Improved radiative corrections to $(e,e^{\text{'}}p)$ experiments: Explicit treatment of kinematical corrections in multiphoton bremsstrahlung

Radiative processes lead to important corrections to $(e,e^{\text{'}}p)$ experiments. While radiative corrections can be calculated exactly in QED and to a good accuracy also including hadronic corrections, these corrections cannot be included into data analyses to arbitrary orders exactly. Nevertheless consideration of multiphoton bremsstrahlung above the low-energy cutoff is important for many $(e,e^{\text{'}}p)$ experiments. To date, higher-order bremsstrahlung effects concerning electron scattering experiments have been implemented approximately by employing the soft-photon approximation (SPA). In this paper we propose a novel approach to multiphoton emission which partially removes the SPA from $(e,e^{\text{'}}p)$ experiments. In this combined approach one hard photon is treated exactly; and additional (softer) bremsstrahlung photons are taken into account resorting to the soft-photon approximation. This partial removal of the soft-photon approximation is shown to be relevant for the missing-energy distribution for several kinematic settings at MAMI and TJNAF energies.

Figure 1

Feynman diagrams beyond the leading order. The Feynman diagrams ${\mathcal{M}}_{\mathrm{Born}}^{(1)}$ and ${\mathcal{M}}_{\mathrm{Born}}^{(2)}$ constitute the Born approximation. ${\mathcal{M}}_{\mathrm{TPE}}$ contains the two-photon exchange (TPE) diagrams. Together, the vacuum polarization, the vertex corrections, and the self-energy diagrams contained in ${\mathcal{M}}_{\mathrm{Born}}^{(2)}$, and the two diagrams from ${\mathcal{M}}_{\mathrm{TPE}}$ are referred to as internal radiative corrections. ${\mathcal{M}}_{\mathrm{brems}}$ contains the external radiative corrections.

Figure 2

The $1\gamma$ matrix element ${\mathcal{M}}_{1\gamma }={\mathcal{M}}_{\mathrm{ei}}+{\mathcal{M}}_{\mathrm{ef}}$ squared.

Figure 3

Missing energy distribution for multiphoton bremsstrahlung. The solid curve (1) (green) was obtained using the modified SPA weight (17). The dotted curve (2) (red) represents the combined approach using weight (16). The inset graph shows the deviation between the two curves in percent, normalized to the combined result. At $E_{\mathrm{m}}=100\hspace{0.5em}\mathrm{MeV}$ the modified SPA calculation overestimates the radiative tail by 3.9%, at $E_{\mathrm{m}}=500\hspace{0.5em}\mathrm{MeV}$ the deviation is 29%. The momentum transfer is $Q^2=2.0\hspace{0.5em}{\mathrm{GeV}}^2$.

Figure 4

Missing-energy distribution for multiphoton bremsstrahlung. The attribution of the curves is as above in Fig. 3. At $E_{\mathrm{m}}=100\hspace{0.5em}\mathrm{MeV}$ the modified SPA calculation overestimates the radiative tail by 3.0%, at $E_{\mathrm{m}}=500\hspace{0.5em}\mathrm{MeV}$ the deviation is 17%. The momentum transfer is $Q^2=8.0\hspace{0.5em}{\mathrm{GeV}}^2$.

Figure 5

Missing-energy distribution for multiphoton bremsstrahlung simulated with SIMC. The solid curve (1) (black) represents the standard SIMC modified SPA radiative corrections. The dotted curve (2) (red) shows the $E_{\mathrm{m}}$ distribution obtained by inserting our combined radiative correction approach into SIMC. The latter one has more strength in the radiative tail. The total yield differs by 2.3%. The momentum transfer is $Q^2=2.0\hspace{0.5em}{\mathrm{GeV}}^2$, as in Fig. 3.

Figure 6

Missing-energy distribution for multiphoton bremsstrahlung simulated with SIMC. The attribution of the curves is the same as in Fig. 5. The total yield differs by 1.4%. The momentum transfer is $Q^2=8.0\hspace{0.5em}{\mathrm{GeV}}^2$, as in Fig. 4.

Figure 7

Angular bremsstrahlung photon distribution. The solid curve (1) (green) represents the modified SPA, the dotted curve (2) (red) shows the combined approach. The SPA distribution deviates from the combined calculation especially in the vicinity of the incident electron, around $0^{°}$. The kinematic settings are as in Fig. 3.

Figure 8

Angular bremsstrahlung photon distribution. The attribution of the curves is as in Fig. 7. The SPA distribution deviates from the combined calculation especially in the vicinity of the incident electron, around $0^{°}$. The kinematic settings are as in Fig. 4.

Figure 9

Angular bremsstrahlung photon distribution generated with SIMC. The solid curve (1) (black) represents the standard SIMC modified SPA calculation. The dotted curve (2) (red) shows the angular distribution simulated with the combined approach. The largest difference between the standard SIMC treatment and the data occurs in the middle between the $e$ and $e^{\text{'}}$ directions and in the height of the two peaks. This is due to the peaking approximation which is used by the standard SIMC code; it overestimates the peaks. The comparison with experimental data can be found in Fig. 11. The kinematics are the same as in Figs. 3 and 7.

Figure 10

Angular bremsstrahlung photon distribution generated with SIMC. The attribution of the curves is the same as in Fig. 9 but there was no data available for these kinematic settings. They are the same as in Figs. 4 and 8.

Figure 11

Angular bremsstrahlung photon distribution for $\mathrm{H}(e,e^{\text{'}}p)$ scattering generated with SIMC and compared to data. The bump on the right hand side at the proton direction is an artefact due to protons punching through the spectrometers. The dash-dotted curve (1) (green) and the (2) (red) curve show standard SIMC photon distributions, both using different versions of the peaking approximation. The dotted curve (3) (black) represents the data. The solid line (4) (magenta) is the new combined approach implemented into SIMC. Additionally there is a blue line (5) from Ref. [10] showing a full angular SPA simulation. The kinematics are the same as in Figs. 3 and 7. The combined approach (4) fits the data well.

Figure 12

Photon angular distribution with (1) (solid curve, red) and without (2) (dotted curve, black) SPA proton bremsstrahlung. The simulation neglecting proton bremsstrahlung overestimates the electron peaks slightly. In the rest of the photon angle domain the two curves coincide. The momentum transfer $Q^2=2.0\hspace{0.5em}{\mathrm{GeV}}^2$.

Figure 13

Missing energy with (1) (solid curve, red) and without (2) (dotted curve, black) SPA proton bremsstrahlung. The two curves are hardly distinguishable. The momentum transfer $Q^2=2.0\hspace{0.5em}{\mathrm{GeV}}^2$.