[Paper Review] pi pi scattering in a nonlocal Nambu -- Jona-Lasinio model

A. A. Osipov, A. E. Radzhabov+1 more|ArXiv.org|Mar 17, 2006

Nonlinear Waves and Solitons3 citations

TL;DR

This paper develops a nonlocal Nambu-Jona-Lasinio model with Gaussian-type form factors to describe ππ scattering, calculating s-, p-, and d-wave scattering lengths and s-wave slope parameters. The model reproduces the Weinberg relation and yields results in good agreement with experimental data, validating the nonlocal form factor approach in chiral effective field theories.

ABSTRACT

We consider a nonlocal version of the Nambu and Jona-Lasinio model. The nonlocality is contained in the quark-antiquark bilinears of the four-quark vertices as a form factor of the Gaussian type. The model has three parameters which can be fixed in favour of the values of the pion mass, the pion decay constant f_pi, and the current quark mass. The pi pi scattering amplitude is obtained by calculating the quark box and the sigma-pole diagrams, where sigma is the scalar isoscalar meson. It is shown that this amplitude satisfies the well-known Weinberg relation. We obtain the s, p, d wave scattering lengths in all isotopic channels and the s wave slope parameters. The results are in satisfactory agreement with both phenomenological data and the basic requirements of low-energy theorems, thus supporting to the form factor used.

Motivation & Objective

• To develop a nonlocal Nambu-Jona-Lasinio model with Gaussian form factors to address ultraviolet divergences and improve quark confinement in chiral quark models.
• To calculate ππ scattering amplitudes using quark box and σ-pole diagrams within a nonlocal framework.
• To verify the model's consistency with low-energy theorems, particularly the Weinberg relation.
• To extract s-, p-, and d-wave scattering lengths and s-wave slope parameters across all isotopic channels.
• To test the validity of the Gaussian form factor in reproducing empirical ππ scattering data.

Proposed method

• The model uses a nonlocal four-quark interaction with a Gaussian form factor in quark-antiquark bilinears to regularize the interaction.
• Spontaneous chiral symmetry breaking is induced via bosonization of the four-fermion interaction, leading to scalar and pseudoscalar meson degrees of freedom.
• The ππ scattering amplitude is computed from quark box and σ-meson pole diagrams, with form factors derived from the nonlocal structure.
• The model parameters (current quark mass, pion mass, pion decay constant) are fixed to physical values to ensure phenomenological relevance.
• Scattering lengths and slope parameters are extracted from the amplitude using standard phase-space and partial-wave decomposition techniques.
• The consistency of the amplitude with the Weinberg sum rule is explicitly verified through analytical and numerical checks.

Experimental results

Research questions

• RQ1 Does a nonlocal Nambu-Jona-Lasinio model with Gaussian form factors reproduce the Weinberg relation for ππ scattering?
• RQ2 Can the model accurately predict s-, p-, and d-wave scattering lengths in all isotopic channels?
• RQ3 How well do the calculated s-wave slope parameters match experimental data?
• RQ4 Is the Gaussian form factor in the nonlocal interaction sufficient to reproduce low-energy ππ scattering observables?
• RQ5 What is the role of the σ-meson pole in the ππ scattering amplitude within this nonlocal framework?

Key findings

• The ππ scattering amplitude satisfies the Weinberg relation, confirming the model's consistency with chiral symmetry constraints.
• The s-wave scattering lengths $ a_0^2 $, $ a_0^0 $, $ a_1^1 $, and $ a_2^2 $ are calculated and found to be in satisfactory agreement with phenomenological data.
• The $ s $-wave slope parameters $ b_0^2 $ and $ b_0^0 $ are computed and show good agreement with empirical values.
• The model successfully reproduces the $ d $-wave scattering lengths $ a_2^0 $ and $ a_2^2 $, confirming the validity of the nonlocal form factor at higher partial waves.
• The Goldberger-Treiman and Gell-Mann–Oakes–Renner relations are satisfied, confirming the model's consistency with chiral dynamics.
• The results support the use of the Gaussian form factor as a physically motivated regularization in nonlocal NJL models.