Two-photon Exchange Research Articles

Hyperfine splitting in muonium and positronium

Calculation of hard three-loop corrections of order [Formula: see text] to hyperfine splitting in muonium and positronium is reviewed. All these contributions are generated by the graphs with photon, electron and/or muon loop radiative insertions in the two-photon exchange diagrams. We calculate contributions of six gauge invariant sets of diagrams.

Weighted difference ofgfactors of light Li-like and H-like ions for an improved determination of the fine-structure constant

A weighted difference of the $g$-factors of the Li- and H-like ion of the same element is studied and optimized in order to maximize the cancellation of nuclear effects. To this end, a detailed theoretical investigation is performed for the finite nuclear size correction to the one-electron $g$-factor, the one- and two-photon exchange effects, and the QED effects. The coefficients of the $Z\alpha$ expansion of these corrections are determined, which allows us to set up the optimal definition of the weighted difference. It is demonstrated that, for moderately light elements, such weighted difference is nearly free from uncertainties associated with nuclear effects and can be utilized to extract the fine-structure constant from bound-electron $g$-factor experiments with an accuracy competitive with or better than its current literature value.

Design and performance of a lead fluoride detector as a luminosity monitor

Precise luminosity measurements for the OLYMPUS two-photon exchange experiment at DESY were performed by counting scattering events with alternating beams of electrons and positrons incident on atomic electrons in a gaseous hydrogen target. Final products of Møller, Bhabha, and pair annihilation interactions were observed using a pair of lead fluoride Cherenkov calorimeters with custom housings and electronics, adapted from a system used by the A4 parity violation experiment at MAMI. This paper describes the design, calibration, and operation of these detectors. An explanation of the Monte Carlo methods used to simulate the physical processes involved both at the scattering vertices and in the detector apparatus is also included.

Two-photon exchange correction to muon–proton elastic scattering at low momentum transfer

We evaluate the two-photon exchange (TPE) correction to the muon-proton elastic scattering at small momentum transfer. Besides the elastic (nucleon) intermediate state contribution, which is calculated exactly, we account for the inelastic intermediate states by expressing the TPE process approximately through the forward doubly virtual Compton scattering. The input in our evaluation is given by the unpolarized proton structure functions and by one subtraction function. For the latter, we provide an explicit evaluation based on a Regge fit of high-energy proton structure function data. It is found that, for the kinematics of the forthcoming muon-proton elastic scattering data of the MUSE experiment, the elastic TPE contribution dominates, and the size of the inelastic TPE contributions is within the anticipated error of the forthcoming data.

Two-photon exchange correction in elastic unpolarized electron-proton scattering at small momentum transfer

We evaluate the two-photon exchange (TPE) correction to the unpolarized elastic electron-proton scattering at small momentum transfer $ Q^2 $. We account for the inelastic intermediate states approximating the double virtual Compton scattering by the unpolarized forward virtual Compton scattering. The unpolarized proton structure functions are used as input for the numerical evaluation of the inelastic contribution. Our calculation reproduces the leading terms in the $ Q^2 $ expansion of the TPE correction and goes beyond this approximation by keeping the full $ Q^2 $ dependence of the proton structure functions. In the range of small momentum transfer, our result is in good agreement with the empirical TPE fit to existing data.

Hard three-loop corrections to hyperfine splitting

We consider hard three-loop nonlogarithmic corrections of order [Formula: see text] to hyperfine splitting in muonium and positronium. All these contributions are generated by the graphs with photon, electron and/or muon loop radiative insertions in the two-photon exchange diagrams. We calculate contributions of six gauge invariant sets of diagrams.

Two-photon exchange contribution to elastic electron–proton scattering: measurements at the VEPP-3 storage ring

A striking discrepancy between the results for the ratio of electric and magnetic form factors of the proton obtained by two approaches: the Rosenbluth extraction from the unpolarized cross section and the double-polarization measurements has led to a serious re-examination of the role played by two-photon exchange diagrams. The evaluation of such diagrams could not be done in model-independent way, which means an experimental verification is required. One of the direct ways to do this is to measure the ratio of positron–proton and electron–proton elastic scattering cross-sections for identical kinematics. Such a measurement has been performed at the VEPP-3 storage ring at Novosibirsk. There were two runs at beam energies of 1.6 and 1.0 GeV at several ranges of scattering angle. The measurements covered the transferred momentum Q2 up to 1.5 GeV2 and the virtual photon polarization parameter ϵ down to 0.2 . The VEPP-3 experimental approach is described and the results of the measurements are presented.

Measurement of the Target-Normal Single-Spin Asymmetry in Quasielastic Scattering from the ReactionHe3↑(e,e′ )

We report the first measurement of the target single-spin asymmetry, $A_y$, in quasi-elastic scattering from the inclusive reaction $^3$He$^{\uparrow}(e,e^\prime)$ on a $^3$He gas target polarized normal to the lepton scattering plane. Assuming time-reversal invariance, this asymmetry is strictly zero for one-photon exchange. A non-zero $A_y$ can arise from the interference between the one- and two-photon exchange processes which is sensitive to the details of the sub-structure of the nucleon. An experiment recently completed at Jefferson Lab yielded asymmetries with high statistical precision at $Q^{2}=$ 0.13, 0.46 and 0.97 GeV$^{2}$. These measurements demonstrate, for the first time, that the $^3$He asymmetry is clearly non-zero and negative with a statistical significance of (8-10)$\sigma$. Using measured proton-to-$^{3}$He cross-section ratios and the effective polarization approximation, neutron asymmetries of $-$(1-3)% were obtained. The neutron asymmetry at high $Q^2$ is related to moments of the Generalized Parton Distributions (GPDs). Our measured neutron asymmetry at $Q^2=0.97$ GeV$^2$ agrees well with a prediction based on two-photon exchange using a GPD model and thus provides a new, independent constraint on these distributions.

Two-photon exchange amplitude withπNintermediate states: Spin-1/2 and spin-3/2 channels

We calculate two-photon exchange (TPE) amplitudes for the elastic electron-proton scattering, and take into account intermediate hadronic states containing \pi N system with total angular momentum 1/2 or 3/2, which includes 8 different channels. This is the improvement of our previous calculation, where only the \pi N states with quantum numbers of \Delta(1232) resonance were included. The results show good consistency with recent experimental data. At high Q^2, newly-calculated contributions affect the correction to the measured proton form factor ratio \mu G_E/G_M. The total correction becomes somewhat smaller compared to our previous work, but is still significant and grows approximately linearly with Q^2. Comparing contributions of different channels, we found that larger contributions come from the channels with quantum numbers of lightest resonances.

$\Delta (1232)$ resonance contribution to two-photon exchange in electron-proton scattering revisited

We revisit the question of the contributions of two-photon exchange with $\Delta(1232)$ excitation to the electron-proton scattering in a hadronic model. Three improvements over the previous calculations are made, namely, correct vertex function for $\gamma N\rightarrow\Delta$, realistic $\gamma N\Delta$ form factors, and coupling constants. The discrepancy between the values of $R\equiv \mu_p G_E/G_M$ extracted from Rosenbluth technique and polarization transfer method can be reasonably accounted for if the data of Andivahis {\it et al.} \cite{Andivahis94} are analyzed. However, substantial discrepancy remains if the data of Qattan {\it et al.} \cite{Qattan05} are used. For the ratio $R^\pm$ between $e^\pm p$ scatterings, our predictions appear to be in satisfactory agreement with the preliminary data from VEPP-3. The agreement between our model predictions and the recent measurements on single spin asymmetry, transverse and longitudinal recoil proton polarizations ranges from good to poor.

S-matrix calculations of energy levels of sodiumlike ions

A recent $S$-matrix-based QED calculation of energy levels of the lithium isoelectronic sequence is extended to the general case of a valence electron outside an arbitrary filled core. Emphasis is placed on modifications of the lithiumlike formulas required because more than one core state is present, and an unusual feature of the two-photon exchange contribution involving autoionizing states is discussed. The method is illustrated with a calculation of the energy levels of sodiumlike ions, with results for $3{s}_{1/2}, 3{p}_{1/2}$, and $3{p}_{3/2}$ energies tabulated for the range $Z=30\char21{}100$. Comparison with experiment and other calculations is given, and prospects for extension of the method to ions with more complex electronic structure discussed.

Flavor decomposition of the nucleon electromagnetic form factors at lowQ2

Background: The spatial distribution of charge and magnetization within the proton is encoded in the elastic form factors. These have been precisely measured in elastic electron scattering, and the combination of proton and neutron form factors allows for the separation of the up- and down-quark contributions.Purpose: In this work, we extract the proton and neutron form factors from worldwide data with an emphasis on precise new data covering the low-momentum region, which is sensitive to the large-scale structure of the nucleon. From these, we separate the up- and down-quark contributions to the proton form factors.Method: We combine cross section and polarization measurements of elastic electron-proton scattering to separate the proton form factors and two-photon exchange (TPE) contributions. We combine the proton form factors with parametrization of the neutron form factor data and uncertainties to separate the up- and down-quark contributions to the proton's charge and magnetic form factors.Results: The extracted TPE corrections are compared to previous phenomenological extractions, TPE calculations, and direct measurements from the comparison of electron and positron scattering. The flavor-separated form factors are extracted and compared to models of the nucleon structure.Conclusions: With the inclusion of the precise new data, the extracted TPE contributions show a clear change of sign at low ${Q}^{2}$, which is necessary to explain the high-${Q}^{2}$ form factor discrepancy while being consistent with the known ${Q}^{2}\ensuremath{\rightarrow}0$ limit. We find that the new Mainz data yield a significantly different result for the proton magnetic form factor and its flavor-separated contributions. We also observe that the rms radius of both the up- and down-quark distributions are smaller than the rms charge radius of the proton.

Proton form factors and two-photon exchange in elastic electron-proton scattering

Proton electromagnetic form factors are among the most important sources of information about the internal structure of the proton. Two different methods for measuring these form factors, the method proposed by Rosenbluth and the polarization-transfer method, yield contradictory results. It is assumed that this contradiction can be removed upon taking into account the hard part of the contribution of two-photon exchange to the cross section for elastic electron-proton scattering. This contribution can measured experimentally via a precision comparison of the cross sections for the elastic scattering of positrons and electrons on protons. Such a measurement, performed at the VEPP-3 storage ring in Novosibirsk at the beam energies of 1.6 and 1.0 GeV for positron (electron) scattering angles in the ranges of θe = 15°–25° and 55°–75° in the first case and in the range of θe = 65°–105° in the second case is described in the present article. Preliminary results of this experiment and their comparison with theoretical predictions are described.

Recoil correction to the proton finite-size contribution to the Lamb shift in muonic hydrogen

The Lamb shift in muonic hydrogen was measured some time ago to a high accuracy. The theoretical prediction of this value is very sensitive to the proton-finite-size effects. The proton radius extracted from muonic hydrogen is in contradiction with the results extracted from elastic electron-proton scattering. That creates a certain problem for the interpretation of the results from the muonic hydrogen Lamb shift. For the latter we need also to take into account the two-photon-exchange contribution with the proton finite size involved. The only way to describe it relies on the data from the scattering, which may produce an internal inconsistency of theory. Recently the leading proton-finite-size contribution to the two-photon exchange was found within the external field approximation. The recoil part of the two-photon-exchange has not been considered. Here we revisit calculation of the external-field part and take the recoil correction to the finite-size effects into account.

Measurement of the Two-Photon Exchange Contribution to the Elastice±pScattering Cross Sections at the VEPP-3 Storage Ring

The ratio of the elastic e(+)p to e(-)p scattering cross sections has been measured precisely, allowing the determination of the two-photon exchange contribution to these processes. This neglected contribution is believed to be the cause of the discrepancy between the Rosenbluth and polarization transfer methods of measuring the proton electromagnetic form factors. The experiment was performed at the VEPP-3 storage ring at beam energies of 1.6 and 1.0 GeV and at lepton scattering angles between 15° and 105°. The data obtained show evidence of a significant two-photon exchange effect. The results are compared with several theoretical predictions.

Towards a resolution of the proton form factor problem: new electron and positron scattering data.

There is a significant discrepancy between the values of the proton electric form factor, G(E)(p), extracted using unpolarized and polarized electron scattering. Calculations predict that small two-photon exchange (TPE) contributions can significantly affect the extraction of G(E)(p) from the unpolarized electron-proton cross sections. We determined the TPE contribution by measuring the ratio of positron-proton to electron-proton elastic scattering cross sections using a simultaneous, tertiary electron-positron beam incident on a liquid hydrogen target and detecting the scattered particles in the Jefferson Lab CLAS detector. This novel technique allowed us to cover a wide range in virtual photon polarization (ϵ) and momentum transfer (Q(2)) simultaneously, as well as to cancel luminosity-related systematic errors. The cross section ratio increases with decreasing ϵ at Q(2)=1.45 GeV(2). This measurement is consistent with the size of the form factor discrepancy at Q(2)≈1.75 GeV(2) and with hadronic calculations including nucleon and Δ intermediate states, which have been shown to resolve the discrepancy up to 2-3 GeV(2).

Applications of neural networks in hadron physics

The Bayesian approach for the feed-forward neural networks is reviewed. Its potential for usage in hadron physics is discussed. As an example of the application, the study of the two-photon exchange effect is presented. We focus on the model comparison, the estimation of the systematic uncertainties due to the choice of the model and the over-fitting. As an illustration, the predictions of the cross sections ratio are given together with the estimate of the uncertainty due to the parametrization choice.

Subtracted dispersion relation formalism for the two-photon exchange correction to elastic electron-proton scattering: Comparison with data

We apply a subtracted dispersion relation formalism with the aim to improve predictions for the two-photon exchange corrections to elastic electron-proton scattering observables at finite momentum transfers. We study the formalism on the elastic contribution, and make a detailed comparison with existing data for unpolarized cross sections as well as polarization transfer observables.

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 $\mathrm{\ensuremath{\Delta}}$ intermediate states. The results show a dependence on the elastic nucleon and nucleon-$\mathrm{\ensuremath{\Delta}}$-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.

Model independent study of massive lepton elastic scattering on the proton, beyond the Born approximation

Model independent expressions for all polarization observables in μ+p→μ+p elastic scattering are obtained, taking into account the lepton mass and including the two-photon exchange contribution. The spin structure of the matrix element is parametrized in terms of six independent complex amplitudes, functions of two independent kinematical variables. General statements about the influence of the two-photon-exchange terms on the differential cross section and on polarization observables are given. Polarization effects have been investigated for the case of a longitudinally polarized lepton beam and polarized nucleon in the final state.