1,-\textbf{k}\,),\\
p_2^\mu &=&  (E_2,\textbf{k}\,).
\end{eqnarray}
The heavy quarkonium momentum becomes purely timelike while the relative momentum is   spacelike in the rest frame. $\alpha=E_1/(E_1+E_2)$  and $\beta=1-\alpha$ with the on-shell conditions $E_1=\sqrt{m_1^2-k^2}$, $E_2=\sqrt{m_2^2-k^2}$, and $k^2=-\textbf{k}^2$. $m_1$ and $m_2$ denote the masses  of the heavy quark $Q$ and anti-quark $\bar{Q^\prime}$, respectively.

The Dirac spinors for the heavy quark $Q$ and anti-quark $\bar{Q^\prime}$ can be written as
\begin{eqnarray}
u_1(p_1,\lambda) &=&  \sqrt{\frac{E_1+m_1}{2E_1}}\left(
                                           \begin{array}{ll}
                             ~~~~\xi_\lambda \\
\frac{\vec{\sigma}\cdot \overrightarrow{p_1}}{E_1+m_1}\xi_\lambda
                                           \end{array}
                                         \right)\,,
\end{eqnarray}
\begin{eqnarray}
v_2(p_2,\lambda) &=&  \sqrt{\frac{E_2+m_2}{2E_2}}\left(
                                           \begin{array}{ll}
\frac{\vec{\sigma}\cdot \overrightarrow{p_2}}{E_2+m_2}\xi_\lambda\\
                             ~~~~\xi_\lambda\end{array}
                                         \right)\,,
\end{eqnarray}
where $\xi_\lambda$ is the two-component Pauli spinors and $\lambda$ is the polarization quantum number.

One can easily get the covariant expressions for the spin-singlet and spin-triplet combinations
of spinor bilinearities.  The corresponding projection operators are
\begin{widetext}
\begin{eqnarray}
\Pi_S(k) &=&  -i\sum_{\lambda_1,\lambda_2} u_b(p_1,\lambda_1)\bar{v}_c(p_2,\lambda_2)\langle\frac{1}{2}\lambda_1\frac{1}{2}\lambda_2|S S_z\rangle\otimes \frac{\textbf{1}_c}{\sqrt{N_c}}\nonumber\\
&=&\frac{i}{4\sqrt{2 E_1 E_2}\omega}(\alpha \,p\!\!\!\slash_{H}-k\!\!\!\slash+m_1)\frac{p\!\!\!\slash_{H}+E_1+E_2}{E_1+E_2}\Gamma_S
(\beta\,p\!\!\!\slash_{H}+k\!\!\!\slash-m_2)\otimes \frac{\textbf{1}_c}{\sqrt{N_c}}\,,\label{projection}
\end{eqnarray}
\end{widetext}
where the auxiliary parameter $\omega=\sqrt{E_1+m_b}\sqrt{E_2+m_c}$ and $\textbf{1}_c$ is the unit matrix in the fundamental representation of the color SU(3) group. $\Gamma_S=\gamma^5$ for spin-singlet combination with spin $S=0$,
 while $\Gamma_S=\varepsilon\!\!\!\slash_{H}=\varepsilon_\mu(p_H) \gamma^\mu$ for spin-triplet combination with spin $S=1$.

To get the relativistic corrections to the form factors of $B_c$ into $S$-wave charmonium, one may perform the  Taylor expansion of the amplitudes in powers of $k^\mu$
\begin{eqnarray}
{\cal A}(k)&=& {\cal A}(0)+\frac{\partial {\cal A}(0)}{\partial k^\mu}\mid_{k=0}k^\mu\nonumber\\
&&+\frac{1}{2!}\frac{\partial^2 {\cal A}(0)}{\partial k^\mu\partial k^\nu}\mid_{k=0}k^\mu k^\nu+\ldots,
\end{eqnarray}
where the terms linear in $k$ should be dropped since they do not contribute to the matching coefficients.
In this paper,  we will consider the  contributions at the ${\cal O}(|\textbf{k}|^2)$ level.  One can use the following replacement to simplify the calculation~\cite{Guo:2011tz}
\begin{eqnarray}
k^\mu k^\nu&\rightarrow& \frac{|\textbf{k}|^2}{D-1}(-g^{\mu\nu}+\frac{P_{H}^\mu P_{H}^\nu}{P_{H}^2}).
\end{eqnarray}

The treatment of the final state phase space integrations at the ${\cal O}(|\textbf{k}|^2)$ level
is slightly different from the leading order calculation. We will adopt the following rescaling
transformation for the external momenta in order to get their relativistic corrections~\cite{Guo:2011tz}
\begin{eqnarray}
p_H=p_H'\frac{E_1+E_2}{m_1+m_2}.\label{rescale}
\end{eqnarray}

\subsection{Form factors}

The form factors of $B_c$ into $S$-wave charmonium, i.e. $f_{+}$,
$f_{0}$, $V$, $A_{0}$, $A_{1}$, and $A_{2}$  are defined in common~\cite{Wirbel:1985ji}
\begin{eqnarray}
\langle \eta_{c}(p)\vert \bar c \gamma^{\mu}b\vert B_{c}(P)\rangle
&=&f_{+}(q^{2})(P^{\mu}+p^{\mu}-\frac{m_{B_{c}}^{2}-
m_{\eta_{c}}^{2}}{q^{2}}q^{\mu})\nonumber\\ &&+f_{0}(q^{2})
\frac{m_{B_{c}}^{2}-m_{\eta_{c}}^{2}}{q^{2}}q^{\mu}\,,
\end{eqnarray}
\begin{eqnarray}
 && \langle J/\psi(p,\varepsilon^{*})\vert \bar c \gamma^{\mu}b\vert
B_{c}(P)\rangle =\frac{2 i
V(q^{2})}{m_{B_{c}}+m_{J/\psi}}\epsilon^{\mu\nu\rho\sigma}
\varepsilon_{\nu}^{*}p_{\rho}P_{\sigma}\,,
\nonumber\\
&& \langle J/\psi(p,\varepsilon^{*})\vert \bar c
\gamma^{\mu}\gamma_{5}b\vert B_{c}(P)\rangle =2
m_{J/\psi}A_{0}(q^{2})\frac{\varepsilon^{*}\cdot q}{q^{2}}q^{\mu}
\nonumber\\&&~~~~~-A_{2}(q^{2})\frac{\varepsilon^{*}\cdot
q}{m_{B_{c}}+m_{J/\psi}}(
P^{\mu}+p^{\mu}-\frac{m_{B_{c}}^{2}-m_{J/\psi}^{2}}{q^{2}}q^{\mu})
\nonumber\\&&~~~~~+(m_{B_{c}}+m_{J/\psi})A_{1}(q^{2})
(\varepsilon^{*\mu}-\frac{\varepsilon^{*}\cdot q}{q^{2}} q^{\mu})\,,
\end{eqnarray}
where the momentum transfer is defined as $q=P-p$ with the $B_c$ meson momentum $P$ and the charmonium momentum $p$.

\begin{figure}[th]
\begin{center}
\includegraphics[width=0.45\textwidth]{formfactors.pdf}
\end{center}
    \vskip -0.7cm \caption{Feynman diagrams for the form factors of $B_c$ into $S$-wave charmonium, where ``$\oplus$'' denotes certain current operators.}\label{Fig-formfactors}
\end{figure}

The leading order  results at ${\cal O}(\alpha_s )$ and the NLO QCD corrections at ${\cal O}(\alpha_s^2 )$
of the form factors can be found in Refs.~\cite{Bell:2005gw,Qiao:2011yz,Qiao:2012vt}. The leading order  results are obtained
from the Feynman diagrams in Fig.~\ref{Fig-formfactors}. For completeness,
we list the leading order results of the form factors here
\begin{eqnarray} f_+^{LO}(q^{2})&=&8 \sqrt{2} C_A C_F \pi  \sqrt{z+1}
 \alpha _s \psi(0)_{B_c}
\psi(0)_{\eta_c}\nonumber\\&&\times\frac{
\left(-y^2+3 z^2+2 z+3\right) }{\left((1-z)^2-y^2\