Sub-femtometer scale color charge fluctuations in a proton made of three quarks and a gluon

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 $1/x \sim 10 - 100$.