Abstract
Exploiting recently obtained analytic solutions of classical Yang-Mills equations for higher order perturbations in the field of the dilute object (proton), we derive the complete first saturation correction to the single inclusive semi-hard gluon production in high energy proton-nucleus collisions by applying the Lehmann-Symanzik-Zimmermann reduction formula. We thus finalize the program started by Balitsky (see ref. [1]) and independently by Chirilli, Kovchegov and Wertepny (see ref. [2]) albeit using a very different approach to carry out our calculations. We extracted the functional dependence of gluon spectrum on the color charge densities of the colliding objects; thus our results can be used to evaluate complete first saturation correction to the double/multiple inclusive gluon productions.
Highlights
We extracted the functional dependence of gluon spectrum on the color charge densities of the colliding objects; our results can be used to evaluate complete first saturation correction to the double/multiple inclusive gluon productions
The single inclusive gluon productions in high energy nuclear collisions can be expressed as a function of two dimensionless parameters Q2s,P /k⊥2 and Q2s,T /k⊥2, involving the saturation scales of the projectile and the target respectively
Note that since n(4)(k) is cubic in ρP, it does not result in any nontrivial contribution to the single inclusive gluon production after averaging over the projectile color charge densities in the McLerran-Venugopalan model
Summary
We briefly recapitulate the general setup of our calculations and the results obtained in ref. [3]. In order to facilitate this, we consider a subgauge transformation defined by the condition ∂iβi(τ = 0, x) = 0 to which we will refer to as the initial time Coulomb gauge, see e.g. refs. The subgauge transformations, not affecting the FS gauge, preserve the form of the Yang-Mills equations and only modify the initial conditions. We will present solutions of the classical Yang-Mills equations as power series expansion in the coupling constant g in the initial time Coulomb subgauge (keeping all power of gρT ). By analogy with ζi(x), we introduce the eikonally rotated projectile color charge density ρR(x) = U †(x)∂iαP(1),iU (x) = g U †(x)ρP (x)U (x) In terms of these auxiliary quantities, we have order-g initial conditions bη(k) = 2β(1)(τ = 0, k) = −ikiζ(1),i(k) − gρR(k), b⊥(k) = −i ijkiβj(1)(τ = 0, k) = −i ijkiζ(1),j(k). D2p d2q p × q (2π) (2π) p2⊥q⊥2 b (p − q), b⊥(q) , bη(k − p) d2p d2q p · q (2π) (2π)2 2p2⊥
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