Motivated by the fact that a polarized ${}^3$He nucleus behaves as an `effective' neutron target, we examine manifestations of neutron electromagnetic polarizabilities in elastic Compton scattering from the Helium-3 nucleus. We calculate both unpolarized and double-polarization observables using chiral perturbation theory to next-to-leading order (${\mathcal O}(e^2 Q)$) at energies, $ω\leq m_π$, where $m_π$ is the pion mass. Our results show that the unpolarized differential cross section can be used to measure neutron electric and magnetic polarizabilities, while two double-polarization observables are sensitive to different linear combinations of the four neutron spin polarizabilities.
[Note added in 2018] The qualitative conclusions and analytic formulae presented in this paper are correct, but several of the numerical results are wrong: see the erratum posted as arXiv:1804.01206 for further details. A full suite of corrected numerical results for cross sections and asymmetries can be found in Margaryan et al., arXiv:1804.00956. They can also be obtained as an interactive Mathematica notebook by emailing hgrie@gwu.edu.
Differential cross section of deuteron Compton scattering has been calculated using the Bonn NN-potential, a consistent set of meson-exchange currents and seagulls, and lowest- and higher-order electromagnetic polarizabilities of the nucleon. Estimates of the polarizabilities of the neutron are obtained from recent experimental data.
We provide updated predictions for elastic γ${}^3$He cross sections and asymmetries that correct erroneous results we published in Phys. Rev. Lett. 98, 232303 (2007) and Nucl. Phys. A 819, 98 (2009).
In recent years, real and virtual Compton scattering off the nucleon have attracted considerable interest from both the experimental and theoretical sides. Real Compton scattering gives access to the so-called electromagnetic polarizabilities containing the structure information beyond the global properties of the nucleon such as its charge, mass, and magnetic moment. These polarizabilities have an intuitive interpretation in terms of induced dipole moments and thus characterize the response of the constituents of the nucleon to a soft external stimulus. The virtual Compton scattering reaction…
We examine manifestations of neutron electromagnetic polarizabilities in coherent Compton scattering from the Helium-3 nucleus. We calculate $γ^3$He elastic scattering observables using chiral perturbation theory to next-to-leading order (${\mathcal O}(e^2 Q)$). We find that the unpolarized differential cross section can be used to measure neutron electric and magnetic polarizabilities, while two double-polarization observables are sensitive to different linear combinations of the four neutron spin polarizabilities. [Note added in 2018] An erratum for this paper has been posted as arXiv:1804.0…
A microscopic calculation of Compton scattering on the nucleon is presented which encompasses the lowest energies -- yielding nucleon polarizabilities -- and extends to energies of the order of 600 MeV. We have used the covariant "Dressed K-Matrix Model" obeying the symmetry properties which are appropriate in the different energy regimes. In particular, crossing symmetry, gauge invariance and unitarity are satisfied. The extent of violation of analyticity (causality) is used as an expansion parameter.
A calculation of deuteron Compton scattering using non-relativistic perturbation theory is presented,with the primary motivation of investigating the feasibility of determining the neutron polarizabilities from this type of experiment. This calculation is expected to be valid for energies below 100 MeV. Pion-exchange, relativistic, and recoil corrections are also included. The low-energy theorem for gauge invariance is shown to be satisfied. The relative effects of the different terms and their effects on the determinations of the polarizabilities are discussed at energies of 49, 69, and 95 Me…
Goal of this work is the consistent description of elastic Compton scattering from the single nucleon and the deuteron. The theoretical framework chosen is Chiral Perturbation Theory, which is the low-energy formulation of Quantum Chromodynamics, where we extend the spectrum of active degrees of freedom from only pions and nucleons to also the Delta(1232) resonance. In the deuteron sector, we treat the nucleon-nucleon interaction non-perturbatively. Besides the Compton cross sections, our main concern is with the nucleon polarizabilities, which are a useful tool to describe the reaction of the…
Deuteron Compton scattering is calculated to $\mathcal{O}(Q^2)$ in pionless effective field theory using a dibaryon approach. The vector amplitude, which was not included in the previous pionless calculations, contributes to the cross section at $\mathcal{O}(Q^2)$ and influences significantly the extracted values of nucleon electric polarizability at incident photon energy 49 MeV. We recommend future high precision deuteron compton scattering experiments being performed at 25-35 MeV photon energy where the nucleon polarizability effects are appreciable and the pionless effective field theory i…
Threshold photon electroproduction off the proton allows one to measure new electromagnetic observables which generalise the usual polarisabilities. There are -- a priori -- ten "generalised polarisabilities", functions of the virtual photon mass. The purpose of this paper is to lay down the appropriate formalism to extract these quantities from the photon electroproduction cross sections. We also give a first estimate of the generalised polarisabilities in the non relativistic quark model.
A calculation of deuteron Compton scattering using non-relativistic diagrammatic perturbation theory is presented, with the primary motivation of investigating the feasibility of determining the neutron polarizabilities from this type of experiment. This calculation is expected to be valid for energies below 100 MeV. Previous theoretical and experimental estimates for the polarizabilities are given. All diagrams are calculated by expanding the photon wavefunctions into partial waves and using realistic deuteron wavefunctions. The Green's function for the intermediate state in the dispersive st…
Compton scattering processes are ideal to study electric and magnetic dipole polarizabilities of nucleons. These fundamental quantities parametrize the responses of the internal degrees of freedoms to a monochromatic photon probe. In this work, the inelastic channel $γd\rightarrow γnp$ is treated in $χ$EFT, with a focus on the kinematic region of the neutron quasi-free peak, where the momentum of the outgoing proton is small enough that it is considered to remain at rest. This provides access to the Compton scattering process $γn \rightarrow γn$, from which the scalar polarizabilites of the ne…
Deuteron Compton scattering below pion photoproduction threshold is considered in the framework of the nonrelativistic diagrammatic approach with the Bonn OBE potential. A complete gauge-invariant set of diagrams is taken into account which includes resonance diagrams without and with NN-rescattering and diagrams with one- and two-body seagulls. The seagull operators are analyzed in detail, and their relations with free- and bound-nucleon polarizabilities is discussed. It is found that both dipole and higher-order polarizabilities of the nucleon are needed for a quantitative description of rec…
We reexamine the low-energy expansion of polarized Compton scattering off the proton and show that the leading non-Born contribution to the beam asymmetry of low-energy Compton scattering is given by the magnetic polarizability alone, the electric polarizability cancels out. Based on this fact we propose to determine the magnetic dipole polarizability of the proton from the beam asymmetry. We also present the low-energy expansion of doubly-polarized observables, from which the spin polarizabilities can be extracted.
We discuss the low-energy behavior of the spin-averaged amplitude of virtual Compton scattering (VCS) off a nucleon. Based on gauge invariance, Lorentz invariance and the discrete symmetries, it is shown that to first order in the frequency of the final real photon only two generalized polarizabilities appear. Different low-energy expansion schemes are discussed and put into perspective.
Virtual Compton scattering off nucleons and pions at low energies is discussed. Predictions for the generalized polarizabilities of the nucleon are presented within the framework of heavy-baryon chiral perturbation theory and the linear sigma model. First results for the generalized polarizabilities of the charged pion in chiral perturbation theory at ${\cal O}(p^4)$ are shown.
As guideline for forthcoming experiments, we present predictions from Chiral Effective Field Theory for polarized cross sections in low energy Compton scattering for photon energies below 170 MeV, both on the proton and on the neutron. Special interest is put on the role of the nucleon spin polarizabilities which can be examined especially well in polarized Compton scattering. We present a model-independent way to extract their energy dependence and static values from experiment, interpreting our findings also in terms of the low energy effective degrees of freedom inside the nucleon: The pola…
Virtual Compton scattering off the nucleon has been studied in the one-loop approximation of the linear sigma model. The three generalized polarizabilities of the nucleon have been calculated and compared with the existing theoretical predictions. In particular, we find that only two of the three scalar polarizabilities are independent observables.
I give a summary of recent results on nucleon polarizabilities, with emphasis on chiral perturbation theory. The predictive calculations of Compton scattering off the nucleon are compared to recent empirical determinations and lattice QCD calculations of the polarizabilities, thereby testing chiral perturbation theory in the single-baryon sector.
We calculate the $N^*$ contributions to the generalized polarizabilities of the proton in virtual Compton scattering. The following nucleon excitations are included: $N^*(1535)$, $N^*(1650)$, $N^*(1520)$, $N^*(1700)$, $Δ(1232)$, $Δ^*(1620)$ and $Δ^*(1700)$. The relationship between nucleon structure parameters, $N^*$ properties and the generalized polarizabilities of the proton is illustrated.
Virtual Compton scattering off the nucleon (VCS) is studied in the regime of low energy of the outgoing real photon. This regime allows one to directly access the generalized polarizabilities of the nucleon in a VCS experiment. In the derivation of the low energy theorem for VCS that exists in the literature, the low energy limit taken for virtual initial photons does not match on that for real photons, when one approaches the initial photon's mass shell. While this problem has for a long time been attributed to the non-analyticity of the Compton amplitude with respect to the photon virtuality…
We present the first attempt to extract the scalar dipole dynamical polarizabilities from proton real Compton scattering data below pion-production threshold. The theoretical framework combines dispersion relations technique, low-energy expansion and multipole decomposition of the scattering amplitudes. The results are obtained with statistical tools that have never been applied so far to Compton scattering data and are crucial to overcome problems inherent to the analysis of the available data set.
This review gives an update on virtual Compton scattering (VCS) off the nucleon, $γ^* N \to N γ$, in the low-energy regime. We recall the theoretical formalism related to the generalized polarizabilities (GPs) and model predictions for these observables. We present the GP extraction methods that are used in the experiments: the approach based on the low-energy theorem for VCS and the formalism of Dispersion Relations. We then review the experimental results, with a focus on the progress brought by recent experimental data on proton generalized polarizabilities, and we conclude by some perspect…
We propose to determine the magnetic dipole polarizability of the proton directly from the beam asymmetry of low-energy Compton scattering based on the fact that the leading non-Born contribution to the asymmetry is due to the magnetic polarizability alone; the electric polarizability cancels out. The beam asymmetry thus provides the simplest observable with a clean separation of the magnetic polarizability from the electric one. Introducing polarizabilities in a Lorentz-invariant fashion we compute the higher-order (recoil) effects of polarizabilities on beam asymmetry and show that these eff…
We outline a microscopic framework to calculate nucleon Compton scattering from the level of quarks and gluons within the covariant Faddeev approach. We explain the connection with hadronic expansions of the Compton scattering amplitude and discuss the obstacles in maintaining electromagnetic gauge invariance. Finally we give preliminary results for the nucleon polarizabilities.
In a previous paper virtual Compton scattering off the nucleon has been investigated in the one-loop approximation of the linear sigma model in order to determine the 3 scalar generalized polarizabilities. We have now extended this work and calculated the 7 vector polarizabilities showing up in the spin-dependent amplitude of virtual Compton scattering. The results fulfill 3 model-independent constraints recently derived. Compared to the constituent quark model there exist enormous differences for some of the vector polarizabilities. At vanishing three-momentum of the virtual photon, the analy…
We present a projector formalism which allows to define dynamical polarizabilities of the nucleon from a multipole expansion of the nucleon Compton amplitudes. We give predictions for the energy dependence of these dynamical polarizabilities both from dispersion theory and from leading-one-loop chiral effective field theory. Based on the good agreement between the two theoretical frameworks, we conclude that the energy dependence of the dynamical polarizabilities is dominated by chiral dynamics, except in those multipole channels where the first nucleon resonance Delta(1232) can be excited. Bo…
The spin-independent part of the virtual Compton scattering (VCS) amplitude from the nucleon is calculated within the framework of heavy baryon chiral perturbation theory (HBChPT). The calculation is performed to third order in external momenta according to chiral power counting. The relation of the tree-level amplitudes to what is expected from the low-energy theorem is discussed. We relate the one-loop results to the structure coefficients of a low-energy expansion for the model-dependent part of the VCS amplitude recently defined by Fearing and Scherer. Finally we discuss the connection of …
We review the current state of knowledge of the nucleon polarizabilities and of their role in nucleon Compton scattering and in hydrogen spectrum. We discuss the basic concepts, the recent lattice QCD calculations and advances in chiral effective-field theory. On the experimental side, we review the ongoing programs aimed to measure the nucleon (scalar and spin) polarizabilities via the Compton scattering processes, with real and virtual photons. A great part of the review is devoted to the general constraints based on unitarity, causality, discrete and continuous symmetries, which result in m…
Below pion threshold virtual Compton scattering off the nucleon gives access to the generalized electromagnetic polarizabilities. Different theoretical results for the generalized polarizabilities have been compared. In particular, the influence of the generalized polarizabilities on the unpolarized cross section and on double polarization observables has been investigated. Predictions for these observables have been obtained in the linear sigma model and chiral perturbation theory.
Compton scattering on light nuclei ($A=2,3$) has emerged as an effective avenue to search for signatures of neutron polarizabilities, both spin--independent and spin--dependent ones. In this discussion I will focus on the theoretical aspect of Compton scattering on light nuclei; giving first a brief overview and therafter concentrating on our Compton scattering calculations based on Chiral effective theory at energies of the order of pion mass. These elastic $γ$d and $γ$He-3 calculations include nucleons, pions as the basic degrees of freedom. I will also discuss $γ$d results where the $Δ$-iso…
We present a calculation of the cross section and the event generator of the reaction $πe\to πe γ$. This reaction is sensitive to the pion generalized dipole polarizabilities, namely, the longitudinal electric $α_L(q^2)$, the transverse electric $α_T(q^2)$, and the magnetic $β(q^2)$ which, in the real-photon limit, reduce to the ordinary electric and magnetic polarizabilities $\barα$ and $\barβ$, respectively. The calculation of the cross section is done in the framework of chiral perturbation theory at ${\cal O}(p^4)$. A pion VCS event generator has been written which is ready for implementat…
The electric ($α_π$) and magnetic ($β_π$) Compton polarizabilities of both the charged and the neutral pion are of fundamental interest in the low-energy sector of quantum chromodynamics (QCD). Pion polarizabilities affect the shape of the $γπ\toγπ$ Compton scattering angular distribution at back scattering angles and $γγ\toππ$ absolute cross sections. Theory derivations are given of the $γπ\toγπ$ Compton scattering differential cross section, dispersion relations, and sum rules in terms of the polarizabilities. We review experimental charged and neutral polarizability studies and theoretical …
We perform an expansion of the virtual Compton scattering amplitude for low energies and low momenta and show that this expansion covers the transition from the regime to be investigated in the scheduled photon electroproduction experiments to the real Compton scattering regime. We discuss the relation of the generalized polarizabilities of virtual Compton scattering to the polarizabilities of real Compton scattering.
We derive two relations involving spin polarizabilities of a spin-1/2 particle and consider their empirical implications for the proton. Using the empirical values of the proton anomalous magnetic moment, electric and magnetic charge radii, moments of the spin structure functions $g_1$, $g_2$, and of two spin polarizabilities, the present relations constrain the low-momentum behavior of generalized polarizabilities appearing in virtual Compton scattering. In the case of the proton, the dispersive model evaluations of the spin and generalized polarizabilities appear to be consistent with these …
An effective field theory is used to give a model-independent description of Compton scattering at energies comparable to the pion mass. The amplitudes for scattering on the proton and the deuteron, calculated to fourth order in small momenta in chiral perturbation theory, contain four undetermined parameters that are in one-to-one correspondence with the nucleon polarizabilities. These polarizabilities are extracted from fits to data on elastic photon scattering on hydrogen and deuterium. For the proton we find: alpha_p = 12.1 +/- 1.1 +/- 0.5, beta_p = 3.4 +/- 1.1 +/- 0.1, both in units of 10…
We examine elastic Compton scattering off the deuteron for photon energies between 50 MeV and 100 MeV in the framework of chiral effective field theories to next-to-leading order. We compare one theoretical scheme with only pions and nucleons as explicit degrees of freedom to another in which the Delta(1232) resonance is treated as an explicit degree of freedom. Whereas pion degrees of freedom suffice to describe the experimental data measured at about 70 MeV, the explicit Delta(1232) gives important contributions that help to reproduce the angular dependence at higher energies. The static iso…
We derive two new sum rules for the unpolarized doubly virtual Compton scattering process on a nucleon, which establish novel low-$Q^2$ relations involving the nucleon's generalized polarizabilities and moments of the nucleon's unpolarized structure functions $F_1(x,Q^2)$ and $F_2(x,Q^2)$. These relations facilitate the determination of some structure constants which can only be accessed in off-forward doubly virtual Compton scattering, not experimentally accessible at present. We perform an empirical determination for the proton and compare our results with a next-to-leading-order chiral pert…
Virtual Compton scattering in the $Δ$-resonance region is considered in the case of a target nucleus. The discussion involves generalized polarizabilities and is developed for zero-spin nuclei, focusing on the new information coming from virtual Compton scattering in comparison with real Compton scattering.
The subject of nucleon polarizabilities in real, virtual, and doubly virtual Compton scattering is discussed with respect to what is known and how such quantities can be extracted from data.
We examine elastic Compton scattering from the deuteron for photon energies ranging from zero to 100 MeV, using state-of-the-art deuteron wave functions and NN-potentials. Nucleon-nucleon rescattering between emission and absorption of the two photons is treated by Green's functions in order to ensure gauge invariance and the correct Thomson limit. With this Green's-function hybrid approach, we fulfill the low-energy theorem of deuteron Compton scattering and there is no significant dependence on the deuteron wave function used. Concerning the nucleon structure, we use Chiral Effective Field T…
We study low-energy virtual Compton scattering off the proton within the framework of a nonrelativistic constituent quark model. The Compton tensor is divided into two separately gauge-invariant contributions. The first consists of the groundstate propagation in the direct and crossed channels together with an appropriately chosen term to satisfy gauge invariance. The residual part contains the relevant structure information characterized by the so-called generalized polarizabilities. We discuss two different schemes to obtain the generalized polarizabilities from the residual term. Explicit p…
The concept of Compton scattering by even-even nuclei from giant-resonance to nucleon-resonance energies and the status of experimental and theoretical researches in this field are outlined. Nuclear Compton scattering in the giant-resonance energy-region provides information on the dynamical properties of the in-medium mass of the nucleon. The electromagnetic polarizabilities of the nucleon in the nuclear medium can be extracted from nuclear Compton scattering data obtained in the quasi-deuteron energy-region. Recent results are presented for two-body effects due to the mesonic seagull amplitu…
We are studying the electron scattering process e p to e' p' gamma in order to obtain information on the genuine virtual Compton scattering (VCS) process gamma^* N to gamma N. In addition to the two kinematical variables of real Compton scattering, e.g. the scattering angle theta and the energy omega' of the outgoing photon, the invariant amplitude for VCS depends on a third kinematical variable, which we choose as the absolute value of the three-momentum transfer to the nucleon. The structure-dependent coefficients in the VCS amplitude therefore acquire a momentum dependence and are termed ``…
The Compton scattering on a charged pion is studied using the dispersion relations. Unknown mass, full width and decay width into γγof the sigma meson are found from a fit to the experimental data for the process γ+γ\to π^0+π^0. These parameters are used for the calculation of the cross section of the Compton scattering on the charged pion. The analysis of the obtained results shows that the data on the elastic γ\pion scattering in the energy region up to \sqrt{s}=1 GeV (together with the data on process γγ\to π^0π^0) could be used both for a determination of the pion polarizability values wit…
Polarized Compton scattering on the deuteron is studied in nuclear effective field theory. A set of tensor structures is introduced to define 12 independent Compton amplitudes. The scalar and vector amplitudes are calculated up to ${\cal O}((Q/Λ)^2)$ in low-energy power counting. Significant contribution to the vector amplitudes is found to come from the spin-orbit type of relativistic corrections. A double-helicity dependent cross section $Δ_1 σ= (σ_{+1-1}-σ_{+1+1})/2$ is calculated to the same order, and the effect of the nucleon isoscalar spin-dependent polarizabilities is found to be small…
The subject of low energy polarized Compton scattering from the proton, which is characterized by phenomenological spin-polarizabilities, is introduced and connection is made to new theoretical and experimental developments which were reported to this meeting.
We perform a fit of the real Compton scattering (RCS) data below pion-production threshold to extract the electric ($α_{E1}$) and magnetic ($β_{M1}$) static scalar dipole polarizabilities of the proton, using fixed-$t$ subtracted dispersion relations and a bootstrap-based fitting technique. The bootstrap method provides a convenient tool to include the effects of the systematic errors on the best values of $α_{E1}$ and $β_{M1}$ and to propagate the statistical errors of the model parameters fixed by other measurements. We also implement various statistical tests to investigate the consistency …
Compton scattering offers in principle an intriguing new window on nucleon structure. Existing experiments and future programs are discussed and the state of theoretical understanding of such measurements is explored.
We review recent developments in the theoretical investigation of the nucleon polarizabilities. We first report on the static polarizabilities as measured in real Compton scattering, comparing and interpreting the results from various theoretical approaches. In a second step, we extend the discussion to the generalized polarizabilities which can be accessed in virtual Compton scattering, showing how the information encoded in these quantities can provide a spatial interpretation of the induced polarization densities in the nucleon.
A status report on the topic Compton scattering and polarizabilities is presented with emphasis on the scalar t-channel as entering into dispersion theory. Precise values for the polarizabilities are obtained leading to $α_p = 12.0\pm 0.6$ $(12.0)$, $β_p=1.9\mp 0.6$ $(1.9)$, $α_n= 12.5\pm 1.7$ $(13.4)$, $β_n= 2.7 \mp 1.8$ $(1.8)$ in units of $10^{-4}$ fm$^3$ and $γ^{(p)}_π= -36.4 \pm 1.5$ $(-36.6)$, $γ^{(n)}_π= 58.6 \pm 4.0$ $(58.3)$, $(γ^{(p)}_0= -0.58\pm 0.20)$, $(γ^{(n)}_0 = +0.38\pm 0.22)$ in units of $10^{-4}$ fm$^4$, for the proton (p) and neutron (n), respectively. The data given with a…
Sum rules for parity violating spin polarizabilities are derived and discussed. They hold both for hadron and nuclear stable targets of arbitrary spin and are exact in strong interactions. Examples of applications to the cases of proton and deuteron targets are given. The legitimacy of the dispersive approach in the Standard Model is discussed.
The low energy virtual Compton scattering process $eN\to e'Nγ$ offers a new and potentially high resolution window on nucleon sturcture via measurement of so-called generalized polarizabilities (GPs). We present calculations of GPs within heavy baryon chiral perturbation theory and discuss present experimental efforts.
We analyze polarized Compton scattering which provides information on the spin-structure of the nucleon. For scattering processes with photon energies up to 100 MeV the spin-structure dependence can be encoded into four independent parameters-the so-called spin-polarizabilities $γ_i, i=1...4$ of the nucleon, which we calculate within the framework of the "small scale expansion" in SU(2) baryon chiral perturbation theory. Specific application is made to "forward" and "backward" spin- polarizabilities.
We evaluate the forward Compton scattering off the proton, based on Kramers-Kronig kind of relations which express the Compton amplitudes in terms of integrals of total photoabsorption cross sections. We obtain two distinct fits to the world data on the unpolarized total photoabsorption cross section, and evaluate the various spin-independent sum rules using these fits. For the sum of proton electric and magnetic dipole polarizabilities, governed by the Baldin sum rule, we obtain the following average (between the two fits): $α_{E1}+β_{M1}=14.0(2)\times 10^{-4}\,\mathrm{fm}^3$. An analogous su…
Using hydrogen and deuterium targets, Compton scattering by the proton and neutron have been studied at the tagged photon beam of the MAMI (Mainz) accelerator using different experimental setups. The theoretical tools for the analysis of the experimental data have been investigated, as there are the nonsubtracted dispersion theory and the theory of quasi-free reactions on the proton and neutron bound in the deuteron. Experimental Compton scattering data are understood in the first and second resonance region with good precision. Precise electromagnetic polarizabilities and spin polarizabilitie…
Compton scattering offers a unique opportunity to study the dynamical structure of hadrons over a wide kinematic range, with polarizabilities characterizing the hadron active internal degrees of freedom. We present calculations and detailed analysis of electric and magnetic and the spin-dependent dynamical polarizabilities for the lowest in mass SU(3) octet of baryons. These extensive calculations are made possible by the recent implementation of semi-automatized calculations in chiral perturbation theory which allows evaluating polarizabilities from Compton scattering up to next-to-the-leadin…
Low-energy Compton scattering off the proton is used for determination of the proton polarizabilities. However, the present empirical determinations rely heavily on the theoretical description(s) of the experimental cross sections in terms of polarizabilities. The most recent determinations are based on either the fixed-$t$ dispersion relations (DR) or chiral perturbation theory in the single-baryon sector ($χ$PT). The two approaches obtain rather different results for proton polarizabilities, most notably for $β_{M1}$ (magnetic dipole polarizability). We attempt to resolve this discrepancy by…
We discuss the virtual Compton scattering amplitude $γ^\ast+π\to γ+π$ which enters the reaction $π^-+e^-\toπ^-+e^-+γ$. The low-energy scattering amplitude is divided into a model-independent part, involving only the electromagnetic form factor, and a residual part which contains structure information related specifically to (virtual) Compton scattering. In the limit of vanishing final-photon energy, the residual amplitude can be expressed in terms of three generalized dipole polarizabilities which are functions of the squared virtual-photon four-momentum. We study the VCS amplitude in the fram…
