1 Introduction

Elastic lepton-proton scattering is a fundamental process that allows us to study the structure of the proton. It is described theoretically in the Standard Model by a perturbative expansion in \(\alpha \approx \frac{1}{137}\) with terms beyond leading order commonly called radiative corrections. Calculating such radiative corrections have been extensively described in the paper by Mo and Tsai [1], which also stressed the importance of electron-proton and positron-proton elastic scattering experiments, and subsequent work by Maximon and Tjon [2], and others.

In the Born or single photon exchange approximation the elastic \(e^\pm p\) scattering cross section is given by the reduced Rosenbluth cross section,

$$\begin{aligned} \frac{d\sigma _{e^\pm p}}{d\Omega }={\frac{d\sigma }{d\Omega }}_{Mott} \frac{\tau {G_M^p}^2 +\epsilon {G_E^p}^2}{\epsilon (1+\tau )}, \end{aligned}$$
(1)

where \(\tau =\frac{Q^2}{4 M_p^2}\) and \(\epsilon =(1 +2(1+\tau ) \tan ^2{\frac{\theta _l}{2}} )^{-1}\).

Measurements using the unpolarized Rosenbluth separation technique yielded values for \(G_{E}^{p}\) and \(G_{M}^{p}\). Their ratio, \(\mu ^{p}\,G_{E}^{p}/G_{M}^{p}\), was found to be close to unity over a broad range in \(Q^{2}\) (shown by the blue data points in Fig. 1) leading to the proton form factors being envisaged as very similar and often modeled by the same dipole form factor.

Fig. 1
figure 1

Proton form factor ratio measured using unpolarized [3,4,5,6,7,8,2.5 Beamdump/Faraday cup

A new extracted beam facility from DESY II will need a beamdump. Figure 13 shows the conceptual design of the beamdump for TPEX experiment. Assuming a maximum current of 100 nA and a beam energy of 7 GeV the maximum power to be handled is 700 W. To contain the showering. the beamdump used to have order of 5 Molière radii laterally and order of 25 radiation lengths longitudinally.

Fig. 13
figure 13

Schematic view of a possible beamdump/Faraday cup for TPEX

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To augment the luminosity measurement proposed in Sect. 2.4 it is considered to modify the beamdump to also function as a Faraday cup to integrate the charge that passes through the target. Then, assuming the length of the target cell and density of liquid hydrogen are known, a measure of the luminosity can be obtained online. As shown in Fig. 13, an insulated ring held at negative voltage of a few hundred volts is needed to suppress secondary emission from back scattering out of the Faraday cup.

3 Plans and expectations

We propose to commission the experiment using 2 GeV electrons. We do this to commission the electronics, detectors, and data acquisition system taking advantage of the relatively high cross section at 2 GeV. About 2 weeks of beam time is required for this commissioning after the experiment was installed and surveyed. We would also like a brief run (few days) with positrons to verify that the beam alignment and performance do not change with positron running. The commissioning run (including a few days with positrons) would also allow a crosscheck of the OLYMPUS data at \(30^\circ \), \(50^\circ \), and \(70^\circ \) and give a modest extension in \(Q^2\) up to 2.7 (GeV/c)\(^2\).

Table 1 shows \(Q^{2}\), \(\epsilon \), differential cross section, and event rate expected for one day of running for the proposed left/right symmetric configuration with 2 GeV lepton beams averaging 40 nA on a 20 cm liquid hydrogen target and using just the central \(3\times 3\) array of crystals to calculate the acceptance area.

The main TPEX run would be made at 3.0 GeV and would require approximately 6 weeks (2 weeks with electrons and 4 weeks with positrons in total) to collect the required data. Table 2 shows \(Q^{2}\), \(\epsilon \), differential cross section, and event rate expected for one day of running for the proposed configuration with 3 GeV lepton beams. This would extend the measurements to \(Q^2=4.57\) (GeV/c)\(^2\) where the form factor ratio discrepancy is large. The 6 weeks could be divided into shorter periods if that fit better with the DESY synchrotron schedule though longer, uninterrupted runs would be preferable. To minimize systematics we would like to switch between positron and electron running as frequently as possible (e.g. 1 day positron, 1 day electron, and 1 day positron repeating).

Table 1 Kinematics, cross section, and events expected in one day for an incident lepton beam of 2 GeV and 40 nA averaged current on a 20 cm liquid hydrogen target
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Table 2 Kinematics, cross section, and events expected in one day for an incident lepton beam of 3 GeV and 40 nA averaged current on a 20 cm liquid hydrogen target and 3.6 msr acceptance and a left/right symmetric detector configuration
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The \(Q^2\) range that the proposed TPEX experiment would be capable of reaching is shown in Fig. 14 for the 2 and 3 GeV runs of this proposal. The reach with TPEX can be seen in relation to the discrepancy in the form factor ratio. With additional crystals at back angles the 4 GeV runs would also be possible in a reasonable time frame.

Fig. 14
figure 14

Proton form factor ratio as before but also showing the \(Q^2\) range accessible with the proposed TPEX configuration at 2 and 3 GeV. The 4 GeV range would be possible with additional crystals

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Fig. 15
figure 15

Charge-averaged cross section divided by the dipole cross section from OLYMPUS and expected uncertainties and coverage from TPEX at 2 and 3 GeV

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The TPEX experiment at DESY would also measure the charge-averaged cross section just like the recent result from OLYMPUS (see Fig. 6). As mentioned above this cross section is insensitive to charge-odd radiative corrections including “hard” two-photon exchange terms. Thus, it provides a more robust measure of the proton form factors. The expected charge-averaged cross section uncertainties (assuming dipole cross section) are shown in Fig. 15 for TPEX assuming 6 days of running at 2 GeV and 6 weeks of running at 3 GeV with only 50% data collection efficiency. The recent OLYMPUS results are also shown.

4 Conclusion

The observed discrepancy in the proton form factor ratio presents a fundamental challenge in nuclear physics and quantum electrodynamics (QED). Despite the inclusion of leading order QED radiative corrections, these corrections alone have proven insufficient to resolve the discrepancy. This suggests that higher order corrections might be necessary to achieve a comprehensive understanding of the phenomenon. Additionally, it is plausible that more detailed models for the intermediate hadronic state could be required to accurately account for the observed deviation. Furthermore, it is crucial to consider the possibility of alternative processes that may contribute to the observed discrepancy.

To address this issue and gain further insights into the proton form factors at higher momentum transfers, the establishment of an extracted positron and electron beam facility at Deutsches Elektronen-Synchrotron (DESY) would offer a unique opportunity. Such a facility would enable the measurement of the two-photon exchange contribution to elastic lepton-proton scattering across a kinematic range where the evident discrepancy is prominent. The proposed TPEX experiment outlines an initial plan for an experimental configuration that could help resolve this issue and provide insight to the radiative corrections needed to understand the proton form factors at higher momentum transfers.