Sum rules across the unpolarized Compton processes involving generalized polarizabilities and moments of nucleon structure functions

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 $\mathrm{\Delta }(1232)$-resonance contribution to the muonic-hydrogen $2P-2S$ Lamb shift yields $-1\pm 1\mu \mathrm{eV}$, confirming the previously conjectured smallness of this effect.

Figure 1

Diagram of the Compton scattering process showing the 4-momenta of the particles.

Figure 2

$Q^2$ dependence of the proton structure moment $M_1^{(2)}$ according to the empirical BC fit (black solid curve) [34], in comparison with the $\pi N+\mathrm{\Delta }+\pi \mathrm{\Delta }$ BChPT calculation. For the latter, we also show the result when an additional form factor dependence is included in the $\mathrm{\Delta }$-exchange contribution as given by Eq. (56); blue dashed (magenta dashed-dotted) curves show the results with (without) the form factor. The blue band shows the uncertainty of the BChPT result with the form factor, estimated as in Ref. [26]. At the real photon point, the observable yields the Baldin sum-rule value for ${\alpha }_{E1}+{\beta }_{M1}$ [5]. The data point at $Q^2=0.3{\mathrm{GeV}}^2$ is from JLab/HallC [28].

Figure 3

$Q^2$ dependence of the proton structure moment $M_2^{(1)}$ according to the empirical BC fit (black solid curve) [34], in comparison with the $\pi N+\mathrm{\Delta }+\pi \mathrm{\Delta }$ BChPT calculation. For the latter, we also show the result when an additional form factor dependence is included in the $\mathrm{\Delta }$-exchange contribution as given by Eq. (56); blue dashed (magenta dashed-dotted) curves show the results with (without) the form factor. The blue band shows the uncertainty of the BChPT result with the form factor, estimated as in Ref. [26]. At the real photon point, the observable yields the Baldin sum-rule value for ${\alpha }_{E1}+{\beta }_{M1}$ [5].

Figure 4

The low-$Q^2$ behavior of the non-Born piece of the subtraction function. Shown are the HBChPT calculation [42] (dark yellow band), the BChPT calculation of this work (blue dashed and magenta dashed-dotted curves show the results with and without the form factor, respectively; the wider blue band shows the uncertainty of the BChPT result with the form factor, estimated in Ref. [26]), and the empirical superconvergence relation estimate of Ref. [40] (black solid curve). At the real photon point, the PDG 2016 value of ${\beta }_{M1}=(2.5\pm 0.4)\times {10}^{-4}{\mathrm{fm}}^3$ [41] is shown. Note that the HBChPT curve is shifted to reproduce that value, whereas Ref. [42] uses a larger value ${\beta }_{M1}=(3.15\pm 0.50)\times {10}^{-4}{\mathrm{fm}}^3$ found in the most recent HBChPT fit [43].

Figure 5

Two-photon-exchange diagram with intermediate $\mathrm{\Delta }(1232)$ excitation.