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bvv@jinr.ru

egle.tomasi@cea.fr

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We discuss recent experimental results concerning the cross section ratio of positron over electron elastic scattering on protons, and compare with the predictions of a pre-existent calculation. The deviation from unity of this ratio, i.e., a charge asymmetry different from zero, is the signature of contributions beyond the Born approximation. After reviewing the published results, we compare the elastic data to a calculation which includes the diagram corresponding to two-photon exchange. It turns out that all the data on the cross section ratio, in the limit of their precision, do not show evidence of enhanced two-photon contribution beyond the expected percent level. Our results confirm that experimental evidence for a large contribution of two-photon exchange is not yet found.

Elastic electron-proton scattering has been the subject of large experimental and theoretical efforts for many decades. Since the works that valued the Nobel Prize to Hofstadter in 1967, it is a privileged way to learn about the proton internal structure. Assuming that the interaction occurs through the exchange of one virtual photon with four-momentum

In recent years large experimental and theoretical work has been devoted to this subject due to the possibility of very precise measurements at large transferred momentum. The development of 100% duty cycle electron machines at Jefferson Laboratory (JLab), with highly polarized electron beams, the construction of large solid angle spectrometers, and detectors, the development of proton polarimetry in the GeV region made possible to apply the polarization method suggested by Akhiezer and Rekalo at the end of the 1960s

Earlier, the privileged method to extract FFs was based to the ‘Rosenbluth separation’

The data on the FFs ratio, collected mostly by the GEp collaboration at JLab (

It is fair to remind the reader that in the 1970s the presence of a possible

the charge

Model independent statements, derived from symmetry properties of the strong and electromagnetic interactions, give reliable predictions of the

FFs acquire an imaginary part, and one additional charge-odd amplitude, of the order or

instead of two FFs, functions only of

nonlinearities arise in the Rosenbluth fit, i.e., in the unpolarized (reduced) cross section versus

due to the charge-odd (C-odd) terms, a nonvanishing charge-asymmetry should be observed in

Summarizing, it can be stated from these general features that, in the presence of a sizable

Note that for the crossed channels (the annihilation channels

In Ref.

In the total contribution from hard

In this work we compile and discuss the results of three recent experiments that were especially built to detect a possible charge asymmetry through the measurement of the cross section ratio of electron and positron elastic scattering on the proton. This observable is sensitive to the real part of the

Assuming one photon exchange, the unpolarized elastic cross section

Note that the Born elastic cross section is intended to be the measured cross section,

Radiative correction factor as a function of

A deviation from unity of the ratio

A C-odd effect is enhanced in the ratio of

In Eq. (5) of Ref.

The charge asymmetry that includes soft and hard

By correcting the data for the contributions of the vertex-type corrections

The data on

The odd radiative correction term is usually splitted in the following parts:

Bremstrahlung process with emission of a real photon: this part of the contribution is strictly dependent on the experimental cuts over measured energy and angles of detected particles. This term is large, and contains the infrared singularities which cancels with one from virtual two photon contribution.

virtual two photon corrections, which are splitted in two parts due to the uncertainties of the calculation with respect to proton form factors and intermediate proton state contributions:

soft part of two photon virtual contribution, that includes the case when one of the virtual photons is soft. In this case the intermediate proton and electron are almost on mass shell, and one can treat this term as one photon exchange contribution, with some factor of additional soft virtual photon. This part of the contribution can be exactly calculated in QED, and contains infrared singularities which cancels with the real soft photon contribution;

hard part, where both virtual photons are hard. In this case one has to consider six proton form factors instead of two, where one of the protons is off-shell, and in addition, some intermediate proton states, as

The splitting of the two photon contribution into the soft and hard parts is not uniquely defined and may differ from one author to another. The answers differ by some finite expression, which depends on kinematical invariants, and can be explained by different methods of calculation. The generally adopted approach is the splitting that was considered in the works of two groups

Three results from recent experiments that measured the ratio

Radiatively corrected ratio of positron to electron cross sections

Weighted average of the ratio

In the analysis of the experimental data, the radiative correction codes are embedded in the Monte Carlo used to analyze the data, and it is not straightforward to unfold the effects from the acceptance and the efficiency of the setup. Note that the

It may be difficult to evaluate the size of the applied radiative corrections and their dependence on the relevant kinematical variables from the published results. However, it is always possible to calculate radiative corrections for different energy cuts to compare different models and study, at least qualitatively, their effect in comparison to the data. In particular we consider the calculations from Refs.

As radiative corrections applied to the data may differ from one paper to another by some finite expression (which depends on kinematical invariants), in order to be less sensitive to model corrections, we consider the total odd contribution from Ref.

We calculate the asymmetry

The experiment, published in Refs.

The measured (uncorrected) ratio

Ratio of cross sections

In order to compare separately the

This experiment, published in Ref.

Ratio of cross sections

Four options of radiative corrections were implemented, following Mo-Tsai at first order (solid red circles) (a) or including approximately high orders by exponentiation (solid red squares) (b), and following Maximon-Tjon

The statistical error is evaluated to be

Weighted average of the difference

The calculation is shown in Fig.

The total odd contribution from Ref.

The calculations after subtraction, for both choices of the radiative correction ansatz, fall within the errors of most data points.

The CLAS experiment

The data are plotted in Fig.

Ratio of cross sections

Also in this case the calculations after subtraction, for both choices of the radiative correction ansatz, fall within the errors of most data points.

The previous analysis wants to evidence the general

Point to point difference between the calculation from Eq.

Same as Fig.

The calculation of the hard

The difference point by point between data and theory shows in general a discrepancy at per-thousand level in most cases, what is beyond the theoretical and experimental precisions, with a slight

For the OLYMPUS data, the sensitivity to different ansatz of radiative corrections is shown in Table

Experimental data for the four OLYMPUS analysis. The weighted average of the ratio

Difference between the data and the calculation, for the four OLYMPUS analyses: weighted average of the ratio

This paper compares the calculation from Ref.

This work completes (and is consistent with) the analysis published in Ref.

We stress that the extraction of the ‘hard’ two photon contribution is somewhat ambiguous as it depends on the model used for the implemented radiative corrections, the main problem being the subtraction of the infrared divergent part. If this subtraction may be straightforward in the calculation, it definitely originates differences in the Monte Carlo implementation. Even if the same model for the radiative correction is used in the different experiments, which is not the case, the data are corrected with dedicated Monte Carlo, implemented for the specific experiment. The numerical approximations and cuts, that depend on the relevant kinematical variables, are handled differently by the different collaborations. Moreover radiative corrections are implemented together with acceptance and efficiency corrections, that are specific to the individual set-up, making impossible a quantitatively precise comparison.

Nevertheless, we would like to stress that

We minimized this effect by subtracting the applied radiative corrections and replacing with the calculation from Ref.

The procedure of subtracting two models of radiative corrections, in the same kinematical conditions, enhances the model-dependent difference. A similar procedure was validated in previous works, for example

The conclusions of the recent experimental papers are far from being definite statements. The common issue is that measurements at large

In the OLYMPUS paper it is clearly stated that “We do not agree with the conclusions of earlier papers

The VEPP publication concludes “on a significant two-photon exchange effect”, nuanced by a discussion on the used normalization and by the statement that the data are “in moderate agreement with several TPE predictions explaining the form factor discrepancy at high

In the CLAS publications one finds the following statement in the abstract: “Our results .. demonstrate a nonzero contribution from TPE effects and are in excellent agreement with the calculations that include TPE effects and largely reconcile the form-factor discrepancy up to

We do not enter here in the comparison and the virtues of the model dependent

We confirm the conclusions of that paper of no evident enhancement of the

We acknowledge V. Fadin and D. Nikolenko for interest in this work and useful discussions. Thanks are due to B. Raue for clarifying issues concerning the JLab CLAS Collaboration results and providing data in a tabulated form.