Pion interferometry of nuclear collisions. I. Theory

The topic of pion interferometry (identical pion correlations) is analyzed in detail in the context of relativistic nuclear collisions. Through an exactly solvable field theoretic model specified by an ensemble of classical pion source currents, $J_i\mathrm{}\left(x\right)\mathrm{}$, we calculate the ${\pi }^{-}{\pi }^{-}$ correlation function $R({\tilde{k}}_1,{\tilde{k}}_2)$ for chaotic, coherent, and partially coherent pion fields. We analyze how $R$ can be used to determine the degree of coherence of the produced pion field as well as the geometric structure of the source of the chaotic field component. With this model we are able to distinguish between those correlations due to Bose-Einstein symmetrization (the Hanbury-Brown and Twiss or Goldhaber effect) and those due to specific multiparticle production dynamics. In particular we show that Bose-Einstein symmetrization dominates the form of $R({\tilde{k}}_1,{\tilde{k}}_2)$ only for chaotic pion fields produced over a time scale large compared to ${m_{\pi }}^{-1\mathrm{}}$. If, due to collective phenomena, there is some coherence of the pion field, then the intercept $R(\tilde{k},\tilde{k})=2-D^2\left(\tilde{k}\right)$ is shown to measure mode by mode that degree of coherence $D\left(\tilde{k}\right)$. Geometric information about the source of the chaotic field component may be extracted from $R({\tilde{k}}_1,{\tilde{k}}_2)$ only after $D\left(\tilde{k}\right)$ has been determined. Expressions are also derived that incorporate distortions of $R$ due to one-body and two-body final state interactions. These expressions will be numerically evaluated in a subsequent paper. Relative ${\pi }^{-}{\pi }^{-}$ interactions lead to a penetration factor $G({\tilde{k}}_1,{\tilde{k}}_2)$ that modulates the form of $R({\tilde{k}}_1,{\tilde{k}}_2)$. An expression for $G$ is obtained to all orders in the one-body optical potential but to first order in the two-body potential. This penetration factor must be evaluated before data for $R$ can be used to determine $D\left(\tilde{k}\right)$.