Theory of deeply virtual Compton scattering on the nucleon

TL;DR

Belitsky, Müller and Kirchner develop a comprehensive twist‑three DVCS formalism on the nucleon, including gluon transversity and BH–DVCS interference, and derive a full azimuthal cross‑section with Fourier harmonics that separate twist‑two and twist‑three GPD contributions. They implement a WW‑based GPD modeling via double distributions and D‑terms, assess radiative corrections and propose a DVCS scheme to mitigate gluonic NLO effects, while providing quantitative predictions for fixed‑target and collider kinematics. The work demonstrates how charge and spin asymmetries, together with harmonic projections, can in principle extract the full set of GPDs and access parton angular momentum and gluon‑spin content, though it highlights substantial model dependence and the need for higher‑order and mass corrections. Overall, the paper supplies a detailed analytical blueprint and phenomenological guidance for extracting GPDs from DVCS data across diverse experimental setups, with explicit attention to twist‑three effects and small‑$x_B$ behavior.

Abstract

We compute the cross section for leptoproduction of the real photon off the nucleon, which is sensitive to the deeply virtual Compton scattering amplitude with power accuracy. Our considerations go beyond the leading twist and involve the complete analysis in the twist-three approximation. We discuss consequences of the target and lepton beam polarizations for accessing the generalized parton distributions from experimental measurements of the azimuthal angular dependence of the final state photon or nucleon. We introduce several sets of asymmetries, defined as Fourier moments with respect to the azimuthal angle, which allow for a clear separation of the twist-two and -three sectors. Relying on a simple ansatz for the generalized parton distributions, we give quantitative estimates for azimuthal and spin asymmetries, discuss the uncertainties of these predictions brought in by radiative corrections, and compare them with experimental data as well as other theoretical expectations. Furthermore, we derive a general parametrization of the DVCS amplitudes in the region of small Bjorken variable.

Figures (16)

• Figure 1: The kinematics of the leptoproduction in the target rest frame. The $z$-direction is chosen counter-along the three-momentum of the incoming virtual photon. The lepton three-momenta form the lepton scattering plane, while the recoiled proton and outgoing real photon define the hadron scattering plane. In this reference system the azimuthal angle of the scattered lepton is $\phi_l = 0$, while the azimuthal angle between the lepton plane and the recoiled proton momentum is $\phi_N = \phi$. When the hadron is transversely polarized (in this reference frame) $S_\perp = (0, \cos {\mit\Phi}, \sin {\mit\Phi}, 0)$, the angle between the polarization vector and the scattered hadron is denoted as $\varphi = {\mit\Phi} - \phi_N$.
• Figure 2: The real (a,c) and imaginary (b,d) part of the u-quark CFFs $\xi^{\alpha} {\cal H}$ in LO (solid) and NLO (dash-dotted). It displays also the twist-three functions in the WW approximation $\xi^{\alpha} {\cal H}^{\rm eff-WW}$ with ${\cal H}^{\perp} = 0$ (dashed), and the additive contribution $\xi^{\alpha} C^{3(-)} \otimes H$ (dotted), which enter the CFF ${\cal H}^{\rm eff}$ multiplied by a kinematical factor singular at $\Delta^2 = \Delta_{\rm min}^2$. The valence and sea $u$-quark contributions are shown in (a,b) and (c,d), respectively, as a function of $x_{\rm B}$ with $x_{\rm B} \ge 0.05$. The GPDs are taken at $\Delta^2=0$ and ${\cal Q}^2= 4\ \hbox{GeV}^2$ with $b_{\rm val}=b_{\rm sea}=1$.
• Figure 3: The relative deviation of the sea $u$-quark CFF from its asymptotic value is displayed for $\Im{\rm m} {\cal H}$ and $\Re{\rm e} {\cal H}/\Im{\rm m} {\cal H}$ as the solid and dashed line, respectively, in (a). The same quantities are plotted for ${\cal H}^{\rm eff-WW}$ with ${\cal H}^\perp =0$ as dash-dotted and dotted lines, respectively. The right panel (b) shows the same as in (a) but for the valence $u$-quark CFF $\widetilde{\cal H}$. The GPD-parameters are the same as in Fig. \ref{['Fig-CFF-Vec']}.
• Figure 4: The relative NLO corrections ${\cal R}$ (see Eq. \ref{['RelRadCorr']}) to the absolute value (a,c) and the phase (b,d) of the CFFs ${\cal H}$ (a,b) and $\widetilde{\cal H}$ (c,d). The magnitude of radiative corrections with neglected gluon GPD contributions are displayed as dashed and solid curves for the factorization scale setting $\mu_F = {\cal Q}$ and $\mu_F \approx \mu_F^{\rm DVCS}$, respectively. The complete (quark + gluon) NLO result is given for these both scale settings by dotted and dash-dotted lines, respectively. The coupling is set to $\alpha_s/\pi = 0.1$, the quark GPD-parameters are the same as in Fig. \ref{['Fig-CFF-Vec']}, and $b_G=2$.
• Figure 5: The real (a) and imaginary (b) part of ${\cal Q}^2\frac{d}{d{\cal Q}^2} \ln {\cal H}$ for MRS A' parametrization with $b_{\rm val} = b_{\rm sea} = 1$ at $\Delta^2 = 0$. The dash-dotted (dotted) line shows the scale violation due to the evolution of quarks, while the solid (dashed) one includes also the evolution of gluons for the scale setting $\mu_F = 3.8 {\cal Q}$ ($\mu_F = {\cal Q}$). The parameters are set as in Fig. \ref{['Fig-NLO-Cor']}.