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

We examine the polarized doubly-virtual Compton scattering (VVCS) off the nucleon using chiral perturbation theory ($\chi$PT). The polarized VVCS contains a wealth of information on the spin structure of the nucleon which is relevant to the calculation of the two-photon-exchange effects in atomic spectroscopy and electron scattering. We report on a complete next-to-leading-order (NLO) calculation of the polarized VVCS amplitudes $S_1(\nu, Q^2)$ and $S_2(\nu, Q^2)$, and the corresponding polarized spin structure functions $g_1(x, Q^2)$ and $g_2(x,Q^2)$. Our results for the moments of polarized structure functions, partially related to different spin polarizabilities, are compared to other theoretical predictions and"data-driven"evaluations, as well as to the recent Jefferson Lab measurements. By expanding the results in powers of the inverse nucleon mass, we reproduce the known"heavy-baryon"expressions. This serves as a check of our calculation, as well as demonstrates the differences between the manifestly Lorentz-invariant baryon $\chi$PT (B$\chi$PT) and heavy-baryon (HB$\chi$PT) frameworks.

Traditionally, its general properties, such as unitarity, analyticity and crossing, are used to establish various useful sum rules for the nucleon magnetic moment (Gerasimov-Drell-Hearn [5,6] and Schwinger sum rules [7][8][9]) and polarizabilities (e.g., Baldin [10] and Gell-Mann-Goldberger-Thirring sum rules [11]). More recently, the interest in nucleon VVCS has been renewed in connection with precision atomic spectroscopy, where this amplitude enters in the form of two-photon exchange (TPE) corrections. As the TPE corrections in atomic domain are dominated by low-energy VVCS, it makes sense to calculate them systematically using chiral perturbation theory (χPT), which is a low-energy effective-field theory of the Standard Model.
In this paper, we present a state-of-the-art χPT calculation of the polarized nucleon VVCS, relevant to TPE corrections to hyperfine structure of hydrogen and muonic hydrogen. This will extend the leading-order χPT evaluation of the TPE effects in hyperfine splittings [12][13][14][15][16][17]. Here, we however, do not discuss the TPE evaluation, but rather focus on testing the χPT framework against the available empirical information on low-energy spin structure of the nucleon.
Our present calculation is extending Ref. [30] to the case of polarized VVCS. We therefore use a manifestly-covariant extension of SU (2) χPT to the baryon sector called Baryon χPT (BχPT). First attempts to calculate VVCS in the straightforward BχPT framework (rather than the heavy-baryon expansion or the "infrared regularization") were done by Bernard et al. [31] and our group [32]. The two works obtained somewhat different results, most notably for the proton spin polarizability δ LT . Here we improve on [32] in three important aspects appreciable at finite Q 2 : 1) inclusion of the Coulomb-quadrupole (C2) N → ∆ transition [33,34], 2) correct inclusion of the elastic form-factor contributions to the integrals Γ 1 (Q 2 ), I 1 (Q 2 ) and I A (Q 2 ) (see Sections III C and III D for details), and 3) cancellations between different orders in the chiral prediction and their effect on the convergence of the effective-field-theory calculation, and thus, the error estimate. These improvements, however, do not bring us closer to the results of [31], and the source of discrepancies most likely lies in the different counting and renormalization of the π∆-loop contributions. Bernard et al. [31] use the so-called small-scale expansion [35] for the ∆(1232) contributions, whereas we are using the δ-counting scheme [36] (see also Ref. [37,Sec. 4] for review).
This paper is organized as follows. In Sec. II A, we introduce the polarized VVCS amplitudes and their relations to spin structure functions. In Sec. II B, we introduce the spin polarizabilities appearing in the low-energy expansion (LEX) of the polarized VVCS amplitudes. In Sec. II C, we briefly describe our χPT calculation, focusing mainly on the uncertainty estimate. In Sec. III, we show our predictions for the proton and neutron polarizabilities, as well as some interesting moments of their structure functions. In Sec. III G, we summarize the results obtained herein, comment on the improvements done with respect to previous calculations, and give an outlook to future applications. In App. B, we discuss the structure functions, in particular, we define the longitudinal-transverse response function, discuss the ∆-pole contribution, and give analytical results for the tree-level πN -and ∆-production channels of the photoabsorption cross sections. In App. C, we give analytical expressions for the πN -loop and ∆-exchange contributions to the static values and slopes of the polarizabilities and moments of structure functions. The complete expressions, also for the π∆-loop contributions, can be found in the Supplemented material.  [38] at the order they appear first. The πN ∆ coupling constant h A is fit to the experimental Delta width and the γ * N ∆ coupling constants g M , g E and g C are taken from the pion photoproduction study of Ref. [33]. The polarized part of forward VVCS can be described in terms of two independent Lorentz-covariant and gauge-invariant tensor structures and two scalar amplitudes [3]: Here, q and p are the photon and nucleon four-momenta (cf. Fig. 1), ν is the photon lab-frame energy, Q 2 = −q 2 is the photon virtuality, and γ µν = 1 2 [γ µ , γ ν ] and γ µνα = 1 2 (γ µ γ ν γ α − γ α γ ν γ µ ) are the usual Dirac matrices. Alternatively, one can use the following laboratory-frame amplitudes: introduced in Eq. (A2). The optical theorem relates the absorptive parts of the forward VVCS amplitudes to the nucleon structure functions or the cross sections of virtual photoabsorption: Im S 2 (ν, Q 2 ) = 4π 2 αM N ν 2 g 2 (x, Q 2 ) = M 2 N K(ν, Q 2 ) with α the fine structure constant, and K(ν, Q 2 ) the photon flux factor. Note that the photon flux factor in the optical theorem and the cross sections, measured in electroproduction processes, is a matter of convention and has to be chosen in both quantities consistently. In what follows, we use Gilman's flux factor: The helicity-difference photoabsorption cross section is defined as where the photons are transversely polarized, and the subscripts on the right-hand side indicate the total helicities of the γ * N states. The cross section σ LT corresponds to a simultaneous helicity change of the photon and nucleon spin flip, such that the total helicity is conserved. A detailed derivation of the connection between this response function and the photoabsorption cross sections can be found in App. B. The forward VVCS amplitudes satisfy dispersion relations derived from the general principles of analyticity and causality: 1 with ν el = Q 2 /2M N the elastic threshold.

B. Low-energy expansions and relations to polarizabilities
The VVCS amplitudes naturally split into nucleon-pole (S pole i ) and non-pole (S nonpole i ) parts, or Born (S Born i ) and non-Born (S i ) parts: The Born amplitudes are given uniquely in terms of the nucleon form factors [1]: The same is true for the nucleon-pole amplitudes, which are related to the Born amplitudes in the following way: Here, we used the elastic Dirac and Pauli form factors F 1 (Q 2 ) and F 2 (Q 2 ), related to the electric and magnetic Sachs form factors G E (Q 2 ) and G M (Q 2 ) through: A low-energy expansion (LEX) of Eq. (5), in combination with the unitarity relations given in Eq. (3), establishes various sum rules relating the nucleon properties (electromagnetic moments, polarizabilities) to experimentally observable response functions [1,3]. The leading terms in the LEX of the RCS amplitudes are determined uniquely by charge, mass and anomalous magnetic moment, as the global properties of the nucleon. These lowestorder terms represent the celebrated low-energy theorem (LET) of Low, Gell-Mann and Goldberger [39,40]. The polarizabilities, related to the internal structure of the nucleon, enter the LEX at higher orders. They make up the non-Born amplitudes, and can be related to moments of inelastic structure functions.
The process of VVCS can be realized experimentally in electron-nucleon scattering, where a virtual photon is exchanged between the electron and the nucleon. This virtual photon acts as a probe whose resolution depends on its virtuality Q 2 . In this way, one can access the so-called generalized polarizabilities, which extend the notion of polarizabilities to the case of response to finite momentum transfer. The generalized forward spin polarizability γ 0 (Q 2 ) and the longitudinal-transverse polarizability δ LT (Q 2 ) are most naturally defined via the LEX of the non-Born part of the lab-frame VVCS amplitudes [1]: Their definitions in terms of integrals over structure functions are postponed to Eqs. (19) and (22). Here, we only give the definition of the moment I 3 (Q 2 ): which is related to the Schwinger sum rule in the real photon limit [7-9, 41, 42]. The LEX of the non-pole part of the covariant VVCS amplitudes can be described entirely in terms of moments of inelastic spin structure functions (up to O(ν 4 ) [43]): I 1 (Q 2 ) and I A (Q 2 ) are generalizations of the famous Gerasimov-Drell-Hearn (GDH) sum rule [5,6] from RCS to the case of virtual photons [1]. Their definitions are given in Eqs. (26) and (32). I 2 (Q 2 ) is the well-known Burkhardt-Cottingham (BC) sum rule [44]: which can be written as a "superconvergence sum rule": The latter is valid for any value of Q 2 provided that the integral converges for x → 0.
Combining Eq. (5) with the above LEXs of the VVCS amplitudes, we can relate I A (Q 2 ), , γ 0 (Q 2 ) and δ LT (Q 2 ) to moments of inelastic structure functions, see Sec. III. It is important to note that only γ 0 (Q 2 ) and δ LT (Q 2 ) are generalized polarizabilities. The relation of the inelastic moments I A (Q 2 ) and I 1 (Q 2 ) to polarizabilities will be discussed in details in Secs. III C and III D. The difference between S 1 (ν, Q 2 ) and S nonpole 1 (ν, Q 2 ), cf. Eq. (8a), will be important in this context. where power counting, predictive orders (Sec. III A), and the renormalization procedure (Sec. III B) are discussed.
A few remarks are in order for the inclusion of the ∆(1232) and the tree-level ∆-exchange contribution. In contrast to Ref. [32], we include the Coulomb-quadrupole (C2) N → ∆ transition described by the LEC g C . The relevant Lagrangian describing the non-minimal γ * N ∆ coupling [33,34] (note that in these references the overall sign of g C is inconsistent between the Lagrangian and Feynman rules) reads: with M + = M N + M ∆ and the dual of the electromagnetic field strength tensorF µν = 1 2 µνρλ F ρλ . Even though the Coulomb coupling is subleading compared with the electric and magnetic couplings (g E and g M ), its relatively large magnitude, cf. Table I, makes it numerically important for instance in γ 0 (Q 2 ). Furthermore, we study the effect of modifying the magnetic coupling using a dipole form factor: where Λ 2 = 0.71 GeV 2 . The inclusion of this Q 2 dependence mimics the form expected from vector-meson dominance. It is motivated by observing the importance of this form factor for the correct description of the electroproduction data [33].
To estimate the uncertainties of our NLO predictions, we definẽ such that the neglected next-to-next-to-leading order terms are expected to be of relative sizeδ 2 [33]. The uncertainties in the values of the parameters in Table I have a much smaller impact compared to the truncation uncertainty and can be neglected. Unfortunately, ∆I A (Q 2 ), γ 0 (Q 2 ) andγ 0 (Q 2 ), i.e., the sum rules involving the cross section σ T T (ν, Q 2 ), as well as the polarizability ∆I 1 (Q 2 ), turn out to be numerically small. Their smallness suggests a cancellation of leading orders (which can indeed be confirmed by looking at separate contributions as shown below). Therefore, an error ofδ 2 (Q 2 )P (Q 2 ), where P (Q 2 ) is a generalized polarizability, might underestimate the theoretical uncertainty for some of the NLO predictions. To avoid this, we estimate the uncertainty of our NLO polarizability predictions by: where P LO (Q 2 ) is the πN -loop contribution, P NLO (Q 2 ) are the ∆-exchange and π∆-loop contributions, and P (Q 2 ) = P LO (Q 2 ) + P NLO (Q 2 ). This error prescription is similar to the one used in, e.g., Refs. [46][47][48][49]. Here, since we are interested in the generalized polarizabilities, we added in quadrature the error due to the static piece P (0) and the Q 2 -dependent remainder P (Q 2 ) − P (0). Note that the static values of I A (0) and I 1 (0) are given by the elastic Pauli form factor, which is not part of our BχPT prediction and is considered to be exact.
Note that our result for the spin polarizabilities (and the unpolarized moments [30]) are NLO predictions only at low momentum transfers Q m π . At larger values of Q ∆, they become incomplete LO predictions. Indeed, in this regime the ∆ propagators do not carry additional suppression compared to the nucleon propagators, and the π∆ loops are promoted to LO. In general, we only expect a rather small contribution from omitted π∆ loops to the Q 2 dependence of the polarizabilities, since π∆ loops show rather weak dependence on Q 2 compared with the ∆ exchange or πN loops. Nevertheless, this issue has to be reflected in the error estimate. Since the static polarizabilities P (0) are not affected, it is natural to separate the error on the Q 2 -dependent remainder P (Q 2 ) − P (0), as done in Eq. (18). To accommodate for the potential loss of precision above Q ∆, we define the relative error δ(Q 2 ) as growing with increasing Q 2 , see Eq. (17).
Upon expanding our results in powers of the inverse nucleon mass, M −1 N , we are able to reproduce existing results of heavy-baryon χPT (HBχPT) at LO. We, however, do not see a rationale to drop the higher-order M −1 N terms when they are not negligible (i.e., when their actual size exceeds by far the natural estimate for the size of higher-order terms).
Comparing our BχPT predictions to HBχPT, we will also see a deficiency of HBχPT in the description of the Q 2 behaviour of the polarizabilities. Note that the O(p 4 ) HBχPT results from Ref. [50,51], which we use here for comparison, do not include the ∆. These references studied the leading effect of the latter in the HBχPT framework, using the smallscale expansion [35], observing no qualitative improvement in the HBχPT description of the empirical data [50,51] when including it. We therefore choose to use the O(p 4 ) results as the representative HBχPT curves.
Another approach used in the literature to calculate the polarizabilities in χPT is the infrared regularization (IR) scheme, introduced in Ref. [52]. This covariant approach tries to solve the power counting violation observed in Ref. [53] by dropping the regular parts of the loop integrals that contain the power-counting-breaking terms. However, this subtraction scheme modifies the analytic structure of the loop contributions and may lead to unexpected problems, as was shown in Ref. [54]. As we will see in the next section, the IR approach also fails to describe the Q 2 behaviour of the polarizabilities.

III. RESULTS AND DISCUSSION
We now present the NLO BχPT predictions for the nucleon polarizabilities and selected moments of the nucleon spin structure functions. Our results are obtained from the calculated non-Born VVCS amplitudes and the LEXs in Eqs. (10) and (12). For a cross-check, we used the photoabsorption cross sections described in App. B. In addition to the full NLO results, we also analyse the individual contributions from the πN loops, the ∆ exchange, and the π∆ loops.
A. γ 0 (Q 2 ) -generalized forward spin polarizability The forward spin polarizability, provides information about the spin-dependent response of the nucleon to transversal photon probes. The RCS analogue of the above generalized forward spin polarizability sum rule is sometimes referred to as the Gell-Mann, Goldberger and Thirring (GGT) sum rule [11]. At Q 2 = 0, the forward spin polarizability is expressed through the lowest-order spin The forward spin polarizability of the proton is relevant for an accurate knowledge of the (muonic-)hydrogen hyperfine splitting, as it controls the leading proton-polarizability correction [16,62].  [50]; note that the corresponding proton curve is outside of the plotted range. The black dotted line is the MAID model prediction [55][56][57], which is taken from Ref. [1] (proton) and Ref. [20] (neutron). The pink band is the IR+∆ result from Ref. [58], and the gray band is the BχPT+∆ result from Ref. [31]. Empirical extractions for the proton: Ref. [18] (blue dots), Ref. [59] (purple square) and Ref. [60] (orange triangle; uncertainties added in quadrature); and neutron: Ref. [20] (blue diamonds) and Ref. [61] (green dots; statistical and systematic uncertainties added in quadrature). Lower Panel: Longitudinal-transverse spin polarizability for the proton (left) and neutron (right). The orange dot-dashed and purple shortdashed lines are the O(p 3 ) and O(p 4 ) HB results from Ref. [50]. The pink band is the IR result from Ref. [58] and the gray band is the covariant BχPT+∆ result from Ref. [31]. The black dotted line is the MAID model prediction [55][56][57]; note that for the proton we use the updated estimate from Ref. [1] obtained using the π, η, ππ channels.
the static forward spin polarizability amount to, in units of 10 −4 fm 4 : while the slope is composed as follows, in units of 10 −4 fm 6 : further Jefferson Laboratory data [18,60] at very low Q 2 . For the neutron, only data at finite Q 2 are available [20,61]. The experimental data for the proton are fairly well reproduced in the whole Q 2 range considered here, while for the neutron the agreement improves with increasing Q 2 . The HB limit of our πN -loop contribution reproduces the results published in Refs. [50,63] for arbitrary Q 2 . In addition, our prediction is compared to the MAID model [1,20], the IR+∆ calculation of Ref. [58] and the BχPT+∆ result of Ref. [31].
The πN -production channel gives a positive contribution to the photoabsorption cross section σ T T (ν, Q 2 ) at low Q 2 , cf. Fig. 10. Accordingly, one observes that the πN loops give a sizeable positive contribution to γ 0 (Q 2 ). The Delta, on the other hand, has a very large effect by cancelling the πN loops and bringing the result close to the empirical data. From Fig. 3 {upper panel}, one can see that it is the ∆ exchange which dominates, while π∆ loops are negligible. This was expected, since the forward spin polarizability sum rule is an integral over the helicity-difference cross section, in which σ 3/2 is governed by the Delta at low energies (the relevant energy region for the sum rule).
To elucidate the difference between the present calculation and the one from Ref. [31], we note that the two calculations differ in the following important aspects. Firstly, Ref. [31] uses the small-scale counting [64] that considers ∆ and m π as being of the same size, ∆ ∼ m π .
In practice, this results in a set of π∆-loop graphs which contains graphs with one or two γ∆∆ couplings and hence two or three Delta propagators. Such graphs are suppressed in the δ-counting and thus omitted from our calculation while present in that of Ref. [31].
Secondly, the Lagrangians describing the interaction of the Delta are constructed differently and assume slightly different values for the coupling constants. In particular, we employ (where possible) the so-called "consistent" couplings to the Delta field, i.e., those couplings that project out the spurious degree of freedom, see Refs. [37,65,66]. The authors of Ref. [31], on the other hand, use couplings where the consistency in this sense is not enforced.
The effects of these differences are of higher order in the δ-counting expansion, and their contribution to the Q 2 dependence of the considered polarizabilities is expected to be rather small; however, the differences at Q 2 = 0 could be noticeable [67].
Finally, as mentioned in Sec. II C, the inclusion of the dipole form factor in the magnetic coupling g M is expected to be important to generate the correct Q 2 behaviour of the polarizabilities. Comparing our predictions for the forward spin polarizability with and without inclusion of the form factor, see Fig. 3 {upper panel}, confirms this. Without the dipole our results for the proton and neutron are closer to the ones from Ref. [31], where the form factor is not included. For the neutron, our prediction without the dipole is able to describe the experimental points at very low Q 2 , deviating from the data with increasing Q 2 . The π∆-loop contribution does not modify the Q 2 behavior of γ 0 (Q 2 ), and only differs from Ref. [31] by a small global shift. Note also the relatively large effect of g C , which generates a sign change for virtualities above ∼ 0.2 GeV 2 , see Fig. 3 {upper panel}.
The longitudinal-transverse spin polarizability, contains information about the spin structure of the nucleon, and is another important input in the determination of the (muonic-)hydrogen hyperfine splitting [16,62]. It is also relevant in studies of higher-twist corrections to the structure function g 2 (x, Q 2 ), given by puzzle. The IR calculation in Ref. [58] also showed a deviation from the data and predicted a rapid rise of δ LT (Q 2 ) with growing Q 2 . The problem is solved by keeping the relativistic structure of the theory, as the BχPT+∆ result of Ref. [31] showed.
As expected, already the leading πN loops provide a reasonable agreement with the experimental data, cf. Fig. 2 {lower panel}. Since the ∆-exchange contribution to δ LT (Q 2 ) is small, the effect of the g M form factor is negligible in this polarizability, as is that of the g C coupling, cf. Fig. 3 {lower panel}. In fact, we predict both the ∆-exchange and the π∆-loop contributions to be small and negative. This is in agreement with the MAID model, which predicts a small and negative contribution of the P 33 wave to δ LT (Q 2 ). However, in the calculation of Ref. [31], which is different from the one presented here only in the way the ∆(1232) is included, the contribution of this resonance to δ LT p (Q 2 ) is sizeable and positive. can produce a substantial shift of the δ LT (Q 2 ) as a whole. A higher-order calculation should resolve the discrepancy between the two covariant approaches, however, it will partially lose the predictive power since the LECs appearing at higher orders will have to be fitted to experimental data.
The πN -loop, ∆-exchange, and π∆-loop contributions to the NLO BχPT prediction of the static longitudinal-transverse polarizability are, in units of 10 −4 fm 4 : while the slopes are, in units of 10 −4 fm 6 : The helicity-difference cross section σ T T exhibits a faster fall-off in ν than its spin-averaged counterpart σ T . This is due to a cancellation between the leading (constant) terms of σ 1/2 and σ 3/2 at large ν. 3 The resulting 1/ν fall-off of the helicity-difference cross section allows one to write an unsubtracted dispersion relation for the VVCS amplitude g nonpole Eq. (10a). This is the origin of the GDH sum rule [5,6], which establishes a relation to the anomalous magnetic moment κ. It is experimentally verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [71,72].
There are two extensions of the GDH sum rule to finite Q 2 : the generalized GDH integrals  [50,51]. The gray band is the BχPT+∆ result from Ref. [31]. The black dotted line is the MAID model prediction [69]. Experimental extractions for the proton: Ref. [ I A (Q 2 ) and I 1 (Q 2 ). The latter will be discussed in Sec. III D. The former is defined as: 4 Due to its energy weighting, the integral in Eq. (26) converges slower than the one in the generalized forward spin polarizability sum rule (19). Therefore, knowledge of the cross section at higher energies is required and the evaluation of the generalized GDH integral is not as simple as the evaluation of γ 0 (Q 2 ).
The generalized GDH integral I A (Q 2 ) is directly related to the non-pole amplitude (ν, Q 2 ), which differs from non-Born amplitude g T T (ν, Q 2 ) by a term involving the elastic Pauli form factor: cf. Eqs. (2a) and (8a). Consequently, I A (Q 2 ) is not a pure polarizability, but also contains an elastic contribution. The "non-polarizability" or the Born part of I A (Q 2 ) is given by: where we refer to the polarizability part as ∆I A (Q 2 ). The same is true for the generalized GDH integral I 1 (Q 2 ), which is directly related to S nonpole 1 (ν, Q 2 ): In the following, we will add the Born parts to our LO and NLO BχPT predictions for the polarizabilities ∆I A (Q 2 ) and ∆I 1 (Q 2 ), employing an empirical parametrization for the elastic Pauli form factor [73]. This allows us to compare to the experimental results for I A (Q 2 ) and I 1 (Q 2 ), cf. The E97-110 experiment at Jefferson Lab has recently published their data for I An (Q 2 ) in the region of 0.035 GeV 2 < Q 2 < 0.24 GeV 2 [27]. In addition, there are results for I An (Q 2 ) from the earlier E94-010 experiment [21], and for I Ap (Q 2 ) from the E08-027 experiment [60].
The O(p 4 ) HB calculation gives a large negative effect [51], which does not describe the data.
The BχPT+∆ result from Ref. [31], which mainly differs from our work by the absence of the dipole form factor in g M , looks similar to this HB result and only describes the data points at lowest Q 2 . Our NLO prediction, however, follows closely the Q 2 evolution of the data. In Fig. 5 {upper panel}, we show the polarizability ∆I A (Q 2 ), whose Q 2 evolution is clearly dominated by the ∆ exchange. Similar to the case of γ 0p (Q 2 ), inclusion of the dipole in g M and the Coulomb coupling g C is very important in order to describe the experimental data. The LO prediction, on the other hand, slightly overestimates the data, cf. Fig. 4 {upper panel}.
At the real-photon point: I A (0) = − κ 2 4 and ∆I A (0) = 0. Therefore, we give only the slope of the polarizability ∆I A (Q 2 ) [showing also the separate contributions from πN loops, ∆ exchange and π∆ loops] in units of GeV −2 : d∆I An (Q 2 ) dQ 2 Including the empirical Pauli form factor [73], we find, in units of GeV −2 : D. Γ 1 (Q 2 ) and I 1 (Q 2 ) -the first moment of the structure function g 1 (x, Q 2 ) The second variant for a generalization of the GDH sum rule to finite Q 2 is defined as: where I 1 (0) = − κ 2 4 . This generalized GDH integral directly stems from the amplitude S nonpole 1 (ν, Q 2 ) with the LEX from Eq. (12a). It is given by the first moment of the structure function g 1 (x, Q 2 ), Γ 1 (Q 2 ) = x 0 0 dx g 1 (x, Q 2 ), as follows: . The isovector combination: is related to the axial coupling of the nucleon through the Bjorken sum rule [74,75]: As explained in Eq. (28), the moment I 1 (Q 2 ) splits into a polarizability part ∆I 1 (Q 2 ) and a Born part I Born 1 (Q 2 ). Figure 4 {lower panel} shows the Q 2 dependence of I 1 (Q 2 ) which, in contrast to I A (Q 2 ) shown in Figure 4  panel}, one sees that ∆I 1 (Q 2 ) is less sensitive to g C and the dipole form factor in g M than ∆I A (Q 2 ).
For the proton, our NLO BχPT prediction gives a very good description of the experimental data [18,60] and is in reasonable agreement with the MAID prediction [69].
For the neutron, one observes good agreement with the empirical evaluations including extrapolations to unmeasured energy regions starting from Q 2 > 0.1 GeV 2 [27,61]. In the region of Q 2 < 0.05 GeV 2 , one observes an interesting tension between the recent E97-110 experiment [27] and the data from CLAS [61]. While the newest measurement finds I 1n (0.035 GeV 2 ) < κ 2 n /4, thus suggesting a negative slope at low Q, the older measurement found a rather large value for I 1n (0.0496 GeV 2 ). A similar but milder behaviour is seen in the E97-110 [27] and E94-010 [21] data for I An . The MAID predictions do not agree with the CODATA recommended values for the anomalous magnetic moments of the proton and neutron [70], which in our work are imposed by using empirical parametrizations for the elastic Pauli form factors [73]. The slope of the HB result from Ref. [51] is too large and therefore only reproduces the data at very low Q 2 . Figure 6 shows the moment Γ 1 (Q 2 ) for the proton and neutron, while Fig. 7 shows the isovector combination Γ 1, p−n (Q 2 ). The LO and NLO BχPT predictions are identical, because our calculation produces the same Delta contributions for the proton and the neutron.
For the isovector combination, the MAID model only agrees with the data at very low Q 2 < 0.10 GeV 2 . The same is true for the IR result [58,76], while all other chiral results describe the data: NLO BχPT (this work), BχPT+∆ [31] and HBχPT [51].

∆ exchange and π∆ loops] in units of GeV
Including the empirical Pauli form factor [73], we find, in units of GeV −2 : E.d 2 (Q 2 ) -a measure of color polarizability Another interesting moment to consider is d 2 (Q 2 ), which is related to the twist-3 part of the spin structure function g 2 (x, Q 2 ) [79,80]: where g W W 2 (x, Q 2 ) is the twist-2 part of g 2 (x, Q 2 ). Using the Wandzura-Wilczek relation [81], one can relate d 2 (Q 2 ) to moments of the spin structure functions g 1 (x, Q 2 ) and g 2 (x, Q 2 ): This relation, however, only holds for asymptotically large Q 2 . It is also in the high-Q 2 region, where d 2 (Q 2 ) is a measure of color polarizability [82,83], through its relation to the gluon field strength tensor [80]. We refer to Ref. [84] for a recent review on the spin structure of the nucleon, including a discussion of sum rules for deep inelastic scattering and color polarizabilities.
What we consider in the following is the inelastic part of d 2 (Q 2 ), defined as the moment of g 1 (x, Q 2 ) and g 2 (x, Q 2 ) spin structure functions, cf. Eq. (38): This moment provides another testing ground for our BχPT predictions through comparison with experiments on the neutron [22]. Going towards the low-Q 2 region, the interpretation ofd 2 (Q 2 ) in terms of color polarizabilities will fade out. The above definition, however, implies it is related to other VVCS polarizabilities: Note thatd 2 (Q 2 ) and its first two derivatives with respect to Q 2 vanish at Q 2 = 0. The considerations in Eqs. (28) and (29) have no effect ond 2 (Q 2 ), since the Born contribution from I A (Q 2 ) and I 1 (Q 2 ) cancel out. Therefore,d 2 (Q 2 ) is a pure polarizability.
In Fig. 8 {upper panel}, we show our NLO BχPT prediction and other results ford 2 (Q 2 ).
While MAID [69] and BχPT describe the experimental data for the neutron [22] very well, the HB limit [50,51] is showing a fast growth with Q 2 . This illustrates the importance of keeping the relativistic result. Note also that, even though the πN -loop contribution is dominant, both g C and the form factor in g M are essential to obtain a curvature that reproduces the data, cf. Fig. 9 {upper panel}. For the proton there are, to our knowledge, no experimental results to compare with. However, the agreement between the NLO BχPT prediction and the MAID prediction at low energies is reasonable.
F. γ 0 (Q 2 ) -fifth-order generalized forward spin polarizability It is interesting to compare the generalized fifth-order forward spin polarizability sum rule,γ to the sum rule integrals for I A (Q 2 ) and γ 0 (Q 2 ), since they differ merely by their energy weighting of σ T T (ν, Q 2 ) and a constant prefactor, cf. Eqs. (19), (26) and (41).
Therefore, the description of γ 0 (Q 2 ) should be easiest in a low-energy effective-field theory such as χPT, whereas γ 0 (Q 2 ) and I A (Q 2 ) receive larger contributions from higher energies.
In Fig. 8  bringing it into a better agreement with data. In general, the BχPT curves start above the empirical data points at the real-photon point, and then decrease asymptotically to zero above Q 2 > 0.1 GeV 2 . On the other hand, the MAID prediction reproduces the empirical data at the real-photon point, then decreases to negative values until about Q 2 > 0.06 GeV 2 , from where it also starts to asymptotically approach zero. Consequently, our NLO BχPT prediction of γ 0 (Q 2 ) is consistently above the MAID prediction. This is very different to what we saw for I A (Q 2 ) in Fig. 4 {upper panel}, where the MAID prediction at the realphoton point is above the experimental value. While the agreement of our predictions with the empirical data is in general quite good for all moments of σ T T (ν, Q 2 ), one should point out that both for γ 0n (Q 2 ) and γ 0p (Q 2 ) we overestimate the data at low Q 2 . For I A (Q 2 ) such observation cannot be made because ∆I A (0) = 0, and thus, I A (0) is given by the empirical Pauli form factor only. From I A (Q 2 ), γ 0 (Q 2 ) and γ 0 (Q 2 ), the latter has the smallest, however, non-negligible dependence on g C and the dipole in g M , cf. Fig. 9 {lower panel}.
The  [78,85]) is almost one order of magnitude larger than the empirical value, and therefore not shown in Fig. 8.

G. Summary
Our results are summarized in Table II, Table III. We can see that the inclusion of the Delta turns out to be very important for all moments of the helicity-difference cross section. To describe the Q 2 behavior of the polarizabilities, the magnetic coupling of the N → ∆ transition should be modified by a dipole form factor, as has been observed previously in the description of electroproduction data [33]. This dipole form factor effectively takes account of vector-meson exchanges. The Coulomb-quadrupole N → ∆ transition, despite its subleading order, is important in the description of some moments of spin structure functions. This is contrary to what we saw for the moments of unpolarized structure functions [30], where the Coulomb coupling had a negligible effect.
The π∆ loops are mainly relevant for the generalized GDH integrals.

IV. CONCLUSIONS
We have presented a complete NLO calculation of the polarized non-Born VVCS amplitudes in covariant BχPT, with pion, nucleon, and ∆(1232) fields. The dispersion relations between the VVCS amplitudes and the tree-level photoabsorption cross sections served as a cross-check of these calculations.
The obtained moments of the proton and neutron spin structure functions, related to generalized polarizabilities and GDH-type integrals, agree well with the available experimental data. The description of their Q 2 evolution is improved compared to the previous χPT predictions. In particular, the NLO BχPT predictions obtained here give a better description of the empirical data (e.g., from the Jefferson Laboratory "Spin Physics Program") than the HB [50,51] and IR [58] calculations.
The demonstrated predictive power of the χPT framework amplitudes makes it well suited for extending the χPT evaluation of the TPE effect in the hyperfine structure of (muonic-)hydrogen [15][16][17] to next-to-leading order. FH gratefully acknowledges financial support from the Swiss National Science Foundation.

Appendix A: Tensor decompositions of the VVCS amplitudes
In this appendix, we review the decomposition of the forward VVCS process into tensor structures and scalar amplitudes. In particular, we consider the connection between the covariant and the semi-relativistic decomposition in the lab frame that is defined in terms of the conventional transverse, longitudinal, transverse-transverse, and transverse-longitudinal amplitudes.
As explained in Sec. II A, the process of forward VVCS off the nucleon can be described in terms of four explicitly covariant amplitudes S 1, 2 and T 1, 2 [3]: where µ ( * µ ) are the incoming (outgoing) photon polarization vectors, ν is the photon lab-frame energy and Q 2 is the photon virtuality. Alternatively, the decomposition in the laboratory frame (which in the forward case coincides with the Breit frame) is parametrized in terms of the nucleon Pauli matrices σ and the four scalar functions f L , f T , g T T , and g LT : Here, q andq = q/| q | are the photon three-momentum in the lab system and its unit vector. The modified polarization vector components are given by:

Appendix B: Photoabsorption cross sections
In the forward kinematics, the spin-dependent VVCS amplitudes and the spin polarizabilities can be described in terms of the polarized structure functions g 1 (x, Q 2 ) and g 2 (x, Q 2 ), or equivalently, the helicity-difference cross section σ T T (ν, Q 2 ) and the longitudinal-transverse response function σ LT (ν, Q 2 ), with the help of dispersion relations (5) and the optical theorem (3). In this way, the photoabsorption cross sections, measured in electroproduction processes, form the basis for most empirical evaluations shown throughout Sec. III. In the following, we present the BχPT predictions for the tree-level cross sections of πN -, π∆-and ∆-production through photoabsorption on the nucleon, cf. Figs. 8, 9 and 10 in Ref. [30]. In Secs. B 1 and B 2, we will discuss the leading πN -production channel and the ∆-production channel, respectively. We used these cross sections to verify the polarizability predictions obtained otherwise from the calculated non-Born VVCS amplitudes. Due to the bad high-energy behavior of the π∆-production cross sections in BχPT, cf. Fig. 10, the dispersion relations in Eq. (5) require further subtractions for a reconstruction of the π∆-loop contribution to the spin-dependent VVCS amplitudes. Therefore, not all polarizabilities could be verified, but only those appearing as higher-order terms in the LEX of the VVCS amplitudes, such asγ 0 [16].

πN -production channel
In order to extract the response function σ LT (ν, Q 2 ), we have developed a method similar to the one used to calculate σ T T (ν, Q 2 ), see, for example, Ref. [86]. For σ LT (ν, Q 2 ), however, the calculation is more complicated because one has to take into account that the associated Compton process involves a spin-flip of the nucleon, as illustrated in Fig. 11. When calculating the cross section, the product of the incoming nucleon spinors has to reflect this flip.
Let us now consider the related photoabsorption process and, in particular, the tree-level γ * N → FIG. 11. Relation between the forward Compton process and the photoabsorption process given by the optical theorem. In particular, we show the longitudinal-transverse contribution. The double-line arrows represent the spin of the external particles, while the dot represents the scalar (longitudinal) polarization of the incoming photon. Inside the blob the intermediate states are represented: e.g., nucleons with spins r (which are averaged in the calculation of the cross section) and pions.
πN channel, see diagrams in Fig. 8 of Ref. [30]. We define the πN -production amplitude as: with the Dirac structures: where u h A (P A ) and u † h B (P B ) are the Dirac spinors, and P A and P B are the four-momenta of the incoming and outgoing nucleons, respectively. When calculating the photoabsorption cross section, related to the VVCS amplitude in Eq. (B1), the nucleon spin flip should be implemented by u h (P A ) in T † and u h (P A ) in T , together with the appropriate transverse and longitudinal photon polarization vectors * ± and L .
However, if one wants to use the properties of the Dirac matrices, it is more useful to construct an operator to produce this spin flip in the external nucleons of Fig. 11. This is accomplished by introducing the projector Γ LT ≡ 1 2 √ 2 (γ 1 + iγ 2 )γ 5 , which also takes into account the extra factor √ 2 in Eq. (B1). We checked that with this projector one correctly extracts δ LT by comparing the HB limit of our result to the HB result of Ref. [50], where the authors calculate this polarizability from the Compton amplitude directly. With all those ingredients, the longitudinal-transverse cross section is calculated in the following way: with where θ is the scattering angle in the center-of-mass (cm) frame, and | p i | cm (| p f | cm ) is the threemomentum of an incoming (outgoing) particle in the cm frame. An explicit calculation of the matrix X ij leads to: where | q i | cm (| q f | cm ) is the relative three-momentum of the incoming (outgoing) particles in the cm frame. Here, s, t and u are the usual Mandelstam variables. For the different γ * N → πN channels, we obtain the following amplitudes A i , where we introduce q A as the four-momentum of the incoming photon and q B as the four-momentum of the outgoing pion: (B7b) • γ * p → π + n The analytical expressions shown above were checked with the amplitudes given in Ref. [87]. Analytical expressions for the tree-level γ * N → πN channel of the σ LT (ν, Q 2 ) and σ T T (ν, Q 2 ) cross sections are given below (proton channels: π + n and π 0 p; neutron channel π − p). We checked that they reproduce the known results in the real-photon limit [45,86]. To shorten the final expressions for the cross sections, which are considerably longer for finite Q 2 than in the real-photon limit, we define the following dimensionless kinematic variables: Here, (E N i ) cm and (E N f ) cm are the energies of the incoming and outgoing nucleon, E γ cm is the energy of the incoming photon, E π cm is the energy of the outgoing pion, all in the cm frame.

∆-production channel
The tree-level ∆-exchange diagram in Fig. 2 of Ref. [30] contributes to the non-Born part of the VVCS amplitudes. The contribution of the ∆ exchange to the VVCS amplitudes can be split into [17]: (ν, Q 2 ), and similarly for the unpolarized VVCS amplitudes discussed in Ref. [30]. Here, we introduced the ∆-pole contributions S ∆-pole i and the ∆-non-pole contributions S ∆-exch.

∆-exchange contribution
Here, we give analytical expressions for the tree-level ∆-exchange contributions to the nucleon spin polarizabilities and their slopes at Q 2 = 0. Note that the ∆-exchange contributes equally to proton and neutron polarizabilities. Recall that for the magnetic γ * N ∆ coupling we introduced a dipole form factor to mimic vector-meson dominance: g M → g M /(1 + Q 2 /Λ 2 ) 2 .