Compton Scattering and Generalized Polarizabilities

In recent years, real and virtual Compton scattering off the nucleon have attracted considerable interest from both the experimental and theoretical sides. Real Compton scattering gives access to the so-called electromagnetic polarizabilities containing the structure information beyond the global properties of the nucleon such as its charge, mass, and magnetic moment. These polarizabilities have an intuitive interpretation in terms of induced dipole moments and thus characterize the response of the constituents of the nucleon to a soft external stimulus. The virtual Compton scattering reaction e−p → e−pγ allows one to map out the local response to external fields and can be described in terms of generalized electromagnetic polarizabilities. A simple classical interpretation in terms of the induced electric and magnetic polarization densities is proposed. We will discuss experimental results for the polarizabilities of the proton and compare them with theoretical predictions. INTRODUCTION AND OVERVIEW Real Compton scattering (RCS), γ(q,ε(λ ))+ N(p,s) → γ(q′,ε ′(λ ′))+ N(p′,s′), has a long history of providing important theoretical and experimental tests for models of nucleon structure (see, e.g., Refs. [1, 2, 3] for an introduction). Based on the requirement of gauge invariance, Lorentz invariance, crossing symmetry, and the discrete symmetries, the famous low-energy theorem of Low [4] and Gell-Mann and Goldberger [5] uniquely specifies the terms in the low-energy scattering amplitude up to and including terms linear in the photon momentum. The coefficients of this expansion are expressed in terms of global properties of the nucleon: its mass, charge, and magnetic moment. In principle, any model respecting the symmetries entering the derivation of the LET should reproduce the constraints of the LET. It is only terms of second order which contain new information on the structure of the nucleon specific to Compton scattering. For a general target, these effects can be parameterized in terms of two constants, the electric and magnetic polarizabilities α and β , respectively [6]. The scattering amplitude may be parameterized in terms of six independent functions Ai depending on the photon energy ω and the scattering angle, T =~ε ′∗ ·~ε A1 +~ε ′∗ · q̂~ε · q̂A2 + i~σ ·~ε ×~ε A3 + · · · . (1) In the forward and backward directions only two functions, namely, A1 and A3, contribute. For example, the Taylor series expansion of A1, for the proton, is given by A1 = − e2 m +4π(α +β z)ω2 − e2 4m3 (1− z)ω2 +[ω4] , (2) where z = cos(θ). The leading-order term is given by the Thomson term and the forward and backward amplitudes are sensitive to the combinations α + β and α −β , respectively. The sum of the polarizabilities is constrained by the Baldin sum rule [7],