[Paper Review] Wandzura-Wilczek approximation for the twist-3 DVCS amplitude

N. Kivel, M. V. Polyakov+2 more|University of Regensburg Publication Server (University of Regensburg)|Jul 27, 2000

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper derives Wandzura-Wilczek-type relations for twist-3 skewed parton distributions in deeply virtual Compton scattering (DVCS) on the pion, showing that these distributions develop discontinuities at $x = /pm heta$ in the WW approximation. Although this leads to formally divergent terms in the amplitude for transversely polarized photons, these contributions are suppressed at twist-3 accuracy and do not affect physical observables, while the longitudinally polarized amplitude remains finite and free of such divergences.

ABSTRACT

We present a derivation of Wandzura-Wilczek (WW) like relations for skewed parton distributions. It is demonstrated for photon-pion scattering that the skewed twist-3 parton distributions contributing to the DVCS amplitude have discontinuities at the points $x=\pmξ$ in the WW approximation. This may lead to a violation of factorisation for the twist-3 DVCS amplitude with transverse polarization of the virtual photon. We show, however, that the contribution of the divergencies to the scattering of a transversely polarized virtual photon affects DVCS observables only at order $1/Q^2$ and can be neglected at twist-3 accuracy. For the scattering of a longitudinally polarized photon the twist-3 amplitude is free of such divergencies.

Motivation & Objective

• To derive Wandzura-Wilczek-type relations for twist-3 skewed parton distributions (SPDs) in DVCS.
• To investigate the behavior of twist-3 SPDs in the WW approximation, particularly their analytic structure.
• To assess whether discontinuities at $x = heta$ and $x = - heta$ lead to physical divergences in the DVCS amplitude.
• To determine the impact of such divergences on observables, especially for transversely and longitudinally polarized virtual photons.
• To clarify the theoretical implications and potential for experimental verification of the observed behavior.

Proposed method

• Derives WW-like relations for twist-3 SPDs using operator-level analysis, focusing on matrix elements of non-symmetric quark bilinears.
• Applies the Wandzura-Wilczek approach by expressing twist-3 distributions in terms of twist-2 SPDs through the elimination of total-derivative operators.
• Uses integral equations derived from operator identities involving the covariant derivative and momentum transfer to solve for SPDs.
• Simplifies the general solution by neglecting terms of order $\Delta_{\perp}^2$, corresponding to the $1/Q^2$ power corrections.
• Analyzes the analytic structure of $H_3(x,\xi)$ in the WW approximation, identifying poles at $x = \pm \xi$.
• Evaluates the physical impact of these discontinuities on DVCS amplitudes for both transverse and longitudinal photon polarizations.

Experimental results

Research questions

• RQ1 Do twist-3 skewed parton distributions in the Wandzura-Wilczek approximation exhibit discontinuities at $x = \pm \xi$?
• RQ2 What is the physical significance of these discontinuities in the DVCS amplitude for transversely polarized virtual photons?
• RQ3 Are the divergent contributions from these discontinuities observable at twist-3 accuracy in the $1/Q$ expansion?
• RQ4 Why is the amplitude for longitudinally polarized photons free of such divergencies in the WW approximation?
• RQ5 Can the absence of divergences in the longitudinal case be experimentally verified in future DVCS measurements?

Key findings

• The twist-3 distribution $H_3(x,\xi)$ in the Wandzura-Wilczek approximation develops discontinuities at $x = \pm \xi$, leading to formally divergent terms in the DVCS amplitude.
• These divergences affect the amplitude for transversely polarized virtual photons but contribute only at order $1/Q^2$, which is beyond the accuracy of twist-3 calculations.
• As a result, the divergent contributions can be neglected in physical observables at twist-3 accuracy, preserving the validity of the WW approximation.
• For longitudinally polarized virtual photons, the twist-3 amplitude remains finite and free of such discontinuities due to cancellation mechanisms.
• The theoretical structure suggests that the WW approximation remains consistent for twist-3 DVCS, with no unphysical singularities affecting leading-twist observables.
• The findings support the use of the WW approximation for twist-3 SPDs in DVCS, with potential for experimental validation via spin asymmetries in future JLab, DESY, and CERN experiments.