How to Parameterize a Light and Broad Resonance (the Σ Meson) * )

The evidence for the existence of a light and broad resonance in I=J=0 ππ scattering process – named as the σ resonance – has been accumulated, and more and more physicists working on the subject accepted the renaissance of the σ resonance. Among those works, model independent analysis 1), 2) are vital in clarifying the issue whether the σ resonance exists or not, since one has to distinguish and separate the contribution of a broad resonance to the phase motion from background contributions, and the latter is often very difficult, if not impossible, to estimate. However, there are different opinions in the literature against the existence of the light σ resonance 3), based upon the following observation: the ππ phase shift does not pass 90 at low energies, which would have occurred, according to Eq. (1.7) given below, at s = M2 with M the Breit–Wigner mass which should be light. This question is actually already answered, in one aspect, by Ishida and collaborators. 4) who pointed out that chiral symmetry requires the background contribution to the phase shift in I=J=0 ππ scatterings to be negative,∗∗) Ishida and collaborators argue that the background phase cancels the large phase motion generated by the Breit–Wigner resonance and therefore there is no contradiction between the observed ππ phase shift and the existence of a light and broad resonance. In this talk I will further point out that a conventional Breit-Wigner description to the σ resonance actually violates chiral symmetry and should be abandoned, and it is not even necessary for a light and broad resonance to develop a phase shift passing 90.