Generalized dipole polarizabilities and the spatial structure of hadrons

Abstract

We present a phenomenological discussion of spin-independent, generalized dipole polarizabilities of hadrons entering the virtual Compton scattering process ${\ensuremath{\gamma}}^{*}\stackrel{\ensuremath{\rightarrow}}{h}\ensuremath{\gamma}h.$ We introduce a new method of obtaining a tensor basis with appropriate Lorentz-invariant amplitudes which are free from kinematical singularities and constraints. The result is summarized in terms of a compact effective Lagrangian. We then motivate a gauge-invariant separation into a generalized Born term containing ground-state properties only and a residual contribution describing the model-dependent internal structure. The generalized dipole polarizabilities are defined in terms of Lorentz-invariant residual amplitudes. Particular emphasis is laid on a physical interpretation of these quantities as characterizing the spatial distributions of the induced electric polarization and magnetization of hadrons. It is argued that three dipole polarizabilities---namely, the longitudinal electric ${\ensuremath{\alpha}}_{L}{(q}^{2}),$ the transverse electric ${\ensuremath{\alpha}}_{T}{(q}^{2}),$ and the magnetic $\ensuremath{\beta}{(q}^{2})$ ones---are required in order to fully reconstruct local polarizations induced by soft external fields in a hadron. One of these polarizabilities, ${\ensuremath{\alpha}}_{T}{(q}^{2}),$ describes an effect of higher order in the soft final-photon momentum ${q}^{\ensuremath{'}}.$ We argue that the associated spatial distributions obtained via Fourier transforms in the Breit frame are meaningful even for such a light particle as the pion. The spatial distributions are determined at large distances $r\ensuremath{\sim}{1/m}_{\ensuremath{\pi}}$ for pions, kaons, and octet baryons by the use of chiral perturbation theory.