Electromagnetic moments of quasi-stable particle

2010

We deal with the problem of assigning electromagnetic moments to a quasi-stable particle (i.e., a particle with mass located at particle's decay threshold). In this case, an application of a small external electromagnetic field changes the energy in a non-analytic way, which makes it difficult to assign definitive moments. On the example of a spin-1/2 field with mass $M_{*}$ interacting with two fields of masses $M$ and $m$, we show how a conventionally defined magnetic dipole moment diverges at $M_{*}=M+m$. We then show that the conventional definition makes sense only when the values of the applied magnetic field $B$ satisfy $|eB|/2M_{*}\ll|M_{*}-M-m|$. We discuss implications of these re…

Separation of proton polarizabilities with the beam asymmetry of Compton scattering

2013

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 effects are most suppressed in forward kinematics. With the prospects of precision Compton experiments…

Forward doubly-virtual Compton scattering off the nucleon in chiral perturbation theory: II. Spin polarizabilities and moments of polarized structure…

2020

We examine the polarized doubly-virtual Compton scattering (VVCS) off the nucleon using chiral perturbation theory ($\chi$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 $S_1(\nu, Q^2)$ and $S_2(\nu, Q^2)$, 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 structure functions, partially related to different spin polarizabilities, are compared to other th…

The lifetime of unstable particles in electromagnetic fields

2007

We show that the electromagnetic moments of unstable particles (resonances) have an absorptive contribution which quantifies the change of the particle's lifetime in an external electromagnetic field. To give an example we compute here the imaginary part of the magnetic moment for the cases of the muon and the neutron at leading order in the electroweak coupling. We also consider an analogous effect for the strongly-decaying $\Delta$(1232) resonance. The result for the muon is Im$ \mu = e G_F^2 m^3/768 \pi^3$, with $e$ the charge and $m$ the mass of the muon, $G_F$ the Fermi constant, which in an external magnetic field of $B$ Tesla give rise to the relative change in the muon lifetime of $…

Partial-wave analysis of proton Compton scattering data below the pion-production threshold

2018

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 ($\chi$PT). The two approaches obtain rather different results for proton polarizabilities, most notably for $\beta_{M1}$ (magnetic dipole polarizability). We attempt to resolve this discrepancy by performing a partial-wave analysis of the world data on proton Compton scattering below thr…

The subtraction contribution to the muonic-hydrogen Lamb shift: a point for lattice QCD calculations of the polarizability effect

2020

The proton-polarizability contribution to the muonic-hydrogen Lamb shift is a major source of theoretical uncertainty in the extraction of the proton charge radius. An empirical evaluation of this effect, based on the proton structure functions, requires a systematically improvable calculation of the "subtraction function", possibly using lattice QCD. We consider a different subtraction point, with the aim of accessing the subtraction function directly in lattice calculations. A useful feature of this subtraction point is that the corresponding contribution of the structure functions to the Lamb shift is suppressed. The whole effect is dominated by the subtraction contribution, calculable o…

Forward doubly-virtual Compton scattering off the nucleon in chiral perturbation theory: The subtraction function and moments of unpolarized structur…

2020

The forward doubly-virtual Compton scattering (VVCS) off the nucleon contains a wealth of information on nucleon structure, 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 low-energy VVCS in chiral perturbation theory ($\chi$PT). Here we focus on the unpolarized VVCS amplitudes $T_1(\nu, Q^2)$ and $T_2(\nu, Q^2)$, and the corresponding structure functions $F_1(x, Q^2)$ and $F_2(x,Q^2)$. Our results are confronted, where possible, with "data-driven" dispersive evaluations of low-energy structure quantities, such as nucleon polarizabilities. We find significant dis…

Hadronic light-by-light scattering contribution to the muon $g-2$ on the lattice

2018

International Workshop on “Flavour Changing and Conserving Processes, FCCP2017, Anacapri, Italy, 7 Sep 2017 - 9 Sep 2017; The European physical journal / Web of Conferences 179, 01017 (2018). doi:10.1051/epjconf/201817901017

Breakdown of the expansion of finite-size corrections to the hydrogen Lamb shift in moments of charge distribution

2015

We quantify a limitation in the usual accounting of the finite-size effects, where the leading $[(Z\alpha)^4]$ and subleading $[(Z\alpha)^5]$ contributions to the Lamb shift are given by the mean-square radius and the third Zemach moment of the charge distribution. In the presence of any non-smooth behaviour of the nuclear form factor at scales comparable to the inverse Bohr radius, the expansion of the Lamb shift in the moments breaks down. This is relevant for some of the explanations of the "proton size puzzle". We find, for instance, that the de R\'ujula toy model of the proton form factor does not resolve the puzzle as claimed, despite the large value of the third Zemach moment. Withou…

Sum rules across the unpolarized Compton processes involving generalized polarizabilities and moments of nucleon structure functions

2018

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 perturbation theory prediction. We also show how these relations may be useful for a model-independent …

Hadron structure in the description of electromagnetic reactions

2002

The description of electromagnetic reactions at intermediate energies, such as pion electroproduction or (virtual) Compton scattering, traditionally starts from covariant tree-level Feynman diagrams (Born or pole terms). Internal hadron structure is included by means of (on-shell) form factors in the vertices while free propagators are used. To overcome problems with gauge invariance, simple prescriptions, such as, choosing ${F}_{1}^{V}{(q}^{2}{)=F}_{\ensuremath{\pi}}{(q}^{2})$ in pion electroproduction or the ``minimal substitution,'' are used. We discuss the inherent assumptions of such approaches and study the general structure of electromagnetic vertices and propagators for pions and nu…

Proton polarizabilities: status, relevance, prospects

2013

This is a brief review of the status of understanding the proton polarizabilities in chiral perturbation theory and of their relevance to the `proton charge radius puzzle'.

Evaluation of the forward Compton scattering off protons: Spin-independent amplitude

2015

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): $\alpha_{E1}+\beta_{M1}=14.0(2)\times 10^{-4}\,\mathrm{fm}^3$. An analogous sum rule involving the quadrupole polarizabilities of the proton is evaluated too. The spin-…

Spin-dependent sum rules connecting real and virtual Compton scattering verified

2017

We present a detailed derivation of the two sum rules relating the spin polarizabilities measured in real, virtual, and doubly-virtual Compton scattering. For example, the polarizability $\delta_{LT}$, accessed in inclusive electron scattering, is related to the spin polarizability $\gamma_{E1E1}$ and the slope of generalized polarizabilities $P^{(M1,M1)1}-P^{(L1,L1)1}$, measured in, respectively, the real and the virtual Compton scattering. We verify these sum rules in different variants of chiral perturbation theory, discuss their empirical verification for the proton, and prospect their use in studies of the nucleon spin structure.

Nucleon polarizabilities: From Compton scattering to hydrogen atom

2016

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 model-independent relations involving nucleon polarizabilities. We (re-)derive a variety of such rel…

Chiral perturbation theory of hyperfine splitting in muonic hydrogen

2016

We present the leading-order prediction of baryon chiral perturbation theory for the proton polarizability contribution to the 2S hyperfine splitting in muonic hydrogen, and compare with the results of dispersive calculations.

Moments of nucleon structure functions at next-to-leading order in baryon chiral perturbation theory

2014

We obtain leading- and next-to-leading-order predictions of chiral perturbation theory for several prominent moments of nucleon structure functions. These parameter-free results turn out to be in overall agreement with the available empirical information on nearly all of the considered moments, in the region of low momentum transfer $({Q}^{2}l0.3\phantom{\rule{4pt}{0ex}}{\mathrm{GeV}}^{2})$. Especially surprising is the situation for the spin polarizability ${\ensuremath{\delta}}_{LT}$, which thus far was not reproducible in chiral perturbation theory for proton and neutron simultaneously. This problem, known as the ``${\ensuremath{\delta}}_{LT}$ puzzle,'' is not seen in the present calcula…

Generalized polarizabilities of the nucleon in baryon chiral perturbation theory

2017

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 ($ep\to ep\gamma$) 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$\chi$PT), and confront the results with the empirical information. At this order our results are predictions, in the sense that all the parameters are well-known from elsewhere. Within the relatively large unc…

Dissecting the Hadronic Contributions to (g−2)μ by Schwinger’s Sum Rule

2018

The theoretical uncertainty of $(g\ensuremath{-}2{)}_{\ensuremath{\mu}}$ is currently dominated by hadronic contributions. In order to express those in terms of directly measurable quantities, we consider a sum rule relating $g\ensuremath{-}2$ to an integral of a photoabsorption cross section. The sum rule, attributed to Schwinger, can be viewed as a combination of two older sum rules: Gerasimov-Drell-Hearn and Burkhardt-Cottingham. The Schwinger sum rule has an important feature, distinguishing it from the other two: the relation between the anomalous magnetic moment and the integral of a photoabsorption cross section is linear, rather than quadratic. The linear property makes it suitable …

Quark transverse charge densities in the from lattice QCD

2009

Abstract We extend the formalism relating electromagnetic form factors to transverse quark charge densities in the light-front frame to the case of a spin-3/2 baryon and calculate these transverse densities for the Δ ( 1232 ) isobar using lattice QCD. The transverse charge densities for a transversely polarized spin-3/2 particle are characterized by monopole, dipole, quadrupole, and octupole patterns representing the structure beyond that of a pure point-like spin-3/2 particle. We present lattice QCD results for the Δ-isobar electromagnetic form factors for pion masses down to approximatively 350 MeV for three cases: quenched QCD, two-degenerate flavors of dynamical Wilson quarks, and three…

Evaluation of the forward Compton scattering off protons. II. Spin-dependent amplitude and observables

2016

The forward Compton scattering off the proton is determined by substituting the empirical total photoabsorption cross sections into dispersive sum rules. In addition to the spin-independent amplitude evaluated previously [Phys. Rev. D 92, 074031 (2015)], we obtain the spin-dependent amplitude over a broad energy range. The two amplitudes contain all the information about this process, and we, hence, can reconstruct the nonvanishing observables of the proton Compton scattering in the forward kinematics. The results are compared with predictions of chiral perturbation theory where available. The low-energy expansion of the spin-dependent Compton scattering amplitude yields the Gerasimov-Drell…

Light-by-light scattering sum rules constraining meson transition form factors

2012

Relating the forward light-by-light scattering to energy weighted integrals of the \gamma* \gamma -fusion cross sections, with one real photon (\gamma) and one virtual photon (\gamma*), we find two new exact super-convergence relations. They complement the known super-convergence relation based on the extension of the GDH sum rule to the light-light system. We also find a set of sum rules for the low-energy photon-photon interaction. All of the new relations are verified here exactly at leading order in scalar and spinor QED. The super-convergence relations, applied to the \gamma* \gamma -production of mesons, lead to intricate relations between the \gamma \gamma -decay widths or the \gamma…

Lower bound on the proton charge radius from electron scattering data

2019

The proton charge-radius determinations from the electromagnetic form-factor measurements in electron-proton scattering require an extrapolation to zero momentum transfer ($Q^2=0$) which is prone to model-dependent assumptions. We show that the data at finite momentum transfer can be used to establish a rigorous lower bound on the proton charge radius. Using the available $ep$ data at low $Q^2$, we obtain $R_E > 0.850(1)$ fm as the lower bound on the proton radius. This reaffirms the discrepancy between the $ep$ and muonic-hydrogen values, while bypassing the model-dependent assumptions that go into the fitting and extrapolation of the $ep$ data.

Nucleon andΔ(1232)form factors at low momentum transfer and small pion masses

2012

An expansion of the electromagnetic form factors of the nucleon and $\ensuremath{\Delta}(1232)$ in small momentum transfer and pion mass is performed in a manifestly covariant EFT framework consistent with chiral symmetry and analyticity. We present the expressions for the nucleon and $\ensuremath{\Delta}(1232)$ electromagnetic form factors, charge radii, and electromagnetic moments in the framework of $SU(2)$ baryon chiral perturbation theory, with nucleon and $\ensuremath{\Delta}$-isobar degrees of freedom, to next-to-leading order. Motivated by the results for the proton electric radius obtained from the muonic-hydrogen atom and electron-scattering process, we extract values for the seco…

Light-by-light forward scattering amplitudes in Lattice QCD

2017

We present our preliminary results on the calculation of hadronic light-by-light forward scattering amplitudes using vector four-point correlation functions computed on the lattice. Using a dispersive approach, forward scattering amplitudes can be described by $\gamma^* \gamma^* \to$ hadrons fusion cross sections and then compared with phenomenology. We show that only a few states are needed to reproduce our data. In particular, the sum rules considered in this study imply relations between meson$-\gamma\gamma$ couplings and provide valuable information about individual form factors which are often used to estimate the meson-pole contributions to the hadronic light-by-light contribution to …

Reply to "Comment on `Breakdown of the expansion of finite-size corrections to the hydrogen Lamb shift in moments of charge distribution'"

2016

To comply with the critique of the Comment [J. Arrington, arXiv:1602.01461], we consider another modification of the proton electric form factor, which resolves the "proton-radius puzzle". The proposed modification satisfies all the consistency criteria put forward in the Comment, and yet has a similar impact on the puzzle as that of the original paper. Contrary to the concluding statement of the Comment, it is not difficult to find an ad hoc modification of the form factor at low $Q$ that resolves the discrepancy and is consistent with analyticity constraints. We emphasize once again that we do not consider such an ad hoc modification of the proton form factor to be a solution of the puzzl…

Direct calculation of hadronic light-by-light scattering

2015

We report calculations of hadronic light-by-light scattering amplitudes via lattice QCD evaluation of Euclidean four-point functions of vector currents. These initial results include only the fully quark-connected contribution. Particular attention is given to the case of forward scattering, which can be related via dispersion relations to the $\gamma^* \gamma^* \to$ hadrons cross section, and thus allows lattice data to be compared with phenomenology. We also present a strategy for computing the hadronic light-by-light contribution to the muon anomalous magnetic moment.

Predictions of covariant chiral perturbation theory for nucleon polarisabilities and polarised Compton scattering

2015

We update the predictions of the SU(2) baryon chiral perturbation theory for the dipole polarisabilities of the proton, $\{\alpha_{E1},\,\beta_{M1}\}_p=\{11.2(0.7),\,3.9(0.7)\}\times10^{-4}$fm$^3$, and obtain the corresponding predictions for the quadrupole, dispersive, and spin polarisabilities: $\{\alpha_{E2},\,\beta_{M2}\}_p=\{17.3(3.9),\,-15.5(3.5)\}\times10^{-4}$fm$^5$, $\{\alpha_{E1\nu},\,\beta_{M1\nu}\}_p=\{-1.3(1.0),\,7.1(2.5)\}\times10^{-4}$fm$^5$, and $\{\gamma_{E1E1},\,\gamma_{M1M1},\,\gamma_{E1M2},\,\gamma_{M1E2}\}_p=\{-3.3(0.8),\,2.9(1.5),\,0.2(0.2),\,1.1(0.3)\}\times10^{-4}$fm$^4$. The results for the scalar polarisabilities are in significant disagreement with semi-empirical …

Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization

2023

Lattice QCD calculations of the hadronic vacuum polarization (HVP) have reached a precision where the electromagnetic (e.m.) correction can no longer be neglected. This correction is both computationally challenging and hard to validate, as it leads to ultraviolet (UV) divergences and to sizeable infrared (IR) effects associated with the massless photon. While we precisely determine the UV divergence using the operator-product expansion, we propose to introduce a separation scale $\Lambda\sim400\;$MeV into the internal photon propagator, whereby the calculation splits into a short-distance part, regulated in the UV by the lattice and in the IR by the scale $\Lambda$, and a UV-finite long-di…

Causality constraint on bound states and scattering with zero-range force, or do perturbative pions deserve another chance?

2016

Hadronic light-by-light scattering amplitudes from lattice QCD versus dispersive sum rules

2017

The hadronic contribution to the eight forward amplitudes of light-by-light scattering ($\gamma^*\gamma^*\to \gamma^*\gamma^*$) is computed in lattice QCD. Via dispersive sum rules, the amplitudes are compared to a model of the $\gamma^*\gamma^*\to {\rm hadrons}$ cross sections in which the fusion process is described by hadronic resonances. Our results thus provide an important test for the model estimates of hadronic light-by-light scattering in the anomalous magnetic moment of the muon, $a_\mu^{\rm HLbL}$. Using simple parametrizations of the resonance $M\to \gamma^*\gamma^*$ transition form factors, we determine the corresponding monopole and dipole masses by performing a global fit to …

Sum rules for light-by-light scattering

2010

We derive a set of sum rules for the light-by-light scattering and fusion: $\gamma\gamma \to all$, and verify them in lowest order QED calculations. A prominent implication of these sum rules is the superconvergence of the helicity-difference total cross-section for photon fusion, which in the hadron sector reveals an intricate cancellation between the pseudoscalar and tensor mesons. An experimental verification of superconvergence of the polarized photon fusion into hadrons is called for, but will only be possible at $e^+ e^-$ and $\gamma\gamma$ colliders with both beams polarized. We also show how the sum rules can be used to measure various contributions to the low-energy light-by-light …

The anomalous magnetic moment of the muon in the Standard Model

2020

We are very grateful to the Fermilab Directorate and the Fermilab Theoretical Physics Department for their financial and logistical support of the first workshop of the Muon g -2 Theory Initiative (held near Fermilab in June 2017) [123], which was crucial for its success, and indeed for the successful start of the Initiative. Financial support for this workshop was also provided by the Fermilab Distinguished Scholars program, the Universities Research Association through a URA Visiting Scholar award, the Riken Brookhaven Research Center, and the Japan Society for the Promotion of Science under Grant No. KAKEHNHI-17H02906. We thank Shoji Hashimoto, Toru Iijima, Takashi Kaneko, and Shohei Nis…

Lattice QCD calculation of hadronic light-by-light scattering

2015

We perform a lattice QCD calculation of the hadronic light-by-light scattering amplitude in a broad kinematical range. At forward kinematics, the results are compared to a phenomenological analysis based on dispersive sum rules for light-by-light scattering. The size of the pion pole contribution is investigated for momenta of typical hadronic size. The presented numerical methods can be used to compute the hadronic light-by-light contribution to the anomalous magnetic moment of the muon. Our calculations are carried out in two-flavor QCD with the pion mass in the range of 270 to 450MeV, and contain so far only the diagrams with fully connected quark lines.

Analytic structure ofϕ4theory using light-by-light sum rules

2013

Abstract We apply a sum rule for the forward light-by-light scattering process within the context of the ϕ 4 quantum field theory. As a consequence of the sum rule a stringent causality criterion is presented and the resulting constraints are studied within a particular resummation of graphs. Such resummation is demonstrated to be consistent with the sum rule to all orders of perturbation theory. We furthermore show the appearance of particular non-perturbative solutions within such approximation to be a necessary requirement of the sum rule. For a range of values of the coupling constant, these solutions manifest themselves as a physical bound state and a K-matrix pole. For another domain …

A dispersion relation for the pion-mass dependence of hadron properties

2010

We present a dispersion relation in the pion-mass squared, which static quantities (nucleon mass, magnetic moment, etc.) obey under the assumption of analyticity in the entire complex $m_\pi^2$ plane modulo a cut at negative $m_\pi^2$ associated with pion production. The relation is verified here in a number of examples of nucleon and $\Delta$-isobar properties computed in chiral perturbation theory up to order $p^3$. We outline a method to obtain relations for other mass-dependencies, and illustrate it on a two-loop example.

Dissecting the Hadronic Contributions to (g−2)μ by Schwinger’s Sum Rule

2018

The theoretical uncertainty of (g−2)μ is currently dominated by hadronic contributions. In order to express those in terms of directly measurable quantities, we consider a sum rule relating g−2 to an integral of a photoabsorption cross section. The sum rule, attributed to Schwinger, can be viewed as a combination of two older sum rules: Gerasimov-Drell-Hearn and Burkhardt-Cottingham. The Schwinger sum rule has an important feature, distinguishing it from the other two: the relation between the anomalous magnetic moment and the integral of a photoabsorption cross section is linear, rather than quadratic. The linear property makes it suitable for a straightforward assessment of the hadronic c…

The two-photon decay of X(6900) from light-by-light scattering at the LHC

2022

The LHCb Collaboration has recently discovered a structure around 6.9 GeV in the double-$J/\psi$ mass distribution, possibly a first fully-charmed tetraquark state $X(6900)$. Based on vector-meson dominance (VMD) such a state should have a significant branching ratio for decaying into two photons. We show that the recorded LHC data for the light-by-light scattering may indeed accommodate for such a state, with a $\gamma \gamma$ branching ratio of order of $10^{-4}$, which is larger even than the value inferred by the VMD. The spin-parity assignment $0^{-+}$ is in better agreement with the VMD prediction than $0^{++}$, albeit not significantly at the current precision. Further light-by-light…

Polarisabilities of the nucleon in baryon chiral perturbation theory and beyond

2019

We review the recent baryon chiral perturbation theory results for the nucleon polarisabilities that describe the different regimes of nucleon Compton scattering --- real, virtual, and doubly virtual. We stress the importance of the empirical verification of the theory in the context of the calculation of the inelastic nucleon structure corrections, such as the two-photon exchange contributions. We also discuss the recently obtained constraints that relate the different regimes of nucleon Compton scattering and can provide additional information on the nucleon structure.

Pseudoscalar-Meson Contributions to g-2 via Schwinger's Sum Rule

2019

The Schwinger sum rule is presented as a new promising tool to study the hadronic contributions to the muon anomalous magnetic moment. In particular, we show preliminary results for the light-by-light scattering contribution of pseudoscalar mesons.

Gold-plated moments of nucleon structure functions in baryon chiral perturbation theory

2014

We obtain leading- and next-to-leading order predictions of chiral perturbation theory for several prominent moments of nucleon structure functions. These free-parameter free results turn out to be in overall agreement with the available empirical information on nearly all of the considered moments, in the region of low-momentum transfer ($Q^2 < 0.3$ GeV$^2$). Especially surprising is the situation for the spin polarizability $\delta_{LT}$, which thus far was not reproducible in chiral perturbation theory for proton and neutron simultaneously. This problem, known as the "$\delta_{LT}$ puzzle," is not seen in the present calculation.