Electromagnetic pion radius and dispersion sum rules

A general set of dispersion sum rules is considered for the pion form factor G(t) in the mixed-modulus phase representation. The connection between the distribution of the G(t) complex zeros and the sizes of dispersion integrals along the cut is stated. Under the assumption that G(t) ～ t^α at large t the following restriction on the asymptotic behaviour of G(t) is obtained: |G(t)| ? at^?2 (a>0). Using present experimental data we evaluate the electromagnetic mean squared radius, r_π = 0.71 ± 0.30 fm.     

Hypergeometric Solutions of Soliton Equations

We consider multivariable hypergeometric functions related to Schur functions and show that these hypergeometric functions are tau functions of the KP hierarchy and are simultaneously the ratios of Toda lattice tau functions evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts the parameters of the hypergeometric functions. We construct the determinant representation and the integral representation of a special type for the KP tau functions. We write a system of linear differential and difference equations on these tau functions, which play the role of string equations.