Abstract
We derive two new sum rules for the unpolarized doubly virtual Compton scattering process on a nucleon, which establish novel low-$Q^2$ relations involving the nucleon's generalized polarizabilities and moments of the nucleon's unpolarized structure functions $F_1(x,Q^2)$ and $F_2(x,Q^2)$. These relations facilitate the determination of some structure constants which can only be accessed in off-forward doubly virtual Compton scattering, not experimentally accessible at present. We perform an empirical determination for the proton and compare our results with a next-to-leading-order chiral perturbation theory prediction. We also show how these relations may be useful for a model-independent determination of the low-$Q^2$ subtraction function in the Compton amplitude, which enters the two-photon-exchange contribution to the Lamb shift of (muonic) hydrogen. An explicit calculation of the $\Delta(1232)$-resonance contribution to the muonic-hydrogen $2P-2S$ Lamb shift yields $-1 \pm 1$ $\mathsf\mu$eV, confirming the previously conjectured smallness of this effect.
Highlights
Besides the charge and magnetization distributions in a nucleon, accessed in the elastic lepton-nucleon scattering process, the low-energy nucleon structure is characterized by its polarizability distributions, which are accessed in Compton scattering (CS) processes with real and virtual photons; see Refs. [1,2,3,4,5] for some reviews.The CS process is the starting point for deriving sum rules for various electromagnetic structure quantities [6]
The Baldin sum rule for the sum of the dipole polarizabilities [7] and the Gerasimov-Drell-Hearn (GDH) sum rule for the anomalous magnetic moment [8,9] are derived by considering the real Compton scattering (RCS) process
Note that in the most general case one has to use the basis consisting of all 21 tensor amplitudes introduced in Ref. [24] in order to avoid kinematic constraints; as long as only the non-Born part of the VVCS amplitude is important, one can use the minimal decomposition of Eq (2); see Refs. [24,25]
Summary
Besides the charge and magnetization distributions in a nucleon, accessed in the elastic lepton-nucleon scattering process, the low-energy nucleon structure is characterized by its polarizability distributions, which are accessed in Compton scattering (CS) processes with real and virtual photons; see Refs. [1,2,3,4,5] for some reviews. The Baldin sum rule for the sum of the dipole polarizabilities [7] and the Gerasimov-Drell-Hearn (GDH) sum rule for the anomalous magnetic moment [8,9] are derived by considering the real Compton scattering (RCS) process These sum rules all relate a measured low-energy observable to an integral over a photoabsorption cross section on the nucleon and are model-independent. Further sum rules involving the spin structure functions were derived by Schwinger [11] Another important relation by Cottingham [12] connects the unpolarized VVCS with the electromagnetic correction to the proton-neutron mass difference. Which extend the GDH, Burkhard-Cottingham, and Schwinger family of sum rules These new sum rules allow us to connect the moments of the nucleon’s low-Q2 spin-dependent structure functions g1;2, respectively, as measured in inclusive electron scattering [17,18], to lowenergy electromagnetic structure quantities of the nucleon.
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