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-1/2 quarks bound by the exchange of a massive vector meson. We use the ladder-model kernel, which has the same p^-4 scaling behavior as the true kernel in a theory with a fixed point of the renormalization group at g≠0. For a bound state with the quantum numbers of the pion, the leading asymptotic behavior is χ(q^μ)~cq^-4+ε(g)γ₅, where ε(g)=1-(1-g²π²)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 cq^-a(lnq)ⁿφ(^q^μ), where c is an unknown constant, a may be integral or nonintegral, n is an integer, and φ(^q^μ) is a representation function of the rotation group in four dimensions.