Two-photon Exchange Amplitude Research Articles

Calculating the two-photon exchange contribution to KL→μ+μ− decay

We present a theoretical framework within which both the real and imaginary parts of the complex, two-photon exchange amplitude contributing to KL→μ+μ− decay can be calculated using lattice quantum chromodynamics. The real part of this two-photon amplitude is of approximately the same size as that coming from a second-order weak strangeness-changing neutral-current process. Thus a test of the standard model prediction for this second-order weak process depends on an accurate result of this two-photon amplitude. A limiting factor of our proposed method comes from low-energy three-particle ππγ states. The contribution from these states will be significantly distorted by the finite volume of our calculation—a distortion for which there is no available correction. However, a simple estimate of the contribution of these three-particle states suggests their contribution to be at most a few percent allowing their neglect in a lattice calculation with a 10% target accuracy. Published by the American Physical Society 2024

Target normal single-spin asymmetry in inclusive electron-nucleon scattering in the 1/Nc expansion

The target normal single-spin asymmetry in inclusive electron-nucleon scattering is studied in the low-energy regime that includes the $\mathrm{\ensuremath{\Delta}}$ resonance. The particular interest in the asymmetry resides in that it is driven by two-photon exchange effects. It probes the spin-dependent absorptive part of the two-photon exchange amplitude, which is free of infrared and collinear singularities and represents the most pristine expression of two-photon exchange dynamics. The study presented here uses the $1/{N}_{c}$ expansion of QCD, which combines the $N$ and $\mathrm{\ensuremath{\Delta}}$ through the emergent $SU(4)$ spin-flavor symmetry in the baryon sector and allows for a systematic construction of the transition electromagnetic (EM) currents. The analysis includes the first subleading corrections in the $1/{N}_{c}$ expansion and presents results for elastic and inelastic final states. The asymmetry is found to be in the range ${10}^{\ensuremath{-}3}\ensuremath{-}{10}^{\ensuremath{-}2}$. The $\mathrm{\ensuremath{\Delta}}$ resonance plays an important role as an intermediate state in the elastic asymmetry and as a final state in the inclusive asymmetry.

Target-normal single spin asymmetries measured with positrons

Two-photon exchange and the larger class of hadronic box diagrams are difficult to calculate without a large degree of model-dependence. At the same time, these processes are significant radiative corrections in parity-violating electron scattering, in neutron decay, and may even be responsible for the proton’s form factor ratio discrepancy. New kinds of experimental data are needed to help constrain models and guide future box-diagram calculations. The target-normal single spin asymmetry, $$A_n$$ , formed with an unpolarized beam scattering from a target that is polarized normal to the scattering plane, is sensitive to the imaginary part of the two-photon exchange amplitude, and can provide a valuable constraint. A measurement with both electrons and positrons can reduce sources of experimental error, and distinguish between the effects of two-photon exchange and those of time-reversal symmetry violation. This article describes a proposed experiment in Hall A, using the new Super Big-Bite Spectrometer that can cover a momentum transfer range in the critical zone of uncertainty between where hadronic calculations and those based on partonic degrees of freedom are expected to be accurate.

Exact relations for two-photon-exchange effect in elastic ep scattering by dispersion relation and hadronic model * *Supported in part by the National Natural Science Foundation of China (11375044,11975075)

The two-photon-exchange (TPE) effect plays a key role in extracting the form factors (FFs) of the proton. In this work, we discuss several exact properties of the TPE effect in the elastic scattering. By taking four low energy interactions as examples, we analyze kinematical singularities, asymptotic behaviors, and branch cuts of TPE amplitudes. The analytical expressions clearly indicate several exact relations between the dispersion relation (DR) and hadronic model (HM) methods. This suggests that the two methods must be modified to general forms, while novel forms yield the same results. After the modification, new DRs include a non-trivial term with two singularities. Furthermore, new DRs automatically include contributions due to the seagull interaction, meson-exchange effect, contact interactions, and off-shell effect. To analyze the elastic scattering data sets, the new forms must be used.

Two-Photon Exchange in Elastic Electron Scattering on Hadronic Systems

In the present review, we discuss different aspects of the two-photon exchange (TPE) physics in elastic ep scattering at high Q2, as well as at low Q2. The imaginary part of the TPE amplitude gives rise to beam and target single-spin asymmetries. Different theoretical approaches to the calculation of these observables are considered. The real part of the TPE amplitude influences the unpolarized cross-section and double-spin observables and is, most likely, responsible for the discrepancy between two methods of measurements of the proton form factors. We review different methods of calculations of the TPE amplitudes in the framework of the “hadron” and “quark-gluon” approaches. We discuss the dispersion approach suitable for low and intermediate Q2, which includes elastic and inelastic intermediate hadronic states, as well as the connection of TPE with the proton radius puzzle. The present situation with direct experimental searches for the TPE amplitude in the e+p/e−p charge asymmetry is also discussed, as well as attempts to extract the TPE amplitudes from existing experimental data obtained by the Rosenbluth and double polarization techniques. The TPE physics in other processes such as elastic мp, e-nucleus, and еп scattering is also reviewed.

NUCLEAR STRUCTURE CORRECTIONS IN THE HYPERFINE STRUCTURE OF P-STATES IN LIGHT MUONIC IONS

The effects of the nuclear structure from two-photon exchange amplitudes in the hyperfine structure of P-states in muonic ions of lithium, beryllium, and boron are calculated. The operator of particle interaction in momentum space is constructed using the method of projection operators on states with certain total momenta of the atom F and muon J. Corrections to the nucleus structure are expressed in terms of electromagnetic form factors of nuclei and nuclear charge radii.

Nuclear Structure Effects in the Hyperfine Structure of Muonic Lithium, Beryllium, and Boron

Nuclear structure corrections to hyperfine splitting of energy levels in muonic lithium, beryllium, and boron ions are calculated. Account is taken of the contributions from the two-photon exchange amplitudes, including the vacuum polarization effect, and from the one-photon amplitude, which provide corrections to the nuclear structure and recoil of order $${{\alpha }^{5}}$$ and $${{\alpha }^{6}}$$ . Numerical results for different parametrizations of electromagnetic form factors are compared.

Target-normal single-spin asymmetry in elastic electron-nucleon scattering

We estimate the target-normal single-spin asymmetry at nearly forward angles in elastic electron-nucleon scattering. In the leading-order approximation, this asymmetry is proportional to the imaginary part of the two-photon exchange (TPE) amplitude, which can be expressed as an integral over the doubly virtual Compton scattering (VVCS) tensor. We develop a model that parametrizes the VVCS tensor for the case of nearly forward scattering angles. Our parametrization ensures a proper normalization of the imaginary part of the TPE amplitude on the well-known forward limit expression, which is given in terms of nucleon structure functions measurable in inelastic electron-nucleon scattering experiments. We discuss applicability limits of our theory and provide target-normal single-spin asymmetry predictions for both elastic electron-proton and electron-neutron scattering.

Corrections of two-photon interactions in the fine and hyperfine structure of the P-energy levels of muonic hydrogen

In the framework of the quasipotential method in quantum electrodynamics we calculate corrections to the nuclear structure proportional to $r_N^2$ from two-photon exchange amplitudes in the fine and hyperfine structure of P-states in muonic hydrogen, as well as the photon-photon interaction amplitudes, leading to the exchange of the axial vector meson. In constructing the quasipotential of the muon-nucleus interaction, we use the method of projection operators on states of two particles with a definite spin and total angular momentum. Analytical calculation of the matrix elements is performed and contributions to the fine and hyperfine structure of the $2P_{1/2}$ and $2P_{3/2}$ levels are obtained.

First Measurement of the Q^{2} Dependence of the Beam-Normal Single Spin Asymmetry for Elastic Scattering off Carbon.

We report on the first Q^{2}-dependent measurement of the beam-normal single spin asymmetry A_{n} in the elastic scattering of 570MeV vertically polarized electrons off ^{12}C. We cover the Q^{2} range between 0.02 and 0.05 GeV^{2}/c^{2} and determine A_{n} at four different Q^{2} values. The experimental results are compared to a theoretical calculation that relates A_{n} to the imaginary part of the two-photon exchange amplitude. The result emphasizes that the Q^{2} behavior of A_{n} given by the ratio of the Compton to charge form factors cannot be treated independently of the target nucleus.

Improved extraction of two-photon exchange amplitudes from elastic electron-proton scattering cross section data up to Q2=5.20(GeV/c)2

In this work, we improve on and extend to low- and high-${Q}^{2}$ values the extractions of the two-photon-exchange (TPE) amplitudes and the ratio ${P}_{l}/{P}_{l}^{\mathrm{Born}}(\ensuremath{\varepsilon},{Q}^{2})$ using world data on electron-proton elastic scattering cross section ${\ensuremath{\sigma}}_{R}(\ensuremath{\varepsilon},{Q}^{2})$ with an emphasis on data covering the high-momentum region, up to ${Q}^{2}=\phantom{\rule{4pt}{0ex}}5.20\phantom{\rule{0.16em}{0ex}}{(\mathrm{GeV}/\mathrm{c})}^{2}$, to better constrain the TPE amplitudes in this region. We provide a new parametrization of the TPE amplitudes, along with an estimate of the fit uncertainties. We compare the results to several previous phenomenological extractions and hadronic TPE predictions. We use the new parametrization of the TPE amplitudes to extract the ratio ${P}_{l}/{P}_{l}^{\mathrm{Born}}(\ensuremath{\varepsilon},{Q}^{2})$, and then compare the results to previous extractions, several theoretical calculations, and direct measurements at ${Q}^{2}=\phantom{\rule{4pt}{0ex}}2.50\phantom{\rule{0.16em}{0ex}}{(\mathrm{GeV}/\mathrm{c})}^{2}$.

Beam normal spin asymmetry for the ep→eΔ(1232) process

We calculate the single spin asymmetry for the $e p \to e \Delta(1232)$ process, for an electron beam polarized normal to the scattering plane. Such single spin asymmetries vanish in the one-photon exchange approximation, and are directly proportional to the absorptive part of a two-photon exchange amplitude. As the intermediate state in such two-photon exchange process is on its mass shell, the asymmetry allows one to access for the first time the on-shell $\Delta \to \Delta$ as well as $N^\ast \to \Delta$ electromagnetic transitions. We present the general formalism to describe the $e p \to e \Delta$ beam normal spin asymmetry, and provide a numerical estimate of its value using the nucleon, $\Delta(1232)$, $S_{11}(1535)$, and $D_{13}(1520)$ intermediate states. We compare our results with the first data from the Qweak@JLab experiment and give predictions for the A4@MAMI experiment.

Two-photon exchange contribution to elastic e− -proton scattering: Full dispersive treatment of πN states and comparison with data

We evaluate the two-photon exchange correction to the elastic electron-proton scattering cross section within a dispersive framework. Besides the elastic contribution, we account for all $\pi N$ intermediate state contributions using the phenomenological MAID fit as an input. We develop a novel method for the analytical continuation of the two-photon exchange amplitudes into the unphysical region and generalize our previous work to the momentum transfer region $ 0.064~\mathrm{GeV}^2 \lesssim Q^2 \lesssim 1~\mathrm{GeV}^2$. We compare our results with recent OLYMPUS, CLAS and VEPP-3 data as well as with empirical fits and estimates in the forward angular region.

Forward two-photon exchange in elastic lepton\u2013proton scattering and hyperfine-splitting correction

We relate the forward two-photon exchange (TPE) amplitudes to integrals of the inclusive lepton–proton scattering cross sections. These relations yield an alternative way for the evaluation of the TPE correction to hyperfine-splitting (HFS) in the hydrogen-like atoms with an equivalent to the standard approach (Iddings, Drell and Sullivan) result implying the Burkhardt–Cottingham sum rule. For evaluation of the individual effects (e.g., elastic contribution) our approach yields a distinct result. We compare both methods numerically on examples of the elastic contribution and the full TPE correction to HFS in electronic and muonic hydrogen.

Phenomenological extraction of two-photon exchange amplitudes from elastic electron-proton scattering cross section data

In this work, I improve on and extend to low and high $Q^2$ values the extractions of the $\varepsilon$ dependence of the real parts of the two-photon exchange (TPE) amplitudes relative to the magnetic form factor, as well as the ratio $P_l/P_l^{Born} (\varepsilon,Q^2)$ by using world data on $\sigma_R(\varepsilon,Q^2)$ with an emphasis on precise new data covering the low-momentum region which is sensitive to the large-scale structure of the nucleon. I provide simple parametrizations of the TPE amplitudes, along with an estimate of the fit uncertainties. The extracted TPE amplitudes are compared with previous phenomenological extractions and TPE calculations. The $P_l/P_l^{Born}$ ratio is extracted by using the new parametrizations of the TPE amplitudes and compared to previous extractions, TPE calculations, and direct measurements at $Q^2$ = 2.50 (GeV/c)$^2$. The extracted TPE amplitudes are on the few-percentage-points level and behave roughly linearly with increasing $Q^2$ where they become nonlinear at high $Q^2$. Contrary to $Y_M$, which is influenced mainly by elastic contributions, I find $Y_E$ to be influenced by inelastic contributions at large $Q^2$ values. While $Y_E$ and $Y_3$ differ in magnitude, they have opposite sign and tend to partially cancel each other. This suggests that the TPE correction to $\sigma_R(\varepsilon,Q^2)$ is driven mainly by $Y_M$ and to a lesser extent by $Y_3$ in agreement with previous phenomenological extractions and hadronic TPE calculations.

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.

Forward sum rule for the2γ-exchange correction to the charge-radius extraction from elastic electron scattering

Two-photon exchange (TPE) contributions to elastic electron-proton scattering in the forward regime and in leading log $\sim t\ln|t|$ approximation in the momentum transfer $|t|$ are considered. The imaginary part of TPE amplitude in these kinematics is related to the total photo absorption cross section. The real part of the TPE amplitude is obtained from the imaginary part by means of a fixed-$t$ dispersion relation. The dispersion integrals for the relevant elastic $ep$-scattering amplitude converge and do not need subtraction. This allows us to make clean a prediction for the real part of the TPE amplitude at forward angles with the leading term $\sim t\ln|t|$. Numerical estimates are comparable to the experimental precision in extracting the charge radius from experimental data.

Two-photon exchange in electron-trinucleon elastic scattering

We discuss two-photon exchange (TPE) in elastic electron scattering off the trinucleon systems, 3He and 3H. The calculations are done in the semirelativistic approximation with the trinucleon wave functions obtained with the Paris and CD-Bonn nucleon-nucleon potentials. An applicability area of the model is wide enough and includes the main part of kinematical domain where experimental data exist. All three TPE amplitudes (generalized form factors) for electron 3He elastic scattering are calculated. We find that the TPE amplitudes are few times more significant in the scattring of electrons off 3He then in the electron proton scattering.

Two-photon exchange corrections to elastic electron-proton scattering at large momentum transfer within the SCET approach

We calculate the two-photon exchange (TPE) corrections in the region where the kinematical variables describing the elastic $ep$ scattering are moderately large momentum scales relative to the soft hadronic scale. For such kinematics we use the QCD factorization approach formulated in the framework of the soft-collinear effective theory (SCET). Such technique allows us to develop a description for the soft-spectator scattering contribution which is found to be important in the region of moderately large scales. Together with the hard-spectator contribution we present the complete factorization formulas for the TPE amplitudes at the leading power and leading logarithmic accuracy. The momentum region where both photons are hard is described by only one new nonperturbative SCET form factor. It turns out that the same form factor also arises for wide-angle Compton scattering which is also described in the framework of the SCET approach. This allows us to estimate the soft-spectator contribution associated with the hard photons in a model independent way. The main unknown in our description of the TPE contribution is related with the configuration where one photon is soft. The nonperturbative dynamics in this case is described by two unknown SCET amplitudes. We use a simple model in order to estimate their contribution. The formalism is then applied to a phenomenological analysis of existing data for the reduced cross section as well as for the transverse and longitudinal polarization observables.

Δ-resonance contribution to the two-photon exchange amplitude

We consider two-photon exchange (TPE) in the elastic electron-proton scattering and study the contribution arising from the production of Delta(1232) resonance in the intermediate state. We calculate all three TPE amplitudes (generalized form factors), and find that the Delta contribution mainly influences generalized electric form factor (contrary to the elastic contribution, which affects magnetic form factor), and the effect grows with Q^2. If the corresponding correction is applied to the recent polarization transfer measurements of proton form factors, their results will change markedly. Thus we suggest that TPE corrections due to inelastic intermediate states are important to polarization experiments at high Q^2, and should not be neglected.