the quark charge.

It follows from translational invariance of the matrix element in [Eq. (\[ee\])]{}, that $E$ depends only on the relative coordinates, $\z_1 - \y_1$ and $\z_2 - \y_2$. We therefore introduce the momentum space representation, $$\begin{aligned}
E(\z_1,\y_1;\z_2,\y_2) & = & \int {{d\p} \over {(2\pi)^3}} 
  {{d\q} \over {(2\pi)^3}} e^{i\p \cdot (\z_1 - \y_1)  
 -i \q\cdot (\z_2 - \y_2) } \nonumber \\ 
& &  \qquad\times d(p) d(q) E(\p;\q), \label{ft}\end{aligned}$$ and define $F_\L(\l) \equiv E(\l+\L/2; \l-\L/2)$ with $\l \equiv (\p + \q)/2$ and $\L  \equiv \p - \q$. The Dyson equation for $F$ can be derived in the rainbow-ladder approximation which, as shown in Ref. [@as1; @as2], sums up the dominant infrared and ultra-violet contributions to the expectation value of the inverse of two FP operators,

$$F_\L(\l)
 = 1 + N_c \int {{d\k}\over {(2\pi)^3 }} 
{{\left[ (\k-\L/2)\delta_T(\k + \l)(\k+\L/2) \right] } 
 \over  {2\omega(\k + \l)}}
 {{d(\k - \L/2)}\over {(\k - \L/2)^2}}
 {{d(\k + \L/2)}\over {(\k + \L/2)^2}} F_\L(\k), \label{F}$$

It follows from [Eq. (\[ft\])]{} that $\L$ and $\l$ are conjugate to the [*center of mass*]{}, $\R \equiv [ (\z_1 - \y_1) + (\z_2 - \y_2)
 ]/2$ and the [*relative*]{}, $\r \equiv [ (\z_1 - \y_1) - (\z_2 - \y_2)
 ]$ coordinate 