Theoretical constraints and systematic effects in the determination of the proton form factors

We calculate the two-photon exchange corrections to electron-proton scattering with nucleon and $\mathrm{\Delta }$ intermediate states. The results show a dependence on the elastic nucleon and nucleon-$\mathrm{\Delta }$-transition form factors used as input which leads to significant changes compared to previous calculations. We discuss the relevance of these corrections and apply them to the most recent and precise data set and world data from electron-proton scattering. Using this, we show how the form factor extraction from these data is influenced by the subsequent inclusion of physical constraints. The determination of the proton charge radius from scattering data is shown to be dominated by the enforcement of a realistic spectral function. Additionally, the third Zemach moment from the resulting form factors is calculated. The obtained radius and Zemach moment are shown to be consistent with Lamb shift measurements in muonic hydrogen.

Figure 1

Box graph, calculated here with different form factor parametrizations, crossed box implied.

Figure 2

Dependence of the TPE with nucleon intermediate state on the nucleon form factors at $Q^2=3{\mathrm{GeV}}^2$. The correction factor ${\delta }_{2\gamma ,N}$ is calculated once with dipole Sachs FFs and once with the simplified pole fit from Ref. [25]. Left panel: Difference of our calculation to the soft-photon approximation by Mo and Tsai [24]. Right panel: Difference of our calculation to the soft-photon approximation by Maximon and Tjon [23].

Figure 3

Individual contributions to TPE with $\mathrm{\Delta }$ intermediate state at $Q^2=3{\mathrm{GeV}}^2$.

Figure 4

Dependence of the TPE with $\mathrm{\Delta }$ intermediate state on the nucleon form factors at $Q^2=3{\mathrm{GeV}}^2$. Left panel: $N\mathrm{\Delta }\gamma$ vertex as given by Kondratyuk. Right panel: $N\mathrm{\Delta }\gamma$ vertex directly matched to helicity amplitudes from electroproduction of nucleon resonances.

Figure 5

The impact of TPE corrections on the electron-proton scattering cross sections with the highest quoted precision [2]. From the original data, we subtract the Feshbach approximation and add our calculations. We display the cross section divided by that one calculated by dipole Sachs FFs to make the deviations clearer.

Figure 6

The analytic structure of the FFs: shown is the continuation of $t=q^2$ into the complex plane, where the FFs are analytic functions except for the cut on the real axis $t>4M_{\pi }^2$. The physical FFs from scattering are defined on the negative real axis, those from creation/annihilation ($t>t_{\text{phys}}=4m_N^2$) just above the real axis as $F(t^{\prime }=t+i\epsilon )$ for infinitesimal $\epsilon$. This $\epsilon$-prescription also holds in the region $4M_{\pi }^2<t<4m_N^2$.

Figure 7

Left panel: The spectral function generated by the constrained $z$-expansion fit, as given in Table 3. Right panel: The spectral function expected from unitarity as fulfilled in Eq. (26).

Figure 8

The DR approach as given in Table 4.

Figure 9

Result from the DR approach as given in Table 4: The prediction for the form factor ratio, compared to polarization measurements [59, 60] that are not included in the fit here.

Figure 10

The distributions of the deviations between a DR-fit and the individual MAMI data sets, that correspond to six different energy settings of the incoming electron beam and three different spectrometers.

Figure 11

Results from the DR approach as given in Table 4. Close-up of the cross sections with specified spectrometers A (red, squares), B (blue, circles) and C (green, triangles). Energies of the incoming electron beam given.