Two-proton correlation function: a gentle introduction

Abstract

Abstract. The recent COSY-11 collaboration measurement of the two-proton correlation function in the pp → ppη reaction, reported at this meeting [1], arouse some interest in a simple theoretical description of the correlation function. In these notes we present a pedagogical introduction to the practical methods that can be used for calculating the correlation function.Keywords: particle correlations, nucleon-nucleon interaction PACS: 25.75. Gz, 13.75.Cs We are going here to deal with low-energy phenomena and in order to avoid unnecessary complications our approach will be non-relativistic. We wish to develop a practical scheme for calculating the two-particle correlation function C(k), with k denoting their relative momentum, choosing as the departure point the familiar formula(referred to in the literature as the Koonin-Pratt model [2]), expressing C(k) as an overlap of two distributions. The first distribution, D(r), is the effective source density function that is usually for convenience assumed to be a Gaussianwith a single parameter d reflecting the size of the source. The second factor entering the overlap integral (1), is a probability density involving the square of the wave function Φ k k k (r r r) describing two-particle system in the continuum. Actually, formula (1) is only a static approximation to the correlation function derived under the assumption that the final-state interaction between the two detected particles dominates, while all other interactions are negligible. Furthermore, it is assumed that the correlation functions are determined by the two-body densities of states (corrected for their mutual interactions) and that the single particle phase space distribution function of the emitted particle varies slowly with momentum. Admittedly, the question concerning the validity of these assumptions is far from settled but since they result in a manageable calculational scheme it is worthwhile to examine in some detail its consequences.Adopting hereafter units in whichh = c = 1, for non-interacting particles Φ k k k (r r r) takes the form of a plane wave Φ k k k (r r r) = e ik k k · r r r and in this case the correlation function is equal to unity, which means no correlation. However, when the particles are non-interacting identical bosons(fermions), the plane

Two-proton correlation function for thepp→pp+ η andpp→pp+pionsreactions

“…At present it is not possible to draw a solid quantitative conclusion about the size of the system since, for example, in the case of the pp → ppη reaction it would require the solution of a three-body problem where pp and pη [30] interactions are not negligible and both contribute significantly to the proton-proton correlation. However, based on semi-quantitative predictions [31], one can estimate that the system must be unexpectedly large with a radius in the order of 4 fm. This makes the result interesting in context of the predicted quasi-bound ηNN state [32] and in view of the hypothesis [14] that at threshold for the pp → ppη reaction the proton-proton pair may be emitted from a large Borromean-like object whose radius is about 4 fm.…”