Form factors of the isovector scalar current and the ηπ scattering phase shifts

Abstract

A model for S-wave $$\eta \pi $$ηπ scattering is proposed which could be realistic in an energy range from threshold up to above 1 GeV, where inelasticity is dominated by the $$K{\bar{K}}$$KK¯ channel. The T-matrix, satisfying two-channel unitarity, is given in a form which matches the chiral expansion results at order $$p^4$$p4 exactly for the $$\eta \pi \rightarrow \eta \pi $$ηπ→ηπ, $$\eta \pi \rightarrow K{\bar{K}}$$ηπ→KK¯ amplitudes and approximately for $$K{\bar{K}}\rightarrow K{\bar{K}}$$KK¯→KK¯. It contains six phenomenological parameters. Asymptotic conditions are imposed which ensure a minimal solution of the Muskhelishvili–Omnès problem, thus allowing one to compute the $$\eta \pi $$ηπ and $$K{\bar{K}}$$KK¯ form factor matrix elements of the $$I=1$$I=1 scalar current from the T-matrix. The phenomenological parameters are determined such as to reproduce the experimental properties of the $$a_0(980)$$a0(980), $$a_0(1450)$$a0(1450) resonances, as well as the chiral results of the $$\eta \pi $$ηπ and $$K{\bar{K}}$$KK¯ scalar radii, which are predicted to be remarkably small at $$O(p^4)$$O(p4). This T-matrix model could be used for a unified treatment of the $$\eta \pi $$ηπ final-state interaction problem in processes such as $$\eta '\rightarrow \eta \pi \pi $$η′→ηππ, $$\phi \rightarrow \eta \pi \gamma $$ϕ→ηπγ, or the $$\eta \pi $$ηπ initial-state interaction in $$\eta \rightarrow 3\pi $$η→3π.