A New Decomposition of pion-pion S-wave Interaction

Abstract

The experimental isoscalar $\pi \pi \to \pi \pi$ S-wave amplitude squared below 1.75 GeV is characterized by a very broad structure $f_0(400-1200)$ with two narrow dips due to its destructive interference with the $f_0(980)$ and $f_0(1500)$. The $f_0(1370)$ and $f_0(1710)$ do not show up clearly in the $\pi \pi \to \pi \pi$ S-wave amplitude due to their weak coupling to $\pi \pi$. This paper is about the controversial nature of the broad $f_0(400-1200)$. We decompose it into two parts, i.e., t-channel $\rho$ meson exchange plus an additional s-channel resonance $f_0(X)$. With the t-channel $\rho$ meson exchange fixed by the isotensor $\pi \pi \to \pi \pi$ S-wave scattering, we re-fit the CERN-Munich (CM) data on $\pi \pi$ scattering to get parameters for the $f_0(X)$. We find that the $f_0(X)$ is very broad with a nearby pole at (1.67-0.26i) GeV, while the t-channel $\rho$ meson exchange part produces a pole at (0.36-0.53i) GeV. The implication of our results for the study of $\sigma$ and glueball is discussed.

Comments: 11 pages, 3 figures