Two-photon Exchange Contribution Research Articles

Charge-asymmetric correlations in elastic lepton- and antilepton-proton scattering from real photon emission

Observation of charge asymmetry by comparing electron and positron, or muon and anti-muon, scattering on a hadronic target presently serves as an experimental tool to study two-photon exchange effects. In addition to two-photon exchange, real photon emission also contributes to the charge asymmetry. We present a theoretical formalism, explicit expressions, and a numerical analysis of hard photon emission for the charge asymmetry in lepton- and antilepton-proton scattering. Different kinematic conditions are considered, namely, either fixed transferred momentum squared or a fixed lepton scattering angle. The infrared divergence from real photon emission is treated by the Bardin-Shumeiko technique and canceled with the soft part of the two-photon exchange contribution extracted and calculated using Tsai approach. All final expressions are obtained beyond the ultrarelativistic approximation with respect to the lepton mass that allows to evaluate numerically of the considered effects not only for ultrarelativistic leptons (JLab) and but for moderately relativistic (MUSE) kinematics, too.

Two-photon-exchange effect in ep→enπ+ at small −t with the hadronic model and dispersion relation approach

In this paper, the two-photon-exchange (TPE) effect in $ep\ensuremath{\rightarrow}en{\ensuremath{\pi}}^{+}$ at small $\ensuremath{-}t$ is discussed. In the previous work, the TPE contribution with one $\ensuremath{\pi}$ intermediate state is estimated numerically within a hadronic model under the pion-dominance approximation. Here we extend the discussion to include one $\ensuremath{\rho}$ intermediate state. The TPE contribution can be described by one scalar function in the limit ${m}_{e}\ensuremath{\rightarrow}0$, the dispersion relation (DR) satisfied by this scalar function is analyzed. The analytic expressions for the imaginary parts of the TPE contributions from one $\ensuremath{\pi}$ or one $\ensuremath{\rho}$ intermediate state are given within the hadronic model. Combining these analytic expressions and the DR, the corresponding real parts of the TPE contributions can be estimated easily at any available region. This can help the further experimental analysis to include the TPE contributions in a convenient way. The numeric results show that the TPE correction with one $\ensuremath{\rho}$ intermediate state is much smaller than that with one $\ensuremath{\pi}$ intermediate state in the current energy region. These results suggest that the TPE contribution with an elastic state is the main TPE contribution in $ep\ensuremath{\rightarrow}en{\ensuremath{\pi}}^{+}$ at small $\ensuremath{-}t$.

Lattice QCD Calculation of the Two-Photon Exchange Contribution to the Muonic-Hydrogen Lamb Shift.

We develop a method for lattice QCD calculation of the two-photon exchange contribution to the muonic-hydrogen Lamb shift. To demonstrate its feasibility, we present the first lattice calculation with a gauge ensemble at m_{π}=142 MeV. By adopting the infinite-volume reconstruction method along with an optimized subtraction scheme, we obtain with statistical uncertainty ΔE_{TPE}=-28.9(4.9) μeV+93.72 μeV/fm^{2}·⟨r_{p}^{2}⟩, or ΔE_{TPE}=37.4(4.9) μeV, which is consistent with the previous theoretical results in a range of 20-50 μeV.

Form Factors and Two-Photon Exchange in High-Energy Elastic Electron-Proton Scattering.

We present new precision measurements of the elastic electron-proton scattering cross section for momentum transfer (Q^{2}) up to 15.75 (GeV/c)^{2}. Combined with existing data, these provide an improved extraction of the proton magnetic form factor at high Q^{2} and double the range over which a longitudinal or transverse separation of the cross section can be performed. The difference between our results and polarization data agrees with that observed at lower Q^{2} and attributed to hard two-photon exchange (TPE) effects, extending to 8 (GeV/c)^{2} the range of Q^{2} for which a discrepancy is established at >95% confidence. We use the discrepancy to quantify the size of TPE contributions needed to explain the cross section at high Q^{2}.

Contribution of hard photon emission to the charge asymmetry in elastic lepton- and antilepton-proton scattering

The influence of the hard photon emission on the charge asymmetry in the lepton- and antilepton-proton elastic scattering was evaluated for the first time beyond the ultrarelativistic limit, while retaining the lepton mass at all steps of the calculation. This contribution---responsible for the charge asymmetry---is induced by interference between real photon emission from the lepton and proton. During the calculation any excited states of the intermediated proton were not considered, allowing us to use the standard fermionic propagator for this particle. The infrared divergence extracted using Lorenz-invariant approach of Bardin-Shumeiko is canceled by the corresponding soft part of the two-photon exchange contribution. Numerical analysis was performed within kinematic conditions of Jefferson Lab measurements and MUSE experiment in PSI.

Two-photon-exchange effects in e+e−→π+π− at small s

In this work, the two-photon-exchange (TPE) effects in $e^+e^- \rightarrow \pi^+ \pi^-$ at small $\sqrt{s}$ are discussed within a hadronic model. In the limit $m_e\rightarrow 0$, the TPE contribution to the amplitude can be described by one scalar function $\overline{c}_{1}^{(2\gamma)}$. The ratio between this function and the corresponding contribution in one-photon exchange $c_{1}^{(1\gamma)}$ reflects all the information of the TPE corrections. The numerical results on this ratio are presented and an artificial function is used to fit the numerical results. The latter can be used conveniently in the further experimental data analysis. The numerical results show the asymmetry of the differential cross sections in $e^+e^- \rightarrow \pi^+ \pi^-$ is about $-4\%$ at $\sqrt{s}\sim 0.7$ GeV.

Two-photon-exchange effects in ep→enπ+ and their corrections to separated cross sections σL,T,LT,TT at small −t

In this work, the two-photon-exchange (TPE) effects in $ep\rightarrow en\pi^+$ at small $-t$ are discussed within a hadronic model. Under the pion dominance approximation the TPE contribution to the amplitude can be described by a scalar function in the limit $m_e\rightarrow 0$. The TPE contributions to the amplitude and the unpolarized differential cross section are both estimated when only the elastic intermediate state is considered. We find that the TPE corrections to the unpolarized differential cross section are about from $-4\%$ to $-20\%$ at $Q^2=1$--$ 1.6$ GeV$^2$. After considering the TPE corrections to the experimental data sets of unpolarized differential cross section, we analyze the TPE corrections to the separated cross sections $\sigma_{\textrm{L,T,LT,TT}}$. We find that the TPE corrections (at $Q^2=1$--$1.6$ GeV$^2$) to $\sigma_{\textrm{L}}$ are about from $-10\%$ to $-30\%$, to $\sigma_{\textrm{T}}$ are about $20\%$, and to $\sigma_{\textrm{LT,TT}}$ are much larger. By these analysis, we conclude that the TPE contributions in $ep\rightarrow en\pi^+$ at small $-t$ are important to extract the separated cross sections $\sigma_{\textrm{L,T,LT,TT}}$ and the electromagnetic magnetic form factor of $\pi^+$ in the experimental analysis.

Dispersive corrections in elastic electron-nucleus scattering: an investigation in the intermediate energy regime and their impact on the nuclear matter

Measurements of elastic electron scattering data within the past decade have highlighted two-photon exchange contributions as a necessary ingredient in theoretical calculations to precisely evaluate hydrogen elastic scattering cross sections. This correction can modify the cross section at the few percent level. In contrast, dispersive effects can cause significantly larger changes from the Born approximation. The purpose of this experiment is to extract the carbon-12 elastic cross section around the first diffraction minimum, where the Born term contributions to the cross section are small to maximize the sensitivity to dispersive effects. The analysis uses the LEDEX data from the high resolution Jefferson Lab Hall A spectrometers to extract the cross sections near the first diffraction minimum of 12C at beam energies of 362 MeV and 685 MeV. The results are in very good agreement with previous world data, although with less precision. The average deviation from a static nuclear charge distribution expected from linear and quadratic fits indicate a 30.6% contribution of dispersive effects to the cross section at 1 GeV. The magnitude of the dispersive effects near the first diffraction minimum of 12C has been confirmed to be large with a strong energy dependence and could account for a large fraction of the magnitude for the observed quenching of the longitudinal nuclear response. These effects could also be important for nuclei radii extracted from parity-violating asymmetries measured near a diffraction minimum.

Lepton-proton two-photon exchange in chiral perturbation theory

We use heavy baryon chiral perturbation theory to evaluate the two-photon exchange corrections to the low-energy elastic lepton-proton scattering at next-to-leading order accuracy, i.e., ${\mathcal O}(\alpha, M^{-1})$, including a non-zero lepton mass. We consider the elastic proton intermediate state in the two-photon exchange together in the soft photon approximation. The infrared singular contributions are projected out using dimensional regularization. The resulting infrared singularity-free two-photon exchange contribution is in good numerical agreement with existing predictions based on standard diagrammatic soft photon approximation evaluations.

Updated analysis of recent results on electron and positron elastic scattering on the proton

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.

Subtractions and the effective Salpeter term for the Lamb shift in muonic atoms with the nuclear spin I \u2260 1\u22152

While taking into account the nuclear-structure contributions to the Lamb shift, one has to make various subtractions for the two-photon exchange contributions. Such subtractions should be consistent with the structureless part of theory. We study here the subtractions for a two-body atomic systems which consist of a pointlike lepton (an electron or a muon) and a nucleus with spin 0, 1/2, and 1, and find the recoil contribution in order (Zα)5 due to the subtractions for I = 0, 1. (The related contribution to the energy levels for I = 1∕2 of order (Zα)5m is called the Salpeter term.)Graphical abstract

Two-Photon Exchange Correction in Elastic Lepton–Proton Scattering

We present the dispersion relation approach based on unitarity and analyticity to evaluate the two-photon exchange contribution to elastic electron-proton scattering. The leading elastic and first inelastic $\pi N$ intermediate state contributions are accounted for in the region of small momentum transfer $Q^2 < 1~\mathrm{GeV}^2$ based on the available data input. The novel methods of analytical continuation allow us to exploit the MAMI form factor data and the MAID parameterization for the pion electroproduction amplitudes as input in the calculation. The results are compared to the recent CLAS, VEPP-3 and OLYMPUS data as well as to the full two-photon exchange correction in the near-forward approximation, which is based on the Christy and Bosted unpolarized structure functions fit. Additionally, predictions are given for a forthcoming muon-proton scattering experiment.

Two-photon exchange correction to the hyperfine splitting in muonic hydrogen

We reevaluate the Zemach, recoil and polarizability corrections to the hyperfine splitting in muonic hydrogen expressing them through the low-energy proton structure constants and obtain the precise values of the Zemach radius and two-photon exchange (TPE) contribution. The uncertainty of TPE correction to S energy levels in muonic hydrogen of 105 ppm exceeds the ppm accuracy level of the forthcoming 1S hyperfine splitting measurements at PSI, J-PARC and RIKEN-RAL.

Measurement of two-photon exchange effect by comparing elastic e±p cross sections

[Background] The electromagnetic form factors of the proton measured by unpolarized and polarized electron scattering experiments show a significant disagreement that grows with the squared four momentum transfer ($Q^{2}$). Calculations have shown that the two measurements can be largely reconciled by accounting for the contributions of two-photon exchange (TPE). TPE effects are not typically included in the standard set of radiative corrections since theoretical calculations of the TPE effects are highly model dependent, and, until recently, no direct evidence of significant TPE effects has been observed. [Purpose] We measured the ratio of positron-proton to electron-proton elastic-scattering cross sections in order to determine the TPE contribution to elastic electron-proton scattering and thereby resolve the proton electric form factor discrepancy. [Methods] We produced a mixed simultaneous electron-positron beam in Jefferson Lab's Hall B by passing the 5.6 GeV primary electron beam through a radiator to produce a bremsstrahlung photon beam and then passing the photon beam through a convertor to produce electron/positron pairs. The mixed electron-positron (lepton) beam with useful energies from approximately 0.85 to 3.5 GeV then struck a 30-cm long liquid hydrogen (LH$_2$) target located within the CEBAF Large Acceptance Spectrometer (CLAS). By detecting both the scattered leptons and the recoiling protons we identified and reconstructed elastic scattering events and determined the incident lepton energy. A detailed description of the experiment is presented.

Nucleon spin-averaged forward virtual Compton tensor at large Q2

The nucleon spin-averaged forward virtual Compton tensor determines important physical quantities such as electromagnetically-induced mass differences of nucleons, and two-photon exchange contributions in hydrogen spectroscopy. It depends on two kinematic variables: $\nu$, the virtual photon energy in the proton rest frame, and $Q^2$, the photon's invariant four-momentum squared. Using the operator product expansion, we calculate the tensor's large-$Q^2$ behavior for $\nu=0$, including for the first time the full spin-2 contribution and correcting a previous result in the literature for the spin-0 contribution. Implications for the proton radius puzzle are discussed.

Model-independent determination of the two-photon exchange contribution to hyperfine splitting in muonic hydrogen

We obtain a model-independent prediction for the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen. We use the relation of the Wilson coefficients of the spin-dependent dimension-six four-fermion operator of NRQED applied to the electron-proton and to the muon-proton sectors. Their difference can be reliably computed using chiral perturbation theory, whereas the Wilson coefficient of the electron-proton sector can be determined from the hyperfine splitting in hydrogen. This allows us to give a precise model-independent determination of the Wilson coefficient for the muon-proton sector, and consequently of the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen, which reads delta {overline{E}}_{pmu, mathrm{H}mathrm{F}}^{mathrm{TPE}}(nS)=-frac{1}{n^3}1.161(20) meV. Together with the associated QED analysis, we obtain a prediction for the hyperfine splitting in muonic hydrogen that reads Epμ,HFth(1S) = 182.623(27) meV and Epμ,HFth(2S) = 22.8123(33) meV. The error is dominated by the two-photon exchange contribution.

Hard Two-Photon Contribution to Elastic Lepton-Proton Scattering Determined by the OLYMPUS Experiment

The OLYMPUS Collaboration reports on a precision measurement of the positron-proton to electron-proton elastic cross section ratio, R_{2γ}, a direct measure of the contribution of hard two-photon exchange to the elastic cross section. In the OLYMPUS measurement, 2.01GeV electron and positron beams were directed through a hydrogen gas target internal to the DORIS storage ring at DESY. A toroidal magnetic spectrometer instrumented with drift chambers and time-of-flight scintillators detected elastically scattered leptons in coincidence with recoiling protons over a scattering angle range of ≈20° to 80°. The relative luminosity between the two beam species was monitored using tracking telescopes of interleaved gas electron multiplier and multiwire proportional chamber detectors at 12°, as well as symmetric Møller or Bhabha calorimeters at 1.29°. A total integrated luminosity of 4.5 fb^{-1} was collected. In the extraction of R_{2γ}, radiative effects were taken into account using a MonteCarlo generator to simulate the convolutions of internal bremsstrahlung with experiment-specific conditions such as detector acceptance and reconstruction efficiency. The resulting values of R_{2γ}, presented here for a wide range of virtual photon polarization 0.456<ε<0.978, are smaller than some hadronic two-photon exchange calculations predict, but are in reasonable agreement with a subtracted dispersion model and a phenomenological fit to the form factor data.

Two-photon-exchange effects in the unpolarized μp scattering within a hadronic model

In this work, the two-photon-exchange (TPE) effects in the unpolarized $\mu p$ scattering are discussed within the hadronic model where the intermediate states $N,\Delta$ and $\sigma$ are considered. The contribution from the $N$ intermediate is close to the results given by Ref. \cite{Tomalak2014} at the small $Q$ and there is a sizeable difference when $Q>0.25$GeV (where $Q^2$ is the four momentum transfer). The contributions from the $\Delta$ and the $\sigma$ intermediate states are much smaller than that from the $N$ intermediate at the small $Q$. In the kinematic region with $k_i\subseteq [0.01,0.3]$ GeV and $Q \leq0.4$GeV (where $k_i$ is the three momentum of initial muon at Lab frame), a naive expression for the TPE contributions is given, which can be used directly for other analysis.

Effective field theories for muonic hydrogen

Experimental measurements of muonic hydrogen bound states have recently started to take place and provide a powerful setting in which to study the properties of QCD. We profit from the power of effective field theories (EFTs) to provide a theoretical framework in which to study muonic hydrogen in a model independent fashion. In particular, we compute expressions for the Lamb shift and the hyperfine splitting. These expressions include the leading logarithmic O (mμ α 6 ) terms, as well as the leading hadronic effects. Most remarkably, our analyses include the determination of the spin-dependent and spin-independent structure functions of the forward virtualphoton Compton tensor of the proton to O (p 3 ) in HBET and including the Delta particle. Using these results we obtain the leading hadronic contributions to the Wilson coeffcients of the lepton-proton four fermion operators in NRQED. The spin-independent coeffcient yields a pure prediction for the two-photon exchange contribution to the muonic hydrogen Lamb shift, which is the main source of uncertainty in our computation. The spindependent coeffcient yields the prediction of the hyperfine splitting. The use of EFTs crucially helps us organizing the computation, in such a way that we can clearly address the parametric accuracy of our result. Furthermore, we review in the context of NRQED all the contributions to the energy shift of O (mμ α 5 , as well as those that scale like mr α 6 × logarithms.

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.