Evaluation of the forward Compton scattering off protons: Spin-independent amplitude

We evaluate the forward Compton scattering off the proton, based on Kramers-Kronig kind of relations which express the Compton amplitudes in terms of integrals of total photoabsorption cross sections. We obtain two distinct fits to the world data on the unpolarized total photoabsorption cross section and evaluate the various spin-independent sum rules using these fits. For the sum of proton electric and magnetic dipole polarizabilities governed by the Baldin sum rule, we obtain the following average (between the two fits): ${\alpha }_{E1}+{\beta }_{M1}=14.0(2)\times 10^{-4}{\mathrm{fm}}^3$. An analogous sum rule involving the quadrupole polarizabilities of the proton is evaluated too. The spin-independent forward amplitude of proton Compton scattering is evaluated in a broad energy range. The results are compared with previous evaluations and the only experimental data point for this amplitude (at 2.2 GeV). We remark on sum rules for the elastic component of polarizabilities.

Figure 1

Fits of the experimental data for the total photoabsorption cross section on the proton. Fit I is obtained using maid [12] results below the $2\pi$ production and data from LEGS [13] and Armstrong et al. [6] above it. Fit II uses said [14] and the data of MacCormick et al. [15]. Both fits use Bartalini et al. [16] and the high-energy data [17, 18, 19] displayed in the inset.

Figure 2

Our evaluation of $\mathrm{Re}f$ based on the two fits of the photoabsorption cross section compared with previous evaluations [5, 6, 9]. The experimental data point is from Ref. [10].

Figure 3

Our evaluations of $f(\nu )$ compared to the $\chi \mathrm{PT}$ calculation of Ref. [22].

Figure 4

The fourth-order sum rule constraint for ${\alpha }_{E\nu }+\frac{1}{12}{\alpha }_{E2}$ and ${\beta }_{M\nu }+\frac{1}{12}{\beta }_{M2}$ combinations of polarizabilities compared to results from dispersion relation approaches (DR) [11, 24], baryon chiral perturbation theory ($\mathrm{B}\chi \mathrm{PT}$) [25], and heavy baryon chiral perturbation theory ($\mathrm{HB}\chi \mathrm{PT}$) [26]. The errors of the $\chi \mathrm{PT}$ derive from our crude estimate of the next-order corrections.

Figure 5

Tree-level CS diagrams.

Figure 6

One-loop graphs contributing to the forward CS. Diagrams obtained from these by crossing of the photon lines are included too.