Title:

Dispersion relations for scattering by spindependent potentials

The understanding of the two nucleon interaction is the central problem in the theory of nuclear forces. Whether specifically many body forces turn out to be important or not, the investigation of the two body interaction is bound to give us some insight into the mechanism responsible for the strongest binding known to man. So far, the only attempts, in theory, to go beyond pure guesswork (or phenomenology, as it is politely called) have been along two essentially different directions. The first type of computations, which were initiated by Yukawa and have since been assiduously followed up by many physicists (e.g. Taketani et al.) may be collectively classed as "meson theoretical calculations". The main idea behind these efforts if the belief (perhaps quite true) that nuclear forces are mediated by a meson cloud and, naturally, the closer together the nucleons are, the larger the number of pions taking part. This rather loose inverse relationahip between the range of a force and the corresponding mass of the intermediate state is now widely accepted among elementary particle physicists. However, the quantitative realisation of the Taketani programme takes the form of computing one perturbation graph after another in the framework of field theory. The one pion exchange graph, of course, yielded the famous Yukawa potential which, as expected, was very successful in explaining the long range effect of the nucleon force. Higher order effects have not had the same success in the "medium" range, with the consequent flourishing of phenomenology. Apart from the increasing technical difficulty of calculating higher order graphs, it is very probable that the perturbation series is divergent in which case the whole project is essentially illusory, and any occasional agreements with experiments must be regarded as fortuitous. There have been many variations from straight expansion in the coupling strength or the number of intermediate particles. A comprehensive survey is given in the review of Y oravcsik & Noyes(16). A more hopeful approach has appeared in recent years, following Fandelstam's proposal of double dispersion relations for the scattering amplitude in a twoparticlesin, twoparticlesout process. It was initiated by Champ & Fubini(4), who indicated how one can obtain a useful "potential ", starting from the Mandelstam representation. But he could do it reliably only for the case of scalar particles, mainly because no one had derived a Kandelstam representation, or even a single variable dispersion relation, for two particles of spin 0.5 interacting through a general spin and velocity dependent potential. Up to now, there have appeared only two papers on this subject and both have touched the problem only partially. Hamilton(9) has proved dispersion relations which are those (12) similar to those that Khuri obtained for central potentials, considering only an additional tensor term. The other attempt has been by Buslayev(3) who considers a particle scattered by a spinorbit potential. The present work is intended to fill this gap; we obtain dispersion relations (in energy, for fixed momentum transfer) for two spin 2 particles interacting through a complete potential subject only to reasonable physical requirements. The relations obtained turn out to be substantially different in form from those derived or postulated as yet; and the methods used have the virtue of being generalisable to the scattering between systems of arbitrary spin, as indicated in the last chapter. The derivation of these dispersion relations is also a step forward in the fulfilment of the CharapFubini programme of deducing a realistic twonucleon potential. We will now give a brief outline of this manuscript. In Chapter II, we discuss the form of the potential for a twonucleon system, and then go on to derive some properties of the resolvent of the Hamiltonian. We also have a brief look at the spectral decomposition of the total Hamiltonian. Most of the matter contained in this chapter is of a preliminary character, and no originality is claimed. Chapter III forms the bulk of the thesis in which we obtain the analytic properties and asymptotic behaviour of the Hamiltonian Green's function in the complex energy plane. We start by deriving an integral equation for the Green's function incorporating the outgoing boundary condition. This equation does not have a bounded or square integrable kernel so that the usual methods of solution do not apply. However, it is seen that the kernel is only "weakly" singular (to use the terminology of Yikhlin(15)), and Fredholm's theorems can still be applied. This is the central point of the argument, which permits us to infer that the Green's function is analytic except on the spectrum of the Hamiltonian. We then proceed to investigate the spectrum of the total Hamiltonian, and the approach is in part borrowed from Povzner(19). The chapter concludes with a detailed investigation of the Green's function at high energy. The analyticity and asymptotics obtained in Chapter III help us in writing down an explicit form for the scattering amplitude in Chapter IV. The amplitude is treated in detail, and split into five parts with different spininvariants, as e.g. in Goldberger, Nambu & Oehme(8) . Each coefficient separately obeys a dispersion relation which is then written down. We close the chapter with a few physical remarks of interest. Chapter V contains a generalisation of the above results to systems with arbitrarily high spins. It gives only a sketch of the arguments without going into much cumbersome detail.
