Takesi Ogimoto

Continuation of S-Matrix into Second Riemann Sheet

In the framework of relativistic dispersion theory, it is shown that the S matrix continued into the second Riemann sheet is just the inverse of the one on the first Riemann sheet. This formula is used for deriving a dispersion-like relation between the real and imaginary parts of the scattering phase shift and the product expansion for the S matrix. These results are generalizations of van Kampens formulas in the theory of nonrelativistic potential scattering to those in the relativistic field theory. The Castillejo-Dalitz-Dyson ambiguity is discussed on the basis of the S matrix. Relations connecting the sum of the oscillator strengths with scattering lengths are derived in generalized forms. (auth)

Analyticity in Coupling Constant, Angular Momentum and Energy of the S-Matrix for Potential Scattering

It is shown that the amplitude for scattering by a superposition of the Yukawa potential is a meromorphic function of coupling constant g, angular momentum I and linear momentum k. Regge poles for bound states are seen to be non-decreasing function of g. In the Case of l real, poles in the the g-plane are shown, as a function of energy (E=k2), to be real analytic R-functions with no left-hand cut. Finally, a holomorphy domain of the S-matrix in three variables is obtained.

Maximum Analyticity in Angular Momentum and Energy of the Bethe-Salpeter Scattering Amplitudes

It is shown that under a certain mass relation the partial wave ~cattering amplitude in the ladder approximation is regular in the usual cut plane of the total energy squared and in the whole angular momentum plane except at poles. At the negative integers in the angular mcmentum plane the amplitude is concluded to be regular. The purpose of this paper is to consider the analytic continuation of Lee-Sawyers domain into the left half of the !-plane, thus proving the validity of the maximum analyticity hypothesis at least in the ladder approximation. In §2 the partial wave projection of the integral equation for the B-S amplitude is obtained formally. In §3, for Rel>-3/2, analyticity in s and l, s being the c.m. energy squared, is studied rigorously by using a similar technique to Wicks simultaneous rotation of the integral path. The amplitude for Rel> -3/2 is shown to be analytic in s X l space, except on the usual cuts in s and except for poles in l, if m<p,, m being the mass of the colliding particles and p, the mass of the exchanged particle. This mass relation may be caused due to our simple rotation adopted here. In §4 the B-S integral equation is transformed into the ccordinate representation, in preparation for the analytic continuation with respect to l, and the singular part of the kernel is separated out. In §5 the analytic continuation into the region Rel<-3/2 is carried out. In §§6 and 7 it is shown that the partial wave

On Levinson's Theorem in the Theory of Multi-Channel Scattering

Using a Ree-type model, Revinsons theorem was investigated in the case of multi-channel scatterings. In this case, contrary to the case of potential scatterings or Dysons mcdel, it seems that Revinsons theorem becomes ambiguous in interpreting the physical meaning of the result obtained. (auth)

Subtraction of Dispersion Relation in Multi-Channel Scattering

A minimum number of subtractions is determined for the dispersion relation of the inverse scattering amplitude in the multi-channel case, by using the symmetric property of the scattering amplitude and the multi-channel character. (auth)

Relations among Pole-Residues of Scattering Amplitudes in Multi-Channel Scattering

A model with two scattering channels is considered, for which the particles are spiniess and neutral. The reactions pi + Y yields pi + Y, pi + Y yields K + N, and K + N yields K + N are considered as an example of this model. The partial wave amplitudes of these reactions are the boundary values of analytic functions, which can be continued into the second, third, and fourth Riemann surfaces. These partial wave amplitudes are regnlar in the complex z-plane, except for the branch iines and possible one-particle singularities. Expressions are found relating the residues, at the complex poles, of the scattering amplitudes on each Riemann surface considered. (T.F.H.)

Macroscopic Causality and Lower Limit for the Energy Derivative of the Scattering Phase Shift : Relativistic Case

A relation between the macroseopic causality and lower bound of the energy derivative of the scattering phase shift is studied in the case of relativistic quantum field theory. From the requirement of the macroscopic causality the energy derivative of the real part of the phase shift must be non- negative in such an energy region that the imagt-nary part of phase shift does not vary rapidly with the energy, whereas in another region such an inequality is not generally valid.