Forward dispersion relation constraints on the pion-nucleon p-wave and d-wave scattering lengths

Samaranayake, V.K.; Woolcock, W.S.

doi:10.1016/0550-3213(72)90590-1

Cited by 13 publications

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“…The general form of the pion-nucleon T-matrix in a state of orbital angular momentum £ produced by a potential of the separable form The earliest models using the nonrelativistic Lippmann-Schwinger equation were those constructed by Desgrolard and Hammann [Desl] by fitting the scat tering lengths and volumes [Sam3,Bug7] and the phase shifts [Alml,Car2] …”

Section: Al Separable Potential Modelsmentioning

confidence: 99%

Back Matter

1990

ΠNN Systems

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The following two appendices are aimed at providing the reader of some basic features and ingredients of two-body subsystems, which may be used as input to the description of the 7rNN system. Of course, we could have limited these appendices to include only those aspects which have eventually been used in the 7rNN equations: the separable models, and perhaps the chiral model which may be used in Born approximations to describe some 7rNN processes. However, we have gone a little further to also include the results of other approaches. This is just to give the reader some feeling of what has been going on in those subjects. Some of them are in fact quite fashionable these days, and many articles are still coming out for which one needs extended review articles! The interested reader should go ahead and look up the current publications to understand the frontier of those subjects. 251 A.l. Separable Potential ModelsSeparable potentials are the interactions most widely used in the numerical solution of the three-body problem. As we have shown throughout the theoretical chapters 1-4, they simplify enormously the integral equations by reducing them from two continuous variables to only one continuous variable.In the case of the 7rN subsystem a large number of separable potential models exist for the six dominant partial waves with I = 0 and 1, S\\ , Sn , P\\ , JP31 , P\$ , and P33 . The reason for this proliferation of models being, that the 7rN subsystem is treated sometimes nonrelativistically, or only the pion is considered a relativistic particle (RPK), while some other times both particles are treated relativistically. Even in this last case, the choice of relativistic two-body equation is essentially a matter of taste, since models exist based on the Lippmann-Schwinger equation with relativistic kinematics, the Blankenbeckler-Sugar equation, the Kadyshevski equation, or the Bethe-Salpeter equation. Also a very popular prescription in the case of the Pu and P33 partial waves, has been to parametrize them by means of energy-dependent separable potentials.The general form of the pion-nucleon T-matrix in a state of orbital angular momentum £ produced by a potential of the separable form The ranges of the Su and S31 waves are 2.63 and 3.04 /ra -1 . In the case of the 531 wave they found that in order to fit the Almehed-Lovelace phase [Alml], the form factor must have a hole at the origin, so that in this case they multiplied Eq. (A.ll)by l-0.5048e"P 2 .The and in the case of the Pu partial wave a rank-two separable potential. These models, however, have never been applied to any physical problem. A.2. Chew-Low ModelsChew-Low theory is the simplest possible meson field theory which is real istic enough to predict some of the main features of the 7rN subsystem (like the P33 resonance) and yet also simple enough that it can be solved by numerical methods or even for some special cases in analytical form. This theory is based on two assumptions: a) The only mechanism contributing to 7rN scattering is the emission and a...

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