Short-distance behavior of the Bethe-Salpeter wave function in the ladder model

We investigate the short-distance behavior of the (Wick-rotated) Bethe-Salpeter wave function for two spin-$\frac{1}{2}$ quarks bound by the exchange of a massive vector meson. We use the ladder-model kernel, which has the same $p^{-4\mathrm{}}$ scaling behavior as the true kernel in a theory with a fixed point of the renormalization group at $g\ne 0$. For a bound state with the quantum numbers of the pion, the leading asymptotic behavior is $\chi \left(q^{\mu }\right)\sim {\mathrm{cq}}^{-4+\epsilon \mathrm{}\left(g\right)\mathrm{}}{\gamma }_5$, where $\epsilon \mathrm{}\left(g\right)=1-{(1-\frac{g^2}{{\pi }^2})}^{\frac{1}{2}}$. Our method also provides the full asymptotic series, although it should be noted that the nonleading terms will depend on the nonleading behavior of the ladder-model kernel. A general term has the form ${\mathrm{cq}}^{-a}{\left(\ln q\right)}^n\phi \left({\hat{q}}^{\mu }\right)$, where $c$ is an unknown constant, $a$ may be integral or nonintegral, $n$ is an integer, and $\phi \left({\hat{q}}^{\mu }\right)$ is a representation function of the rotation group in four dimensions.