All-path-length and sub-eikonal corrections to momentum broadening in the opacity expansion approach

Abstract

We present a detailed study of momentum broadening for high-energy partons traversing the Quark-Gluon Plasma (QGP), extending the Gyulassy-Levai-Vitev (GLV) formalism to include both all-path-length (APL) and sub-eikonal corrections. Traditional GLV calculations rely on the large separation distance and large formation time approximations, which are valid for large systems, but whose applicability in small systems such as pp and p/dA may fail. We derive analytic expressions for the momentum broadening distributions to first order in the opacity expansion, and perform a numerical investigation to quantify their impact. Our results show that the APL result reduces the low-momentum broadening and the sub-eikonal correction enhances the high-momentum broadening. The combined APL and sub eikonal correction show that the sub-eikonal correction mitigates the effect of the APL correction.

Figures (5)

• Figure 1: The relevant Feynman diagrams for momentum broadening at leading order in the opacity.
• Figure 2: The broadening distribution to first order in the opacity expansion $d^{(1)}N/d^3 \vec{p}$ is shown for three correction schemes---(GLV + APL), sub-eikonal $(\mathrm{GLV})_{\mathrm{sub}}$, and the combined correction $(\mathrm{GLV} + \mathrm{APL})_{\mathrm{sub}}$. Each corrected distribution is plotted both individually (top panel) and as a ratio to the unmodified GLV baseline (bottom panel), as a function of $\boldsymbol{p}$ for fixed initial parton momentum $P^+ = 40$ GeV and system size $L = 4$ fm.
• Figure 3: The ratios of the various corrected broadening distributions — the all path-length–corrected $(\mathrm{GLV} + \mathrm{APL})$, the sub-eikonal corrected $(\mathrm{GLV})_{\mathrm{SUB}}$, and the combined all path-length and sub-eikonal corrected $(\mathrm{GLV} + \mathrm{APL})_{\mathrm{SUB}}$ distributions — to the standard GLV result are plotted as functions of $\boldsymbol{p}$. For the path-length–corrected case (a), results for different system sizes $L = 1, 4, 10~\mathrm{fm}$ at fixed initial parton momentum $P^+ = 10$ GeV are shown, while for the sub-eikonal case (b), results for different initial momenta $P^{+} = 7, 8, 10~\mathrm{GeV}$ at fixed system size $L = 4$ fm are shown.
• Figure 4: The ratios of the various corrected broadening distributions — the all path-length–corrected $(\mathrm{GLV} + \mathrm{APL})$, the sub-eikonal corrected $(\mathrm{GLV})_{\mathrm{SUB}}$, and the combined all path-length and sub-eikonal corrected $(\mathrm{GLV} + \mathrm{APL})_{\mathrm{SUB}}$ distributions — to the standard GLV result are plotted as function of system size L in (a), for a fixed initial parton momentum of $P^+ = 10$ GeV. As a function of initial parton momentum $P^+$ in (b), for a fixed system size $L=4$ fm.
• Figure 5: Double scattering diagram with crossed gluon exchanges.