The singlet effective-range parameters for the Mongan separable potentials are shown to be more than trivially in error. The implications of this result on triton calculations using these potentials are discussed.The most extensive separable-potential fits to the two-nucleon data have been done by Mongan", for convenience, we will refer to these as his "old" and "new" fits, respectively. The singlet
Using an off-mass-shell approach, we develop a method for the 0(4) expansion of the scattering amplitude for two unequal-mass, arbitrary-spin particles. General-spin spherical harmonics for 0(4) are constructed. With the aid of these spherical harmonics, the M amplitude is decomposed into 0 (4) partial waves. The prescription for obtaining the on-mass-shell helicity states from the 0(4) partial waves is given. This 0(4) expansion, which is valid even away from J-0, is useful in simplifying the Bethe-Salpeter equation. L INTRODUCTIONT HE work of Toller 1 and collaborators and of Freedman and Wang 2 has demonstrated that 0(3,1) or 0(4) symmetry is a powerful tool for classifying Regge trajectories at /=0. Their approach was limited to equal-mass scattering, however, and applied only at the point /= 0. The purpose of the present work is to generalize 0(4) expansions to unequal-mass twobody reactions at small L We adopt an off-shell approach in order to deal with the problems of unequal masses. The amplitudes which we expand in 0(4) basis states are generalized M-functions, which satisfy a Bethe-Salpeter equation. The methods developed in this paper are applied to two specific Bethe-Salpeter equation problems, i WV" and irp scattering, in two separate papers. 3 ' 4 By using off-shell amplitudes we are able to discuss the properties of these amplitudes at vanishing total four-momentum £^=0. Moreover, we can discuss the nature of the symmetry breaking for ty^O, where t=-k 2 . The expansion of the generalized M-function in 0(4) basis states is diagonal in the 0(4) quantum numbers at & M =0. Near /=0 one can expand M in powers of \/L To any given order in t only a finite number of off-diagonal terms contribute. This permits us to use the 0(4) expansion to analyze Regge poles near /=0. Moreover, the expansion is useful in simplifying the Bethe-Salpeter equation for small t from an integral equation in two variables to a set of a small number of coupled single-variable integral equations. It seems readily feasible to use this technique to construct dynamical models by numerical solution of the BetheSalpeter equation. Such work is in progress. Some general results concerning the Regge trajectory of the pion are being published separately. 5 In Sec. II we review the representations of 0(4). In Sec. Ill we expand the generalized M functions in states which transform according to irreducible representations of 0(4), and in Sec. IV we establish the connection between these states and physical helicity states. In the Appendix some useful properties of the 0(4) representation matrices are gathered together. II. REPRESENTATIONS OF THE GROUP 0(4)We label the ordinary three-dimensional space axes xi, X2, and x z , with x± the "time" coordinate. The matrix representing a rotation through the angle 0 in the #2X3 plane is called Ri(6), and the matrix representing a rotation through 0 in the xix^ plane (a "boost") is called Si (d), etc. These matrices are £1= R 2 = n 0 0 0 0 cos0 --sin0 0 0 sin0 cos0 0 1. 0 0 0 I cos0 0 sin0 0 0 10 0 ...
An off-shell generalization of 0 (4) symmetry is developed, which is applicable to unequal-mass reactions involving arbitrary spins. The symmetry breaking at 15*0 is also investigated. These methods are used to analyze the hypothesis that the pion Regge trajectory has the Toller quantum number M -\ at zero momentum transfer, as seems to be implied by high-energy photoproduction data. We find that if this hypothesis is correct, then the pion trajectory is necessarily quite complicated. The M= 1 trajectory must mix with another trajectory. The NNTT vertex function then shows a zero near t = Q, in agreement with some fits to high-energy data, but this zero is factorizable. Moreover, a model is exhibited that seems consistent with the hypothesis of partially conserved axial-vector current.
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