The electromagnetic polarizabilities of the nucleon are shown to be essentially composed of the nonresonant $α_p(E_{0+})=+3.2$, $α_n(E_{0+})=+4.1$,the $t$-channel $α^t_{p,n}=-β^t_{p,n}=+7.6$ and the resonant $β_{p,n}(P_{33}(1232))=+8.3$ contributions (in units of $10^{-4}$fm$^3$. The remaining deviations from the experimental data $Δα_p=1.2\pm 0.6$, $Δβ_p=1.2\mp 0.6$, Δα_n=0.8\pm 1.7$ and $Δβ_n=2.0\mp 1.8$ are contributed by a larger number of resonant and nonresonant processes with cancellations between the contributions. This result confirms that dominant contributions to the electric and magnetic polarizabilities may be represented in terms of two-photon couplings to the $σ$-meson having the predicted mass $m_σ=666$ MeV and two-photon width $Γ_{γγ}=2.6$ keV.
In addition to the E2/M1 ratio of the N->$Δ$ transition, the electromagnetic polarizabilities and spin-polarizabilities are important structure constants of the nucleon which serve as sensitive tests of chiral perturbation theory and of models of the nucleon. Recently, these quantities have been investigated experimentally at MAMI (Mainz) by high-precision Compton scattering using hydrogen and deuterium targets, where for the latter the method of quasi-free scattering has been applied.
I discuss recent results regarding the influence of pion-nucleon and and nucleon-resonance physics on the (spin-) polarizabilities of the nucleon as measured in (polarized) Compton scattering.
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.
Fixed-$t$ subtracted dispersion relations are presented for Compton scattering off the nucleon at energies $E_γ\leq 500$ MeV, as a formalism to extract the nucleon polarizabilities from the data with a minimum of model dependence. The scalar polarizabilities difference $α- β$ and the backward spin polarizability $γ_π$ enter directly as fit parameters in this formalism.
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.
We present the first heavy baryon chiral perturbation theory calculation at ${\mathcal O}(p^4)$ for the spin-dependent amplitudes for virtual Compton scattering off the nucleon, and extract the ${\cal O}(p^4)$ order results for the generalized spin polarizabilities of the nucleon.
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.
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…
The nucleon polarizabilities alpha(E) and beta(M) are studied in the context of the perturbative chiral quark model. We demonstrate that meson cloud effects are sufficient to explain the electric polarizability of nucleon. Contributions of excite quark states to the paramagnetic polarizability are dominant and cancel the diamagnetic polarizability arising from the chiral field. The obtained results are compared to data and other theoretical predictions.
The status of Compton scattering by the nucleon at energies of the first and second resonance is summarized. In addition to a general test of dispersion theories and a precise determination of polarizabilities, the validities of four fundamental sum rules are explored. {\it Recommended} averages of experimental values for the electromagnetic polarizabilities and spin-polarizabilities of the nucleon are determined: $α_p=12.0\pm 0.6$, $β_p=1.9\mp 0.6$, $α_n= 12.5 \pm 1.7$, $β_n= 2.7 \mp 1.8$ (unit $10^{-4}$fm$^3$), $γ^{(p)}_π= -38.7 \pm 1.8$, $γ^{(n)}_π= 58.6 \pm 4.0$ (unit $10^{-4}$fm$^4$).
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…
A dispersion relation formalism for the virtual Compton scattering (VCS) reaction on the proton is presented, which for the first time allows a dispersive evaluation of 4 generalized polarizabilities at a four-momentum transfer $Q^2 \leq$ 0.5 GeV$^2$. The dispersive integrals are calculated using a state-of-the-art pion photo- and electroproduction analysis. The dispersion formalism provides a new tool to analyze VCS experiments above pion threshold, thus increasing the sensitivity to the generalized polarizabilities of the nucleon.
Following up on the plenary talk given by Ulf Meißner, I discuss a generalization of the concept of nucleon spin-polarizabilities in terms of multipole excitation/de-excitation and present an overview of the available theoretical predictions for these elusive but interesting structure parameters of the nucleon.
The polarizabilities $α$ (electric), $β$ (magnetic) and $γ_π$ (backward spin) of the nucleon are investigated in terms of degrees of freedom of the nucleon using recent results for the CGLN amplitudes and resonance couplings $A_{1/2}$ and $A_{3/2}$. The photon excitation strengths of the excited states are given in terms of partial integrated photoabsorption cross sections and resonant contributions of the Gerasimov-Drell-Hearn (GDH) sum rule. As a test of the present predictions, cross section differences $(σ_{3/2}-σ_{1/2})$ of the excited states are compared with data recently measured at MA…
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.
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.
Virtual Compton Scattering on the nucleon: $γ^* N \to γN$ is a new and rapidly developing field at low and high energies. This lecture is about the low energy part, i.e. for energies in the $(γp)$ center-of-mass mainly up to the $Δ(1232)$ resonance region. I review the concept of Generalized Polarizabilities of the Nucleon, and the experiments dedicated to their measurement.
We propose a new method that fixes the coupling to two photons of the recently found lightest QCD resonance, the sigma. This coupling provides crucial information for discriminating the yet unknown nature of this special state. Our method uses available data on the nucleon polarizabilities together with analyticity and unitarity. Taking into account all the uncertainties, our result is Gamma_{pole}= 1.2 +- 0.4 keV.
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…
We calculate the nucleons' electromagnetic polarizabilities in heavy baryon chiral perturbation theory including all terms to order ${\cal O} (q^4)$. The chiral prediction of the electric polarizabilities for the neutron and the proton are in good agreement with the data. In the case of the magnetic polarizabilities the big positive contribution from the $Δ(1232)$ resonance is largely cancelled by a non--analytic loop contribution of the $\ln M_π$ type. This novel effect helps to understand the rather small empirical value of the nucleons' magnetic polarizability.
We present a complete one-loop analysis of the four nucleon spin polarizabilities in the framework of heavy baryon chiral perturbation theory. The first non-vanishing contributions to the isovector and first corrections to the isoscalar spin polarizabilities are calculated. No unknown parameters enter these predictions. We compare our results to various dispersive analyses. We also discuss the convergence of the chiral expansion and the role of the delta isobar.
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.
The signs and values of the two-photon couplings $F_{Mγγ}$ of mesons $(M)$ and their couplings $g_{MNN}$ to the nucleon as entering into the $t$-channel parts of the difference of the electromagnetic polarizabilities $(α-β)$ and the backward angle spin polarizabilities $γ_π$ are determined. The excellent agreement achieved with the experimental polarizabilities of the proton makes it possible to make reliable predictions for the neutron. The results obtained are $α_n=13.4\pm 1.0$, $β_n=1.8\mp 1.0$ ($10^{-4}$ fm$^3$), and $γ^{(n)}_π=57.6\pm 1.8$ ($10^{-4}$ fm$^4$). New empirical information on …
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 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.
Based on Compton scattering and meson photoproduction data the polarizabilities of the nucleon are precisely studied and well understood due to recent experimental and theoretical work based on nonsubtracted dispersion relations. The {\it recommended} experimental values are $α_p=12.0\pm 0.6$, $(12.0)$, $β_p=1.9\mp 0.6$, $(1.9)$, $α_n=12.5\pm 1.7$, $(12.7\pm 0.9)$, $β_n=2.7\mp 1.8$, $(2.5\mp 0.9)$ 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$^4$ [1]. The nu…
We discuss the virtual Compton scattering reaction $e^-p\to e^-pγ$ at low energies. We present results for the generalized polarizabilities of the nucleon obtained in heavy baryon chiral perturbation theory at $O(p^3)$.
The nucleon generalized polarizabilities (GPs), probed in virtual Compton scattering (VCS), describe the spatial distribution of the polarization density in a nucleon. They are accessed experimentally via the process of electron-proton bremsstrahlung ($ep\to epγ$) at electron-beam facilities, such as MIT-Bates, CEBAF (Jefferson Lab), and MAMI (Mainz). We present the calculation of the nucleon GPs and VCS observables at next-to-leading order in baryon chiral perturbation theory (B$χ$PT), and confront the results with the empirical information. At this order our results are predictions, in the s…
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 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…
The lightest QCD resonance, the sigma, has been recently fixed in the pi-pi scattering amplitude. The nature of this state remains nowadays one of the most intriguing and difficult issues in particle physics. Its coupling to photons is crucial to discriminate its structure. We propose a new method that fixes this coupling using only available precise experimental data on the proton electromagnetic polarizabilities together with analyticity and unitarity. Taking into account the uncertainties in the analysis and in the parameter values, our result is Gamma_{pole}(sigma -> gamma gamma) = (1.2 +-…
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…
We calculate the spin-polarizabilities of the nucleon by means of dispersion relation, where the pion photoproduction amplitude predicted by chiral soliton model is utilized. We consider the $Nπ$ and $Δπ$ channels in the pion-photoproduction amplitude, and also evaluate the contribution of the anomalous term of the $π^0γγ$ process. In a narrow decay-width limit of the $Δ$ particle the result coincides with that in heavy baryon chiral perturbation theory (HBChPT) except for the interference part of the electric and magnetic amplitudes. A comparison with the multipole analysis is given.
Nucleon generalized polarizabilities are investigated within a relativistic framework, defining such quantities through a Lorentz covariant multipole expansion of the amplitude for virtual Compton scattering. The key physical ingredients in the calculation of the nucleon polarizabilities are the Lorentz invariant reduced matrix elements of the electromagnetic transition current, which can be evaluated from off-energy-shell helicity amplitudes. The evolution of the proton paramagnetic polarizability, $β_{para}(q)$, as a function of the virtual-photon three-momentum transfer $ q,$ is explicitly …
Electromagnetic properties of the nucleon are explored with lattice QCD using a novel technique. Focusing on background electric fields, we show how the electric polarizability can be extracted from nucleon correlation functions. A crucial step concerns addressing contributions from the magnetic moment, which affects the relativistic propagation of nucleons in electric fields. By properly handing these contributions, we can determine both magnetic moments and electric polarizabilities. Lattice results from anisotropic clover lattices are presented. Our method is not limited to the neutron; we …
Using the techniques of chiral effective field theories we evaluate the so called generalized polarizabilities of the nucleon, which characterize the structure dependent components in virtual Compton scattering (VCS) as probed in the electron scattering reaction e N \to e' N gamma. Results are given for both spin-dependent and spin-independent structure effects to O(p^3) in SU(2) Heavy Baryon Chiral Perturbation Theory and to O(epsilon^3) in the SU(2) Small Scale Expansion. Finally we compare our calculations with results from the pioneering VCS experiment on the proton from Mainz.
Spin-polarizabilities are predicted by calculating the cross-section difference $σ_{3/2}-σ_{1/2}$ from available data for the resonance couplings $A_{3/2}$ and $A_{1/2}$ and CGLN amplitudes. The forward spin-polarizabilities are predicted to be $γ^{(p)}_0=-0.58\pm 0.20$ and $γ^{(n)}_0=+0.38\pm 0.22$ in units of $10^{-4}$fm$^4$ where the different signs are found to be due to the isospin dependencies of the $E_{0+}$ and the $(M,E)^{(1/2)}_{1+}$ amplitudes. The backward spin-polarizabilities are predicted to be $γ^{(p)}_π=-36.6$ and $γ^{(n)}_π=+58.3$, to be compared with the experimental values …
Nucleon properties are investigated in background electric fields. As the magnetic moments of baryons affect their relativistic propagation in constant electric fields, electric polarizabilities cannot be determined without knowledge of magnetic moments. This is analogous to the experimental situation, for which determination of polarizabilities from the Compton amplitude requires subtraction of Born terms. With the background field method, we devise combinations of nucleon correlation functions in constant electric fields that isolate magnetic moments and electric polarizabilities. Using an e…
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 difference between the electromagnetic self-energies of proton and neutron can be calculated with the Cottingham formula, which expresses the self-energies as an integral over the electroproduction cross sections---provided the nucleon matrix elements of the current commutator do not contain a fixed pole. We show that, under the same proviso, the subtraction function occurring in the dispersive representation of the virtual Compton forward scattering amplitude is determined by the cross sections. The representation in particular leads to a parameter-free sum rule for the nucleon polarizabi…
A unified presentation is given on the use of dispersion relations in the real and virtual Compton scattering processes off the nucleon. The way in which dispersion relations for Compton scattering amplitudes establish connections between low energy nucleon structure quantities, such as polarizabilities or anomalous magnetic moments, and the nucleon excitation spectrum is reviewed. We discuss various sum rules for forward real and virtual Compton scattering, such as the Gerasimov-Drell-Hearn sum rule and its generalizations, the Burkhardt-Cottingham sum rule, as well as sum rules for forward n…
The $t$-channel contribution to the difference of electromagnetic polarizabilities of the nucleon, $(α-β)^t$, can be quantitatively understood in terms of a $σ$-meson pole in the complex $t$-plane of the invariant scattering amplitude $A_1(s,t)$ with properties of the $σ$ meson as given by the quark-level Nambu--Jona-Lasinio model (NJL). Equivalently, this quantity may be understood in terms of a cut in the complex $t$-plane where the properties of the $σ$ meson are taken from the $ππ-> σ-> ππ$, $γγ-> σ-> ππ$ and $N\bar{N} -> σ-> ππ$ reactions. This equivalence may be understood as a sum rule …
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 extract the next-to-next-to-leading order results for spin-flip generalized polarizabilities (GPs) of the nucleon from the spin-dependent amplitudes for virtual Compton scattering (VCS) at ${\cal O}(p^4)$ in heavy baryon chiral perturbation theory. At this order, no unknown low energy constants enter the theory, allowing us to make absolute predictions for all spin-flip GPs. Furthermore, by using constraint equations between the GPs due to nucleon crossing combined with charge conjugation symmetry of the VCS amplitudes, we get a next-to-next-to-next-to-leading order prediction for one of th…
The status of the experimental and theoretical investigations on the polarizabilities of the nucleon is presented. This includes a confirmation of the validity of the previously introduced recommended values of the polarizabilities [1,2]. It is shown that the only meaningful approach to a prediction of the polarizabilities is obtained from the nonsubtracted dispersion theory, where the appropriate degrees of freedom taken from other precise experimental data are taken in account. The present values of the recommended polarizabilities are $α_p= 12.0 \pm 0.5$, $β_p= 1.9 \mp 0.5$, $α_n= 12.6 \pm …
Previous calculations of the polarizabilities of the nucleon within the framework of heavy baryon chiral perturbation theory have included the contribution of the $Δ$(1232) only its effect on various contact terms or have been performed in chiral SU(3) where systematic errors are difficult to control. Herein we perfrom a corresponding calculation in chiral SU(2) wherein $Δ$(1232) is treated as an explicit degree of freedom and the expansion is taken to third order in soft momenta, the pion mass and the quantity $M_Δ-M_N$, collectively denoted by $ε$. We present the results of a systematic $O(ε…
Dispersion theory of nucleon (nucleus) Compton scattering spin polarizabilities and quasi-optical $γ$-ray polarization plane rotation and birefringence effect in a matter with polarized protons (nuclei)
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 diquark model is used to calculate the electromagnetic polarizabilities and charge radii of the nucleons for three different potentials. Making the scalar diquark lower in mass introduces a mixing angle $θ$ between the $\left| 56\right\rangle $ and $\left| 70\right\rangle $ states ,which allows an improvement in value of all 6 properties. Generalizing the Gamov-Teller matrix and the magnetic moment operator to the diquark model gives constraints on this mixing. We obtain for the Richardson potential $θ=23.2^{\circ },$ $\overline{α}_p=7.9_{-0.9}^{+1.0}\times 10^{-4}fm^3,$ $\overline{α}_n=7.…
We examine the polarized doubly-virtual Compton scattering (VVCS) off the nucleon using chiral perturbation theory ($χ$PT). The polarized VVCS contains a wealth of information on the spin structure of the nucleon which is relevant to the calculation of the two-photon-exchange effects in atomic spectroscopy and electron scattering. We report on a complete next-to-leading-order (NLO) calculation of the polarized VVCS amplitudes $S_1(ν, Q^2)$ and $S_2(ν, Q^2)$, and the corresponding polarized spin structure functions $g_1(x, Q^2)$ and $g_2(x,Q^2)$. Our results for the moments of polarized structu…
We study spin-dependent sum rules for forward virtual Compton scattering(VVCS) off the nucleon in heavy baryon chiral perturbation theory at order $O(p^4)$. We show how these sum rules can be evaluated from low energy expansions (in the virtual photon energy) of the forward VVCS amplitudes. We study in particular the Burkhardt -Cottingham sum rule in HBChPT and higher terms in the low energy expansion, which can be related to the generalized forward spin polarizabilities of the nucleon. The dependence of these observables on the photon virtuality $Q^2$ can be accessed, at small and intermediat…
Recent experimental results on the proton and neutron polarizabilities are examined from the point of view of backward dispersion relations. Results are found to be in reasonable agreement with the measured values. A rigorous relationship between the nucleon and pion polarizabilities is derived and shown to be in excellent agreement with several models.
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…
We study the response of the nucleon, as a system of three bound (constituent) non relativistic quarks, to external (quasi static) electric and magnetic fields. The approach, based on a sum rule technique, is applied to a large class of two and three-body interquark potentials. Lower and upper bounds to the electric polarizability and para-magnetic susceptibility are explicitly calculated within a large variety of constituent models and their values related to the features of the interquark interaction picture. The rôle of three-body forces is investigated in details as well as the effects of …
