# Comments on the dispersion relation method to vector-vector interaction

## TL;DR

This paper critically examines a dispersion relation method for vector-vector interactions, revealing its limitations in predicting bound states at low energies due to artificial singularities, despite some improvements when folding with mass distributions.

## Contribution

The study provides a detailed analysis of the dispersion relation method's applicability to $
ho-
ho$ interactions, highlighting its shortcomings and proposing modifications to improve its extrapolation capabilities.

## Key findings

- The method cannot reliably predict the $f_2(1270)$ resonance due to artificial singularities.
- Folding with the $
ho$ mass distribution removes singularities and allows low-energy extrapolation.
- The method produces unphysical large imaginary parts in the $D$ function.

## 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 $s-s_\mathrm{th}$, 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.