Comments on the dispersion relation method to vector-vector interaction

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 |$s-s_\mathrm{th}$|⁠, matching the expression to |$1-VG$| at threshold. We study this in detail for the |$\rho\rho$| interaction and to see the convergence of the method we make an extra subtraction matching |$1-VG$| at threshold up to |$(s-s_\mathrm{th})^3$|⁠. We show that the method cannot be used to extrapolate the results down to 1270 MeV where the |$f_2(1270)$| resonance appears, due to the artificial singularity stemming from the “on-shell” factorization of the |$\rho$| exchange potential. In addition, we explore the same method but folding this interaction with the mass distribution of the |$\rho$|⁠, and we show that the singularity disappears and the method allows one to extrapolate to low energies, where both the |$(s-s_\mathrm{th})^2$| and |$(s-s_\mathrm{th})^3$| expansions lead to a zero of |$\mathrm{Re}\,D(s)$|⁠, at about the same energy where a realistic approach produces a bound state. Even then, the method generates a large |$\mathrm{Im}\,D(s)$| that we discuss is unphysical.