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The electric (α π) and magnetic (β π) Compton polarizabilities of both the charged and the neutral pion are of fundamental interest in the low-energy sector of quantum chromodynamics (QCD). Pion polarizabilities affect the shape of the γ π → γ π Compton scattering angular distribution at back scattering angles and γ γ → π π absolute cross sections. Theory derivations are given for the γ π → γ π Compton scattering differential cross section, dispersion relations, and sum rules in terms of the polarizabilities. We review experimental charged and neutral polarizability studies and theoretical predictions. The π 0 polarizabilities were deduced from DESY Crystal Ball γ γ → π 0 π 0 data, but with large uncertainties. The charged pion polarizabilities were deduced most recently from (1) radiative pion Primakoff scattering π − Z → π − Z γ at CERN COMPASS, (2) two-photon pion pair production γ γ → π + π − at SLAC Mark II, and (3) radiative pion photoproduction γ p → γ π + n from the proton at MAMI in Mainz. A stringent test of chiral perturbation theory (ChPT) is possible based on comparisons of precision experimental charged pion polarizabilities with ChPT predictions. Only the CERN COMPASS charged pion polarizability measurement has acceptably small uncertainties. Its value α π ± − β π ± = (4. 0 ± 1. 8) × 1 0 − 4 fm 3 agrees well with the two-loop ChPT prediction α π ± − β π ± = (5. 7 ± 1. 0) × 1 0 − 4 fm 3 , strengthening the identification of the pion with the Goldstone boson of chiral symmetry breaking in QCD. [ABSTRACT FROM AUTHOR]

We review the recent progress of the theoretical understanding of the spin structure of the nucleon based on Heavy Baryon Chiral Perturbation Theory (HBChPT). At low Q2 the spin structure of the nucleon is encoded into the generalized spin polarizabilities (spin GP's) extracted from the virtual Compton scattering (VCS) amplitudes. The spin GP's of the nucleon have been calculated up to next-to-leading order in HBChPT. Furthermore, the forward spin generalized polarizabilities, which are related to the nucleon polarized structure functions g1 and g2 through the sum rules based on dispersion relations, also have been calculated in HBChPT up to next-to-leading order. As a summary, the physics content of the existing data is discussed and some perspectives for future theoretical and experimental activities in this field are also presented. [ABSTRACT FROM AUTHOR]