[Paper Review] Compton scattering: from deeply virtual to quasi-real

A.V. Belitsky, A.H. Mueller+1 more|arXiv (Cornell University)|Dec 29, 2012

Quantum Chromodynamics and Particle Interactions12 references4 citations

TL;DR

This paper presents a unified framework for virtual Compton scattering off polarized nucleons, interpolating between deeply virtual and quasi-real regimes using helicity-dependent Compton Form Factors (CFFs). It introduces a kinematic-singularity-free parametrization of the Compton tensor in terms of CFFs, directly linked to Generalized Parton Distributions (GPDs) at high virtuality, and establishes exact relations between CFFs and Generalized Polarizabilities through a minimal Dirac bilinear basis, with explicit Born-level expressions provided.

ABSTRACT

We address the question of interpolation of the virtual Compton scattering process off a polarized nucleon target between the deeply virtual regime for the initial-state photon and its near on-shell kinematics making use of the photon helicity-dependent Compton Form Factors (CFFs) as a main ingredient of the formalism. The five-fold differential cross section for the reaction with all possible polarization options for the lepton and nucleon spins is evaluated in terms of CFFs in the rest reference frame of the initial-state nucleon. We suggest a rather simple parametrization of the Compton hadronic tensor in terms of CFFs which are free from kinematical singularities and are directly related, at large photon virtualities, to Generalized Parton Distributions. We also provide a relation of our basis spanned by a minimal number of Dirac bilinears to the one introduced by Tarrach for the parametrization of the virtual Compton tensor and utilize the former to establish a set of equalities among our CFFs and Generalized Polarizabilities. As a complementary result, we express Compton scattering in the Born approximation in terms of CFFs as well.

Motivation & Objective

• To develop a consistent formalism for interpolating virtual Compton scattering across the deeply virtual and quasi-real regimes.
• To eliminate kinematical singularities in the Compton tensor parametrization using a minimal basis of Dirac bilinears.
• To establish direct connections between Compton Form Factors (CFFs) and Generalized Polarizabilities (GPs) via a new tensor basis.
• To provide explicit Born-level expressions for Compton scattering in terms of CFFs, valid across kinematic regions.

Proposed method

• Parametrizes the hadronic Compton tensor using a minimal set of five Dirac bilinears, ensuring absence of kinematical singularities.
• Expresses the five-fold differential cross section in the nucleon's rest frame using CFFs as the central observable.
• Relates the proposed CFF basis to Tarrach's tensor basis, enabling exact equalities between CFFs and Generalized Polarizabilities.
• Derives low-energy expansions of the Compton amplitude in terms of CFFs, linking them to electric and magnetic polarizabilities.
• Computes the Born-level amplitude for Compton scattering off nucleons using CFFs, with explicit expressions for all helicity amplitudes.
• Utilizes the full kinematical structure, including virtuality $\mathcal{Q}^2$, $t$-channel momentum transfer, and nucleon mass effects via $\epsilon$.

Experimental results

Research questions

• RQ1 How can the virtual Compton scattering amplitude be consistently parametrized across the deeply virtual and quasi-real regimes without kinematical singularities?
• RQ2 What is the precise relation between Compton Form Factors (CFFs) and Generalized Polarizabilities (GPs) in the low-energy limit?
• RQ3 How do the CFFs derived from a minimal Dirac bilinear basis relate to the tensor parametrization introduced by Tarrach?
• RQ4 What are the explicit expressions for the Born-level Compton scattering amplitudes in terms of CFFs, valid for all kinematic regions?
• RQ5 How do the CFFs connect to Generalized Parton Distributions (GPDs) at large virtuality?

Key findings

• The proposed CFF parametrization is free from kinematical singularities and directly related to Generalized Parton Distributions (GPDs) at large photon virtuality.
• Exact equalities are established between the CFFs and Generalized Polarizabilities (GPs), providing a direct link between observables and nucleon structure.
• The Born-level amplitude for Compton scattering is explicitly expressed in terms of CFFs, with all helicity amplitudes derived and verified to be finite, including cancellation of spurious $1/\widetilde{K}$ singularities.
• The low-energy limit of the CFFs reproduces the electric $\alpha$ and magnetic $\beta$ polarizabilities of the nucleon, confirming consistency with known results.
• The tensor parametrization in terms of the minimal Dirac basis allows a complete and unambiguous decomposition of the Compton amplitude, enabling precise extraction of structure functions from experimental data.
• The framework provides a unified description valid from deeply virtual to quasi-real kinematics, with all expressions explicitly given in the nucleon's rest frame and applicable to polarized lepton-nucleon scattering experiments.