Time-Dependent Observables in Heavy Ion Collisions I: Setting up the Formalism

Abstract

We adapt the Schwinger-Keldysh formalism to study heavy-ion collisions in perturbative QCD. Employing the formalism, we calculate the two-point gluon correlation function $G_{22}^{a\mu,b\nu}$ due to the lowest-order classical gluon fields in the McLerran-Venugopalan model of heavy ion collisions and observe an interesting transition from the classical fields to the quasi-particle picture at later times. Motivated by this observation, we push the formalism to higher orders in the coupling and calculate the contribution to $G_{22}^{a\mu,b\nu}$ coming from the diagrams representing a single rescattering between two of the produced gluons. We assume that the two gluons go on mass shell both before and after the rescattering. The result of our calculation depends on the ordering between the proper time of the rescattering $\tau_Z$ and the proper time $\tau$ when the gluon distribution is measured. For (i) $\tau_Z\gg 1/Q_s$ and $\tau-\tau_Z\gg 1/Q_s$ (with $Q_s$ the saturation scale) we obtain the same results as from the Boltzmann equation. For (ii) $\tau-\tau_Z\gg \tau_Z\gg 1/Q_s$ we end up with a result very different from kinetic theory and consistent with a picture of "free-streaming" particles. Due to the approximations made, our calculation is too coarse to indicate whether the ordering (i) or (ii) is the correct one: to resolve this controversy, we shall present a detailed diagrammatic calculation of the rescattering correction in the $\varphi^4$ theory in the second paper of this duplex.