The Inclusive (γ,π + π-) Reaction in Nuclei as a Test of the Pion Dispersion Relation in Nuclear Matter

Abstract

The poles of the free pion propagator, \( {\left( {{\omega^2} - {{\vec{q}}^2} - {\mu^2}} \right)^{{ - 1}}} \), give the free pion dispersion relation, \( \omega = {\left( {{{\vec{q}}^2} + {\mu^2}} \right)^{{\frac{1}{2}}}} \), relating the pion energy to its momentum. Inside of nuclear matter, the pion interacts with the nuclear medium and the pion propagator gets modified to $$ D\left( {\omega, q} \right) = {\left[ {{\omega^2} - {{\vec{q}}^2} - {\mu^2} - \Pi \left( {\omega, q} \right)} \right]^{{ - 1}}} $$ (1) where Π(ω,q) is the pion self-energy, (Π ≡ 2ω Vopt), which depends as well upon the nuclear density. The poles of (1) give the pion dispersion relation in the nuclear medium; $$ {\omega^2} - {\vec{q}^2} - {\mu^2} - {\rm Re} \,\Pi \left( {\omega, q} \right) = 0 \to \omega = \tilde{\omega }(q) $$ (2) in addition, the pion acquires a width due to strong interaction given by Γ = -Im Π/ω), which takes into account the loss of elastic pion flux due to quasi-elastic scattering or pion absorption.