on the transverse part of the quark momentum $k$, i.e. $k_\perp =k-(kp)p/p^2$ is a consequence of a subsidiary condition [@Markov40; @Yukawa50], which as it was shown in [@Lukierski77] is equivalent to a ’gauge principle’ and expresses the redundance of the longitudinal component of the relative momentum $k$ for the physical interaction between the quark-antiquark constituents.

The covariant kernels of non-local interactions are $$\label{ff2}
F^{S,P,V,A}_{a(2)}(k_\perp^2)=F^{S,P,V,A}_{a} c^{S,P,V,A} f (k_\perp^2 )\theta (\Lambda_3-|k_\perp |).$$ The coefficients $c^{S,P,V,A}$ renormalize the couplings of four-quark interactions $G_S$ and $G_V$ increasing a strength of these forces for the non-local interactions. They are fixed from the empirical values of meson masses. The step function $\theta (\Lambda_3-|k_\perp |)$, where $\Lambda_3$ is a covariant cutoff, restricts the integration over relative momentum of a bound quark-antiquark pair to the size of the bag. The function $$\label{fk}
f (k_\perp^2) =1+d |k_\perp |^2, \quad |k_\perp |=\sqrt{-k_\perp^2},$$ being a Lorentz scalar, can be calculated in any convenient reference frame. In the following we use the instantaneous rest frame of the meson. In this case $k_\perp =(0, \vec k)$, and $f (k_\perp^2)= 1+d \vec k^2\equiv f(\vec k^2)$. The slope parameter $d$ is fixed by the requirement that the numerical values of the quark condensate and constituent 