model, the quark coordinates are given by $$\begin{aligned}
& & p_{Q+}=xP_{+} \;\; , \;\; 
{\bf p}_{Q\perp}=x{\bf P_{\perp}}+{\bf k}_{\perp} \;\; , \;\;
p_{\bar{q}+}=(1-x)P_{+} \;\; , \;\; 
{\bf p}_{\bar{q}\perp}=(1-x){\bf P_{\perp}}+{\bf k}_{\perp}, \nonumber \\
& & 0 \leq x \leq 1 \;\;\; , \;\;\; {\bf P}={\bf p}_{Q}+{\bf p}_{\bar{q}}.\end{aligned}$$ where ${\bf k}=(k_{z},{\bf  k}_{\perp})$ is the internal momentum. For ${\bf P}$ (and similarly for other vectors), ${\bf P}=(P_{+},
{\bf P_{\perp}})$ with $P_{+}=P_{0}+P_{z}$ and ${\bf P_{\perp}}=(P_{x},P_{y})$ .

To calculate the form factors, one reasonable and often used assumption for the meson wave function $\phi(x,{\bf  k}_{\perp})$ is a Gaussian-type function

$$\begin{aligned}
\phi(x,{\bf k}_{\perp}) & = & \eta({\bf k}) \sqrt{\frac{d{\bf k_{z}}}{dx}}
\;\;\; , \;\;\; \eta({\bf k})= (\pi \omega^{2})^{-3/4}
\exp{(-{\bf k}^{2}/2\omega^{2})} \label{wf1}\end{aligned}$$

where $\omega $ is a scale parameter and $x$ is defined through $$\begin{aligned}
x & = & \frac{e_{Q}+k_{z}}{e_{Q}+e_{\bar{q}}} \;\;\; , \;\;\; 
e_{i}=\sqrt{m^{2}_{i}+{\bf k}^{2}} \;\; (i=Q,\bar{q}).\end{aligned}$$

The wave function (\[wf1\]) has been used in ref. [@model], [@Jaus90] and also in [@OXT],[@ox1] for various applications of the light-front quark model.

A similar wave function is $$\begin{aligned}
\phi(x,{\bf k}_{\perp}) & = & \eta({\bf k}) \sqrt{\frac{d{\bf k_{z}}}{dx}} 
\;\;\; , \;\;\; \eta({\bf k})= N
\exp{(-M^{2}_{0}/ 8 \omega^{2})} \label{wf2}\end{aligned}$$ Here $N$ is the normalization constant and $M_{0}$ is the invariant mass of the 