On the Validity of the Regge Formula in the Unqeual-Mass Case

Abstract

The validity of the Regge formula in the relativistic scattering theory is examined on the basis of the Green's function formalism. It is shown that in the unequal-mass case some Regge trajectories on the mass shell become ghosts in a certain real interval of the invariant energy squared. This result clarifies how the kinematical unphysical singularities appearing in the Khuri formula are canceled by the daughter trajectories proposed by Freedman and Wang. The order of the daughter trajectory is identified with n-l-1, where nand l are the principal and the azimuthal (angular momentum) quantum number, respectively. The Khuri asymptotic formula in the unequal-mass case is discussed in a ladder model of massless-scalar­ particle exchange, and is explicitly demonstrated to be free from the kinematical unphysical singularities. parent at first sight when one combines the Regge meromorphy with the ordi­ nary analyticity of the scattering amplitude in the relativistic S-matrix theory. The Regge formula cannot give the asymptotic behavior for the backward scat­ tering (u = 0) in the unequal-mass case owing to the simple kinematical reason. Freedman and Wang 2 ) have, therefore, suggested to use the Khuri formula 3 ) instead of the Regge one. In order to evaluate the residue functions of the former, they have made use of the latter, and then found that the reduced resi­ dues of the Khuri satellite poles, 2 ) which appear simply due to the transforma­ tion of the variables, 3 ) have poles at u = 0 in the unequal-mass case. Hence, to be consistent with the analyticity of the scattering amplitude, there should exist other Regge trajectories of alternating signature, which are called daughter trajectories, so as to cancel those unphysical singularities. Furthermore, they have explicitly verified their hypothesis in the Bethe-Salpeter formalism by using the four-dimensional symmetry of the case of the vanishing total four-momentum in the u channel. Now, the purpose of the present paper is to examine the validity of the Regge formula in the relativistic theory on the basis of the Green's function formalism, and to clarify the cancellation mechanism of Freedman and Wang. The Regge formula is quite well established in the nonrelativistic potential