Comments on the dispersion relation method to vector–vector interaction

Abstract

We study in detail the method proposed recently to study the vector–vector interaction using the N/D method and dispersion relations, which concludes that, while, for J = 0, one finds bound states, in the case of J = 2, where the interaction is also attractive and much stronger, no bound state is found. In that work, approximations are done for N and D and a subtracted dispersion relation for D is used, with subtractions made up to a polynomial of second degree in s − sth, matching the expression to 1 − VG at threshold. We study this in detail for the ρρ interaction and to see the convergence of the method we make an extra subtraction matching 1 − VG at threshold up to (s−sth)3.We show that the method cannot be used to extrapolate the results down to 1270 MeV where the f2(1270) resonance appears, due to the artificial singularity stemming from the “on-shell” factorization of the ρ exchange potential. In addition, we explore the same method but folding this interaction with the mass distribution of the ρ, and we show that the singularity disappears and the method allows one to extrapolate to low energies, where both the (s − sth)2 and (s − sth)3 expansions lead to a zero of Re D(s), at about the same energy where a realistic approach produces a bound state. Even then, the method generates a large Im D(s) that we discuss is unphysical.