Analytic strong-coupling solutions of the ABFST equation

Abstract

We obtain approximate, but explicit upper- and lower-bound strong-coupling solutions to thet≠0 ABFST equation. The Fredholm denominatorDι(t) can be expressed as an infinite determinant ofone-dimensional integrals in the narrow-resonance approximation. If expanded in a power series int, the integrals can be performed, and the general structure of the solution of coupledO3.1 equations emerges. The determinant is shown to be highly convergent, implying the effective decoupling ofO3.1 partial waves att≠0, and verifying the approximate validity of the trace approximation. In special mass limits, closed expressions for thet0-coefficient ofDι are obtained. The basic approximation is kinematic rather than dynamic, and is expected to be accurate when the resonance massm0 is greater than all other masses and effective momentum transfers; herem0≫μ (pion mass) andt not too large. This has been shown, to yield reasonable results for the leading trajectory. Even in the casem0=μ, however, reasonable agreement of parent and daughter slopes are obtained with exact numerical results, and them0=μ leading upper-bound intercept turns out to be given by the Tiktopoulos-Treiman upper bound. Results for residue functions are presented and their scaling properties are examined. The lower bound solutions provide a natural continuation fromq2=−∞ toq2=0; the scale breaking mass beingm0. Finally, it is shown that the leading trajectory of the lower-bound solution is very close to that found by Swift and Tucker in their analysis of Polkinghorne's exact perturbation theory results.