Initial conditions for dipole evolution beyond the McLerran-Venugopalan model

TL;DR

The paper extends the MV-based description of high-energy QCD by incorporating the first subleading density correction from a quartic color-charge action, deriving a corrected dipole scattering amplitude $N(r)$ on dense targets. Using a semi-classical Wilson-line expansion, it systematically computes contributions up to order $\mathcal{O}(g^8)$, showing renormalization of the two-point color source density and new $\mu^4$-dependent terms that modify $N(r)$ and its $A$-dependence. The authors connect their results to phenomenology by approximately matching to the AAMQS proton fit and predicting how the quartic correction scales with nuclear size, implying a reduced dipole scattering for finite $A$ and a modified classical bremsstrahlung tail. These findings provide a theoretically grounded initial condition for small-$x$ evolution in nuclei and offer a possible explanation for observed deviations from MV-like behavior in heavy-ion collisions and future electron-ion collisions.

Abstract

We derive the scattering amplitude N(r) for a QCD dipole on a dense target in the semi-classical approximation. We include the first subleading correction in the target thickness arising from ~ρ^4 operators in the effective action for the large-x valence charges. Our result for N(r) can be matched to a phenomenological proton fit by Albacete et al over a broad range of dipole sizes r and provides a definite prediction for the A-dependence for heavy-ion targets. We find a suppression of N(r) for finite A for dipole sizes a few times smaller than the inverse saturation scale, corresponding to a suppression of the classical bremsstrahlung tail.

Figures (8)

• Figure 1: Left: scattering amplitude for an adjoint dipole ($N_A=2N-N^2$) on a proton, assuming $Q_s^2=0.168$ GeV$^2$ and $\Lambda^2=0.0576$ GeV$^2$. Right: same for a nucleus with $A=200$ and $Q_s^2\sim A^{1/3}$, $\beta_A\sim A^{-2/3}$.
• Figure 2: $\langle V\rangle$ at order $g^4/\kappa_4$.
• Figure 3: Order $g^8/\kappa_4$ contribution to $\langle V\rangle$.
• Figure 4: $V(x_\perp) V^\dagger(y_\perp)$ at order $g^8/\kappa_4$ ($g^6$ from $V(\bold x_\perp)$$\times$$g^2$ from $V^\dagger(\bold y_\perp)$).
• Figure 5: Expectation value of $V(\bold x_\perp)$ at order $g^4$ times $V(\bold y_\perp)$ at order $g^4$.