Nuclear $p_\perp$-broadening of an energetic parton pair
Résumé
We revisit the transverse momentum broadening of a fast parton pair crossing a nuclear medium, putting emphasis on the pair global color state, for any number of colors N and within the eikonal limit for parton propagation and the Gaussian approximation for the gluon field of the target. The pair transverse momentum probability distribution is derived in a kinetic equation approach, and is determined by an operator ℬ describing the possible transitions between the pair color states when crossing the medium. The exponential of ℬ encompasses the 4-point correlators of Wilson lines in the saturation formalism. We emphasize the relation of ℬ with the anomalous dimension matrices appearing in the study of soft gluon radiation associated to hard 2 → 2 partonic processes. In a well-chosen, orthonormal basis of the pair color states, we rederive ℬ for any type of parton pair, making maximal use of SU(N) invariants and using ‘birdtrack’ color pictorial notations, providing a quite economical derivation of all previously known 4-point correlators (or equivalently, anomalous dimension matrices for 2 → 2 parton scattering). We discuss some general features of the pair transverse momentum distribution. The latter simplifies in the ‘compact pair expansion’ which singles out the global charges (Casimirs) of the pair color states. This study should provide the necessary tools to address nuclear broadening of n-parton systems in phenomenology while highlighting the color structure of the process.