Generalized parton distributions and gravitational form factors at large momentum transfer

TL;DR

The work develops a SCET-based factorization and Sudakov resummation framework for generalized parton distributions at large momentum transfer $|t|$, deriving a universal hard kernel $K$, jet functions $J$, and soft factor $S$ that capture gluon radiation and infrared dynamics. It demonstrates that the quark-in-quark GPD factorizes into these components and that the same Sudakov structure applies to the Feynman (overlap) contribution to proton GPDs, enabling resummation of large logarithms in two-scale exclusive processes. The authors show that the ratio of Feynman to hard contributions for GPD moments is perturbatively calculable and identify a novel $\ ext{O}(\alpha_s)$ power-law $t$-dependence that can dominate in the phenomenologically relevant region, with implications for the large-$t$ behavior of gravitational form factors. They also present a hybrid overlap representation to model power corrections and discuss the implications for form factors and DVCS, offering a concrete route to incorporate large-$t$ behavior into phenomenological GPD parametrizations.

Abstract

Within the soft collinear effective theory (SCET), we derive a factorization theorem which resums Sudakov logarithms $(α_s\ln^2(-t))^n$ to all orders in the quark-in-quark generalized parton distribution (GPD) at large momentum transfer $t$, and perform a consistency check to one-loop. We show that the same Sudakov factor appears in the `Feynman' contribution to the GPDs of the nucleon. Our result enables the resummation of all the large logarithms $\ln Q^2$ and $\ln^2t$ in exclusive processes with two hard scales $Λ_{\rm QCD}^2\ll |t| \ll Q^2$. We also present a SCET power counting analysis of the Feynman contributions to the GPDs and show that the $x$-dependence of GPDs factorizes at large-$t$ with controlled corrections. This in particular implies that any ratio of GPD moments such as the electromagnetic and gravitational form factors (GFF) is perturbatively calculable in this approximation. Furthermore, we identify a novel order $α_s$ power-law $t$-dependence in the GPD and the $D$-type GFF that will dominate over the standard order $(α_s^2)$ `leading twist' asymptotic contribution in the phenomenologically relevant region of $t$.

Figures (6)

• Figure 1: Left: Box diagram. Right: Sail diagram
• Figure 2: Left: The hard scattering contribution to GPDs that dominates in the asymptotic regime $|t|\to \infty$. Right: The Feynman contribution in the asymptotic regime. The labels $hc$, $c$ and $s$ on quark lines denote 'hard-collinear', 'collinear' and 'soft'.
• Figure 3: Sample tree-level graphs for the SCET$(C,\bar{C}, s)$$\rightarrow$ SCET$(c,\bar{c},s)$ matching for operators (a) $\xi_{hc}$, (b) $A_{hc\perp}$ and (c) $\bar{\xi}_{hc}$.
• Figure 4: Left: Proton GPD in the DGLAP region $\xi < x$. $y$ varies in the range $x\le y\le 1$. Right: Proton GPD in the ERBL region $x<\xi$ when $|t|\lesssim \Lambda^2_{\rm QCD}$. The diagram corresponds to $y<\xi$.
• Figure 5: The Feynman contribution to the quark GPD $H^{\rm Feyn}_q$ at $x=0.6$ (left) and $x=0.4$ (right) with $\xi=0$ as a function of $-t$ in units of GeV$^2$. (The $y$-axis is arbitrary.) Orange curve: Total $H_q^{\rm Feyn}$ without the Sudakov factor. The black and blue curves denote the leading order $\alpha_s^0$ and the next-to-leading order $\alpha_s$ contributions, respectively. The dashed curve denotes the total $H^{\rm Feyn}_q$ with the NNLL Sudakov factor $U$ included.