On the determination of πN phase shifts from isospin constraints and fixed t analyticity

Abstract

This paper treats the problem of the unique and constructive determination of πN scattering amplitudes by means of fixed t dispersion relations and isospin invariance from data on the differential cross section and polarization of the elastic and charge exchange reactions. The conclusion is that the amplitudes can be uniquely constructed in a neighborhood (−α, a) of t = 0 if they are continuous functions of both real s and t for s on the cuts and 4μ2≳t≳−α0, α0≳α and there exists ν0 such that, for ‖ν‖?ν0, both ‖A+(ν,t)‖/‖A+(ν,t)+νB+(ν,t)‖ <const, t∈(−α0,4μ2) (as indicated by spin rotation measurements) and ‖Re(A++νB+)‖/Im(A++νB+)<const, t∈(−α0,0). This result is specific for πN scattering, as it depends on a small number of experimental facts concerning this process, which are enumerated in the text.

Amplitude Analysis

“…Another very readable source is the review of Bowcock and Burkhardt [2]. More mathematical discussions are contained, for example, in the papers of Sabba Stefanescu [3] and the book of Chadan and Sabatier [4].…”

Zero Trajectories and the Problem of the Reconstruction of the Phase

“…I n particular, we present a qualitative description of the results obtained in the analysis of the Crichton ambiguity, in the spinless case. I n Section IV., we show that the requirements of fixed t analyticity and of isospin invariance allow a unique and stable determination of the phase of the X N scattering amplitudes, within a large class of functions [34]. I n Section V., we discuss zero trajectories from a "heuristic" point of view and present methods for resolving their discrete ambiguity, both from [36] and using local two variable analyticity.…”

“…fixed (real) t. Thus, the phase of A(.$, t ) is uniquely determined if the finitc set of complex parameters {s,}>~, may be (reliably) det,erinined. I n Section IV., we shall show essentially that, in x N scattering, the requirement of isospin invariance in conjunction with some experimental information of a general type allows such a determination in some interval ( -tl, 0) of t values, within a rather large and physically quite plaiisible class of amplitudes [34]. This method uses (essentially) the observables at only one fixed value of the momentum transfer 1.…”

“…Further, similarly to 3 2.3.3., the extrapolation of the observables at fixed energy to the interval 0 5 t 5 4p2 does not lead to the modulus of an amplitude (cf. [34]). However, in xN scattering there is the additional information coming from charge exchange scattering and one can show [34] that, using a few supplementary experimental facts and plausible assumptions, there exists an interval of t , 0 2 t 2 ti, where the phase is uniquely determined, in a stable manner, given the extrapolation of the observables.…”

“…The proof of this theorem is not difficult and is given in [34]. In the following, we shall refer to conditions (i)-(iii) of the Theorem as conditions (V) (from validity); in 5 4.3.2., we point out that, if the zeroes of h,, hstay away from each other, they are sufficient for the validity of the reconstruction even in the presence of small errors.…”