On the construction of amplitudes with Mandelstam analyticity from observable quantities

Abstract

It is shown that the problem of the construction of scattering amplitudes with Mandelstam analyticity from knowledge of their modulus in the three physical channels can be reduced, within a rather large class of functions, to the second Cousin problem of the theory of functions of two complex variables. As a consequence, it can be solved completely and explicitly. We derive conditions on the modulus function, under which at least one solution exists, as well as criteria for the correct resolution of the discrete ambiguity at fixed energy.

Zero Trajectories and the Problem of the Reconstruction of the Phase

“…We show namely in Section VII. (following [47]) that it is reducible, within a rather large class of ampli'tudes, to Cousin's second problem in the theory of functions of several variables; this problem was solved (in a brilliant manner) by Cousin already in 1895 (see Appendix C ) . The first study concerning the possible role of two variable analyticity in phase shift analysis is [48].…”

“…As a consequence of (Al), the limiting function A(so, t ) , so E C8+, has a continuous derivative with respect to t along the cuts C,, C, and the Cauchy-Riemann equations hold even for t E Ct (or C,) (see Appendix A). We ignore the possibility of thresholds where aA/as becomes unbounded and refer to [47] for their treatment.…”

“…The main part of the reconstruction is, in faet, the explicit procedure of obtaining the whole set of zeros of A(s, t ) in A from the set of trajectories on 8JZ [47]. This may appear to be a n awful task: the zeros t ( s ) are not holomorphic, but only locally algebraic functions of s. However, we shall perform this construction explicitly in 4 7.3.…”

“…They have to do with certain properties of the zero trajectories of the amplitude on aJl and, as with the other assumptions describing the class 3, are not critical. In fact, for the purpose of brevity, we make even stronger assumptions than in [47], which we otherwise follow closely. 7.2.…”

“…This leads t o a multiple zero of A(sc, t ) a t t = t,. I n this case, i t is still possible t o identify trajectories unambiguously on both sides of s, by continuing analytically aIong a small path around s, in the complex s plane; in [47] we discuss this process in more detail; here, we assume simply that, as long as s E C,+, this does not happen. (This is the "generical" situation : we do not expect the set of four real equations with three real unknowns (s E C.+, Re t, Im t ) : A(s, t ) = 0, aA/at(s, t ) = 0 to have a solution.)…”