Nucleon polarizabilities in covariant baryon chiral perturbation theory with explicit Δ degrees of freedom
NNucleon polarizabilities in covariant baryon chiral perturbationtheory with explicit ∆ degrees of freedom M. Thürmann, ∗ E. Epelbaum, † A. M. Gasparyan,
1, 2, ‡ and H. Krebs § Institut für Theoretische Physik II,Ruhr-Universität Bochum, D-44780 Bochum, Germany NRC “Kurchatov Institute” - ITEP,B. Cheremushkinskaya 25, 117218 Moscow, Russia
Abstract
We compute various nucleon polarizabilities in chiral perturbation theory implementing the ∆ -full ( ∆ -less) approach up to order (cid:15) + q ( q ) in the small-scale (chiral) expansion. The calculationis carried out using the covariant formulation of χ PT by utilizing the extended on-mass shell renor-malization scheme. Except for the spin-independent dipole polarizabilities used to fix the values ofcertain low-energy constants, our results for the nucleon polarizabilities are pure predictions. Wecompare our calculations with available experimental data and other theoretical results. The im-portance of the explicit treatment of the ∆ degree of freedom in the effective field theory descriptionof the nucleon polarizabilities is analyzed. We also study the convergence of the /m expansionand analyze the efficiency of the heavy-baryon approach for the nucleon polarizabilities. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ nu c l - t h ] J u l . INTRODUCTION Understanding the structure of the nucleon is one of the key challenges in the physics ofstrong interactions, and quantum chromodynamics (QCD) in particular. One of the mostdirect ways to access the nucleon structure is to use electromagnetic probes. In the presentwork we focus on the nucleon polarizabilites, which characterize the (second-order) responseof the nucleon to an applied electromagnetic field. In recent decades, the nucleon polariz-abilites have been intensively studied both experimentally and theoretically. At the moment,the dipole scalar (spin-independent) polarizabilites of both the proton and the neutron aredetermined fairly well by various methods [1] as well as the forward and backward spinpolarizabilites of the proton [2–4]. Recent measurements of double-polarized Compton scat-tering at the Mainz Microtron allowed one to extract also other proton spin polarizabilites[5, 6].There are also experimental results for some of the generalized ( Q -dependent) polariz-abilites of the proton and the neutron [7–11].From the theoretical side, a significant progress has been made using lattice simulations[12–18], i.e. by directly solving QCD in the non-perturbative regime on a discrete Euclideanspace-time grid. However, one is not yet in the position to perform an accurate determinationof the nucleon polarizabilites calculated on the lattice for physical pion masses.Another systematic theoretical approach is provided by effective field theories, in partic-ular, by chiral perturbation theory ( χ PT), see [19, 20] for pioneering studies of the nucleon’selectromagnetic polarizabilities in this framework. Chiral perturbation theory is an effectivefield theory of the standard model consistent with its symmetries and the ways they arebroken. It allows one to expand hadronic observables in powers of the small parameter q defined as the ratio of the typical soft scales such as the pion mass M and external-particle3-momenta | (cid:126)p | and the hard scale Λ b of the order of the ρ -meson mass. The effective chiralLagrangian is expanded in powers of derivatives and the pion mass. In the nucleon sector,an additional complication arises due to the presence of an extra mass scale, namely thenucleon mass, which can potentially break the power counting. One way to circumvent thisproblem is to perform the /m expansion on the level of the effective Lagrangian. This leadsto the so-called heavy-baryon approach. The heavy-baryon scheme has been intensively usedfor the analysis of many hadronic reactions including the nucleon Compton scattering (and,therefore, nucleon polarizabilites), see e.g. [21–24] and [21, 25] for review articles. The heavy-baryon expansion is, however, known to violate certain analytic properties of the S -matrix[26], which may lead to a slower convergence of the chiral expansion. This feature has alsobeen observed in the actual calculations of the nucleon polarizabilites.An alternative approach to processes involving nucleons consists in keeping the covariantstructure of the effective Lagrangian and absorbing the power-counting breaking terms bya redefinition of the lower order low-energy constants [26, 27]. In this work, we adopta version of the covariant approach known as the extended on-mass-shell renormalizationscheme (EOMS) [27, 28]. When necessary, we will slightly modify this scheme in order toenable a direct comparison to the heavy-baryon results (see e.g. [29]).Another obstacle for the rapid convergence of the chiral expansion in the single-nucleonsystems is the presence of the ∆ (1232)-resonance that is located close to the pion-nucleonthreshold and is known to strongly couple to the pion-nucleon channel. This introducesanother small scale ∆ ≡ m ∆ − m ≈ M , which leads to the appearance of terms of order O ( M/ ∆) in the expansion of observables. A natural way to improve this situation is to2nclude the ∆ -isobar field explicitly into the effective Lagrangian. We follow here the so-called small-scale-expansion (SSE) scheme by treating the scale ∆ on the same footing as M or | (cid:126)p | [30]. The universal expansion parameter is then called (cid:15) . For recent applicationsof this theoretical approach to various processes in the single-nucleon sector see [31–34]. Inthis work, we compare the efficiency and convergence of both the ∆ -full and ∆ -less schemesby calculating various nucleon polarizabilites up to orders (cid:15) + q and q , respectively. Ouranalysis is particularly instructive since we calculate a set of higher-order polarizabilites,which do not depend on any free parameters. We also perform the /m expansion of ourresults in order to analyze the efficiency of the heavy baryon approach for the nucleonpolarizabilites.There is an alternative scheme for the chiral expansion in the presence of explicit ∆ degreeof freedom [35] called the δ -counting. The main difference from the small-scale expansionis a different power counting assignment for the ∆ -nucleon mass difference ∆ by assumingthe hierarchy of scales M (cid:28) ∆ (cid:28) Λ b . In such an approach, loop diagrams with several ∆ -lines are suppressed in contrast with the calculations within the small-scale expansion,see [36–39] for recent applications. We compare our results with the ones obtained withinthe δ -counting and discuss the importance of such contributions.As a stringent test of our scheme, we also compare our results with the fixed- t dispersion-relations analyses of [40–45]. This method is based solely on the principles of analyticityand unitarity and therefore defines an important benchmark for theoretical approaches.Our paper is organized as follows. The effective Lagrangian and the power countingrelevant for the construction of the Compton scattering amplitude within χ PT as well asthe renormalization of the low-energy constants (LECs) are given in Section II. In Section III,the formalism for the Compton scattering is described and the nucleon polarizabilities areintroduced. The numerical results for the nucleon polarizabilites are presented in Section IV.We summarize our results in Section V. Appendices A-F collect the analytic expressions forthe nucleon polarizabilites.
II. COMPTON SCATTERING IN CHIRAL PERTURBATION THEORYA. Effective Lagrangian
The description of nucleon Compton scattering in χ PT relies on an effective Lagrangian.The effective Lagrangian relevant for the problem at hand to the order we are workingconsists of the following terms L eff = L (2) ππ + L (4) ππ + L (4) W ZW + L (1) πN + L (2) πN + L (3) πN + L (4) πN + L (1) πN ∆ + L (2) πN ∆ + L (1) π ∆∆ , (1)where L W ZW stays for the Wess-Zumino-Witten term [48, 49]. This Lagrangian is built interms of the pion field through the SU(2) matrix U = u = 1 + iF (cid:126)τ · (cid:126)π − F (cid:126)π + . . . ( F is thepion decay constant in the chiral limit), the nucleon field N and the Rarita-Schwinger-spinor ∆ -field ψ µi . The electromagnetic field A µ enters via v µ = − (1 + τ ) eA µ ( e > is the protoncharge). For the application of a scheme that combines effective field theory with dispersion-relations techniquefor the problem under consideration see [46, 47]. L (1) πN = ¯ N ( i /D − m + g A /uγ ) N (2) L (2) πN = c ¯ N (cid:104) χ + (cid:105) N − c m (cid:0) ¯ N (cid:104) u µ u ν (cid:105) D µν N + h.c (cid:1) + c (cid:104) u · u (cid:105) + c i ¯ N [ u µ , u ν ] σ µν N + c m ¯ N F + µν N + c m ¯ N (cid:10) F + µν (cid:11) σ µν N + . . . (3) L (3) πN = d m (cid:16) i (cid:104) D µ , ˜ F + µν (cid:105) D ν + h.c. (cid:17) + d m (cid:0) i (cid:2) D µ , (cid:10) F + µν (cid:11)(cid:3) D ν + h.c. (cid:1) + . . . (4) L (4) πN = ¯ N (cid:16) − e (cid:2) D α , (cid:2) D α , (cid:10) F + µν (cid:11)(cid:3)(cid:3) σ µν − e (cid:104) D α (cid:104) D α , ˜ F + µν (cid:105)(cid:105) σ µν − e (cid:10) F + µν (cid:11) (cid:104) χ + (cid:105) σ µν − e F + µν (cid:104) χ + (cid:105) σ µν + e (cid:10) F + µν (cid:11) (cid:10) F + µν (cid:11) + e ˜ F + µν (cid:10) F + µν (cid:11) + e (cid:68) ˜ F + µν ˜ F + µν (cid:69) + e (cid:10) F − µν F − µν + F + µν F + µν (cid:11) (cid:17) N + (cid:104) ¯ N (cid:16) − e m (cid:10) F + αµ (cid:11) (cid:10) F + αν (cid:11) − e m ˜ F + αµ (cid:10) F + αν (cid:11) − e m (cid:68) ˜ F + αµ ˜ F + αν (cid:69) − e m (cid:10) F − αµ F − αν + F + αµ F + αν (cid:11) (cid:17) { D µ , D ν } N + h.c. (cid:105) , (5)and the terms relevant for the O ( (cid:15) ) calculations from the πN ∆ and π ∆∆ Lagrangians: L (1) πN ∆ = h A (cid:0) ¯Ψ µi (cid:104) τ i u µ (cid:105) N + h.c. (cid:1) , L (2) πN ∆ = b (cid:0) i ¯Ψ µi (cid:10) τ i F + µα (cid:11) γ α γ N + h.c. (cid:1) , L (1) π ∆∆ = ¯Ψ µi (cid:16) i (cid:8) [ γ µ , γ ν ] , γ α (cid:9) D αij − m ∆ γ µ , γ ν ] δ ij (cid:17) Ψ νj . (6)The covariant derivatives and the chiral vielbein are defined as follows: D µ = ∂ µ + Γ µ , D µij = ( ∂ µ + Γ µ ) δ ij − i(cid:15) ijk (cid:104) τ k Γ µ (cid:105) , Γ µ = 12 (cid:2) u † ∂ µ u + u∂ µ u † − i ( u † v µ u + uv µ u † ) (cid:3) ,u µ = i (cid:2) u † ∂ µ u − u ∂ µ u † − i ( u † v µ u − u v µ u † ) (cid:3) , (7)while the vector field strength tensors are given by F ± µν = uv µν u † ± u † v µν u , ˜ F + µν = F + µν − (cid:10) F + µν (cid:11) , v µν = ∂ µ v ν − ∂ ν v µ . (8)Notice that the definition of b differs from the one in [30] by a factor of m but is consistentwith that of [31]. All redundant off-shell parameters in L πN ∆ and L π ∆∆ are set to zero (seethe discussion in [51, 52]).For the remaining terms in Eq. (1) and further notations we refer the reader to [30, 50,53, 54]). 4 . Power counting To calculate the nucleon Compton-scattering amplitude one needs to select the relevantFeynman diagrams according to their order D , which is determined by the power-countingformula [55] D = 1 + 2 L + (cid:88) n (2 n − V M n + (cid:88) d ( d − V Bd , (9)where L is the number of loops, V M n is the number of vertices from L (2 n ) ππ and V Bd is thetotal number of vertices from L ( d ) πN , L ( d ) πN ∆ and L ( d ) π ∆∆ . Note that in the small-scale-expansionscheme, the nucleon and delta lines are counted on the same footing. In this work, we labelpurely nucleonic contributions (containing no ∆ lines) as q D and those involving ∆ ’s as (cid:15) D .The tree-level diagrams are shown in Fig. 1. Most of the nucleon pole diagrams donot contribute to the polarizabilites (as the Born terms are subtracted by definition, seeSection III) but are necessary for the renormalization of subdiagrams. Only the nucleonpole diagrams with the d and d vertices generate a small residual non-pole contributionto the generalized polarizabilites due to the specific form of the corresponding effectiveLagrangian.On the other hand, the ∆ -pole graph provides a very important contribution to thenucleon polarizabilites. The pion t -channel exchange diagram with the anomalous π γγ coupling is not included in the definition of the polarizabilites either and is, therefore, notshown. Also not shown are the γN → γN contact terms from L (4) πN . FIG. 1: Tree-level diagrams for nucleon Compton scattering which are taken into account in ouranalysis. Vertices of order O ( q ) , O ( q ) , O ( q ) and O ( q ) are denoted by dots, circles, squares anddiamonds, respectively. Solid, wavy and double lines refer to nucleons, photons and ∆ -isobars,respectively. Time-reversed and crossed diagrams as well as the diagrams with insertions of thenucleon self-energy contact terms are not shown. Loop diagrams start to contribute at order q ( (cid:15) ). The corresponding sets of diagramsare shown in Fig. 2 for the q -loops and in Fig. 3 for the (cid:15) -loops. The subleading q -loopdiagrams are shown in Fig. 4. 5 IG. 2: O ( q ) loop diagrams for nucleon Compton scattering. Dashed lines refer to pions. Allvertices are from the leading order Lagrangians L (2) ππ and L (1) πN . Time-reversed and crossed diagramsare not shown.FIG. 3: O ( (cid:15) ) loop diagrams for nucleon Compton scattering. All vertices are from the leadingorder Lagrangians L (2) ππ , L (1) πN , L (1) πN ∆ and L (1) π ∆∆ . Double lines denote the ∆ . Time-reversed andcrossed diagrams are not shown. C. Renormalization
The ultraviolet divergencies appearing in loop integrals are treated by means of dimen-sional regularization. Divergent parts of the integrals are cancelled by the correspondingcounter terms of the Lagrangian, and the resulting amplitude is expressed in terms of thefinite quantities such as renormalized low-energy constants, physical masses and coupling6
IG. 4: O ( q ) loop diagrams for nucleon Compton scattering. Dots denote the leading order verticesand circles denote the vertices from L (2) πN . Time-reversed and crossed diagrams are not shown. constants. Due to the presence of an additional hard scale (the nucleon or ∆ mass), bary-onic loops contain power-counting-violating terms [56]. Since such terms are local, they canbe absorbed by a redefinition of the low-energy constants of the effective Lagrangian. Inthis work, we adopt the extended on-mass-shell renormalization scheme (EOMS) [27] in acombination with on-shell renormalization conditions for the nucleon mass and magneticmoments.For the nucleon mass and wave-function renormalization, we impose the on-shell condi-tions Σ N ( m N ) = 0 and Σ (cid:48) N ( m N ) = 0 , (10)with Σ N ( /p ) being the nucleon self-energy. By doing so, we fix the bare nucleon mass m and the field normalization factor Z N . The explicit formulae relating the physical and bareparameters can be found in [32]. In what follows, we will denote the physical nucleon massby m , which will not lead to a confusion since the bare nucleon mass will not be discussedanymore.In a complete analogy with the nucleon field, we renormalize the ∆ field. However, at theorder we are working there are no loop corrections to the ∆ self-energy. For the calculationof the static nucleon polarizabilites, we us the real Breit-Wigner mass of the ∆ . The precise7alue of the renormalized ∆ mass is irrelevant under the kinematic conditions considered.On the other hand, for calculation of the dynamical nucleon polarizabilites, in order tobe able to describe the ∆ region, we implement the complex-mass scheme [57, 58] for the ∆ resonance and use the complex ∆ pole mass taking the resonance width into accountexplicitly.For the renormalized constants ¯ c and ¯ c , we use the on-shell condition for the nucleonmagnetic moments: ¯ c = κ p − κ n and ¯ c = κ n . (11)The explicit relation between ¯ c and ¯ c and the bare constants c and c is given in Ap-pendix E.For the remaining low-energy constants ξ i we employ the EOMS renormalization scheme.The renormalized LECs ¯ ξ i are related to the bare quantities as follows: ξ i = ¯ ξ i − β ξ i F A ( M )2 M + ∆ ξ i π F ,ξ i ∈ { d , d , e , e , e , e , e , e , e , e , e , e , e } , (12)with the β functions: β d = − − g A h A , β d = 5 h A ,β e = 0 , β e = 1 − g A + 4 c m ,β e = β e = 0 , β e + β e + β e = c , β e + β e + β e = − c , (13)and the finite shifts ∆ d = − g A c , ∆ d = 3 g A ( c + 2 c )16 . (14)The constants e i do not receive finite shifts due to the power-counting violation because wedo not consider loop diagrams of order higher than O ( q ) . The finite shifts for d and d reproduce those obtained in [59] (note a different definition of the LECs). The constants e and e do not contribute to the nucleon polarizabilites after subtracting the Born terms.Nevertheless, we provide the corresponding β functions for completeness. The LECs e , e , e , e , e , e enter the nucleon Compton scattering amplitude only in the linearcombinations e + e + e and e + e + e , for which the β functions are given inEq. (13).The pion tadpole function in d ≈ dimensions is equal to (see Eq. (F1)) A ( M ) = − M (cid:18) ¯ λ + 132 π ln (cid:18) M µ (cid:19)(cid:19) , ¯ λ = 116 π (cid:18) d − γ E − ln(4 π ) − (cid:19) . (15)8ere, γ E is the Euler-Mascheroni constant and µ is the renormalization scale. The divergen-cies remaining after the renormalization of the LECs are treated in the (cid:103) M S [27, 53] scheme,i.e. we set ¯ λ = 0 . We have checked that the residual renormalization scale dependence ofthe amplitude is of a higher order than we are working.In what follows, we will omit the bars over the renormalized LECs. III. FORMALISM
We consider nucleon Compton scattering γN → γN with the momenta of the initial(final) proton and photon denoted as p ( p (cid:48) ) and q ( q (cid:48) ), respectively. We study the cases ofreal Compton scattering with q = q (cid:48) = 0 and of double virtual Compton scattering with Q = − q = − q (cid:48) .In order to calculate the nucleon polarizabilites, we decompose the scattering amplitude T ( q , z, ω ) in the Breit frame, where ω and z are the photon energy and scattering angle, interms of twelve functions A i : T ( q , z, ω ) = 2 m (cid:88) i =1 A i ( q , z, ω ) χ i , (16)with χ = (cid:126)(cid:15) · (cid:126)(cid:15) ∗ ,χ = (ˆ q × (cid:126)(cid:15) ) · (ˆ q (cid:48) × (cid:126)(cid:15) (cid:48) ) ,χ = ˆ q · (cid:126)(cid:15) ˆ q · (cid:126)(cid:15) (cid:48)∗ + ˆ q (cid:48) · (cid:126)(cid:15) ˆ q (cid:48) · (cid:126)(cid:15) (cid:48)∗ ,χ = ˆ q · (cid:126)(cid:15) ˆ q (cid:48) · (cid:126)(cid:15) (cid:48)∗ ,χ = iσ · (cid:126)(cid:15) × (cid:126)(cid:15) (cid:48)∗ ,χ = iσ · (ˆ q × (cid:126)(cid:15) ) × (ˆ q (cid:48) × (cid:126)(cid:15) (cid:48) ) ,χ = i (ˆ q · (cid:126)(cid:15) × (cid:126)(cid:15) (cid:48)∗ (cid:126)σ · ˆ q + ˆ q (cid:48) · (cid:126)(cid:15) × (cid:126)(cid:15) (cid:48)∗ (cid:126)σ · ˆ q (cid:48) ) ,χ = i (ˆ q · (cid:126)(cid:15) × (cid:126)(cid:15) (cid:48)∗ (cid:126)σ · ˆ q (cid:48) + ˆ q (cid:48) · (cid:126)(cid:15) × (cid:126)(cid:15) (cid:48)∗ (cid:126)σ · ˆ q ) ,χ = i ˆ q · (cid:126)(cid:15) ˆ q (cid:48) · (cid:126)(cid:15) (cid:48)∗ (cid:126)σ · ˆ q × ˆ q (cid:48) ,χ = i (ˆ q · (cid:126)(cid:15) ˆ q · (cid:126)(cid:15) (cid:48)∗ (cid:126)σ · ˆ q × ˆ q (cid:48) + ˆ q (cid:48) · (cid:126)(cid:15) ˆ q (cid:48) · (cid:126)(cid:15) (cid:48)∗ (cid:126)σ · ˆ q × ˆ q (cid:48) ) ,χ = i (ˆ q (cid:48) · (cid:126)(cid:15) (cid:48)∗ (cid:126)σ · (cid:126)(cid:15) × ˆ q − ˆ q · (cid:126)(cid:15) (cid:126)σ · (cid:126)(cid:15) (cid:48)∗ × ˆ q (cid:48) ) ,χ = i (ˆ q (cid:48) · (cid:126)(cid:15) (cid:48)∗ (cid:126)σ · (cid:126)(cid:15) × ˆ q (cid:48) − ˆ q · (cid:126)(cid:15) (cid:126)σ · (cid:126)(cid:15) (cid:48)∗ × ˆ q ) . (17)The initial (final) photon polarization vector (cid:15) µ ( (cid:15) (cid:48) µ ) is defined in the Coulomb gauge ( (cid:15) = (cid:15) (cid:48) = 0 ). The amplitude (16) is supposed to be sandwiched between the Pauli spinors of theinitial and final nucleon.Given the presence of the Pauli matrices (cid:126)σ in Eq. (17), one can see that there are fourspin-independent structures χ − χ and eight spin-dependent structures χ − χ . All χ i obey crossing-invariance. For real Compton scattering, only χ , χ , χ , χ , χ , χ survive.The Born terms have to be subtracted from the amplitude as explained, e.g., in [60] inorder to exclude the contributions with unexcited nucleons in the intermediate state. Thisprocedure essentially reduces to subtracting the tree-level Q = 0 nucleon-pole diagramswith the nucleon charge and magnetic moments replaced by the full Dirac and Pauli formfactors calculated consistently within our scheme applying the same power counting. The9nomalous pion t -channel exchange diagram is also excluded from the definition of thepolarizabilites.The amplitudes A i can be expressed in terms of the nucleon polarizabilites by performingan expansion in ω around ω = 0 : A ( ω ) = 4 πE N m (cid:20) α E1 ω + ω
12 (2 zα E2 − β M2 + 12 α E1 ν ) + ω (cid:0) (30 z − α E3 − zβ M3 + 450 zα E2 ν − β M2 ν + 2700 α E1 ν ) + O ( ω ) (cid:21) ,A ( ω ) = 4 πE N m (cid:20) β M1 ω + ω
12 (2 zβ M2 − α E2 + 12 β M1 ν ) + ω (cid:0) (30 z − β M3 − zα E3 + 450 zβ M2 ν − α E2 ν + 2700 β M1 ν ) + O ( ω ) (cid:21) ,A ( ω ) = 4 πE N m (cid:20) ( γ E1E1 − γ E1M2 ) ω + ω zγ E2E2 + 5 γ E1E1 ν − zγ E2M3 − γ E1M2 ν + 2 γ M2E3 + 5 γ M2M2 ) + O ( ω ) (cid:21) ,A ( ω ) = 4 πE N m (cid:20) ( γ M1M1 − γ M1E2 ) ω + ω zγ M2M2 + 5 γ M1M1 ν − zγ M2E3 − γ M1E2 ν + 2 γ E2M3 + 5 γ E2E2 ) + O ( ω ) (cid:21) ,A ( ω ) = 4 πE N m (cid:20) γ E1M2 ω + ω zγ E2M3 + 5 γ E1M2 ν − γ M2E3 − γ M2M2 ) + O ( ω ) (cid:21) ,A ( ω ) = 4 πE N m (cid:20) γ M1E2 ω + ω zγ M2E3 + 5 γ M1E2 ν − γ E2M3 − γ E2E2 ) + O ( ω ) (cid:21) , (18)where E N is the nucleon energy. We also introduce the linear combinations correspondingto the forward and backward spin polarizabilites γ and γ π γ = − γ E1E1 − γ M1M1 − γ E1M2 − γ M1E2 ,γ π = − γ E1E1 + γ M1M1 − γ E1M2 + γ M1E2 , (19)the higher-order forward spin polarizabilty ¯ γ = − γ E1E1 ν − γ M1M1 ν − γ M1E2 ν − γ E1M2 ν − γ E2E2 − γ M2M2 − ( γ E2M3 + γ M2E3 ) , (20)as well as the longitudinal-transverse spin polarizability δ LT = − d dω (cid:26) m πE N (cid:104) A ( ω ) + A ( ω ) + A ( ω ) (cid:105)(cid:27) ω =0 . (21)There are similar but different amplitude decompositions used in the literature, whichleads to different relations of those amplitudes to the nucleon polarizabilites. For easeof comparison, we provide the transformation matrix from the vector of amplitudes10 this work = ( A , A , A , A , A , A ) defined in Eq. (17) to the vector of amplitudes A LMP =( A , A , A , A , A , A ) considered in [61] A LMP = LA this work , L = z − z z − −
10 0 0 0 − . (22)In this work, we also analyze the so-called dynamical polarizabilites defined in terms ofthe center-of-mass multipoles as follows (see, e.g., [44, 61–63]): α E l ( ω ) = l (2 l − l + 1) f l + EE + lf l − EE ω l , β M l ( ω ) = l (2 l − l + 1) f l + MM + lf l − MM ω l ,γ E l E l ( ω ) = f l + EE − f l − EE ω l +1 , γ E l M l ± ( ω ) = (2 l ± f l ± EM ω l ± ,γ M l M l ( ω ) = f l + MM − f l − MM ω l +1 , γ M l E l ± ( ω ) = (2 l ± f l ± ME ω l ± , (23)for l = 1 , . Note that in contrast to the equations above, in Eq. (23), ω denotes thecenter-of-mass photon energy. IV. RESULTS
We are now in the position to present our numerical results for various proton and neutronpolarizabilities calculated up to order O ( (cid:15) + q ) . Specifically, we consider the followingpolarizabilities: spin-independent (scalar) dipole, quadrupole, octupole, dispersive dipoleand quadrupole polarizabilities as well as dipole, quadrupole and dispersive dipole spinpolarizabilities. We also discuss selected generalized (i.e. Q -dependent) and dynamical(i.e. energy-dependent) polarizabilities.As already mentioned above, most of the results we present are pure predictions and con-tain no free parameters. The only exceptions are the spin-independent dipole polarizabilities α E1 and β M1 at order O ( q ) or O ( (cid:15) + q ) , which are fitted to the experimental values. Allremaining parameters are taken from other processes and are collected in Tables I, II andIII.In the course of the calculation we have used our own code written in M athematica [65] and FORM [66] for the analytical calculation of Feynman diagrams. The numericalevaluation of loop integrals have been performed with help of the
Mathematica package
Package - X [67]. We have also used our own Fortran code for estimating the theoreticalerrors.For our complete results at order O ( (cid:15) + q ) , we also provide estimations of the theoreticalerrors originating from two sources, namely the uncertainties in the input parameters andthe errors caused by the truncation of the small-scale expansion. For the latter uncertainty,we adopt the Bayesian model used in [68, 69] based on the ideas developed in [70–72], see11 − EM M [MeV] F π [MeV] m [MeV] m ∆ [MeV] g A c c h A b [ m − ]137 .
036 138 .
04 92 .
21 938 . .
27 3 . − .
913 1 . − . TABLE I: Parameters used in the current work. The values of α EM , M , m , m ∆ , g A , F π are takenfrom [1]. The LECs c and c are related to the proton and neutron magnetic moment and d and d with the proton and neutron charge radii [59]. The values of the LECs b and h A are extractedfrom the electromagnetic and strong width of the ∆ -resonance, respectively, see [31] for details andexplicit expressions. For the static polarizabilities we use the real ∆ mass as given in the Table,whereas for the generalized polarizabilities we use the pole mass m ∆ = (1210 − i ) MeV. q q (cid:15) q + (cid:15) c [ m − ] − . ± . − . ± . − . ± . − . ± . c [ m − ] 2 . ± .
03 3 . ± .
03 0 . ± .
11 0 . ± . c [ m − ] − . ± . − . ± . − . ± . − . ± . TABLE II: Numerical values of the low energy constants used in the current work as determined bymatching the solution of Roy-Steiner equations for πN scattering [64] to chiral perturbation theoryin [33]. The values for q + (cid:15) correspond to the (cid:15) calculation of [33]. [34] for a recent application to radiative pion photoproduction. The observables are assumedto be expanded in parameter Q given by Q = max (cid:18) M eff Λ b , (cid:112) Q Λ b , ω Λ b (cid:19) , (24)where Q on the right-hand side is the virtuality of the photon, and ω is the photon energyin the case of dynamical polarizabilities. The soft and hard scales are chosen to be M eff =200 MeV and Λ b = 700 MeV in accordance with [73]. Following [68, 69], we utilize theGaussian prior distribution for the expansion coefficients c i : pr( c i | ¯ c ) = 1 √ π ¯ c e − c i / (2¯ c ) , pr(¯ c ) = 1ln(¯ c > / ¯ c < ) 1¯ c θ (¯ c − ¯ c < ) θ (¯ c > − ¯ c ) , (25)with the cut offs ¯ c < = 0 . and ¯ c > = 10 . Further details on the employed Bayesian modelcan be found in [68, 69].In the following sections, we provide a detailed comparison of our results with the avail-able experimental/empirical data as well as with other theoretical approaches based onchiral perturbation theory and on fixed- t dispersion relations. We also discuss generalizedpolarizabilities, investigate the convergence pattern of the /m -expansion for the calculatedpolarizabilities and compare the results of covariant χ PT with the heavy baryon approach.Last but not least, we emphasize that the resulting large absolute numerical values of theoctupole polarizabilites are merely due to the numerical factors in their definition, whichmakes them consistent with the definition of the polarizabilites for composite systems.12 q (cid:15) q + (cid:15) d [ m − ] − . [59] − . [59] − . ± . − . ± . d [ m − ] − . [59] − . [59] − . ± . − . ± . e [ m − ] . . . − . ± . . . . − . ± . e [ m − ] . . . − . ± . . . . − . ± . e + e + e ) [ m − ] . . . − . ± . . . . − . ± . e + e + e ) [ m − ] . . . − . ± . . . . . ± . TABLE III: Numerical values of the low-energy constants obtained from the fit to the empiricalvalues of the electric radius of the proton and neutron ( d i ) and the proton and neutron spin-independent polarizabilites ( e i ). Note that the e i enter at fourth order and therefore do not havevalues for q and (cid:15) . A. Scalar dipole polarizabilites
We start by considering the spin-independent dipole nucleon polarizabilites α E1 and β M1 .The results of the calculations at order O ( q ) and O ( (cid:15) + q ) as well as the individualcontributions from orders O ( q ) , O ( q ) pion-nucleon loops, O ( (cid:15) ) π ∆ -loops and tree-level ∆ -pole graphs are presented in Table IV. At order O ( q ) , there appear low energy constants Proton Neutron α E1 β M1 α E1 β M1 q (without ∆ ) . − .
85 9 . − . q (without ∆ ) .
16 4 .
35 2 .
09 4 . Total (without ∆ ) .
20 2 .
50 11 .
60 3 . q . − .
85 9 . − . (cid:15) π ∆ loop − .
45 5 .
54 2 .
78 0 . (cid:15) ∆ tree − .
78 11 . − .
78 11 . q . − .
16 3 . − . Total .
20 2 .
50 11 .
60 3 . O ( p ) π N loops [61] . − . . − . O ( p / ) π ∆ loops [61] . − . . − . pole [61] − . . − . . Total [61] . ± . . ± . . ± . . ± . Fixed- t DR [40, 42] . . . . HB χ PT fit [74] . ± .
50 3 . ± .
50 11 . ± . . ± . B χ PT fit [75] . ± . . ± . . . . . . . PDG [1] . ± . . ± . . ± . . ± . TABLE IV: Numerical values for the spin-independent dipole polarizabilities of the proton and theneutron in − fm . The values are compared with the results calculated within the δ -countingscheme and obtained using fixed- t dispersion relations.
13n the effective Lagrangian that contribute to the nucleon Compton scattering. We adjustfour relevant linear combinations of them ( e , e , e + e + e and e + e + e ) insuch a way as to reproduce the empirical values of the proton and neutron spin-independentdipole polarizabilities, see Table III.In the case of the ∆ -less theory, the contribution at order O ( q ) for the electric polariz-ability α E1 of the proton (neutron) is about two (five) times smaller than the one at order O ( q ) , which is an indication of a reasonable convergence of the chiral expansion. For themagnetic polarizabilities β M1 , due to some cancellations among O ( q ) loops, the contribu-tions at order O ( q ) are larger than the ones at order O ( q ) but are, nevertheless, comparablewith those for the α E1 .In the ∆ -full scheme, the O ( q ) terms (that differ from the ones in the ∆ -less case bythe values of c i ’s and e i ’s) are significantly larger. This feature can be traced back to thesizable O ( (cid:15) ) contributions, especially from the ∆ -pole tree-level diagrams, that need tobe compensated by adjusting the relevant contact terms. Such contributions appear to bedemoted to higher orders in the ∆ -less scheme. Their importance for other polarizabiliteswill, however, be demonstrated below. Thus, a seemingly better convergence of the ∆ -lessapproach for the dipole polarizabilites can be argued to be accidental. Notice further thatthe convergence issues are not really relevant for the dipole spin-independent polarizabilitesat the order we are working due to the presence of the corresponding compensating contactterms in the Lagrangian.In Table IV, we also provide for comparison the values for the dipole spin-independentpolarizabilites obtained by analyzing experimental data using fixed- t dispersion relations [40,42], and by fitting experimental data employing various versions of the δ -counting schemes(with the loop diagrams calculated utilizing the covariant [75] or heavy-baryon approach[74]). It is particularly instructive to compare our results with [61], where the individualcontributions calculated within the δ -counting scheme are presented. Such a comparisonallows one to analyze the importance of the explicit ∆ degrees of freedom and the sensitivityof the results to employed counting schemes for the ∆ -nucleon mass difference. There aretwo main sources of differences between our approach and the one used in [61] (apart fromslightly different numerical values of the coupling constants). First, different terms in theeffective Lagrangian corresponding to the γN ∆ vertex are used. The γN ∆ Lagrangian of[61] contains two terms with the so-called magnetic and electric γN ∆ -couplings g M and g E : L γN ∆ = 3 e m ( m ∆ + m ) ¯ N T † ( ig M ˜ F µν − g E γ F µν ) ∂ µ ∆ ν + h.c. , (26)which in our scheme correspond to the b - and h -terms (the contribution from the h -termis of a higher order in our power counting and does not appear in the current calculations).The two prescriptions are identical when both the nucleon and the Delta are on the massshell. Otherwise, the difference is compensated by local contact terms of a higher orderin the /m -expansion, see [51, 52, 76, 77] for a related discussion. Such off-shell effectsmanifest themselves, e.g., in the tree-level ∆ -contribution to the magnetic polarizability β M1 . Although the residue of the ∆ pole in the magnetic channel is the same in bothschemes (the constants b and g M are roughly in agreement with each other when calculatingthe magnetic γN ∆ transition form factor), the full result differs almost by a factor of twodue to the presence of the non-pole (background) terms. The non-vanishing (and sizable)contribution of the ∆ -tree-level diagrams to the electric polarizabilites α E1 is in our scheme apure /m -effect caused by the induced electric γN ∆ coupling stemming from the particular14orm of the effective Lagrangian. On the other hand, the ∆ tree-level contribution to α E1 isnegligible in the δ -counting scheme because of the smallness of the electric γN ∆ coupling g E . Note that terms proportional to g E ( h ) start to contribute only at order (cid:15) in the small-scale-expansion scheme. The observed dependence of the considered polarizabilites on theoff-shell effects might be an indication of the importance of such higher order contributions.Fortunately, such /m effects are strongly suppressed for higher-order polarizabilites as willbe shown below.The second difference between the two schemes is related to power-counting of variousdiagrams with internal ∆ -lines. While the πN loops in [61] at order O ( p ) are identical withthe ones included in our O ( q ) results, the diagrams with two and three ∆ -lines inside theloop are suppressed in the δ -counting and are not included in their leading-order π ∆ -loopamplitude. On the other hand, such diagrams are required by gauge invariance (notice,however, that in the Coulomb gauge, their contribution is suppressed by a factor /m ). Inany case, we observe a significant difference between the size of the (cid:15) π ∆ -loop contributionsin our scheme and the O ( p / ) ones of [61] involving only the π ∆ -loops with a single ∆ -line. B. Dipole spin polarizabilites
Next, we consider the dipole spin polarizabilites γ E1E1 , γ M1M1 , γ E1M2 and γ M1E2 .These quantities are less sensitive to the short range dynamics as the relevant contact termsappear at order O ( q ) . Therefore, one expects a better convergence pattern for them. Atthe order we are working, the spin polarizabilites are predictions and do not depend on anyfree parameters. The numerical values of the spin polarizabilites for the proton and neutronare collected in Table V.We also provide theoretical errors for our complete scheme at order O ( q + (cid:15) ) . Theupper error reflects the uncertainty in the input parameters, whereas the lower value is theBayesian estimate of the error coming from the truncation of the small-scale expansion.The experimental values in Table V are obtained from the dispersion-relation analysisof the double-polarized Compton scattering asymmetries Σ and Σ x [5], and, in a newerexperiment, also Σ z [6].Our predictions for the proton spin polarizabilites at order O ( q + (cid:15) ) agree with theexperimental values of [5] within the errors with only a slight deviation for γ M1M1 . Thedeviation from the values extracted in the recent MAMI experiment [6] are somewhat larger.Note that the ∆ -less approach fails to reproduce γ M1M1 for the proton because of the missing ∆ -pole contribution, which would appear as a contact term at order O ( q ) .The contributions of order O ( q ) are in all cases significantly smaller than the leadingterms of order O ( q + (cid:15) ) in the ∆ -full scheme (except for γ E1M2 where the leading-orderresult is small due to cancellations between individual contributions), which is an indicationof a reasonable convergence of the small-scale expansion. The smallness of the O ( q ) -termscan probably also be traced back to the fact that the diagrams containing c , c and c vertices do not contribute to spin polarizabilites. Our ∆ -full results also agree well with thevalues obtained from the fixed- t dispersion relations for the proton and the neutron, exceptfor γ M1M1 , where our prediction appears to be somewhat larger.In Table VI, we present the results for the forward and backward spin polarizabilites γ and γ π which are the linear combinations of the four spin polarizabilites and can be moreeasily accessed experimentally. For these quantities, the agreement with the experimentalvalues is slightly worse, as can be seen from Table VI.15 ( p ) E1E1 γ ( p ) M1M1 γ ( p ) E1M2 γ ( p ) M1E2 q (without ∆ ) − . − .
13 0 .
57 0 . q (without ∆ ) − .
01 0 . − .
25 0 . Total (without ∆ ) − .
47 0 .
36 0 .
32 1 . q − . − .
13 0 .
57 0 . (cid:15) π ∆ loops − .
11 0 .
58 0 . − . (cid:15) ∆ tree − .
07 3 . − .
88 1 . q − .
01 0 . − .
25 0 . Total − . ± . ± . . ± . ± . − . ± . ± . . ± . ± . O ( p ) π N loops [61] − . − . . . O ( p / ) π ∆ loops [61] . − . . − . pole [61] − . . − . . Total [61] − . ± . . ± . . ± . . ± . Fixed- t DR [40] − . . . . Fixed- t DR [41, 44, 45] − . . − .
02 2 . HB χ PT fit [74, 78] − . ± . . ± . − . ± . . ± . MAMI 2015 [5] − . ± . . ± . − . ± . . ± . MAMI 2018 [6] − . ± .
52 2 . ± . − . ± .
67 1 . ± . γ ( n ) E1E1 γ ( n ) M1M1 γ ( n ) E1M2 γ ( n ) M1E2 q (without ∆ ) − . − .
17 0 .
61 1 . q (without ∆ ) − .
46 1 . − .
59 0 . Total (without ∆ ) − .
32 1 . − .
02 2 . q − . − .
17 0 .
61 1 . (cid:15) π ∆ loops .
22 0 .
12 0 . − . (cid:15) ∆ tree − .
07 3 . − .
88 1 . q − .
46 1 . − .
59 0 . Total − . ± . ± . . ± . ± . − . ± . ± . . ± . ± . O ( p ) π N loops [61] − . − . . . O ( p / ) π ∆ loops [61] . − . . − . pole [61] − . . − . . Total [61] − . ± . . ± . . ± . . ± . Fixed- t DR [40] − . . − . . Fixed- t DR [43, 44, 61] − . . − . . HB χ PT fit [74, 78] − . ± . . ± . − . ± . . ± . TABLE V: Numerical values for the dipole spin polarizabilities of the proton (upper table) and theneutron (lower table) in − fm . The upper errors originate from the uncertainty in the inputparameters, the lower errors come from the truncation of the small-scale expansion. The valuesare compared with the results calculated within the δ -counting scheme and obtained using fixed- t dispersion relations. ( p )0 γ ( p ) π ¯ γ ( p )0 δ ( p ) LT q (without ∆ ) .
08 3 .
72 2 .
20 1 . q (without ∆ ) − .
80 1 . − .
37 0 . Total (without ∆ ) .
28 5 .
03 1 .
83 2 . q .
08 3 .
72 2 .
20 1 . (cid:15) π ∆ loops − . − . − .
01 1 . (cid:15) ∆ tree − .
64 7 . − . − . q − .
80 1 . − .
37 0 . Total − . ± . ± . . ± . ± . . ± . ± . . ± . ± . O ( p ) π N loops [61] . . . −O ( p / ) π ∆ loops [61] − . − . − . − ∆ pole [61] − . . − . − Total [61] − . ± . . ± . . ± . − Fixed- t DR [40] − . . − − Fixed- t DR [41, 44, 45] − . . . − HB χ PT fit [74, 78] − . ± . . ± . − − Experiment [2–4] − . ± .
13 8 . ± . − − B χ PT [31] − . ± . − − . ± . γ ( n )0 γ ( n ) π ¯ γ ( n )0 δ ( n ) LT q (without ∆ ) .
06 5 .
45 3 .
06 2 . q (without ∆ ) − .
13 3 . − .
46 0 . Total (without ∆ ) .
93 8 .
68 2 .
60 2 . q .
06 5 .
45 3 .
06 2 . (cid:15) π ∆ loops − . − . − .
01 0 . (cid:15) ∆ tree − .
64 7 . − . − . q − .
13 3 . − .
46 0 . Total − . ± . ± . . ± . ± . . ± . ± . . ± . ± . O ( p ) π N loops [61] . . . −O ( p / ) π ∆ loops [61] − . − . − . − ∆ pole [61] − . . − . − Total [61] . ± . . ± . . ± . − Fixed- t DR [40] − . . − − Fixed- t DR [43, 44, 61] − . . − − HB χ PT fit [74, 78] . ± . . ± . − − B χ PT [31] − . ± . − − . ± . TABLE VI: Numerical values for the combined polarizabilities γ , γ π , ¯ γ and δ LT of the proton(upper table) and the neutron (lower table). All values except for ¯ γ are given in − fm while ¯ γ is given in − fm . The values are compared with various results either calculated within the δ -counting scheme and obtained using fixed- t dispersion relations. The results of [31] are equivalentwith our calculations without the q -contribution. For remaining notation see Table V.
17s in the case of scalar dipole polarizabilites, we compare our ∆ -tree-level and ∆ -loopcontributions with [61] in order to analyze the differences of the two ∆ -full approaches andthe size of the unphysical off-shell terms. For the spin polarizabilites, the off-shell effects(which we identify with the difference of the ∆ -tree-level terms in two schemes considered)are smaller but, nevertheless, comparable to theoretical errors or even larger. This mightindicate that our theoretical errors are somewhat underestimated. This should not comeas a surprise because the Bayesian model for the error estimation that we implement isnot fully trustworthy as long as only two orders in the expansion in terms of the smallparameter Q are used as an input. Notice further that we treat the order q + (cid:15) resultsas being the full fourth-order predictions when estimating truncation errors. The off-shellcontributions add up constructively for the forward and backward spin polarizabilites (ascan be seen in Table VI), which explains the worse agreement with experiment for theselinear combinations.The ∆ -loop terms are also different in the (cid:15) - and δ -counting schemes, which points tothe non-negligible contribution of the diagrams with multiple ∆ -lines. Note, however, thatthe overall absolute values of the (cid:15) ∆ -loops are, on average, smaller than in the case ofthe scalar dipole polarizabilites and than the typical values of the dipole spin polarizabilites.Therefore, spin polarizabilites appear to be less sensitive to such details. On the other hand,the suppression of the (cid:15) ∆ -loops does not exclude the possibility that the (cid:15) ∆ -loops (withorder O ( q ) γN ∆ and γ ∆∆ vertices), which are not included in the current study, yieldimportant contributions, see also the discussion in subsection IV D. C. Higher-order polarizabilities
In this subsection, we focus on higher-order nucleon polarizabilites including scalarquadrupole, dipole dispersive, octupole and quadrupole dispersive, as well as spin quadrupoleand dipole dispersive polarizabilites. All relevant numerical values are collected in Ta-bles VII-IX (we also provide the values for the higher-order forward spin polarizabilities ¯ γ in Table VI). Note that unnaturally large values of the scalar quadrupole and, especially,octupole polarizabilites are related to the traditional l -dependent normalization factor inthe definition of these polarizabilites and have no physical meaning.We summarize the general features of the higher-order polarizabilites. Both ∆ -less and ∆ -full schemes give roughly the same results, except for the channels where the ∆ -tree-levelcontribution is significant, i.e. for magnetic multipoles. Note that in the ∆ -less approach,such contributions would appear only at extremely high orders, which makes the ∆ -lessframework rather inefficient.The second observation concerns the loop contributions. While for all spin polarizabilites,the (cid:15) - ∆ -loops and the q -loops are strongly suppressed, for scalar polarizabilites the situ-ation is different. In the ∆ -less scheme, the q -loops are comparable with the q -loops orlarger, which spoils convergence. On the other hand, in the ∆ -full scheme, a significantpart of the q -loop contributions is shifted to the (cid:15) - ∆ -loops. This happens due to the ∆ -resonance saturation of the low-energy constants c i , in particular c and c [79, 80], whichdo not contribute to the spin polarizabilites. As a result, the convergence pattern of the ∆ -full scheme looks very convincing for both scalar and spin polarizabilites. The only excep-tions are the γ M2E3 polarizabilites, where the (cid:15) result is unnaturally small due to accidentalcancellations between the q -loops and the ∆ -tree-level contributions.Our predictions at order O ( q + (cid:15) ) for all scalar quadrupole and dipole dispersive polar-18 ( p ) E2 β ( p ) M2 α ( p ) E1 ν β ( p ) M1 ν q (without ∆ ) . − . . . q (without ∆ ) . − . − . . Total (without ∆ ) . − . − . . q . − . . . (cid:15) π ∆ loops . − . − . . (cid:15) ∆ tree . − . − . . q . − . − . . Total . ± . ± . − . ± . ± . − . ± . ± . . ± . ± . O ( p ) π N loops [61] . − . . . O ( p / ) π ∆ loops [61] . − . − . . pole [61] . − . − . . Total [61] . ± . − . ± . − . ± . . ± . Fixed- t DR [40, 42], . − . − . . Fixed- t DR [41, 44] . − . − . . α ( n ) E2 β ( n ) M2 α ( n ) E1 ν β ( n ) M1 ν q (without ∆ ) . − . . . q (without ∆ ) . − . − . . Total (without ∆ ) . − . − . . q . − . . . (cid:15) π ∆ loops . − . − . . (cid:15) ∆ tree . − . − . . q . − . − . . Total . ± . ± . − . ± . ± . − . ± . ± . . ± . ± . O ( p ) π N loops [61] . − . . . O ( p / ) π ∆ loops [61] . − . − . . pole [61] . − . − . . Total [61] . ± . − . ± . . ± . . ± . Fixed- t DR [40] . − . − . . Fixed- t DR [43, 44, 61] . − . − . . TABLE VII: Numerical values for the dispersive and the quadropole polarizabilities for the proton(upper table) and the neutron (lower table) in − fm . The values are compared with the resultscalculated in δ -counting χ PT and obtained using fixed- t dispersion relation. For remaining notationsee Table V. izabilites of the proton and the neutron agree within errors with the results based on fixed- t dispersion relations, see Table VII. Note that the predictions of the δ -counting scheme of [61]do not reproduce the fixed- t dispersion relations values for α E2 and β M2 . The main differenceto our result in this channel comes from the q -loops and (cid:15) - ∆ -loops. On the other hand, thedifference in the tree-level- ∆ contributions appears very small, indicating the insignificance19 ( p ) E3 β ( p ) M3 α ( p ) E2 ν β ( p ) M2 ν α ( p ) E1 ν β ( p ) M1 ν q (without ∆ ) . − . − . . . − . q (without ∆ ) . − . − . . . − . Total (without ∆ ) . − . − . . . − . q . − . − . . . − . (cid:15) π ∆ loops . − . − . . . − . (cid:15) ∆ tree − . . . − . − . . q . − . − . . . − . Total . ± . ± . − . ± . ± . − . ± . ± . . ± . ± . . ± . ± . − . ± . ± . α ( n ) E3 β ( n ) M3 α ( n ) E2 ν β ( n ) M2 ν α ( n ) E1 ν β ( n ) M1 ν q (without ∆ ) . − . − . . . − . q (without ∆ ) . − . − . . . − . Total (without ∆ ) . − . − . . . − . q . − . − . . . − . (cid:15) π ∆ loops . − . − . . . − . (cid:15) ∆ tree − . . . − . − . . q . − . − . . . − . Total . ± . ± . − . ± . ± . − . ± . ± . . ± . ± . . ± . ± . − . ± . ± . TABLE VIII: Numerical values for spin-independent octupole polarizabilities α E3 and β M3 ,quadrupole dispersive polarizabilities α E2 ν and β M2 ν as well as higher dipole dispersive polariz-abilities α E2 ν and β M2 ν the proton (denoted with ( p ) ) and the neutron (denoted with ( n ) ). Allvalues are given in − fm . For remaining notation see Table V. of the off-shell effects, as one would expect for such high-order polarizabilites. D. Generalized polarizabilities
Now are now in the position to discuss the generalized ( Q -dependent) nucleon polar-izabilites. We consider the doubly virtual Compton scattering with the initial and finalvirtuality of the photon equal to Q . In Fig. 5, the scalar and spin dipole polarizabilitesfor the proton and the neutron are plotted as a function of Q , and the ∆ -full and ∆ -lessschemes are compared. The scalar polarizabilites at Q = 0 are adjusted to the empiricalvalues, see subsection IV A. The difference of the ∆ -full and ∆ -less spin polarizabilites at Q = 0 was discussed in subsection IV B and can be considered as a higher-order contact-term contribution. Therefore, we focus here on the Q -dependence of the polarizabilitesrelative to their Q = 0 values. For the spin polarizabilites and for the electric scalar po-larizability, the ∆ -full and ∆ -less curves go almost parallel to each other, whereas for themagnetic scalar polarizabilites the slope and the curvature of the curves are opposite in sign.This is due to a significant contribution of the ∆ -tree-level contribution in this channel. Itshould be emphasized that the scalar generalized polarizabilites contribute to the Lamb shiftof muonic hydrogen, see e.g. [81].We also present the Q -dependence of several combined spin polarizabilites, for some20 ( p ) E2E2 γ ( p ) M2M2 γ ( p ) E2M3 γ ( p ) M2E3 q (without ∆ ) − .
56 1 .
16 5 .
78 4 . q (without ∆ ) − . − .
63 0 . − . Total (without ∆ ) − .
02 0 .
53 6 .
10 3 . q − .
56 1 .
16 5 .
78 4 . (cid:15) π ∆ loops .
30 0 . − . − . (cid:15) ∆ tree − . − .
16 1 . − . q − . − .
63 0 . − . Total − . ± . ± . − . ± . ± . . ± . ± . − . ± . ± . γ ( n ) E2E2 γ ( n ) M2M2 γ ( n ) E2M3 γ ( n ) M2E3 q (without ∆ ) − .
18 1 .
97 5 .
59 3 . q (without ∆ ) − . − .
27 0 . − . Total (without ∆ ) − . − .
30 5 .
98 1 . q − .
18 1 .
97 5 .
59 3 . (cid:15) π ∆ loops .
33 0 . − . − . (cid:15) ∆ tree − . − .
16 1 . − . q − . − .
27 0 . − . Total − . ± . ± . − . ± . ± . . ± . ± . − . ± . ± . γ ( p ) E1E1 ν γ ( p ) M1M1 ν γ ( p ) E1M2 ν γ ( p ) M1E2 ν q (without ∆ ) − .
26 0 . − .
29 0 . q (without ∆ ) . . − .
12 0 . Total (without ∆ ) − .
26 0 . − .
41 1 . q − .
26 0 . − .
29 0 . (cid:15) π ∆ loops . − .
02 0 .
02 0 . (cid:15) ∆ tree − .
49 1 . − .
56 0 . q . . − .
12 0 . Total − . ± . ± . . ± . ± . − . ± . ± . . ± . ± . γ ( n ) E1E1 ν γ ( n ) M1M1 ν γ ( n ) E1M2 ν γ ( n ) M1E2 ν q (without ∆ ) − .
62 0 . − .
29 1 . q (without ∆ ) − .
10 0 . − .
18 0 . Total (without ∆ ) − .
72 0 . − .
47 1 . q − .
62 0 . − .
29 1 . (cid:15) π ∆ loops . − .
03 0 . − . (cid:15) ∆ tree − .
49 1 . − .
56 0 . q − .
10 0 . − .
18 0 . Total − . ± . ± . . ± . ± . − . ± . ± . . ± . ± . TABLE IX: Numerical values for the quadrupole spin polarizabilities γ E2E2 , γ M2M2 , γ E2M3 and γ M2E3 and for the dispersive spin polarizabilities γ E1E1 ν , γ M1M1 ν , γ E1M2 ν and γ M1E2 ν of the proton(indicated with ( p ) ) and the neutron (indicated with ( n ) ). All values are given in − fm . Forremaining notation see Table V. .
05 0 .
10 0 . Q [ GeV ] α E [ − f m ] .
05 0 .
10 0 . − Q [ GeV ] β M [ − f m ] .
05 0 .
10 0 . − . − . − . − . − . − . Q [ GeV ] γ E E [ − f m ] .
05 0 .
10 0 . Q [ GeV ] γ M M [ − f m ] .
05 0 .
10 0 . − − − Q [ GeV ] γ E M [ − f m ] .
05 0 .
10 0 . Q [ GeV ] γ M E [ − f m ] FIG. 5: Q -dependence of the scalar and spin polarizabilites for the proton (dotted and dash-dottedlines) and the neutron (dashed and solid lines). The dotted and dashed lines correspond to the ∆ -less O ( q ) results, whereas the dash-dotted and solid lines correspond to the ∆ -full O ( (cid:15) + q ) results. The bands indicate the theoretical truncation errors.
22f which the experimental data are available, see Fig. 6 (their limiting values for Q = 0 are collected in Table VI). We observe no improvement as compared to [31] (pure O ( (cid:15) ) calculation) due to the inclusion of the O ( q ) contributions. In fact, the description of γ for the proton is even worse. A possible source of such a discrepancy could be a missingcontribution of the ∆ -loop diagrams at order O ( (cid:15) ) , as was suggested in [31]. On the otherhand, taking into account a much better description of the data in [82] (within the δ -countingscheme) and the fact that the disagreement of our result with experiment for the value of γ for the proton at Q = 0 was caused by the large contribution from the induced electric γN ∆ -coupling (as a /m effect), one may expect the improvement to be achieved afterincluding the relevant higher-order γN ∆ -vertices from the effective Lagrangian analogouslyto [82]. E. Dynamical polarizabilities
One can also probe the electromagnetic structure of the nucleon by looking at dynamical(energy-dependent) polarizabilites that describe the response to the nucleon electromagneticexcitations at arbitrary energy. In Figs. 7, 8, we present the energy dependence of the dipoleand spinless quadrupole polarizabilites up to the center-of-mass energy ω CM = 300 MeV. Forcomparison, also shown are the results obtained using the δ -counting scheme [61], the fixed- t dispersion relations [44], and the Computational Hadronic Model [84] . The σ and σ truncation errors corresponding to and degree-of-belief intervals are shown as bandsin the figures. Our results agree rather well with the ones of the fixed- t dispersion relationsat ω CM = 0 (except for γ E M ). Therefore, it is natural to compare the two approaches atnon-zero energies. As can be seen from the figures, the deviation of our results from thoseof the fixed- t dispersion relations increases with energy, which may provide yet anotherindication that our theoretical errors are underestimated (as discussed in subsection IV C),and the convergence of the small-scale expansion becomes slower ω CM (cid:38) − MeV.However, for α E a large discrepancy (beyond σ ) between the two theoretical frameworks isobserved already for ω CM (cid:38) − MeV. This could be due to the aforementioned largeinduced electric γN ∆ -coupling in our scheme, whose effect increases with energy. We have extracted those data points from Ref. [61]. − − γ [ − f m ] − − − δ L T [ − f m ] γ π [ − f m ] .
00 0 .
05 0 .
10 0 . − Q [ GeV ] ¯ γ [ − f m ] .
05 0 .
10 0 . − Q [ GeV ] FIG. 6: Q -dependencies of the forward polarizabilities γ and δ LT , the backward polarizability γ π and the combined higher-order polarizability ¯ γ for the proton (left) and the neutron (right). Thethick solid blue lines indicate our ∆ -full O ( q + (cid:15) ) calculations with a σ truncation error bandcorresponding to degree-of-belief intervals and the thick dashed blue lines show our ∆ -less O ( q ) calculations. The red loosely dashed lines represent the NLO B χ PT calculation from [39]with the red error bands. The black dash-dotted line presents the MAID model predictions from [43](proton) and [8] (neutron). The green double-dash-dotted line is the O ( p ) calculation from [83].Empirical data are: for γ ( p )0 from [10] (triangle) and [7] (squares); for γ ( n )0 from [11] (preliminary,triangles), [8] (square) and [9] (diamonds); for δ ( n ) LT from [8].
50 100 150 200 250 30001020 ω cms [ MeV ] α E [ − f m ] − − ω cms [ MeV ] β M [ − f m ] − − ω cms [ MeV ] γ E E [ − f m ] − − ω cms [ MeV ] γ M M [ − f m ] . . ω cms [ MeV ] γ E M [ − f m ] − − ω cms [ MeV ] γ M E [ − f m ] ω cms [ MeV ] α E [ − f m ] − − − ω cms [ MeV ] β M [ − f m ] FIG. 7: The ω -dependence of the real parts of the dipole polarizabilities and the spinless quadrupolepolarizabilites α E2 and β M2 for the proton. The solid blue lines represent our ∆ -full O ( q + (cid:15) ) result. The inner (outer) blue bands stand for the σ ( σ ) truncation error. The red dashed linesare the B χ PT calculation [61] with the red error bands. The black dash-dotted lines correspond tothe fixed- t dispersion-relations calculation [44] and the green double-dash-dotted lines correspondto the results of [84].
50 100 150 200 250 3000102030 ω cms [ MeV ] α E [ − f m ] − − ω cms [ MeV ] β M [ − f m ] − − ω cms [ MeV ] γ E E [ − f m ] − ω cms [ MeV ] γ M M [ − f m ] − . . . . ω cms [ MeV ] γ E M [ − f m ] − − − ω cms [ MeV ] γ M E [ − f m ] ω cms [ MeV ] α E [ − f m ] − − − ω cms [ MeV ] β M [ − f m ] FIG. 8: The ω -dependence of the real parts of the dipole polarizabilities and the spinless quadrupolepolarizabilites α E2 and β M2 for the neutron. The solid blue lines represent our ∆ -full O ( q + (cid:15) ) result. The inner (outer) blue bands stand for the σ ( σ ) truncation error. The red dashed linesare the B χ PT calculation [61] with the red error bands. The black dash-dotted lines correspond tothe fixed- t dispersion-relations calculation [44]. . Heavy-Baryon Expansion In this subsection, we study the convergence of the /m -expansion of our results (thenucleon- ∆ mass difference ∆ is kept finite and constant) obtained within the covariantframework. analyzing such an expansion we can test the efficiency of the heavy-baryonapproach by reproducing some of its contributions appearing at higher orders. We presentthe /m -expansion for the dipole scalar and spin polarizabilities in Tables X-XV startingfrom the leading order (LO) static ( m ) results up to the order /m (N LO). Obviously,the static results as well as the /m -corrections to the leading-order terms coincide with thecorresponding heavy-baryon calculations, see [21–24]. α ( p ) E1 q q (cid:15) Loop (cid:15) Tree
Full α ( n ) E1 q q (cid:15) Loop (cid:15) Tree
Full
LO 12 .
78 8 .
86 7 . . . . .
50 LO 12 .
78 2 .
67 7 . . . . . .
47 8 .
86 2 . − .
60 14 .
61 NLO 9 .
67 2 .
67 4 . − .
60 11 . LO 6 .
60 9 .
53 0 . − .
98 14 .
13 N LO 9 .
50 3 .
22 3 . − .
98 12 . LO 7 .
01 9 . − . − .
11 11 .
26 N LO 9 .
51 3 .
10 2 . − .
11 11 . LO 7 .
04 9 . − . − .
65 11 .
46 N LO 9 .
51 3 .
09 2 . − .
65 11 . LO 7 .
04 9 . − . − .
83 11 .
12 N LO 9 .
51 3 .
09 2 . − .
83 11 . Full .
04 9 . − . − .
78 11 . Full .
51 3 .
09 2 . − .
78 11 . TABLE X: Numerical values for the /m -expansion of α E1 . Note that the (cid:15) Tree only starts at NLO. β ( p ) M1 q q (cid:15) Loop (cid:15) Tree
Full β ( n ) M1 q q (cid:15) Loop (cid:15) Tree
Full
LO 1 . − .
33 1 .
36 11 .
96 2 .
26 LO 1 . − .
35 1 .
36 11 .
96 7 . . − .
09 1 .
61 11 .
96 2 .
04 NLO − . − .
60 0 .
34 11 .
96 3 . LO − . − .
75 9 .
05 11 .
96 4 .
43 N LO − . − .
27 1 .
28 11 .
96 3 . LO − . − .
23 5 .
67 11 .
96 2 .
45 N LO − . − .
12 0 .
69 11 .
96 3 . LO − . − .
14 6 .
39 11 .
96 3 .
38 N LO − . − .
12 1 .
09 11 .
96 3 . LO − . − .
16 5 .
32 11 .
96 2 .
28 N LO − . − .
12 0 .
90 11 .
96 3 . Full − . − .
16 5 .
54 11 .
96 2 . Full − . − .
12 0 .
96 11 .
96 3 . TABLE XI: Numerical values for the /m -expansion of β M1 . We first consider the convergence of the /m -expansion of the individual contributionsfrom the q -, q - and (cid:15) -loop diagrams, and from the ∆ -tree-level terms. In general, theconvergence is rather slow. The most rapid convergence is observed for the q - and q -loops. Sometimes (e.g. for α E , γ M E ), the expanded value approaches the “exact” onealready at NLO-N LO. In other cases, the expanded values oscillate at lower /m -orders,especially when the resulting value is small due to cancellations among various diagrams.It is natural to expect a slower convergence for the diagrams with ∆ -lines as the formalexpansion parameter ∆ /m is roughly twice as large as M/m . Nevertheless, the expansionfor the tree- ∆ -graphs converges, in general, only slightly worse than the πN -loops ( β M isaccidentally m -independent). On the other hand, for the ∆ π -loops, the convergence is verypoor. This set of diagrams comprises loops with one, two and three ∆ -lines, and cancellations27 ( p ) E1E1 q q (cid:15) Loop (cid:15) Tree
Full γ ( n ) E E q q (cid:15) Loop (cid:15) Tree
Full LO − . . . . . . . . − .
22 LO − . . . . . . . . − . − . . . . . − . − .
05 NLO − . . . . . − . − . LO − .
03 0 . − . − . − .
68 N LO − . − .
37 0 . − . − . LO − . − . − . − . − .
11 N LO − . − .
49 0 . − . − . LO − . − .
02 0 . − . − .
50 N LO − . − .
46 0 . − . − . LO − . − . − . − . − .
66 N LO − . − .
46 0 . − . − . Full − . − . − . − . − . Full − . − .
46 0 . − . − . TABLE XII: Numerical values for the /m -expansion of γ E1E1 . The dots mark entries that do notexist e.g. the (cid:15) Tree starts at NLO and therefore does not have a LO contribution. γ ( p ) M1M1 q q (cid:15) Loop (cid:15) Tree
Full γ ( n ) M M q q (cid:15) Loop (cid:15) Tree
Full LO − . . . . .
21 4 .
03 3 .
08 LO − . . . . .
21 4 .
03 3 . .
39 1 . − .
12 3 .
40 6 .
59 NLO 0 .
31 2 . − .
27 3 .
40 5 . LO 0 .
52 1 .
24 2 .
21 3 .
89 7 .
85 N LO − .
13 1 .
56 0 .
78 3 .
89 6 . LO − .
37 0 . − .
44 3 .
87 3 .
26 N LO − .
17 1 .
39 0 .
04 3 .
87 5 . LO − .
15 0 .
46 0 .
81 3 .
84 4 .
96 N LO − .
17 1 .
42 0 .
23 3 .
84 5 . LO − .
12 0 .
50 0 .
39 3 .
86 4 .
62 N LO − .
17 1 .
42 0 .
05 3 .
86 5 . Full − .
13 0 .
49 0 .
58 3 .
85 4 . Full − .
17 1 .
42 0 .
12 3 .
85 5 . TABLE XIII: Numerical values for the /m -expansion of γ M1M1 . The dots mark entries that donot exist e.g. the q starts at NLO and therefore does not have a LO contribution. among them occur quite often. Some of the values strongly oscillate and one hardly sees asign of convergence even at N LO, e.g. for γ M M , γ E M .Nevertheless, we have checked that the /m -expansion converges in principle (formally)for all diagrams. This is illustrated in Figs. 9, 10, where the logarithm of the remainder inthe /m -series is plotted against the order of expansion. As one can see from the plots, theexpanded ∆ -loops approach their unexpanded values very slowly, making such an expansionimpractical. Note that contributions of the ∆ -loops are smaller for the spin-dependentpolarizabilities.We now consider the /m -expansion of the sum of all contributions to the nucleon polar-izabilities. As one can see in Tables X, XI, the electric and magnetic scalar polarizabilitiesat NLO agree rather well with the unexpanded values (for the absolute difference is largebut the relative difference is small), while the individual contributions in some cases stronglyoscillate. Such an agreement is accidental. Moreover, e.g. the β ( p ) M at N LO deviates sig-nificantly from the full result and approaches it again after several oscillations. Nevertheless,these effects can be compensated by a redefinition of the q contact terms.The situation is different for the spin-dependent polarizabilities, where the NLO values inmost cases deviate rather strongly from the unexpanded result, see Tables XII-XV. It shouldbe emphasized that these differences can be absorbed into contact terms only at order q .Summarizing, we conclude that the /m -expansion (and, hence, the heavy-baryonscheme) is rather inefficient for calculating nucleon polarizabilites in the ∆ -full approach,28 ( p ) E1M2 q q (cid:15) Loop (cid:15) Tree
Full γ ( n ) E M q q (cid:15) Loop (cid:15) Tree
Full
LO 1 . . . . − . . . . .
95 LO 1 . . . . − . . . . . . . . . . − . − .
40 NLO 0 . . . . . − . − . LO 0 .
57 0 .
03 1 . − .
92 1 .
11 N LO 0 . − .
43 0 . − . − . LO 0 . − .
36 0 . − .
88 0 .
15 N LO 0 . − .
63 0 . − . − . LO 0 . − .
26 0 . − . − .
52 N LO 0 . − .
59 0 . − . − . LO 0 . − .
25 0 . − .
88 0 .
14 N LO 0 . − .
59 0 . − . − . Full . − .
25 0 . − . − . Full . − .
59 0 . − . − . TABLE XIV: Numerical values for the /m -expansion of γ E1M2 . The dots mark entries that do notexist e.g. the (cid:15) Tree starts at NLO and therefore does not have a LO contribution. γ ( p ) M1E2 q q (cid:15) Loop (cid:15) Tree
Full γ ( n ) M E q q (cid:15) Loop (cid:15) Tree
Full
LO 1 . . . . − . . . . .
95 LO 1 . . . . − . . . . . .
76 1 . − .
11 1 .
89 3 .
56 NLO 1 .
30 0 . − .
08 1 .
89 4 . LO 0 .
89 0 . − .
32 1 .
59 1 .
82 N LO 1 .
36 0 . − .
33 1 .
59 3 . LO 0 .
98 0 . − .
02 1 .
82 2 .
31 N LO 1 .
37 0 . − .
25 1 .
82 3 . LO 0 .
95 0 . − .
80 1 .
71 2 .
42 N LO 1 .
36 0 . − .
26 1 .
71 3 . LO 0 .
95 0 . − .
86 1 .
76 2 .
40 N LO 1 .
36 0 . − .
29 1 .
76 3 . Full .
95 0 . − .
79 1 .
74 2 . Full .
36 0 . − .
27 1 .
74 3 . TABLE XV: Numerical values for the /m -expansion of γ M1E2 . The dots mark entries that do notexist e.g. the (cid:15) Tree starts at NLO and therefore does not have a LO contribution. which is in line with the results of the heavy-baryon calculations mentioned above. On theother hand, the small-scale expansion seems to converge reasonably well.
V. SUMMARY AND OUTLOOK
In this work, we have presented various nucleon polarizabilities obtained within covariantchiral perturbation theory with explicit ∆ (1232) degrees of freedom, calculated up to order O ( (cid:15) + q ) in the small-scale expansion. The theoretical errors were estimated by combiningthe uncertainties of the input parameters and the errors due to the truncation of the small-scale expansion calculated using a Bayesian model. The results were compared with the ∆ -less approach at order O ( q ) and O ( q ) , as well as with the empirical values and othertheoretical approaches (in particular, with the δ -counting ∆ -full scheme and the fixed- t dispersion-relations method).The general conclusion of this study is that the ∆ -full scheme that we adopt is quiteefficient for analyzing the nucleon polarizabilites. It shows reasonable convergence, and theobtained results agree well with experiment and the fixed- t dispersion-relations values. Theresults obtained in the ∆ -less approach are considerably worse both from the point of viewof convergence and agreement with experiment.The scalar dipole polarizabilites were used as an input to adjust four low energy constantsappearing at order O ( q ) in the effective Lagrangian. Therefore, we were not concerned with29 − − i l n | α ( p ) , i E − α ( p ) E | − − i l n | β ( p ) , i M − β ( p ) M | − − i l n | γ ( p ) , i E E − γ ( p ) E E | − − i l n | γ ( p ) , i M M − γ ( p ) M M | − − i l n | γ ( p ) , i E M − γ ( p ) E M | − − i l n | γ ( p ) , i M E − γ ( p ) M E | FIG. 9: Logarithmic difference of the absolute value between the HB-expanded contributions andthe final, non-expanded value for the spin-independent and spin-dependent dipole polarizabilitiesof the proton. i stands for the i -th order in the HB expansion. The black squares represent the q contribution, the red triangles represent the q contribution, the blue circles represent the (cid:15) -loopcontribution and the green diamonds represent the (cid:15) -tree contributions. The dashed lines standfor the corresponding linear regression. the issue of convergence for these quantities (although the convergence is far from beingsatisfactory).Our predictions for the dipole spin polarizabilites γ E E , γ E M , γ M E obtained in the ∆ -full scheme agree with experimental values of [5] and are slightly larger for γ M M . Theagreement is somewhat worse with the analysis of the recent MAMI experiment [5]. Thesame pattern is observed in the comparison with the fixed- t dispersion-relations results.On the other hand, the predictions for the forward and backward spin polarizabilites γ and γ + π differ noticeably from the empirical values. Such a deviation can be explainedby a sizable contributions of the “induced” electric γN ∆ -coupling observed for these linear30 − − i l n | α ( n ) , i E − α ( n ) E | − − i l n | β ( n ) , i M − β ( n ) M | − − i l n | γ ( n ) , i E E − γ ( n ) E E | − − i l n | γ ( n ) , i M M − γ ( n ) M M | − − i l n | γ ( n ) , i E M − γ ( n ) E M | − − i l n | γ ( n ) , i M E − γ ( n ) M E | FIG. 10: Logarithmic difference of the absolute value between the HB-expanded contributions andthe final, non-expanded value for the spin-independent and spin-dependent dipole polarizabilitiesof the neutron. i stands for the i -th order in the HB expansion. The black squares represent the q contribution, the red triangles represent the q contribution, the blue circles represent the (cid:15) -loopcontribution and the green diamonds represent the (cid:15) -tree contributions. The dashed lines standfor the corresponding linear regression. combinations. This effect if formally suppressed by a factor of /m , but numerically it turnsout to be sizable. In the δ -counting scheme of [61], the electric γN ∆ -coupling constantis adjusted to data and appears to be rather small. This is an indication that includinghigher-order ∆ -pole graphs in our scheme might improve the results.Due to small contributions of the q -loops to spin polarizabilites, a rather rapid conver-gence is achieved for all of them.We have also analyzed several higher-order polarizabilites. A nice convergence rate isobserved for all polarizabilites calculated within the ∆ -full scheme. This is, however, notthe case for the higher-order scalar polarizabilites calculated in the ∆ -less approach. This31attern can be understood in terms of the ∆ -resonance saturation of the low-energy constants c and c . The main effect of the explicit treatment of the ∆ is that some parts of the q -loopsare shifted to the (cid:15) -loops.For the scalar quadrupole and dipole-dispersive polarizabilites, our predictions agree withthe results of the fixed- t dispersion-relations approach. However, our results differ noticeablyfrom the ones obtained in the δ -counting scheme. This difference is caused not only by the q -loop contributions, but also by the (cid:15) -loops with multiple ∆ -lines, which contribute athigher orders in the δ -counting scheme. This points to the importance of such terms alsofor higher-order polarizabilites.We also studied the Q -dependence of the nucleon polarizabilites by considering gener-alized scalar and spin polarizabilities. We found that the Q -dependence of the magneticscalar polarizabilites is significantly different in the ∆ -full and the ∆ -less approach. Wealso observed a substantial deviation of the O ( (cid:15) + q ) results for the Q -dependent polar-izabilities γ for the proton and for the neutron as well as δ LT for the neutron from theavailable experimental data and no improvement compared to the O ( (cid:15) ) -results. We expectthat taking into account the O ( (cid:15) ) terms (in particular the tree-level graphs) might improvethe description of the data.An alternative way to study the electromagnetic structure of the nucleon is to considerdynamical (energy-dependent) polarizabilites. We have analyzed the energy dependence ofthe dipole and spinless quadrupole polarizabilites and compared them with other theoreticalinvestigations. In particular, we observed a rather large deviation from the fixed- t dispersion-relations approach at energies ω CM (cid:38) − MeV (in some cases for ω CM (cid:38) MeV),which indicates the slow convergence of the small-scale expansion in that energy region.Finally, we have analyzed the convergence of the /m -expansion of the results obtainedin the covariant calculation for various polarizabilites. Such an scheme allows one to see howreliable the heavy-baryon expansion is for the evaluation of the nucleon polarizabilites. Weconsidered the expansion up to N LO. Our conclusion is that the heavy-baryon expansionis not efficient for calculating nucleon polarizabilites in the ∆ -full approach. Nevertheless,the small-scale expansion seems to converge reasonably well.A natural extension of the current work towards increasing accuracy of the results followsfrom the discussion above. We expect a better accuracy and a better agreement with theexperimental data after including the ∆ -tree-level graphs of order O ( (cid:15) ) with the electric γN ∆ -coupling as well as the O ( (cid:15) ) -loop diagrams, that is performing a complete O ( (cid:15) ) calculation. Acknowledgments
We are grateful to Jambul Gegelia for helpful discussions and to Veronique Bernard andUlf-G. Meißner for sharing their insights into the considered topics. This work was supportedin part by BMBF (contract No. 05P18PCFP1), by DFG (Grant No. 426661267) and byDFG through funds provided to the Sino-German CRC 110 “Symmetries and the Emergenceof Structure in QCD” (Grant No. TRR110).32 ppendix A: q -Values The analytic expressions for α E1 , β M1 , α E2 , β M2 and a linear combination of the spin-dependent first order polarizabilities for both proton and neutron were already calcu-lated in [61] but are also given here for completeness. For convenience we define Ξ =Ξ ( m , m, M ) / ( µ − where µ = M/m , Ξ = ln ( µ ) and Ξ( p , m , m ) = 1 p (cid:113) λ ( m , m , p ) ln (cid:32) m + m + (cid:112) λ ( m , m , p ) − p m m (cid:33) ,λ ( a, b, c ) = a + b + c − ab − bc − ac . (A1)
1. Proton values a. Spin-independent first order polarizabilities α ( p ) E1 = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 110 µ − µ + 870 µ − µ + 80 (cid:1) Ξ + (cid:0) − µ + 157 µ − µ + 304 µ (cid:1) (cid:105) + e g A (9 µ − µ + 9) Ξ π F m ,β ( p ) M1 = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 471 µ − µ + 2 (cid:1) Ξ − (cid:0) µ − µ + 127 µ (cid:1) (cid:105) + e g A (27 µ − µ + 9) Ξ π F m . b. Spin-independent second order polarizabilities α ( p ) E2 = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 16584 µ − µ +212940 µ − µ + 18624 µ + 1344 (cid:1) Ξ − µ − µ + 26437 µ − µ + 34592 µ − µ (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 170 (cid:1) ,β ( p ) M2 = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 49020 µ − µ +43020 µ − µ + 144 (cid:1) Ξ + 990 µ − µ + 20000 µ − µ + 92 µ (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 170 (cid:1) , ( p ) E1 ν = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 102987 µ − µ +3561462 µ − µ + 9522420 µ − µ + 397824 µ + 6912 (cid:1) Ξ − µ + 85032 µ − µ + 1658251 µ − µ + 1156768 µ − µ (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 1210 (cid:1) ,β ( p ) M1 ν = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 184876 µ − µ +730660 µ − µ + 33056 µ + 896 (cid:1) Ξ + 1890 µ − µ + 102731 µ − µ + 96560 µ − µ (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 1210 (cid:1) . c. Spin-dependent first order polarizabilities γ ( p ) E1E1 = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 134 µ − µ + 1790 µ − µ +264) Ξ − (cid:0) µ − µ + 764 µ − µ + 80 (cid:1) (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 29 (cid:1) ,γ ( p ) M1M1 = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 744 µ − µ + 4970 µ − µ +152) Ξ − (cid:0) µ − µ + 2462 µ − µ + 16 (cid:1) (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 29 (cid:1) ,γ ( p ) E1M2 = − e g A π F µ ( µ − m (cid:104) (cid:0)(cid:0) (cid:0) µ − (cid:1) (cid:0) µ − µ + 34 (cid:1) µ + 46 (cid:1) µ +56) Ξ + (cid:0) − µ + 141 µ − µ + 4 µ + 16 (cid:1) (cid:105) + e g A Ξ π F m (cid:0) µ − µ − (cid:1) ,γ ( p ) M1E2 = e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 141 µ − µ + 6 (cid:1) Ξ + (cid:0) µ − µ + 30 µ − (cid:1) (cid:105) − e g A Ξ π F m (cid:0) µ − µ + 3 (cid:1) . . Spin-dependent second order polarizabilities γ ( p ) E2E2 = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 764796 µ − µ + 4759500 µ − µ + 577056 µ + 36096 (cid:1) Ξ + 6030 µ − µ + 461861 µ − µ + 954200 µ − µ − (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 2530 (cid:1) ,γ ( p ) M2M2 = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 2802636 µ − µ + 10510020 µ − µ + 379296 µ + 7296 (cid:1) Ξ + 29250 µ − µ + 1545103 µ − µ + 1278024 µ − µ − (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 2530 (cid:1) ,γ ( p ) E2M3 = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 26382 µ − µ +407652 µ − µ + 91920 µ + 28032 µ + 3072 (cid:1) Ξ − µ + 20397 µ − µ + 119300 µ − µ − µ − (cid:105) + e g A Ξ π F m (cid:0) µ − µ − (cid:1) ,γ ( p ) M2E3 = e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 42072 µ − µ +19860 µ − µ − (cid:1) Ξ + 990 µ − µ + 15413 µ − µ + 832 µ +128 (cid:105) − e g A Ξ π F m (cid:0) µ − µ + 32 (cid:1) ,γ ( p ) E1E1 ν = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 688446 µ − µ +25026300 µ − µ + 74731920 µ − µ + 4788480 µ +299520) Ξ − µ + 570321 µ − µ + 11906948 µ − µ +10422016 µ − µ − (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 6038 (cid:1) , ( p ) M1M1 ν = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 364014 µ − µ +12320172 µ − µ + 31816880 µ − µ + 1383552 µ +45056) Ξ − µ + 300069 µ − µ + 5684492 µ − µ +3793952 µ − µ − (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 6038 (cid:1) ,γ ( p ) E1M2 ν = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 11444436 µ − µ + 104160132 µ − µ + 48945312 µ − µ +3072) Ξ + 72090 µ − µ + 7339287 µ − µ + 28502616 µ − µ + 243968 µ + 7168 (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 5922 (cid:1) ,γ ( p ) M1E2 ν = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 2436924 µ − µ + 13400220 µ − µ + 1224672 µ + 54912 (cid:1) Ξ + 20790 µ − µ + 1438497 µ − µ + 2403672 µ − µ + 3328 (cid:105) + e g A Ξ π F m (cid:0) µ − µ + 846 (cid:1) .
2. Neutron values a. Spin-independent first order polarizabilities α ( n ) E1 = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 30 µ − µ + 80 (cid:1) Ξ − (cid:0) µ − µ (cid:1) (cid:105) + e g A Ξ π F m ,β ( n ) M1 = − e g A π F µ ( µ − m (cid:104) (cid:0) (cid:0) µ − (cid:1) µ + 2 (cid:1) Ξ − µ (cid:105) + e g A Ξ π F m . . Spin-independent second order polarizabilities α ( n ) E2 = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 8400 µ − µ + 9984 µ +1344) Ξ + 113 µ − µ + 4272 µ − µ (cid:105) + e g A Ξ π F m ,β ( n ) M2 = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 1200 µ − µ + 1500 µ − (cid:1) Ξ − µ + 885 µ − µ (cid:105) + e g A Ξ π F m ,α ( n ) E1 ν = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 4320 µ − µ + 97380 µ − µ + 38784 µ + 6912 (cid:1) Ξ − µ + 4031 µ − µ + 28928 µ − µ (cid:105) + e g A Ξ π F m ,β ( n ) M1 ν = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 5600 µ − µ +7136 µ + 896 (cid:1) Ξ + 73 µ − µ + 3168 µ − µ (cid:105) + e g A Ξ π F m . c. Spin-dependent first order polarizabilities γ ( p ) E1E1 = − e g A π F µ ( µ − m (cid:104) (cid:0) (cid:0) µ − (cid:1) µ + 22 (cid:1) Ξ + (cid:0) − µ (cid:1) (cid:105) + 5 e g A Ξ π F m ,γ ( p ) M1M1 = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 50 µ − µ + 88 (cid:1) Ξ − (cid:0) µ − µ + 8 (cid:1) (cid:105) + 5 e g A Ξ π F m ,γ ( p ) E1M2 = e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 46 µ − (cid:1) Ξ − (cid:0) µ + 10 µ − (cid:1) (cid:105) − e g A Ξ π F m , ( p ) M1E2 = e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ − (cid:1) Ξ + (cid:0) µ − (cid:1) (cid:105) − e g A Ξ π F m . d. Spin-dependent second order polarizabilities γ ( n ) E2E2 = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 31410 µ − µ +29304 µ + 9024 (cid:1) Ξ + 472 µ − µ + 15802 µ − µ − (cid:105) + e g A π F m ,γ ( n ) M2M2 = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 31590 µ − µ +27864 µ + 1824 (cid:1) Ξ + 428 µ − µ + 16486 µ − µ − (cid:105) + e g A π F m ,γ ( n ) E2M3 = e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 2610 µ − µ + 3888 µ +768) Ξ + 14 µ − µ + 1844 µ − µ − (cid:105) − e g A Ξ π F m ,γ ( n ) M2E3 = e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 990 µ + 60 µ − (cid:1) Ξ +58 µ − µ − µ + 32 (cid:105) − e g A Ξ π F m ,γ ( n ) E1E1 ν = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 24210 µ − µ +26664 µ + 3744 (cid:1) Ξ + 276 µ − µ + 11506 µ − µ − (cid:105) + 19 e g A Ξ π F m ,γ ( n ) M1M1 ν = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 2052 µ − µ + 45700 µ − µ + 21984 µ + 8192 (cid:1) Ξ − µ + 1969 µ − µ + 17624 µ − µ − (cid:105) + 19 e g A Ξ π F m , ( n ) E1M2 ν = e g A π F µ ( µ − m (cid:104) (cid:0) − µ + 7074 µ − µ + 174660 µ − µ + 127152 µ + 1536 (cid:1) Ξ − µ + 4282 µ − µ + 68156 µ − µ − (cid:105) + e g A Ξ π F m ,γ ( n ) M1E2 ν = − e g A π F µ ( µ − m (cid:104) (cid:0) µ − µ + 27105 µ − µ +19044 µ + 8784 (cid:1) Ξ + ( µ − (cid:0) µ − µ + 8404 µ − (cid:1) (cid:105) + 131 e g A Ξ π F m . Appendix B: (cid:15) -Tree values We give here the explicit analytic expressions for the ∆ -tree contribution to the nucleonpolarizabilites. α E = − b e ( µ + µ ∆ + 1)18 πm µ ( µ ∆ + 1) ,β M = b e πm ( µ ∆ − ,α E = b e (6 µ + 3 µ ∆ + 1)12 πm ( µ ∆ − µ ( µ ∆ + 1) ,β M = − b e (6 µ − µ ∆ + 1)12 πm ( µ ∆ − µ ( µ ∆ + 1) ,α E ν = − b e (18 µ + 15 µ + 31 µ + 9 µ ∆ + 7)144 πm ( µ ∆ − µ ( µ ∆ + 1) ,β M ν = b e (18 µ − µ + 31 µ − µ ∆ + 7)144 πm ( µ ∆ − µ ( µ ∆ + 1) ,α E = − b e (6 µ + 3 µ ∆ + 1)4 πm ( µ ∆ − µ ( µ ∆ + 1) ,β M = 5 b e (6 µ − µ ∆ + 1)4 πm ( µ ∆ − µ ( µ ∆ + 1) ,α E ν = b e (42 µ + 27 µ + 67 µ + 23 µ ∆ + 13)18 πm ( µ ∆ − µ ( µ ∆ + 1) ,β M ν = − b e (42 µ − µ + 67 µ − µ ∆ + 13)18 πm ( µ ∆ − µ ( µ ∆ + 1) ,α E ν = − b e (162 µ + 141 µ + 243 µ + 188 µ + 258 µ + 71 µ ∆ + 57)360 πm ( µ ∆ − µ ( µ ∆ + 1) , M ν = b e (162 µ − µ + 243 µ − µ + 258 µ − µ ∆ + 57)360 πm ( µ ∆ − µ ( µ ∆ + 1) γ E E = − b e ( µ + µ − πm ( µ ∆ − µ ∆ ( µ ∆ + 1) ,γ M M = b e (2 µ − µ + µ + 3 µ ∆ − πm ( µ ∆ − µ ( µ ∆ + 1) ,γ E M = − b e (2 µ ∆ − πm ( µ − ,γ M E = b e (2 µ + 1)36 πm µ ( µ − ,γ E E = − b e (4 µ + 3 µ − µ − µ + 4 µ + 57 µ ∆ + 13)1728 πm ( µ ∆ − µ ( µ ∆ + 1) ,γ M M = b e (4 µ − µ − µ + 12 µ − µ ∆ + 15)1728 πm ( µ ∆ − ( µ ∆ ) ( µ ∆ + 1) ,γ E M = − b e (cid:0) µ − µ − µ + 8 µ − µ ∆ + 1 (cid:1) πm µ ( µ − ,γ M E = b e (cid:0) µ − µ − µ − µ − µ ∆ − (cid:1) πm µ ( µ − ,γ E E ν = b e (12 µ + 9 µ − µ − µ − µ − µ − µ + 73 µ ∆ + 37)576 πm ( µ ∆ − µ ( µ ∆ + 1) ,γ M M ν = − b e (12 µ − µ − µ + 63 µ − µ + 51 µ − µ − µ ∆ + 31)576 πm ( µ ∆ − µ ( µ ∆ + 1) ,γ E M ν = b e (36 µ − µ − µ + 33 µ − µ − µ − µ ∆ + 67)1440 πm µ ( µ − ,γ M E ν = − b e (36 µ − µ − µ + 21 µ − µ − µ − µ ∆ − πm µ ( µ − . Appendix C: (cid:15) -Loop values Due to the length of the expressions, we only provide here the expressions for the first-order polarizabilities. In addition to the definition from Appendix A, we now have m ∆ asan additional mass scale and it is convenient to introduce µ ∆ = m ∆ /m , Ξ = ln( µ ∆ ) as wellas Ξ = Ξ ( m , m ∆ , M ) / ( µ − . 40 . Proton values a. Spin-independent first order polarizabilities α ( p ) E1 = − e h A Ξ F mπ µ ( − µ + µ − µ ∆ + 1) ( − µ + µ + 2 µ ∆ + 1) (cid:104) − µ +228 µ + (cid:0) µ − (cid:1) µ − (cid:0) µ + 325 (cid:1) µ + (cid:0) − µ − µ +445) µ + 6 (cid:0) µ + 301 µ + 148 (cid:1) µ + 30 (cid:0) µ + 34 µ + 32 µ − (cid:1) µ − (cid:0) µ + 204 µ + 343 µ + 126 (cid:1) µ − (cid:0) µ + 1586 µ + 3036 µ + 666 µ +407) µ + 2 (cid:0) µ − µ + 573 µ + 3 µ + 581 (cid:1) µ + 2 (cid:0) µ + 412 µ +692 µ − µ + 295 µ − (cid:1) µ + 6 (cid:0) µ + 164 µ − µ + 6 µ + 171 µ − µ − (cid:0) µ − (cid:1) (cid:0) µ + 34 µ − µ − µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 90 µ + 83 µ + 91 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 17 µ + 32 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 11 (cid:1) µ ∆ + 14 (cid:0) µ − (cid:1) (cid:0) µ + 2 (cid:1) (cid:105) + e h A Ξ F mπ µ (cid:104) µ − µ − (cid:0) µ + 19 (cid:1) µ + 6 (cid:0) µ + 56 (cid:1) µ + (cid:0) µ +482 µ + 440 (cid:1) µ + 6 (cid:0) µ + 29 µ + 13 (cid:1) µ − (cid:0) µ + 326 µ + 248 µ +131) µ − (cid:0) µ + 138 µ + 22 µ + 52 (cid:1) µ + (cid:0) µ + 148 µ − µ − µ +67) µ + 6 (cid:0) µ + 46 µ − µ + 104 µ − (cid:1) µ + 28 (cid:0) µ + 4 µ − µ +22 µ − (cid:1) µ + 42 (cid:0) µ + 2 µ + 12 µ − µ + 5 (cid:1) µ ∆ − (cid:0) µ − µ − µ − µ + 7 µ − (cid:1) (cid:105) + e h A Ξ F mπ µ (cid:104) − µ + 66 µ + 4 (cid:0) µ + 19 (cid:1) µ − (cid:0) µ + 56 (cid:1) µ − (cid:0) µ + 241 µ + 220 (cid:1) µ − (cid:0) µ + 29 µ + 13 (cid:1) µ + (cid:0) µ + 652 µ +496 µ + 262 (cid:1) µ + 6 (cid:0) µ + 138 µ + 22 µ − (cid:1) µ + (cid:0) − µ − µ + 84 µ +324 µ + 31 (cid:1) µ − (cid:0) µ + 46 µ − µ − µ + 77 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ +12 µ + 17 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 5 (cid:1) µ ∆ + 14 (cid:0) µ − (cid:1) (cid:0) µ + 2 (cid:1) (cid:105) + e h A F mπ µ ( − µ + µ − µ ∆ + 1) (cid:104) µ − µ − (cid:0) µ − (cid:1) µ +168 (cid:0) µ + 11 (cid:1) µ + 3 (cid:0) µ + 253 µ − (cid:1) µ − (cid:0) µ + 325 µ µ − (cid:0) µ + 4209 µ + 4278 µ − (cid:1) µ − (cid:0) µ + 144 µ − µ +598) µ + (cid:0) µ + 1797 µ + 3573 µ − µ + 1501 (cid:1) µ + 2 (cid:0) µ + 1272 µ − µ − µ + 82 (cid:1) µ + 14 (cid:0) µ + 57 µ − µ + 427 µ − (cid:1) µ + 14 (cid:0) µ − µ + 85 µ + 4 µ − (cid:1) µ ∆ − (cid:0) µ − µ + 73 µ − µ + 54 µ − (cid:1) (cid:105) ,β ( p ) M1 = − e h A Ξ F mπ µ ( − µ + µ − µ ∆ + 1) (cid:104) − µ + 456 µ + 2 (cid:0) µ +71) µ − (cid:0) µ + 136 (cid:1) µ − (cid:0) µ + 385 µ − (cid:1) µ + 2 (cid:0) µ + 715 µ +388) µ + 2 (cid:0) µ + 479 µ + 44 µ − (cid:1) µ + 2 (cid:0) µ − µ − µ +477) µ + (cid:0) − µ − µ − µ + 620 µ + 95 (cid:1) µ − (cid:0) µ + 58 µ − µ − µ + 136 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 35 µ − (cid:1) µ − (cid:0) µ − (cid:1) µ ∆ + 14 (cid:0) µ − (cid:1) (cid:105) − e h A Ξ F mπ µ (cid:104) − µ − µ + (cid:0) µ + 568 (cid:1) µ + 6 (cid:0) µ − (cid:1) µ − (cid:0) µ + 619 µ + 256 (cid:1) µ − (cid:0) µ + 59 µ − (cid:1) µ + 2 (cid:0) µ + 308 µ +62 µ + 47 (cid:1) µ + 6 (cid:0) µ + 111 µ − µ − (cid:1) µ + (cid:0) − µ + 224 µ +336 µ + 197 (cid:1) µ − (cid:0) µ − µ − µ − µ + 18 (cid:1) µ − (cid:0) µ − µ − µ − µ + 88 (cid:1) µ − (cid:0) µ − µ − µ + 4 µ − (cid:1) µ ∆ + 14 (cid:0) µ − µ +10 µ + 22 µ − µ + 1 (cid:1) (cid:105) + e h A Ξ F mπ µ (cid:104) − µ − µ + (cid:0) µ + 568 (cid:1) µ + 6 (cid:0) µ − (cid:1) µ − (cid:0) µ + 619 µ + 256 (cid:1) µ − (cid:0) µ + 59 µ − (cid:1) µ + 2 (cid:0) µ + 308 µ +62 µ + 47 (cid:1) µ + 6 (cid:0) µ + 111 µ − µ + 80 (cid:1) µ + (cid:0) − µ + 224 µ + 336 µ − µ − (cid:0) µ − µ − µ + 52 µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 21 µ − µ − (cid:0) µ − (cid:1) µ ∆ + 14 (cid:0) µ − (cid:1) (cid:105) + e h A F mπ µ (cid:104) µ + 180 µ − (cid:0) µ + 893 (cid:1) µ − (cid:0) µ − (cid:1) µ +6 (cid:0) µ + 670 µ + 309 (cid:1) µ + 18 (cid:0) µ + 217 µ − (cid:1) µ + (cid:0) − µ − µ + 1656 µ + 1445 (cid:1) µ − (cid:0) µ + 15 µ + 90 µ − (cid:1) µ − (cid:0) µ µ − µ + 503 (cid:1) µ − (cid:0) µ + 39 µ − µ + 4 (cid:1) µ ∆ + 14 (cid:0) µ − µ − µ + 16 µ − (cid:1) (cid:105) . b. Spin-dependent first order polarizabilities γ ( p ) E1E1 = e h A Ξ F m π µ ( µ − µ + 2 µ ∆ − ( µ − µ − µ ∆ − (cid:0) − µ +282 µ + (cid:0) µ − (cid:1) µ − (cid:0) µ − (cid:1) µ − (cid:0) µ − µ − (cid:1) µ + (cid:0) µ − µ − (cid:1) µ + (cid:0) µ − µ − µ − (cid:1) µ +6 (cid:0) µ + 2281 µ + 3724 µ + 3951 (cid:1) µ + (cid:0) − µ + 8181 µ + 19029 µ +25959 µ + 16267 (cid:1) µ − (cid:0) µ + 10535 µ + 11493 µ + 14259 µ + 19673 (cid:1) µ + (cid:0) µ − µ − µ − µ − µ − (cid:1) µ + 2 (cid:0) µ +6380 µ + 2604 µ + 804 µ + 47 µ + 19122 (cid:1) µ + (cid:0) − µ + 11121 µ − µ + 230 µ − µ + 12487 µ + 6389 (cid:1) µ − (cid:0) µ − µ − µ − µ − µ − µ + 3835 (cid:1) µ + (cid:0) µ − µ + 10752 µ − µ − µ + 4107 µ − µ + 1321 (cid:1) µ + 2 (cid:0) µ − (cid:1) (cid:0) µ +1057 µ + 425 µ − µ − µ + 3901 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ − µ +38 µ + 150 µ − µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 319 µ + 13 µ − µ + 252 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 34 µ − µ + 556 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 5 (cid:1) µ ∆ + 7 (cid:0) µ − (cid:1) (cid:0) µ − µ − (cid:1)(cid:17) − e h A Ξ F m π µ (cid:0) − µ + 120 µ + (cid:0) µ − (cid:1) µ + 12 (cid:0) µ + 29 (cid:1) µ + (cid:0) − µ + 1136 µ + 2954 (cid:1) µ + (cid:0) − µ + 158 µ + 874 (cid:1) µ + (cid:0) µ − µ − µ − (cid:1) µ + 6 (cid:0) µ − µ − µ − (cid:1) µ + (cid:0) − µ +1705 µ − µ − µ + 1030 (cid:1) µ − (cid:0) µ − µ + 26 µ − µ − (cid:1) µ + (cid:0) − µ + 496 µ + 300 µ + 340 µ − (cid:1) µ − (cid:0) µ + 2 µ − µ + 46 µ − µ ∆ + 7 (cid:0) µ − µ + 4 µ + 16 µ + 13 µ − (cid:1)(cid:1) − e h A Ξ F m π µ (cid:0) µ − µ + (cid:0) − µ (cid:1) µ − (cid:0) µ + 29 (cid:1) µ (cid:0) µ − µ − (cid:1) µ + 2 (cid:0) µ − µ − (cid:1) µ + (cid:0) − µ +1987 µ + 1888 µ + 4039 (cid:1) µ + (cid:0) − µ + 996 µ + 702 µ + 3048 (cid:1) µ + (cid:0) µ − µ + 825 µ + 165 µ + 820 (cid:1) µ + 6 (cid:0) µ − µ + 26 µ + 84 µ + 75 (cid:1) µ + (cid:0) µ − µ − µ + 700 µ − (cid:1) µ + 14 (cid:0) µ − (cid:1) (cid:0) µ + 4 µ + 25 (cid:1) µ ∆ − (cid:0) µ − (cid:1) (cid:0) µ − µ − (cid:1)(cid:17) − e h A F m π µ ( µ − µ + 2 µ ∆ − ( µ − µ − µ ∆ − (cid:0) µ − µ − (cid:0) µ − (cid:1) µ + 6 (cid:0) µ − (cid:1) µ + 3 (cid:0) µ − µ − (cid:1) µ − (cid:0) µ − µ − (cid:1) µ + (cid:0) − µ + 13950 µ + 36153 µ + 26911 (cid:1) µ − (cid:0) µ + 16422 µ + 27279 µ + 49276 (cid:1) µ + (cid:0) µ − µ − µ − µ + 225 (cid:1) µ + 2 (cid:0) µ + 8664 µ + 6543 µ + 8679 µ + 46169 (cid:1) µ − (cid:0) µ − µ + 3972 µ + 10382 µ + 13357 µ + 25208 (cid:1) µ + (cid:0) − µ +3630 µ + 9990 µ − µ + 68166 µ − (cid:1) µ + (cid:0) µ − µ +20124 µ − µ + 28369 µ − µ + 6763 (cid:1) µ + 2 (cid:0) µ − µ − µ + 10135 µ − µ − µ + 393 (cid:1) µ + (cid:0) µ − µ +16691 µ − µ + 31935 µ − µ + 5234 (cid:1) µ + 168 (cid:0) µ − (cid:1) (cid:0) µ − µ + 47 µ − (cid:1) µ ∆ − (cid:0) µ − (cid:1) (cid:0) µ − µ − µ + 76 µ − (cid:1)(cid:17) ,γ ( p ) M1M1 = e h A Ξ F m π µ ( µ − µ + 2 µ ∆ − ( µ − µ − µ ∆ − (cid:0) − µ +1446 µ + (cid:0) µ + 1649 (cid:1) µ − (cid:0) µ + 893 (cid:1) µ − (cid:0) µ + 6710 µ +2767) µ + 4 (cid:0) µ + 3432 µ + 2113 (cid:1) µ + 2 (cid:0) µ + 5025 µ + 4973 µ +2336) µ − (cid:0) µ + 4365 µ + 5225 µ + 4044 (cid:1) µ − (cid:0) µ + 6073 µ +10931 µ + 8455 µ + 6140 (cid:1) µ + 6 (cid:0) µ − µ + 208 µ + 339 µ + 1116 (cid:1) µ + (cid:0) µ + 277 µ + 3576 µ + 892 µ + 3154 µ + 3885 (cid:1) µ + 2 (cid:0) µ + 1632 µ − µ + 971 µ + 1116 µ − (cid:1) µ − (cid:0) µ − µ + 343 µ − µ +657 µ + 850 µ + 259 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 489 µ − µ + µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 215 µ + 302 µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 1 (cid:1) µ ∆ (cid:0) µ − (cid:1) (cid:0) µ − (cid:1)(cid:17) + e h A Ξ F m π µ (cid:0) µ − µ − (cid:0) µ + 2273 (cid:1) µ + (cid:0) µ − (cid:1) µ + (cid:0) µ + 5122 µ + 3314 (cid:1) µ + (cid:0) µ + 2062 µ + 2410 (cid:1) µ − (cid:0) µ +3125 µ + 3048 µ + 1481 (cid:1) µ − (cid:0) µ + 353 µ + 361 µ + 304 (cid:1) µ + (cid:0) µ − µ − µ − µ + 986 (cid:1) µ + (cid:0) µ + 364 µ − µ + 324 µ + 612 (cid:1) µ + (cid:0) µ + 88 µ − µ + 692 µ − (cid:1) µ + 14 (cid:0) µ + 16 µ + 12 µ − µ +1) µ ∆ − (cid:0) µ − µ − µ − µ + 21 µ − (cid:1)(cid:1) + e h A Ξ F m π µ (cid:0) − µ + 312 µ + (cid:0) µ + 2273 (cid:1) µ + (cid:0) − µ (cid:1) µ − (cid:0) µ + 2561 µ + 1657 (cid:1) µ − (cid:0) µ + 1031 µ + 1205 (cid:1) µ + (cid:0) µ +3125 µ + 3048 µ + 1481 (cid:1) µ + 6 (cid:0) µ + 353 µ + 361 µ + 458 (cid:1) µ + (cid:0) − µ +31 µ + 159 µ + 321 µ + 1196 (cid:1) µ + (cid:0) − µ − µ + 366 µ + 324 µ +612) µ + (cid:0) − µ − µ + 204 µ + 692 µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ +18 µ − (cid:1) µ ∆ + 7 (cid:0) µ − (cid:1) (cid:0) µ − (cid:1)(cid:17) − e h A F m π µ ( µ − µ + 2 µ ∆ − (cid:0) µ − µ − (cid:0) µ +1597) µ + 18 (cid:0) µ + 1063 (cid:1) µ + 3 (cid:0) µ + 5123 µ + 2666 (cid:1) µ − (cid:0) µ + 3516 µ + 3829 (cid:1) µ − (cid:0) µ + 14565 µ + 18741 µ + 6716 (cid:1) µ + (cid:0) µ − µ + 9408 µ + 23984 (cid:1) µ + (cid:0) µ + 2145 µ + 6003 µ − µ − µ + (cid:0) µ + 4860 µ + 3270 µ − µ + 298 (cid:1) µ + (cid:0) µ − µ − µ − µ + 2090 (cid:1) µ − (cid:0) µ − µ + 28 µ − (cid:1) µ ∆ − (cid:0) µ − µ + 196 µ − µ + 218 µ − (cid:1)(cid:1) ,γ ( p ) E1M2 = − e h A Ξ F m π µ ( µ − µ + 2 µ ∆ − ( µ − µ − µ ∆ − (cid:0) µ − µ − (cid:0) µ + 475 (cid:1) µ + 6 (cid:0) µ + 178 (cid:1) µ + (cid:0) µ + 1714 µ +1583) µ − (cid:0) µ + 642 µ + 428 (cid:1) µ − (cid:0) µ + 1077 µ + 1765 µ +2159) µ + 2 (cid:0) µ + 654 µ − µ + 901 (cid:1) µ + (cid:0) µ + 983 µ + 1733 µ +5733 µ + 5652 (cid:1) µ + (cid:0) − µ + 762 µ + 3840 µ + 1570 µ − (cid:1) µ (cid:0) µ + 47 µ − µ + 1340 µ − µ + 3607 (cid:1) µ + 2 (cid:0) µ − µ − µ + 1003 µ − µ − (cid:1) µ + (cid:0) µ + 48 µ − µ + 1484 µ +483 µ − µ + 1279 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 117 µ + 92 µ + 209 µ − µ − (cid:0) µ − (cid:1) (cid:0) µ + 19 µ + 205 µ − (cid:1) µ +42 (cid:0) µ − (cid:1) (cid:0) µ + 1 (cid:1) µ ∆ + 7 (cid:0) µ − (cid:1) (cid:0) µ + 4 µ + 1 (cid:1)(cid:17) + e h A Ξ F m π µ (cid:0) µ − µ − (cid:0) µ + 571 (cid:1) µ + 4 (cid:0) µ − (cid:1) µ +2 (cid:0) µ + 607 µ + 644 (cid:1) µ + (cid:0) − µ + 758 µ + 1958 (cid:1) µ − (cid:0) µ + 763 µ +300 µ + 501 (cid:1) µ + 2 (cid:0) µ − µ − µ − (cid:1) µ + (cid:0) µ + 175 µ − µ − µ − (cid:1) µ − (cid:0) µ + 20 µ + 75 µ + 270 µ − (cid:1) µ + (cid:0) − µ +24 µ − µ − µ + 300 (cid:1) µ + 28 (cid:0) µ − µ − µ − µ + 2 (cid:1) µ ∆ + 7 (cid:0) µ + µ − µ − µ + µ + 1 (cid:1)(cid:1) − e h A Ξ F m π µ (cid:0) µ − µ − (cid:0) µ + 571 (cid:1) µ + 4 (cid:0) µ − (cid:1) µ +2 (cid:0) µ + 607 µ + 644 (cid:1) µ + (cid:0) − µ + 758 µ + 1958 (cid:1) µ − (cid:0) µ + 763 µ +300 µ + 501 (cid:1) µ + 2 (cid:0) µ − µ − µ − (cid:1) µ + (cid:0) µ + 175 µ − µ − µ + 1182 (cid:1) µ − (cid:0) µ + 20 µ + 75 µ − µ + 96 (cid:1) µ + (cid:0) − µ +24 µ − µ + 584 µ − (cid:1) µ + 28 (cid:0) µ − (cid:1) (cid:0) µ − µ − (cid:1) µ ∆ +7 (cid:0) µ − (cid:1) (cid:0) µ + 4 µ + 1 (cid:1)(cid:17) − e h A F m π µ ( µ − µ + 2 µ ∆ − (cid:0) µ − µ − (cid:0) µ + 707 (cid:1) µ +18 (cid:0) µ + 251 (cid:1) µ + 3 (cid:0) µ + 1285 µ + 2038 (cid:1) µ − (cid:0) µ + 180 µ − µ − (cid:0) µ + 1635 µ + 5055 µ + 23338 (cid:1) µ + 2 (cid:0) µ − µ − µ + 4691 (cid:1) µ + (cid:0) µ + 231 µ − µ + 465 µ + 4651 (cid:1) µ − (cid:0) µ +378 µ + 2154 µ + 11 µ − (cid:1) µ − (cid:0) µ − µ − µ − µ +944) µ + 126 (cid:0) µ + 27 µ − µ + 16 µ − (cid:1) µ ∆ + 7 (cid:0) µ + 3 µ + 160 µ − µ + 102 µ − (cid:1)(cid:1) ,γ ( p ) M1E2 = − e h A Ξ F m π µ ( µ − µ + 2 µ ∆ − ( µ − µ − µ ∆ − (cid:0) − µ µ + (cid:0) µ + 233 (cid:1) µ − (cid:0) µ + 347 (cid:1) µ + (cid:0) − µ − µ +41) µ + 4 (cid:0) µ + 533 µ + 453 (cid:1) µ + 2 (cid:0) µ + 687 µ + 317 µ − (cid:1) µ − (cid:0) µ + 1124 µ + 2069 µ + 442 (cid:1) µ + (cid:0) − µ − µ − µ + 1709 µ +1544) µ + 2 (cid:0) µ + 425 µ + 1725 µ + 455 µ − (cid:1) µ + (cid:0) µ + 61 µ +1324 µ − µ − µ + 613 (cid:1) µ + 2 (cid:0) µ + 14 µ − µ + 360 µ − µ + 924 (cid:1) µ + (cid:0) − µ + 138 µ − µ + 508 µ − µ + 2404 µ − µ − (cid:0) µ − (cid:1) (cid:0) µ + 103 µ + 39 µ + 63 µ + 296 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ − µ − µ + 310 (cid:1) µ + 42 (cid:0) µ − (cid:1) (cid:0) µ + 4 µ − (cid:1) µ ∆ +7 (cid:0) µ − (cid:1) (cid:0) µ + 2 µ + 3 (cid:1)(cid:17) + e h A Ξ F m π µ (cid:0) − µ − µ + (cid:0) µ + 137 (cid:1) µ + 6 (cid:0) µ − (cid:1) µ − (cid:0) µ + 170 µ + 230 (cid:1) µ + (cid:0) − µ + 242 µ + 570 (cid:1) µ + (cid:0) µ + 245 µ +796 µ + 267 (cid:1) µ + 6 (cid:0) µ + 8 µ − µ + 52 (cid:1) µ − (cid:0) µ + 11 µ + 315 µ +285 µ − (cid:1) µ − (cid:0) µ + 10 µ − µ − µ + 59 (cid:1) µ + (cid:0) − µ + 40 µ +66 µ + 520 µ − (cid:1) µ + 28 (cid:0) µ + 3 µ + 12 µ − µ + 3 (cid:1) µ ∆ + 7 (cid:0) µ − µ +4 µ − µ + 3 (cid:1)(cid:1) + e h A Ξ F m π µ (cid:0) µ + 48 µ − (cid:0) µ + 137 (cid:1) µ − (cid:0) µ − (cid:1) µ + (cid:0) µ + 340 µ + 460 (cid:1) µ + (cid:0) µ − µ − (cid:1) µ − (cid:0) µ + 245 µ +796 µ + 267 (cid:1) µ − (cid:0) µ + 8 µ − µ − (cid:1) µ + (cid:0) µ + 11 µ + 315 µ +285 µ + 1282 (cid:1) µ + 6 (cid:0) µ + 10 µ − µ + 60 µ − (cid:1) µ + (cid:0) µ − µ − µ + 520 µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 5 µ − (cid:1) µ ∆ − (cid:0) µ − (cid:1) (cid:0) µ + 2 µ + 3 (cid:1)(cid:17) − e h A F m π µ ( µ − µ + 2 µ ∆ − (cid:0) − µ + 684 µ + 3 (cid:0) µ + 223 (cid:1) µ − (cid:0) µ + 523 (cid:1) µ − (cid:0) µ + 689 µ − (cid:1) µ + 6 (cid:0) µ + 816 µ + 217 (cid:1) µ + (cid:0) µ + 1983 µ + 3315 µ + 1874 (cid:1) µ + 2 (cid:0) µ − µ − µ − (cid:1) µ + (cid:0) − µ − µ − µ + 4659 µ + 9271 (cid:1) µ + (cid:0) − µ − µ + 5916 µ µ − (cid:1) µ + (cid:0) − µ + 711 µ − µ + 4248 µ − (cid:1) µ + 42 (cid:0) µ − µ + 115 µ − µ + 10 (cid:1) µ ∆ + 7 (cid:0) µ − µ − µ + 313 µ − µ + 54 (cid:1)(cid:1) .
2. Neutron values a. Spin-independent first order polarizabilities α ( n ) E1 = − e h A Ξ F mπ µ ( − µ + µ − µ ∆ + 1) ( − µ + µ + 2 µ ∆ + 1) (cid:104) − µ +70 µ + (cid:0) µ − (cid:1) µ + (cid:0) − µ (cid:1) µ + (cid:0) − µ + 936 µ + 429 (cid:1) µ +3 (cid:0) µ − µ − (cid:1) µ + 2 (cid:0) µ − µ − µ − (cid:1) µ + (cid:0) − µ +597 µ + 90 µ + 133 (cid:1) µ + (cid:0) − µ + 868 µ − µ + 98 µ + 217 (cid:1) µ +3 (cid:0) µ − µ + 81 µ + 5 µ − (cid:1) µ + 3 (cid:0) µ − (cid:1) (cid:0) µ − µ + 104 µ +13) µ − (cid:0) µ − (cid:1) (cid:0) µ + 9 µ + 21 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 17 µ + 32 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 11 (cid:1) µ ∆ + 2 (cid:0) µ − (cid:1) (cid:0) µ + 2 (cid:1) (cid:105) + e h A Ξ F mπ µ (cid:104) − µ + 4 µ + 4 µ + 16 µ − µ + 8 µ − µ + (131 − µ (cid:1) µ + 3 (cid:0) µ + 47 (cid:1) µ + (cid:0) µ − µ − (cid:1) µ − (cid:0) µ + 41 µ +33) µ − (cid:0) µ − µ + 126 µ + 35 (cid:1) µ + 6 (cid:0) µ − µ + 4 µ − (cid:1) µ + 4 (cid:0) µ +4 µ − µ + 22 µ − (cid:1) µ + 6 (cid:0) µ + 2 µ + 12 µ − µ + 5 (cid:1) µ ∆ + 4 (cid:105) + e h A Ξ F mπ µ (cid:104) − µ + 54 µ + (cid:0) µ − (cid:1) µ − (cid:0) µ + 47 (cid:1) µ + (cid:0) − µ +238 µ + 31 (cid:1) µ + 3 (cid:0) µ + 41 µ − (cid:1) µ + (cid:0) µ − µ + 126 µ − (cid:1) µ − (cid:0) µ − µ − µ + 3 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 12 µ + 17 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ +5) µ ∆ + 2 (cid:0) µ − (cid:1) (cid:0) µ + 2 (cid:1) (cid:105) + e h A F mπ µ ( − µ + µ − µ ∆ + 1) (cid:104) µ − µ − (cid:0) µ − (cid:1) µ +12 (cid:0) µ − (cid:1) µ + (cid:0) µ − µ + 413 (cid:1) µ + (cid:0) − µ + 1764 µ +62) µ + (cid:0) − µ + 1188 µ − µ + 1 (cid:1) µ + 2 (cid:0) µ − µ − µ +22) µ + 2 (cid:0) µ + 57 µ − µ + 427 µ − (cid:1) µ + 2 (cid:0) µ − µ + 85 µ µ − (cid:1) µ ∆ + 2 (cid:0) − µ + 15 µ − µ + 108 µ − µ + 10 (cid:1) (cid:105) ,β ( n ) M1 = − e h A Ξ F mπ µ ( − µ + µ − µ ∆ + 1) (cid:104) − µ + 70 µ + (cid:0) µ − (cid:1) µ + (cid:0) − µ (cid:1) µ + (cid:0) − µ + 358 µ − (cid:1) µ + (cid:0) µ − µ + 45 (cid:1) µ + (cid:0) µ − µ + 38 µ + 107 (cid:1) µ + (cid:0) − µ + 52 µ + 100 µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 5 µ − (cid:1) µ − (cid:0) µ − (cid:1) µ ∆ + 2 (cid:0) µ − (cid:1) (cid:105) − e h A Ξ F mπ µ (cid:104) − µ + 54 µ + (cid:0) µ − (cid:1) µ + (cid:0) − µ (cid:1) µ + (cid:0) − µ +118 µ + 7 (cid:1) µ + 3 (cid:0) µ − µ − (cid:1) µ + (cid:0) µ − µ + 18 µ − (cid:1) µ +12 µ (cid:0) − µ + 5 µ + 2 (cid:1) µ + (cid:0) − µ + 20 µ + 48 µ + 92 µ − (cid:1) µ − (cid:0) µ − µ − µ + 4 µ − (cid:1) µ ∆ + 2 (cid:0) µ − µ + 10 µ + 22 µ − µ + 1 (cid:1) (cid:105) + e h A Ξ F mπ µ (cid:104) − µ + 54 µ + (cid:0) µ − (cid:1) µ + (cid:0) − µ (cid:1) µ + (cid:0) − µ +118 µ + 7 (cid:1) µ + 3 (cid:0) µ − µ + 41 (cid:1) µ + (cid:0) µ − µ + 18 µ + 19 (cid:1) µ − µ (cid:0) µ − µ + 2 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 3 µ − (cid:1) µ − (cid:0) µ − (cid:1) µ ∆ + 2 (cid:0) µ − (cid:1) (cid:105) + e h A F mπ µ (cid:104) µ − µ − (cid:0) µ − (cid:1) µ + 36 (cid:0) µ + 1 (cid:1) µ + (cid:0) µ − µ + 371 (cid:1) µ − (cid:0) µ + 6 µ − (cid:1) µ − (cid:0) µ − µ − µ + 77 (cid:1) µ − (cid:0) µ + 39 µ − µ + 4 (cid:1) µ ∆ + 2 (cid:0) µ − µ − µ + 16 µ − (cid:1) (cid:105) . b. Spin-dependent first order polarizabilities γ ( n ) E1E1 = e h A Ξ F m π µ ( µ − µ + 2 µ ∆ − ( µ − µ − µ ∆ − (cid:104) − µ +122 µ + (cid:0) µ − (cid:1) µ − (cid:0) µ + 51 (cid:1) µ + (cid:0) − µ + 316 µ − (cid:1) µ +3 (cid:0) µ + 457 µ + 347 (cid:1) µ + (cid:0) µ − µ + 708 µ + 733 (cid:1) µ − (cid:0) µ +744 µ + 1347 µ + 1265 (cid:1) µ − (cid:0) µ − µ + 1116 µ + 507 µ + 625 (cid:1) µ + (cid:0) µ − µ + 2142 µ + 898 µ + 3456 (cid:1) µ + (cid:0) µ − µ + 2858 µ µ + 946 µ + 935 (cid:1) µ − (cid:0) µ − µ + 228 µ − µ − µ +915) µ + (cid:0) − µ + 1728 µ − µ + 1256 µ + 540 µ − µ − (cid:1) µ + (cid:0) µ − (cid:1) (cid:0) µ − µ − µ + 97 µ + 1265 (cid:1) µ + (cid:0) µ − (cid:1) (cid:0) µ +199 µ − µ + 411 µ + 323 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 7 µ − µ + 96 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 34 µ − µ + 88 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 5 (cid:1) µ ∆ + (cid:0) µ − (cid:1) (cid:0) µ − µ − (cid:1) (cid:105) + e h A Ξ F m π µ (cid:104) − µ + 5 µ − µ − µ − µ + 34 µ − µ − (cid:0) µ +2) µ + (cid:0) µ + 41 (cid:1) µ + (cid:0) µ + 46 µ + 451 (cid:1) µ + (cid:0) − µ + 39 µ +159) µ − (cid:0) µ + 81 µ − µ + 151 (cid:1) µ + 6 (cid:0) µ − µ + 12 µ − (cid:1) µ +2 (cid:0) µ + 16 µ − µ − µ + 43 (cid:1) µ + 2 (cid:0) µ + 2 µ − µ + 46 µ − µ ∆ + 5 (cid:105) − e h A Ξ F m π µ (cid:104) µ − µ − (cid:0) µ + 2 (cid:1) µ + (cid:0) µ + 41 (cid:1) µ + (cid:0) µ +46 µ + 451 (cid:1) µ + (cid:0) − µ + 39 µ + 279 (cid:1) µ + (cid:0) − µ − µ + 60 µ +121) µ + 6 (cid:0) µ − µ + 8 µ + 25 (cid:1) µ + 2 (cid:0) µ + 16 µ − µ + 74 µ − µ + 2 (cid:0) µ − (cid:1) (cid:0) µ + 4 µ + 25 (cid:1) µ ∆ − (cid:0) µ − (cid:1) (cid:0) µ − µ − (cid:1) (cid:105) + e h A F m π µ ( µ − µ + 2 µ ∆ − ( µ − µ − µ ∆ − (cid:104) − µ + 732 µ +6 (cid:0) µ − (cid:1) µ − (cid:0) µ + 132 (cid:1) µ + (cid:0) − µ + 2367 µ + 230 (cid:1) µ + (cid:0) µ + 3186 µ + 5678 (cid:1) µ + (cid:0) µ − µ + 2261 µ − (cid:1) µ − (cid:0) µ − µ + 4827 µ + 4253 (cid:1) µ + (cid:0) − µ + 3702 µ − µ +4783 µ + 4820 (cid:1) µ + (cid:0) µ − µ + 5706 µ − µ + 2966 (cid:1) µ + (cid:0) µ − µ + 2605 µ − µ + 6079 µ − (cid:1) µ + (cid:0) − µ +3510 µ − µ + 3220 µ − µ + 402 (cid:1) µ + (cid:0) − µ + 411 µ + 115 µ +1127 µ − µ + 2714 µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ − µ + 47 µ − µ ∆ + (cid:0) µ − (cid:1) (cid:0) µ − µ − µ + 76 µ − (cid:1) (cid:105) , ( n ) M1M1 = e h A Ξ F m π µ ( µ − µ + 2 µ ∆ − ( µ − µ − µ ∆ − (cid:104) − µ +102 µ + (cid:0) µ − (cid:1) µ + (cid:0) − µ (cid:1) µ + (cid:0) − µ + 1114 µ + 613 (cid:1) µ + (cid:0) µ − µ − (cid:1) µ + (cid:0) µ − µ − µ − (cid:1) µ − (cid:0) µ − µ + 62 µ − (cid:1) µ + (cid:0) − µ + 1464 µ − µ + 562 µ + 567 (cid:1) µ +5 (cid:0) µ − µ + 149 µ + 15 µ − (cid:1) µ + (cid:0) µ − µ + 726 µ − µ +149 µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 10 µ + 43 µ − (cid:1) µ − (cid:0) µ − (cid:0) µ + 17 µ + 50 µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 1 (cid:1) µ ∆ + (cid:0) µ − (cid:1) (cid:0) µ − (cid:1) (cid:105) + e h A Ξ F m π µ (cid:104) − µ + 3 µ + 4 µ + 20 µ − µ + 34 µ − µ + (194 − µ (cid:1) µ + (cid:0) µ + 181 (cid:1) µ + (cid:0) µ − µ − (cid:1) µ − (cid:0) µ + 81 µ +59) µ + (cid:0) − µ + 201 µ − µ + 137 (cid:1) µ + 2 (cid:0) µ + 15 µ − µ + 78 (cid:1) µ +2 (cid:0) µ − µ + 6 µ + 46 µ − (cid:1) µ + 2 (cid:0) µ + 16 µ + 12 µ − µ + 1 (cid:1) µ ∆ + 7 (cid:105) + e h A Ξ F m π µ (cid:104) − µ + 34 µ + (cid:0) µ − (cid:1) µ − (cid:0) µ + 181 (cid:1) µ + (cid:0) − µ + 390 µ + 149 (cid:1) µ + 3 (cid:0) µ + 81 µ + 99 (cid:1) µ + (cid:0) µ − µ +96 µ + 167 (cid:1) µ − (cid:0) µ + 15 µ + 6 µ − (cid:1) µ − (cid:0) µ − µ + 6 µ − µ +25) µ − (cid:0) µ − (cid:1) (cid:0) µ + 18 µ − (cid:1) µ ∆ + (cid:0) µ − (cid:1) (cid:0) µ − (cid:1) (cid:105) − e h A F m π µ ( µ − µ + 2 µ ∆ − (cid:104) − µ + 21 µ − µ + 349 µ − µ +204 µ − µ + (cid:0) − µ (cid:1) µ + 12 (cid:0) µ − (cid:1) µ + (cid:0) µ − µ − (cid:1) µ + (cid:0) − µ + 1650 µ + 2390 (cid:1) µ − (cid:0) µ − µ + 285 µ +2024) µ + 2 (cid:0) µ + 501 µ − µ + 11 (cid:1) µ + (cid:0) µ − µ − µ − µ + 422 (cid:1) µ − (cid:0) µ − µ + 28 µ − (cid:1) µ ∆ + 50 (cid:105) ,γ ( n ) E1M2 = − e h A Ξ F m π µ ( µ − µ + 2 µ ∆ − ( µ − µ − µ ∆ − (cid:104) µ +48 µ − (cid:0) µ + 148 (cid:1) µ + (cid:0) − µ (cid:1) µ + (cid:0) µ + 574 µ − (cid:1) µ (cid:0) µ − µ − (cid:1) µ + (cid:0) − µ − µ + 741 µ + 669 (cid:1) µ + (cid:0) − µ +1185 µ − µ + 63 (cid:1) µ + (cid:0) µ + 488 µ − µ + 234 µ − (cid:1) µ + (cid:0) µ − µ + 653 µ − µ − (cid:1) µ + (cid:0) µ − µ + 254 µ − µ − µ +124) µ − (cid:0) µ − (cid:1) (cid:0) µ − µ + 59 µ − (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 13 µ +19 µ − (cid:1) µ + 6 (cid:0) µ − (cid:1) (cid:0) µ + 1 (cid:1) µ ∆ + (cid:0) µ − (cid:1) (cid:0) µ + 4 µ + 1 (cid:1) (cid:105) + e h A Ξ F m π µ (cid:104) µ + µ − µ − µ − (cid:0) µ − µ + 30 (cid:1) µ µ + µ + 2 µ +52 µ − (cid:0) µ + 44 (cid:1) µ − (cid:0) µ − (cid:1) µ + (cid:0) µ + 66 µ − (cid:1) µ + (cid:0) µ − µ − (cid:1) µ + (cid:0) − µ + µ − µ − (cid:1) µ − (cid:0) µ + 12 µ − µ + 64 µ − µ + 4 (cid:0) µ − µ − µ − µ + 2 (cid:1) µ ∆ + 1 (cid:105) − e h A Ξ F m π µ (cid:104) µ + 52 µ − (cid:0) µ + 44 (cid:1) µ − (cid:0) µ − (cid:1) µ + (cid:0) µ + 66 µ − µ + (cid:0) µ − µ − (cid:1) µ + (cid:0) − µ + µ − µ + 165 (cid:1) µ − µ (cid:0) µ − µ + 42 (cid:1) µ − (cid:0) µ + 12 µ − µ − µ + 18 (cid:1) µ + 4 (cid:0) µ − (cid:1) (cid:0) µ − µ − µ ∆ + (cid:0) µ − (cid:1) (cid:0) µ + 4 µ + 1 (cid:1) (cid:105) − e h A F m π µ ( µ − µ + 2 µ ∆ − (cid:104) µ + 3 µ + 160 µ − µ + 102 µ +12 µ + 288 µ − (cid:0) µ + 145 (cid:1) µ + (cid:0) − µ (cid:1) µ + (cid:0) µ + 1659 µ − µ + 4 (cid:0) µ − µ + 328 (cid:1) µ + (cid:0) − µ − µ + 627 µ + 172 (cid:1) µ − (cid:0) µ + 102 µ + 125 µ − (cid:1) µ − (cid:0) µ + 29 µ − µ − µ + 128 (cid:1) µ +18 (cid:0) µ + 27 µ − µ + 16 µ − (cid:1) µ ∆ − (cid:105) ,γ ( n ) M1E2 = − e h A Ξ F m π µ ( µ − µ + 2 µ ∆ − ( µ − µ − µ ∆ − (cid:104) µ +44 µ − (cid:0) µ + 184 (cid:1) µ + (cid:0) − µ (cid:1) µ + (cid:0) µ + 730 µ + 67 (cid:1) µ + (cid:0) µ − µ − (cid:1) µ + (cid:0) − µ − µ + 269 µ + 161 (cid:1) µ + (cid:0) − µ +1119 µ + 46 µ − (cid:1) µ + (cid:0) µ + 712 µ − µ − µ − (cid:1) µ + (cid:0) µ − µ + 333 µ − µ + 291 (cid:1) µ + (cid:0) µ − µ + 274 µ − µ + 169 µ − µ − (cid:0) µ − (cid:1) (cid:0) µ − µ − µ + 68 (cid:1) µ − (cid:0) µ − (cid:1) (cid:0) µ + 3 µ µ + 46 (cid:1) µ + 6 (cid:0) µ − (cid:1) (cid:0) µ + 4 µ − (cid:1) µ ∆ + (cid:0) µ − (cid:1) (cid:0) µ + 2 µ + 3 (cid:1) (cid:105) + e h A Ξ F m π µ (cid:104) µ − µ + 4 µ − µ + 2 µ + 48 µ − (cid:0) µ + 88 (cid:1) µ + (39 − µ (cid:1) µ + (cid:0) µ + 166 µ + 51 (cid:1) µ + 3 (cid:0) µ − µ + 15 (cid:1) µ − (cid:0) µ + 69 µ +24 µ − (cid:1) µ − (cid:0) µ − µ + 4 µ + 1 (cid:1) µ − (cid:0) µ + 4 µ − µ − µ +34) µ + 4 (cid:0) µ + 3 µ + 12 µ − µ + 3 (cid:1) µ ∆ + 3 (cid:105) − e h A Ξ F m π µ (cid:104) µ + 48 µ − (cid:0) µ + 88 (cid:1) µ + (cid:0) − µ (cid:1) µ + (cid:0) µ + 166 µ +51) µ + 3 (cid:0) µ − µ − (cid:1) µ − (cid:0) µ + 69 µ + 24 µ + 187 (cid:1) µ + (cid:0) − µ +72 µ + 6 (cid:1) µ − (cid:0) µ + 4 µ − µ + 44 µ − (cid:1) µ + 4 (cid:0) µ − (cid:1) (cid:0) µ + 5 µ − µ ∆ + (cid:0) µ − (cid:1) (cid:0) µ + 2 µ + 3 (cid:1) (cid:105) − e h A F m π µ ( µ − µ + 2 µ ∆ − (cid:104) µ − µ − µ + 313 µ − µ +12 µ + 264 µ − (cid:0) µ + 181 (cid:1) µ − (cid:0) µ − (cid:1) µ + (cid:0) µ + 2163 µ +8) µ + 4 (cid:0) µ − µ − (cid:1) µ + (cid:0) − µ − µ + 1275 µ + 1444 (cid:1) µ − (cid:0) µ − µ + 61 µ + 299 (cid:1) µ + (cid:0) − µ + 9 µ − µ + 576 µ − (cid:1) µ +6 (cid:0) µ − µ + 115 µ − µ + 10 (cid:1) µ ∆ + 54 (cid:105) . Appendix D: q -Values Here, we use the notation ˜ e ≡ e + 2 e + e + g A c + 3 c mπ F and ˜ e ≡ e + 2 e + e − g A c + 15 c mπ F .
1. Proton values a. Spin-independent first order polarizabilities α pE1 = − e g A π F m (cid:104) c Ξ (cid:0) µ − µ + 4 (cid:1) + 3 c Ξ (cid:0) µ − µ + 4 (cid:1) + 5 c µ +9 c µ (cid:105) − e (˜ e + 2˜ e + e + 2 e ) π − e (4 c + c − c )192 π F + e Ξ π F m (cid:104) g A c µ (cid:0) µ − (cid:1) + 9 g A c µ (cid:0) µ − (cid:1) − c m (cid:105) , pM1 = e g A π F ( µ − m (cid:104) c Ξ (cid:0) − µ + 113 µ − µ + 96 (cid:1) + c Ξ (cid:0) − µ +159 µ − µ + 144 (cid:1) + c (cid:0) µ − µ + 4 (cid:1) + c (cid:0) µ − µ + 8 (cid:1) (cid:105) + 2 e (˜ e + e ) π + e (4 c − c − c )192 π F + e π F m Ξ (cid:104) c g A (cid:0) µ − µ + 2 (cid:1) + 3 c g A (cid:0) µ − µ + 4 (cid:1) − c m (cid:105) . b. Spin-independent second order polarizabilities α ( p ) E2 = e g A π F ( µ − m (cid:104) c Ξ (cid:0) − µ + 340 µ − µ + 276 (cid:1) + c Ξ (cid:0) − µ +628 µ − µ + 636 (cid:1) − c (cid:0) µ − µ + 212 (cid:1) − c (cid:0) µ − µ + 92 (cid:1) (cid:105) + e Ξ π F m (cid:104) c m + g A c (cid:0) µ − µ + 6 (cid:1) + g A c (cid:0) µ − µ + 18 (cid:1) (cid:105) + e ( − c + 15 µ c − c )960 π F µ m + 3 e ˜ e πm ,β ( p ) M2 = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 1208 µ − µ + 7100 µ − µ + 64 (cid:1) + c Ξ (cid:0) − µ + 1772 µ − µ + 10780 µ − µ + 160 (cid:1) + c (cid:0) − µ + 848 µ − µ + 848 (cid:1) + c (cid:0) − µ + 1250 µ − µ + 1400 (cid:1) (cid:105) + e Ξ π F m (cid:104) c m + g A c (cid:0) µ − µ + 26 (cid:1) + g A c (cid:0) µ − µ + 46 (cid:1) (cid:105) + e (32 c + (15 µ + 8) c + 44 c )960 π F µ m + 3 e ˜ e πm ,α ( p ) E1 ν = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 6616 µ − µ + 41620 µ − µ + 576 (cid:1) + c Ξ (cid:0) − µ (cid:0) µ − µ + 8136 µ − µ +6176)) + c (cid:0) − µ + 4624 µ − µ + 4784 (cid:1) + 3 c (cid:0) − µ + 2338 µ − µ + 2568 (cid:1) (cid:105) + e Ξ π F m (cid:104) g A c (cid:0) µ − µ + 190 (cid:1) + 15 g A c (cid:0) µ − µ + 22 (cid:1) − c m (cid:105) e (96 c + (16 − µ ) c + 132 c )11520 π F µ m − e ˜ e πm ,β ( p ) M1 ν = − e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) µ − µ + 57904 µ − µ +196496 µ − µ + 4352 (cid:1) + c Ξ (cid:0) µ − µ + 86300 µ − µ +298416 µ − µ + 6400 (cid:1) − c (cid:0) − µ + 7562 µ − µ + 49112 µ − µ + 256 (cid:1) − c (cid:0) − µ + 11226 µ − µ + 74336 µ − µ + 512 (cid:1) (cid:105) + e Ξ π F m (cid:104) g A c (cid:0) µ − µ + 210 (cid:1) + g A c (cid:0) µ − µ + 326 (cid:1) − c m (cid:105) − e (32 c + (5 µ + 8) c + 44 c )3840 π F µ m − e ˜ e πm . c. Spin-dependent first order polarizabilities γ ( p ) E1E1 = e g A π F ( µ − m (cid:104) − c Ξ (cid:0) µ − µ + 2582 µ − µ +1704) − c Ξ (cid:0) µ − µ + 3488 µ − µ + 2400 (cid:1) + 2 c (cid:0) − µ + 223 µ − µ + 216 (cid:1) + c (cid:0) − µ + 599 µ − µ + 608 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 12 (cid:1) + c (cid:0) µ − µ + 18 (cid:1) (cid:105) ,γ ( p ) M1M1 = e g A π F ( µ − m µ (cid:104) c Ξ (cid:0) − µ + 1651 µ − µ + 9584 µ − µ + 128 (cid:1) + c Ξ (cid:0) − µ + 2053 µ − µ + 11604 µ − µ + 128 (cid:1) + c (cid:0) − µ + 1147 µ − µ + 976 µ (cid:1) + c (cid:0) − µ + 1423 µ − µ + 1120 µ (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 38 (cid:1) + c (cid:0) µ − µ + 42 (cid:1) (cid:105) ,γ ( p ) E1M2 = e g A π F ( µ − m (cid:104) − c Ξ (cid:0) µ − µ + 552 µ − µ + 312 (cid:1) − c Ξ (cid:0) µ − µ + 1592 µ − µ + 672 (cid:1) + c (cid:0) − µ + 291 µ − µ c (cid:0) − µ + 285 µ − µ + 128 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 2 (cid:1) + c (cid:0) µ − µ + 2 (cid:1) (cid:105) ,γ ( p ) M1E2 = e g A π F ( µ − m µ (cid:104) c Ξ (cid:0) − µ + 49 µ − µ + 48 µ (cid:1) + c Ξ (cid:0) µ − µ − µ + 16 (cid:1) − c (cid:0) µ − µ + 4 µ (cid:1) + c (cid:0) µ + 4 µ (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c µ (cid:0) µ − (cid:1) − c (cid:0) µ + 6 (cid:1) (cid:105) . d. Spin-dependent second order polarizabilities γ ( p ) E2E2 = e g A π F µ ( µ − m (cid:104) − c Ξ (cid:0) µ − µ + 274613 µ − µ + 1048500 µ − µ + 17920 (cid:1) − c Ξ (cid:0) µ − µ +448178 µ − µ + 1768680 µ − µ + 38400 (cid:1) − c µ (cid:0) µ − µ + 914298 µ − µ + 742400 (cid:1) − c µ (cid:0) µ − µ +374947 µ − µ + 328480 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 1263 (cid:1) + c (cid:0) µ − µ + 2348 (cid:1) (cid:105) ,γ ( p ) M2M2 = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 189729 µ − µ +3028128 µ − µ + 1458080 µ − (cid:1) − c Ξ (cid:0) µ − µ +844403 µ − µ + 2788060 µ − µ + 61440 (cid:1) + c (cid:0) − µ +666339 µ − µ + 4234376 µ − µ + 23040 (cid:1) + 4 c (cid:0) − µ +220176 µ − µ + 1385404 µ − µ + 7680 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 3017 (cid:1) + 3 c (cid:0) µ − µ + 1284 (cid:1) (cid:105) ,γ ( p ) E2M3 = e g A π F µ ( µ − m (cid:104) c Ξ µ (cid:0) − µ + 14319 µ − µ +222784 µ − µ + 77920 (cid:1) − c Ξ (cid:0) µ − µ + 82397 µ µ + 274020 µ − µ + 2560 (cid:1) − c µ (cid:0) µ − µ + 268578 µ − µ + 148736 (cid:1) − c µ (cid:0) µ − µ + 67237 µ − µ + 25504 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 231 (cid:1) + c (cid:0) µ − µ + 116 (cid:1) (cid:105) ,γ ( p ) M2E3 = e g A π F µ ( µ − m (cid:104) − c Ξ µ (cid:0) µ − µ + 7971 µ − µ + 7020 (cid:1) − c Ξ (cid:0) µ − µ + 2946 µ − µ + 1640 µ + 1920 (cid:1) + c µ (cid:0) − µ + 7803 µ − µ + 11008 (cid:1) + 4 c (cid:0) − µ + 612 µ − µ +1112 µ + 480 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 41 (cid:1) + 3 c (cid:0) µ − µ − (cid:1) (cid:105) ,γ ( p ) E1E1 ν = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 252768 µ − µ +8314020 µ − µ + 19956640 µ − µ + 586240 (cid:1) +6 c Ξ (cid:0) − µ + 343603 µ − µ + 11349720 µ − µ +27512320 µ − µ + 837120 (cid:1) − c (cid:0) µ − µ +7772910 µ − µ + 29783392 µ − µ + 276480 (cid:1) − c (cid:0) µ − µ + 2648475 µ − µ + 10258144 µ − µ + 111616 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 6139 (cid:1) + c (cid:0) µ − µ + 8644 (cid:1) (cid:105) ,γ ( p ) M1M1 ν = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 365512 µ − µ +11734236 µ − µ + 26691840 µ − µ + 690176 µ + 12288 (cid:1) +2 c Ξ (cid:0) − µ + 477637 µ − µ + 15279624 µ − µ +34427520 µ − µ + 843776 µ + 12288 (cid:1) − c µ (cid:0) µ − µ +3691774 µ − µ + 13374368 µ − µ + 145408 (cid:1) − c µ (cid:0) µ − µ + 1203489 µ − µ + 4317072 µ µ + 40448 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 7533 (cid:1) + c (cid:0) µ − µ + 9444 (cid:1) (cid:105) ,γ ( p ) E1M2 ν = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 558752 µ − µ +17712684 µ − µ + 38881120 µ − µ + 787200 (cid:1) +6 c Ξ (cid:0) − µ + 670917 µ − µ + 21003984 µ − µ +44459680 µ − µ + 593920 (cid:1) − c (cid:0) µ − µ +16799538 µ − µ + 58760928 µ − µ + 307200 (cid:1) − c (cid:0) µ − µ + 5005722 µ − µ + 16921472 µ − µ + 28160 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 9569 (cid:1) + c (cid:0) µ − µ + 9404 (cid:1) (cid:105) ,γ ( p ) M1E2 ν = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 275051 µ − µ +4756616 µ − µ + 2653280 µ − µ + 5120 (cid:1) − c Ξ µ (cid:0) µ − µ + 1374373 µ − µ + 5044180 µ − µ + 77440 (cid:1) − c µ (cid:0) µ − µ + 4545762 µ − µ + 3381184 (cid:1) − c µ (cid:0) µ − µ + 1381428 µ − µ + 1010576 µ + 16640 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − µ + 5719 (cid:1) + c (cid:0) µ − µ + 6684 (cid:1) (cid:105) .
2. Neutron values a. Spin-independent first order polarizabilities α ( n ) E1 = − e g A π F m (cid:0) c Ξ (cid:0) µ − (cid:1) + 3 c Ξ (cid:0) µ − (cid:1)(cid:1) − e (˜ e + 2˜ e − e − e ) π + e Ξ π F m (cid:0) c g A µ + 9 c g A µ − c m (cid:1) − e (4 c + c − c )192 π F , ( n ) M1 = − e g A π F ( µ − m (cid:0) c Ξ (cid:0) µ − µ + 8 (cid:1) + 2 c Ξ (cid:0) µ − µ + 24 (cid:1) − c (cid:0) − µ (cid:1) − c (cid:0) − µ (cid:1)(cid:1) + 2 e (˜ e − e ) π + e π F (4 c − c − c )+ e Ξ π F m (cid:0) g A c (cid:0) µ − (cid:1) + 3 g A c (cid:0) µ − (cid:1) − c m (cid:1) . b. Spin-independent second order polarizabilities α ( n ) E2 = − e g A π F ( µ − m (cid:104) c Ξ (cid:0) µ − µ + 60 (cid:1) + 4 c Ξ (cid:0) µ − µ +174) − c (cid:0) − µ (cid:1) − c (cid:0) − µ (cid:1) (cid:105) + e ( − c + 15 µ c − c )960 π F µ m + e Ξ π F m (cid:104) g A c (cid:0) µ − (cid:1) + g A c (cid:0) µ − (cid:1) + c m (cid:105) + 3 e (˜ e − e )2 πm ,β ( n ) M2 = − e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) µ − µ + 70 µ − µ + 8 (cid:1) +4 c Ξ (cid:0) µ − µ + 770 µ − µ + 80 (cid:1) + c (cid:0) µ − µ + 544 µ (cid:1) +2 c (cid:0) µ − µ + 536 µ (cid:1) (cid:105) + e (32 c + (15 µ + 8) c + 44 c )960 π F µ m + 3 e (˜ e − e )2 πm + e Ξ π F m (cid:104) g A c (cid:0) µ − (cid:1) + 11 g A c (cid:0) µ − (cid:1) + c m (cid:105) ,α ( n ) E1 ν = − e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) µ − µ + 710 µ − µ + 184 (cid:1) +4 c Ξ (cid:0) µ − µ + 3390 µ − µ + 656 (cid:1) + c (cid:0) µ − µ + 1408 (cid:1) +6 c (cid:0) µ − µ + 584 (cid:1) (cid:105) + e Ξ π F m (cid:104) g A c (cid:0) µ − (cid:1) + 3 g A c (cid:0) µ − − c m (cid:105) + e (96 c + (16 − µ ) c + 132 c )11520 π F µ m − e (˜ e − e )8 πm ,β ( n ) M1 ν = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 419 µ − µ + 5016 µ − µ + 128 (cid:1) + 4 c Ξ (cid:0) − µ + 1567 µ − µ + 19104 µ − µ +896) − c (cid:0) µ − µ + 8768 µ − µ + 512 (cid:1) − c (cid:0) µ − µ +8240 µ − µ + 512 (cid:1) (cid:105) + e Ξ π F m (cid:2) g A c (cid:0) µ − (cid:1) + g A c (cid:0) µ − c m ] − e (32 c + (5 µ + 8) c + 44 c )3840 π F µ m − e (˜ e − e )8 πm . c. Spin-dependent first order polarizabilities γ ( n ) E1E1 = e g A π F ( µ − m (cid:104) c Ξ (cid:0) − µ + 43 µ − µ + 180 (cid:1) + c Ξ (cid:0) − µ +139 µ − µ + 516 (cid:1) − c (cid:0) µ − µ + 64 (cid:1) − c (cid:0) µ − µ + 160 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − (cid:1) + c (cid:0) µ − (cid:1) (cid:105) ,γ ( n ) M1M1 = e g A π F ( µ − m µ (cid:104) c Ξ (cid:0) − µ + 92 µ − µ + 216 µ (cid:1) − c Ξ (cid:0) µ − µ + 786 µ − µ + 64 (cid:1) − c (cid:0) µ − µ + 56 µ (cid:1) − c (cid:0) µ − µ + 200 µ (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − (cid:1) + c (cid:0) µ − (cid:1) (cid:105) ,γ ( n ) E1M2 = e g A π F ( µ − m (cid:104) c Ξ (cid:0) µ − µ + 110 µ − (cid:1) + c Ξ (cid:0) − µ +31 µ − µ + 84 (cid:1) + c (cid:0) µ − µ + 32 (cid:1) − c (cid:0) µ − µ + 16 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) − µ (cid:1) + c (cid:0) µ − (cid:1) (cid:105) ,γ ( n ) M1E2 = e g A π F m µ (cid:104) c Ξ (cid:0) µ − µ + 2 (cid:1) − c Ξ (cid:0) µ + µ − (cid:1) + (2 c − c ) µ (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) − µ (cid:1) + c (cid:0) µ + 3 (cid:1) (cid:105) . d. Spin-dependent second order polarizabilities γ ( n ) E2E2 = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 13239 µ − µ +175180 µ − µ + 20480 (cid:1) + 6 c Ξ (cid:0) − µ + 11782 µ − µ +156750 µ − µ + 19200 (cid:1) + c µ (cid:0) − µ + 30197 µ − µ +138080) + c µ (cid:0) − µ + 53717 µ − µ + 241760 (cid:1) (cid:105) e g A Ξ π F m (cid:104) c (cid:0) µ − (cid:1) + 2 c (cid:0) µ − (cid:1) (cid:105) ,γ ( n ) M2M2 = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 19929 µ − µ +238100 µ − µ + 17280 (cid:1) + 6 c Ξ (cid:0) − µ + 19482 µ − µ +237490 µ − µ + 22080 (cid:1) − c (cid:0) µ − µ + 159284 µ − µ + 3840 (cid:1) − c (cid:0) µ − µ + 315884 µ − µ + 15360 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − (cid:1) + 2 c (cid:0) µ − (cid:1) (cid:105) ,γ ( n ) E2M3 = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) µ − µ + 9408 µ − µ +20080 µ − (cid:1) − c Ξ µ (cid:0) µ − µ + 3596 µ − µ + 5000 (cid:1) + c µ (cid:0) µ − µ + 15452 µ − (cid:1) + c µ (cid:0) − µ + 2957 µ − µ +7904) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) − µ (cid:1) + 2 c (cid:0) µ − (cid:1) (cid:105) ,γ ( n ) M2E3 = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) µ − µ + 4500 µ − µ +1920) + 6 c Ξ (cid:0) µ − µ + 420 µ − µ + 960 (cid:1) + c (cid:0) µ − µ +4568 µ − (cid:1) + c (cid:0) µ − µ + 248 µ − (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) − µ (cid:1) − c (cid:0) µ − (cid:1) (cid:105) ,γ ( n ) E1E1 ν = e g A π F µ ( µ − m (cid:104) − c Ξ (cid:0) µ − µ + 184860 µ − µ + 1211040 µ − µ + 122880 (cid:1) − c Ξ (cid:0) µ − µ +247860 µ − µ + 1654880 µ − µ + 184320 (cid:1) + c (cid:0) − µ +61305 µ − µ + 833840 µ − µ + 84992 (cid:1) + c (cid:0) − µ +163845 µ − µ + 2283920 µ − µ + 223232 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − (cid:1) + 2 c (cid:0) µ − (cid:1) (cid:105) , ( n ) M1M1 ν = e g A π F µ ( µ − m (cid:104) − c Ξ µ (cid:0) µ − µ + 194436 µ − µ + 1285760 µ − µ + 122880 (cid:1) − c Ξ (cid:0) µ − µ +275292 µ − µ + 1814400 µ − µ + 168448 µ + 6144 (cid:1) + c µ (cid:0) − µ + 21277 µ − µ + 292400 µ − µ + 8192 (cid:1) + c µ (cid:0) − µ + 60273 µ − µ + 818160 µ − µ + 31744 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − (cid:1) + 2 c (cid:0) µ − (cid:1) (cid:105) ,γ ( n ) E1M2 ν = e g A π F µ ( µ − m (cid:104) c Ξ (cid:0) − µ + 29085 µ − µ +667452 µ − µ + 264640 µ + 180480 (cid:1) − c Ξ (cid:0) µ − µ +424836 µ − µ + 2553440 µ − µ + 76160 (cid:1) − c (cid:0) µ − µ + 376280 µ − µ + 285888 µ + 97280 (cid:1) + c (cid:0) − µ +287277 µ − µ + 3681712 µ − µ + 56320 (cid:1) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ + 93 (cid:1) + 2 c (cid:0) µ − (cid:1) (cid:105) ,γ ( n ) M1E2 ν = e g A π F µ ( µ − m (cid:104) − c Ξ (cid:0) µ − µ + 33208 µ − µ + 69680 µ − µ + 5120 (cid:1) − c Ξ (cid:0) µ − µ + 166204 µ − µ + 350200 µ − µ + 2560 (cid:1) + c µ (cid:0) − µ + 13323 µ − µ +51616 µ + 33280 (cid:1) + c µ (cid:0) − µ + 137763 µ − µ + 542176 µ +33280) (cid:105) + e g A Ξ π F m (cid:104) c (cid:0) µ − (cid:1) + 2 c (cid:0) µ − (cid:1) (cid:105) . Appendix E: Renormalization of the nucleon magnetic moments
Below, we provide the expressions for the LECs c , c in terms of the renormalizedquantities ¯ c and ¯ c , see subsection II C. c = ¯ c + δc (3)6 + δc (4)6 , c = ¯ c + δc (3)7 + δc (4)7 , (E1)with δc (3)6 = − g A F (4 m − M ) ((4( (cid:15) − m − (cid:15) − M ) A ( m ) (4(5 − (cid:15) ) m + 3( (cid:15) − M ) A ( M )+ (16 m + 2(4 (cid:15) − M m − (cid:15) − M ) B ( m, M, m )) − h A (cid:15) − (cid:15) − F m m ( − (cid:15) − (cid:15) − (2 (cid:15) − m − (cid:15) − ( (cid:15) − (cid:15) − m ∆ m + (20( (cid:15) − (2 (cid:15) − (cid:15) − M + ( (cid:15) − ( (cid:15) (4 (cid:15) (7 (cid:15) − − m ) m + 2( (cid:15) − (cid:15) − m ∆ (10( (cid:15) − (cid:15) − M + ( (cid:15) ( (cid:15) (14 (cid:15) −
43) + 10)+ 2) m ) m + 2( (cid:15) − (cid:15) − (2 (cid:15) − M + (7 − (cid:15) (8 (cid:15) (( (cid:15) − (cid:15) + 3) + 25)) m M + ( (cid:15) ( (cid:15) (8 (cid:15) − (cid:15) + 71) −
51) + 35) m ) m + 2( (cid:15) − M − m ) m ∆ ((( (cid:15) − (cid:15) (12( (cid:15) − (cid:15) + 47) + 20) m − (cid:15) − M ) m − (cid:15) − (cid:15) − (cid:15) + 6) m ( m − M ) ) A ( M )+ h A (cid:15) − (cid:15) − F m m (10( (cid:15) − (cid:15) − (2 (cid:15) − m + 20( (cid:15) − ( (cid:15) − (cid:15) − m ∆ m + ( − (cid:15) − (cid:15) − (2 (cid:15) − M − ( (cid:15) ( (cid:15) ( (cid:15) (4 (cid:15) (7 (cid:15) −
36) + 207) + 111) − m ) m + 2 m ∆ (( (cid:15) ( (cid:15) ( (cid:15) (4 (cid:15) (13 (cid:15) −
94) + 1035) − − m − (cid:15) − (cid:15) − (cid:15) (2 (cid:15) −
7) + 4) M ) m + 2( (cid:15) − (cid:15) − (2 (cid:15) − M − (cid:15) ( (cid:15) (4( (cid:15) − (cid:15) + 27) −
13) + 4) m M + ( (cid:15) (2 (cid:15) (5 (cid:15) (2 (cid:15) −
11) + 91) − m ) m + 2( (cid:15) − M − m ∆ ) m ∆ ( M + m ∆ )((( (cid:15) − (cid:15) (12( (cid:15) − (cid:15) + 47)+ 20) m − (cid:15) − M ) m − (cid:15) − (cid:15) − (cid:15) + 6) m ( m − M ) ) A ( m ∆ )+ h A (( m + m ∆ ) − M )3(2 (cid:15) − F m m ( − (cid:15) − (2 (cid:15) − m + 20( (cid:15) − m ∆ m + (2 (cid:15) − (cid:15) − M + ( (cid:15) ( (cid:15) (14 (cid:15) − −
39) + 38) m ) m + 2 m ∆ (( (cid:15) (4(22 − (cid:15) ) (cid:15) − m − (cid:15) − (cid:15) − M ) m + 2(2 (cid:15) − − (cid:15) − M + 5( (cid:15) ( (cid:15) (2 (cid:15) −
7) + 7) − m M + (7 − (cid:15) − (cid:15) (5 (cid:15) − m ) m + 2( M − m ) m ∆ (20( (cid:15) − M + ( (cid:15) (4(10 − (cid:15) ) (cid:15) −
59) + 34) m ) m + 3(3 − (cid:15) ) ( (cid:15) − (cid:15) − m ( M − m ) ) B ( M, m ∆ , m ) , (E2) δc (3)7 = − − (cid:15) ) g A m F (4 m − M ) (2 A ( m ) − A ( M )) + 2 g A m (4 m − − (cid:15) ) M ) F (4 m − M ) B ( m, M, m ) − h A F ( (cid:15) − (cid:15) − mm (( (cid:15) − (cid:15) − (2 (cid:15) − m + 2( (cid:15) − ( (cid:15) − (cid:15) − m ∆ m + ( − (cid:15) − (2 (cid:15) − (cid:15) − M − ( (cid:15) − ( (cid:15) (4( (cid:15) − (cid:15) + 5)+ 2) m ) m − ( (cid:15) − (cid:15) − m ∆ (2( (cid:15) − (cid:15) − M + ( (cid:15) (2 (cid:15) (2 (cid:15) −
7) + 11) − m ) m + ( (cid:15) − M − m )(( (cid:15) (2 (cid:15) − (cid:15) − (cid:15) + 3) + 7) m − ( (cid:15) − (2 (cid:15) − M ) m + 4( (cid:15) − (cid:15) − m ∆ ( M − m ) ) A ( M ) − h A F ( (cid:15) − (cid:15) − mm ( − (cid:15) − (cid:15) − m + ( (cid:15) − (cid:15) (2 (cid:15) − (cid:15) − (cid:15) + 3) + 7) mm + ((( (cid:15) − (cid:15) (8 (cid:15) (( (cid:15) − (cid:15) + 12) − − m + 8( (cid:15) − (cid:15) − M ) m + m (( − (cid:15) (2 (cid:15) − (cid:15) ( (cid:15) (2 (cid:15) −
7) + 8) + 2) − m ( (cid:15) − (cid:15) ( (cid:15) (4( (cid:15) − (cid:15) + 27) −
13) + 4) M ) m + 2( (cid:15) − (cid:15) − m − M )(( (cid:15) − (cid:15) − m + 2( (cid:15) − M ) m ∆ + ( (cid:15) − (cid:15) − (2 (cid:15) − m ( m − M ) ( m + M ) ) A ( m ∆ ) − h A (( m + m ∆ ) − M )3 F (2 (cid:15) − mm ( − ( (cid:15) − (2 (cid:15) − m + 2( (cid:15) − m ∆ m + (2 (cid:15) − (cid:15) − M + ((2 (cid:15) − (cid:15) + 2) m ) m + m ∆ (( − (cid:15) + 6 (cid:15) − M + ( (cid:15) ( − (cid:15) − (cid:15) −
39) + 25) m ) m − (2 (cid:15) − M − m )(( (cid:15) − M + ( (cid:15) ( − (cid:15) − (cid:15) − m ) m + 4( (cid:15) − m ∆ ( M − m ) ) B ( m ∆ , M, m ) , (E3) δc (4)6 = − g A c M B ( m, M, m )2 F (4 m − M ) (8 m (2 (cid:15) −
2) + M (5 − (cid:15) ))+ A ( M )2 F m (2 (cid:15) − m − M ) ( c m (2 (cid:15) − m ( g A (2 (cid:15) − −
1) + M (2 − g A (2 (cid:15) − m − M )(2 c m (2 (cid:15) −
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