h_{c},2}^{(b)}+\epsilon_{c}^{2}\Omega_{h_{c},2}^{(c)}\big{]}-\frac{1}{2}\epsilon_{b}\epsilon_{c}\Big{[}\tilde{\Lambda}\Big{(}2\Sigma_{h_{c},1}^{(b)}-(w-1)\Sigma_{h_{c},2}^{(b)}\Big{)}+(w-1)\tilde{\Lambda}^{\prime}\Sigma_{h_{c},2}^{(c)}-\Omega_{h_{c},2}^{(Q),\text{mix}}\Big{]}\;.$
(B.4.22)

##  Appendix C A set of relations among universal functions

In this Appendix we use the definitions in Eqs. (48), (4) and (4), together
with the constraints (52), (57), to relate the various structures entering in
the expressions (B.1.1)-(B.1.7), (B.2.1)-(B.2.14), (B.3.1)-(B.3.11) and
(B.4.1)-(B.4.14). The results allow us to obtain the relations among form
factors in Sec. 5.

$\displaystyle\Sigma_{\chi_{c_{0}}}^{(b)}$ $\displaystyle=$ $\displaystyle
3\Sigma_{\chi_{c_{1}},1}^{(b)}+(w-1)\Sigma_{\chi_{c_{1}},2}^{(b)}$
$\displaystyle\Sigma_{\chi_{c2}}^{(b)}$ $\displaystyle=$
$\displaystyle\Sigma_{\chi_{c1},2}^{(b)}$
$\displaystyle\Sigma_{h_{c},1}^{(b)}$ $\displaystyle=$
$\displaystyle\Sigma_{\chi_{c1},1}^{(b)}+(w-1)\Sigma_{\chi_{c_{1}},2}^{(b)}$
(C.1) $\displaystyle\Sigma_{h_{c},2}^{(b)}$ $\displaystyle=$
$\displaystyle\Sigma_{\chi_{c1},2}^{(b)}$

$\displaystyle\Sigma_{h_{c},1}^{(c)}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\Sigma_{\chi_{c0}}^{(c)}+2\Sigma_{\chi_{c1},1}^{(c)}-(w-1)\Sigma_{\chi_{c1},2}^{(c)}-\frac{w-1}{3}\big{[}3\Sigma_{\chi_{c2},1}^{(c)}-2\Sigma_{\chi_{c2},2}^{(c)}\big{]}$
$\displaystyle\Sigma_{h_{c},2}^{(c)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Sigma_{\chi_{c1},2}^{(c)}+\frac{1}{2}\big{[}3\Sigma_{\chi_{c2},1}^{(c)}-2\Sigma_{\chi_{c2},2}^{(c)}\big{]}$
(C.2)

$\displaystyle\Upsilon_{2,\chi_{c0}}^{(b)}$ $\displaystyle=$ $\displaystyle
3\Upsilon_{2,\chi_{c1},1}^{(b)}-(w+1)\Upsilon_{2,\chi_{c1},2}^{(b)}$
$\displaystyle\Upsilon_{2,\chi_{c2}}^{(b)}$ $\displaystyle=$
$\displaystyle\Upsilon_{2,\chi_{c1},2}^{(b)}$
$\displaystyle\Upsilon_{2,h_{c},1}^{(b)}$ $\displaystyle=$
$\displaystyle\Upsilon_{2,\chi_{c1},1}^{(b)}-(w+1)\Upsilon_{2,\chi_{c1},2}^{(b)}$
(C.3) $\displaystyle\Upsilon_{2,h_{c},2}^{(b)}$ $\displaystyle=$
$\displaystyle\Upsilon_{2,\chi_{c1},2}^{(b)}$

$\displaystyle\Upsilon_{2,h_{c},1}^{(c)}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\Upsilon_{2,\chi_{c0}}^{(c)}+(w+1)\big{[}2\Upsilon_{2,\chi_{c1},1}^{(c)}-\Upsilon_{2,\chi_{c1},2}^{(c)}\big{]}+\frac{w+1}{3}\big{[}3\Upsilon_{2,\chi_{c2},1}^{(c)}+2\Upsilon_{2,\chi_{c2},2}^{(c)}\big{]}$
$\displaystyle\Upsilon_{2,h_{c},2}^{(c)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Upsilon_{2,\chi_{c1},2}^{(c)}+\frac{1}{2}\big{[}3\Upsilon_{2,\chi_{c2},1}^{(c)}+2\Upsilon_{2,\chi_{c2},2}^{(c)}\big{]}$
(C.4)

$\displaystyle\Omega_{\chi_{c_{0}}}^{(b)}$ $\displaystyle=$
$\displaystyle-3\Omega_{\chi_{c_{1}},1}^{(b)}+(w-1)\Omega_{\chi_{c_{1}},2}^{(b)}$
$\displaystyle\Omega_{\chi_{c2}}^{(b)}$ $\displaystyle=$
$\displaystyle\Omega_{\chi_{c1},2}^{(b)}$
$\displaystyle\Omega_{h_{c},1}^{(b)}$ $\displaystyle=$
$\displaystyle-\Omega_{\chi_{c1},1}^{(b)}+(w-1)\Omega_{\chi_{c_{1}},2}^{(b)}$
(C.5) $\displaystyle\Omega_{h_{c},2}^{(b)}$ $\displaystyle=$
$\displaystyle\Omega_{\chi_{c1},2}^{(b)}$

$\displaystyle\Omega_{h_{c},1}^{(c)}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\Omega_{\chi_{c0}}^{(c)}+2\Omega_{\chi_{c1},1}^{(c)}+(w-1)\Omega_{\chi_{c1},2}^{(c)}+\frac{w-1}{3}\big{[}3\Omega_{\chi_{c2},1}^{(c)}+2\Omega_{\chi_{c2},2}^{(c)}\big{]}$
$\displaystyle\Omega_{h_{c},2}^{(c)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Omega_{\chi_{c1},2}^{(c)}-\frac{1}{2}\big{[}3\Omega_{\chi_{c2},1}^{(c)}+2\Omega_{\chi_{c2},2}^{(c)}\big{]}$
(C.6)

$\displaystyle\Omega_{h_{c},1}^{(Q),\text{mix}}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\Omega_{\chi_{c0}}^{(Q),\text{mix}}-2\Omega_{\chi_{c1},1}^{(Q),\text{mix}}+(w+1)\Omega_{\chi_{c1},2}^{(Q),\text{mix}}-\frac{w+1}{3}\big{[}3\Omega_{\chi_{c2},1}^{(Q),\text{mix}}-2\Omega_{\chi_{c2},2}^{(Q),\text{mix}}\big{]}$
$\displaystyle\Omega_{h_{c},2}^{(Q),\text{mix}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Omega_{\chi_{c1},2}^{(Q),\text{mix}}-\frac{1}{2}\big{[}3\Omega_{\chi_{c2},1}^{(Q),\text{mix}}-2\Omega_{\chi_{c2},2}^{(Q),\text{mix}}\big{]}\,.$
(C.7)

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