On the chiral covariant approach to $\rho\rho$ scattering

We examine in detail a recent work (D.~G\"ulmez, U.-G.~Mei\ss ner and J.~A.~Oller, Eur. Phys. J. C 77:460 (2017)), where improvements to make $\rho\rho$ scattering relativistically covariant are made. The paper has the remarkable conclusion that the $J=2$ state disappears with a potential which is much more attractive than for $J=0$, where a bound state is found. We trace this abnormal conclusion to the fact that an "on-shell" factorization of the potential is done in a region where this potential is singular and develops a large discontinuous and unphysical imaginary part. A method is developed, evaluating the loops with full $\rho$ propagators, and we show that they do not develop singularities and do not have an imaginary part below threshold. With this result for the loops we define an effective potential, which when used with the Bethe-Salpeter equation provides a state with $J=2$ around the energy of the $f_2(1270)$. In addition, the coupling of the state to $\rho\rho$ is evaluated and we find that this coupling and the $T$ matrix around the energy of the bound state are remarkably similar to those obtained with a drastic approximation used previously, in which the $q^2$ terms of the propagators of the exchanged $\rho$ mesons are dropped, once the cut-off in the $\rho\rho$ loop function is tuned to reproduce the bound state at the same energy.