Abstract
The light-front wave function of a proton composed of three quarks and a perturbative gluon is computed. This is then used to derive expressions for the color charge density correlator $\langle\rho^a(\vec q_1)\, \rho^b(\vec q_2)\rangle$ at ${\cal O}(g^4)$ due to the emission of a gluon by one of the quarks in light-cone gauge. The correlator exhibits the soft and collinear singularities. Albeit, we employ exact gluon emission and absorption vertices, and hence the gluon is not required to carry very small light-cone momentum, or to be collinear to the emitting quark. We verify that the correlator satisfies the Ward identity and that it is independent of the renormalization scale, i.e. that ultraviolet divergences cancel. Our expressions provide x-dependent initial conditions for Balitsky-Kovchegov evolution of the C-even part of the dipole scattering matrix to higher energies. That is, we determine the first non-trivial moment of the color charge fluctuations which act as sources for soft color fields in the proton with wavelengths greater than approximately $1/x \sim 10 - 100$.
Highlights
The purpose of this paper is to derive expressions for the color charge correlator hρaðq⃗ 1Þρbðq⃗ 2Þi in a proton boosted to large momentum Pþ, on the xþ 1⁄4 0 light front
The charge operator ρa corresponds to the plus component of the color current due to valence quarks, and a perturbative gluon which is not required to carry small momentum fraction
Reference [1] showed that G2ðb⃗ ; q⃗ 1 − q⃗ 2Þ exhibits nontrivial behavior as a function of impact parameter b⃗ and relative transverse momentum q⃗ 1 − q⃗ 2 of the probes, changing from “repulsion” at small b and q⃗ 1 − q⃗ 2 to “attraction” at large b, q⃗ 1 − q⃗ 2. Their analysis restricted to the valence quark state of the proton; here, we derive the corrections to the color charge correlator due to the emission of a gluon by one of the quarks
Summary
The purpose of this paper is to derive expressions for the (light-cone gauge) color charge correlator hρaðq⃗ 1Þρbðq⃗ 2Þi in a proton boosted to large momentum Pþ, on the xþ 1⁄4 0 light front. Reference [1] showed that G2ðb⃗ ; q⃗ 1 − q⃗ 2Þ exhibits nontrivial behavior as a function of impact parameter b⃗ and relative transverse momentum q⃗ 1 − q⃗ 2 of the probes (and their relative angle), changing from “repulsion” at small b and q⃗ 1 − q⃗ 2 to “attraction” at large b, q⃗ 1 − q⃗ 2 Their analysis restricted to the valence quark state of the proton; here, we derive the corrections to the color charge correlator due to the emission of a gluon by one of the quarks.
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