Polarizability relations across real and virtual Compton scattering processes

We derive two relations involving spin polarizabilities of a spin-$1/2$ particle and consider their empirical implications for the proton. Using the empirical values of the proton anomalous magnetic moment, electric and magnetic charge radii, moments of the spin structure functions $g_1$, $g_2$, and of two spin polarizabilities, the present relations constrain the low-momentum behavior of generalized polarizabilities appearing in virtual Compton scattering. In the case of the proton, the dispersive model evaluations of the spin and generalized polarizabilities appear to be consistent with these relations. The ongoing measurements of different electromagnetic observables at the MAMI, Jefferson Lab, and $\mathrm{HI}\gamma \mathrm{S}$ facilities may be able to put these relations to a test, or use them to unravel the low-energy spin structure of the nucleon.

Figure 1

Spin polarizabilities ${\gamma }_{E1M2}$ (left panel) and ${\gamma }_{E1E1}$ (right panel) versus the slope of the VCS GPs for the proton. The (brown) band across the plot is the present sum rule constraint based on the empirical information for $I_1^{\prime }(0)$, $⟨r_2^2⟩$ and ${\delta }_{LT}$, cf. Table 1. The horizontal beige band is the result of polarized RCS experiments [21]. The horizontal light-green band is the empirical extraction based on heavy-baryon chiral perturbation theory (HBChPT) [24]. The horizontal purple band is the baryon chiral perturbation theory (BChPT) prediction [25]. The vertical and horizontal blue bands are the DR evaluations for both the RCS and VCS polarizabilities, cf. Table 1.