One-channel Roy equations revisited

Abstract

The Roy equation in the single channel case is a nonlinear, singular integral equation for the phase shift in the low-energy region. We first investigate the infinitesimal neighborhood of a given solution, and then present explicit expressions for amplitudes that satisfy the nonlinear equation exactly. These amplitudes contain free parameters that render the nonuniqueness of the solution manifest. They display, however, an unphysical singularity at the upper end of the interval considered. This singularity disappears and uniqueness is achieved if one uses analyticity properties of the amplitudes that are not encoded in the Roy equation.

Roy equation analyses of $ππ$ scatterings at unphysical pion masses

“…According to the discussion in Refs. [2,43,44], the multiplicity index in this situation is m = 0 + 1 − 1 = 0, while m = 0 in the physical case. Mathematically, by taking m = 0 there exists a unique solution to Roy equations (2) after taking the subtractions and pole terms k I J (s) as inputs.…”

A possible subthreshold pole in S11 channel from πN Roy-Steiner equation analyses

“…The mathematical properties of Roy (-like) equations, as a group of infinite coupled integral equations, have been thoroughly investigated in refs. [37][38][39][40]. In πN RS analyses, we focus on a solution of the S and P waves of s-channel, within matching point W m .…”

The long and winding road from chiral effective Lagrangians to nuclear structure

“…The free parameters of the approach are subtraction constants, which, in the case of pp scattering, can be directly identified with the scattering lengths [56], while for the solution of the N p system it is more convenient to relate them to subthreshold parameters instead. The resulting system of coupled integral equations corresponds to a self-consistency condition for the lowenergy phase shifts, whose mathematical properties were investigated in detail in [61]. Following [56], the authors of [62] pursued the following solution strategy: the phase shifts are parameterized in a convenient way with a few parameters each, which are matched to input partial waves above s m in a smooth way.…”

“…where W j { } denotes a set of points between threshold and s m , f l I s  are the s-channel partial waves with isospin I s , orbital angular momentum l, and total angular momentum j l l 1 2 =  º , and F f l I s [ ]  the right-hand side of the RS equations. In [62] l 1 m = , N=25 (distributed equidistantly) are taken, and the number of subtraction constants are chosen in such a way as to match the number of degrees of freedom predicted by the mathematical properties of the Roy equations [61]. It should be stressed that the form of the RS equations only reduces to that of Roy equations once the t-channel is solved, see [60].…”