Abstract

We derive a factorization formula for forward production of two jets in dilutedense collisions that is valid for an arbitrary value of the momentum imbalance of the jets, kt . This generalizes the transverse momentum dependent (TMD) factorization formula that has been derived before by Dominguez et al. Their formula is valid only for small values of the transverse momentum of the small-x gluon from the target; it has kt dependent TMD gluon distributions, but on-shell hard matrix elements. We extend the TMD formula to all ranges of kt by including off-shell matrix elements. We also add finite Nc corrections. The new formula encompasses both, the TMD factorization for small kt on the order of the saturation scale, and the High Energy Factorization (HEF) for large kt on the order of the momentum of the jets. The TMD and HEF factorizations can be derived from the Color Glass Condensate (CGC) formula for forward di-jet production in the appropriate limits. We show explicitly the equivalence of HEF and CGC in the dilute target approximation.

Highlights

  • The production of two jets in the forward rapidity direction of ultra-relativistic proton-proton and proton-nucleus collisions is sensitive to the small-x gluon distributions in the target, and the largex partons in the projectile

  • The wave function of the projectile is obtained from perturbative quantum chromodynamics, while the gluon distributions in the target are in the non-linear saturation regime

  • The high-energy factorization (HEF) formula is an ansatz formula which turns out to be valid for large kt values, with Fg/A(x2, kt) equal to the unintegrated gluon distribution entering in the cross section for deep inelastic scattering (DIS)

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Summary

Introduction

The production of two jets in the forward rapidity direction of ultra-relativistic proton-proton and proton-nucleus collisions is sensitive to the small-x gluon distributions in the target, and the largex partons in the projectile. The dilute projectile in the HEF formula for di-jet production is represented with a parton distribution function of collinear factorization, fa/p(x1, μ2), the dense target with one kt-dependent gluon distribution, Fg/A(x2, kt), and the hard part of the scattering with off-shell matrix elements, Mag∗→cd [5], [7]:. The HEF formula is an ansatz formula which turns out to be valid for large kt values, with Fg/A(x2, kt) equal to the unintegrated gluon distribution entering in the cross section for deep inelastic scattering (DIS). This distribution is associated with the S -matrix for a quark-anitquark dipole scattering: Fg/A(x2, kt). We derive the HEF formula from the CGC cross section for a dilute target for all three channels: qg∗ → qg, gg∗ → qqand gg∗ → gg

Simplified TMD factorization for finite Nc
Unifying TMD factorization formula
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