News on Compton Scattering γ X → γ X in Chiral EFT

We review theoretical progress and prospects to understand the nucleon’s static dipole polarisabilities from Compton scattering on few-nucleon targets, including new values; see Refs. [1–5] for details and a more thorough bibliography. Why Compton Scattering? Since the electromagnetic field of a real photon induces radiation multipoles by displacing charged constituents and currents, the energyand angle-dependence of the emitted radiation elucidates the distribution, symmetries and dynamics of the charges and currents which constitute the low-energy degrees of freedom inside the nucleon and nucleus; see e.g. a recent review [1]. Energy-dependent polarisabilities parametrise the stiffness of the nucleon N (spin σ2 ) against transitions Xl → Yl′ of given photon multipolarity at fixed frequency ω (l′ = l ± {0; 1}; X, Y = E,M; Ti j = 2 (∂iT j + ∂ jTi); T = E, B). Up to 500 MeV in photon energy, the relevant terms are: Lpol = 2π N† αE1(ω) E2 + βM1(ω) B2 + γE1E1(ω) σ · ( E × ̇ E) + γM1M1(ω) σ · ( B × ̇ B) − 2γM1E2(ω) σi B j Ei j + 2γE1M2(ω) σi E j Bi j + . . . (photon multipoles beyond dipole)] N (1) Two scalar polarisabilities αE1(ω)/βM1(ω) encode electric/magnetic dipole transitions; 4 spinpolarisabilities γE1E1(ω), γM1M1(ω), γE1M2(ω) and γM1E2(ω) the response of the nucleon’s spinstructure. Intuitively interpreted, the electromagnetic field associated with the spin degrees causes bi-refringence in the nucleon (Faraday-effect). Polarisabilities test our understanding of the subtle interplay between electromagnetic and strong interactions and enter in theoretical determinations of the proton-neutron mass difference, and in the two-photon-exchange contribution to the Lamb shift in muonic hydrogen; see e.g. the introduction of [5]. Finally, nuclear targets provide an opportunity to study not only neutron polarisabilities, but indirectly also the nuclear force, since the photons couple to the charged pion-exchange currents in the nucleus. The values αE1(ω = 0) etc. are often called “the (static) polarisabilities”; they compress the richness of information extrapolated from data in a wide range of energies between about 70 MeV and the Δ resonance region into just a few numbers. In the canonical 10−4 fm3, Chiral Effective Field Theory (χEFT, see below) gives (Fig. 1 and [2, 4]): α (p) E1 = 10.65 ± 0.35stat ± 0.2Baldin ± 0.3th , β M1 = 3.15 ∓ 0.35stat ± 0.2Baldin ∓ 0.3th α E1 = 11.55 ± 1.25stat ± 0.2Baldin ± 0.8th , β M1 = 3.65 ∓ 1.25stat ± 0.2Baldin ∓ 0.8th (2) ae-mail: hgrie@gwu.edu; presenter. DOI: 10.1051/ C © Owned by the authors, published by EDP Sciences, 201 / 0 0 (201 ) 201 epjconf EPJ Web of Conferences , 1 61 6 6 1 1 0 0 3 3 4 4 0 06 6

Polarisabilities test our understanding of the subtle interplay between electromagnetic and strong interactions and enter in theoretical determinations of the proton-neutron mass difference, and in the two-photon-exchange contribution to the Lamb shift in muonic hydrogen; see e.g. the introduction of [5].Finally, nuclear targets provide an opportunity to study not only neutron polarisabilities, but indirectly also the nuclear force, since the photons couple to the charged pion-exchange currents in the nucleus.The values α E1 (ω = 0) etc. are often called "the (static) polarisabilities"; they compress the richness of information extrapolated from data in a wide range of energies between about 70 MeV and the Δ resonance region into just a few numbers.In the canonical 10 −4 fm 3 , Chiral Effective Field Theory (χEFT, see below) gives (Fig. 1 We also predicted the spin values [2,5], prior to the pioneering MAMI results [6] (in 10 −4 fm 4 ): M. Ahmed and L. Myers reported on the concerted ongoing and planned efforts at HIγS [7], MAX-Lab [4,8], and MAMI [9].Interpreting such data requires of course commensurate theory support; cf. the open letter by theorists from a variety of backgrounds Ref. [10].One must carefully evaluate data-consistency for hidden systematic errors; subtract binding effects in few-nucleon systems; extract the polarisabilities; identify their underlying mechanisms and relate them to QCD -and all that with reproducible theoretical uncertainties and minimal theoretical bias.Indeed, χEFT, the low-energy theory of QCD and extension of Chiral Perturbation Theory to few-nucleon systems, has been quite successful in proton and few-nucleon Compton scattering.With we recently derived single-nucleon Compton amplitudes from zero energy to about 400 MeV.For ω m π , they contain all contributions at O(e 2 δ 4 ) (N 4 LO, accuracy δ 5 2%), and for ω ∼ Δ M all at O(e 2 δ 0 ) (NLO, accuracy δ 2 20%) [1,2].A reproducible and systematically improvable a priori estimate of the theoretical accuracies of observables, like in eqs.( 2) and (3) or Fig. 5, is essential to uniquely disentangle chiral dynamics from data; see Ref. [5] and Refs.therein.Figure 2 lists examples of the 3 classes of contributions in few-nucleon systems.Charged exchange currents and rescattering often dominate over the nucleonic structure contributions.Compton photons therefore also provide direct, non-trivial benchmarks how accurately the chiral expansion accounts order-by-order for nuclear binding and its mesonic contributions.For the deuteron, our results are complete at O(e 2 δ 3 ) from the Thomson limit up to about 120 MeV, including the Δ(1232) [1].
In the preceding talk, L. Myers eloquently described the difficulties and successes of adding 22 deuteron points at MAXlab [4,8].This first new data in over a decade effectively doubled the deuteron's world dataset.Our analysis shows that is well consistent with and within the world dataset (χ 2 = 45.2 for 44 degrees of freedom), and with the Baldin sum rule.These data alone slashed the statistical error by 30% .Just to illustrate the data quality, Figure 3 shows that the χ 2 distribution of the new world dataset agrees quite well with the analytic expectation.The future lies in un-, single-and double-polarised experiments of high accuracy and theoretical analyses with reproducible systematic uncertainties.To understand the subtle differences of the pion clouds around the proton and neutron induced by explicit chiral symmetry breaking in QCD, we need the neutron polarisabilities with uncertainties comparable to those of the proton -eq.( 2) shows that this is mostly an issue of better data.Therefore, MAMI, MAXlab and HIγS aim for deuteron data with statistical and systematic uncertainties of better than 5%, and plan extensions to 3 He.In general, heavier nuclei are experimentally better to handle and provide count rates which scale at least linearly with the target charge when photons scatter incoherently from the protons, i.e. for ω 100 MeV.But a theoretical description of their energy levels with adequate accuracy is involved.For the proton, amplitudes on the 2%-level are available; for deuteron and 3 He, we now extend descriptions with similar accuracies into the Delta resonance region.Around 3 He-4 He- 6 Li may well be the "sweetspot" between needs and wants of theorists and experimentalists.Figure 1 shows that not including the Δ(1232) can lead to false signals in high-accuracy extractions of neutron polarisabilities [12].

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Since the 8 spin-polarisabilities are a top priority of experiment and theory alike, we performed a series of sensitivity studies; see Fig. 4 and summary in [1,Sec. 6.1].Recently, the deuteron cross section and asymmetry with arbitrary photon and target polarisations has for example been parametrised via 18 independent observables [3].Particularly attractive are some asymmetries which are sensitive to only 1 or 2 polarisabilities.For spin polarisabilities with an error of ±2 × 10 −4 fm 4 , asymmetries should be measured with an accuracy of 10 −2 , with differential cross sections of a dozen nb/sr at 100 MeV or a few dozen nb/sr at 250 MeV.Relative to single-nucleon Compton scattering, interference with the deuteron's D wave and pion-exchange current increases the sensitivity to the "mixed" spin polarisabilities γ E1M2 and γ M1E2 .A Mathematica file for ω < 120 MeV is available from hgrie@gwu.edu(see screen-shot in Fig. 4), and is finalised for the proton and 3 He.
Finally, χEFT connects data with emerging lattice-QCD simulations by reliable extrapolations from numerically less costly, heavier pion masses within the χEFT regime to the physical point and circumvents a direct lattice simulation of Compton scattering -that would be highly nontrivial.The lattice, in turn, tests to which extent χEFT adequately captures the m π -dependence of the low-energy dynamics, and may predict short-distance (fit) parameters from QCD .W. Detmold's plenary showed intriguing lattice results for the proton's and neutron's β M1 at m π = 806 MeV; see Fig. 5 and Ref. [5].

M1
is nearly identical to the chiral result even well beyond the range in which χEFT should be applicable.This suggests that the experimental finding β (p) M1 ≈ β (n) M1 is something of a coincidence.The agreement with simulations for α E1 is even better [5].Corridors: theoretical uncertainties in the regime where χEFT can be expected to converge [5].

Figure 2 .Figure 3 .
Figure 2. Contributions to deuteron Compton scattering.Ellipse: NN S -matrix:(a): single-nucleon; (b) photon coupling to charged exchange currents which bind the nucleus as dictated by chiral symmetry; (c) rescattering between emission and absorption restores the low-energy Thomson limit and guarantees current conservation.