Nucleon Polarizabilities and Compton Scattering as Playground for Chiral Perturbation Theory

Abstract

I give a summary of recent results on nucleon polarizabilities, with emphasis on chiral perturbation theory. The predictive calculations of Compton scattering off the nucleon are compared to recent empirical determinations and lattice QCD calculations of the polarizabilities, thereby testing chiral perturbation theory in the single-baryon sector.

1. Introduction

The name Chiral Perturbation Theory ($\chi$PT) was first introduced in the seminal works of Pagels [1], who used it to describe a systematic expansion in the pion mass $m_{\pi }$, which is small compared to other hadronic scales. Some years later, in 1979, Weinberg [2] made an enlightening proposal for effective-field theories (EFT) and the $\chi$PT acquired its present meaning by Gasser and Leutwyler [3,4] in this, more powerful, connotation. Since then, $\chi$PT stands for a low-energy EFT of the strong sector of the Standard Model. Written in terms of hadronic degrees of freedom, rather than quarks and gluons, it offers an efficient way of calculating low-energy hadronic physics. Many calculations can be done analytically in a systematic perturbative expansion, in contrast to the ab initio calculations, viz., lattice QCD, Dyson–Schwinger equations, and other non-perturbative calculations in terms of quark and gluon fields.

However, as in any EFT framework, the convergence and the predictive power of $\chi$PT calculations are often of concern. After all, the expansion in energy and momenta is not as clear-cut as usual expansions in a small coupling constant. Moreover, each new order brings more and more free parameters—the low-energy constants (LECs). This is why the cases where $\chi$PT provides true predictions are very valuable. One such case, considered here, is the process of Compton scattering (CS) off the nucleon, see Figure 1. It allows one to study the low-energy properties of the nucleon [5,6].

The nucleon is characterized by a number of different polarizabilities, the most important of which are the electric and magnetic dipole polarizabilities ${\alpha }_{E1}$ and ${\beta }_{M1}$. These quantities describe the size of the electric and magnetic dipole moments induced by an external electric $\overrightarrow{E}$ or magnetic $\overrightarrow{H}$ field:

$$\begin{array}{ccc}\hfill {\overrightarrow{d}}_{\mathrm{ind}}.& =& 4\pi {\alpha }_{E1}\overrightarrow{E},\hfill \end{array}$$

(1a)

$$\begin{array}{ccc}\hfill {\overrightarrow{\mu }}_{\mathrm{ind}}.& =& 4\pi {\beta }_{M1}\overrightarrow{H}.\hfill \end{array}$$

(1b)

In loosely bound systems, such as atoms and molecules, these polarizabilities are roughly given by the volume of the system. The nucleon is apparently a much more rigid object—its polarizabilities are orders of magnitude smaller than its volume ($\sim 1$ fm${}^3$). The most accurate evidence of that comes from the Baldin sum rule (sometimes referred to as the Baldin–Lapidus sum rule) [7,8]. It is a very general relation based on the principles of causality, unitarity, and crossing symmetry akin to the Kramers–Kronig relation (see, e.g., in [9] for a pedagogical review). The Baldin sum rule expresses the sum of dipole polarizabilities in terms of an integral of the total photoabsorption cross section ${\sigma }_T$:

$${\alpha }_{E1}+{\beta }_{M1}=\frac{1}{2{\pi }^2}{\int }_{{\nu }_0}^{\infty }\mathrm{d}\nu \frac{{\sigma }_T\left(\nu \right)}{{\nu }^2}.$$

(2)

Empirical evaluations [10,11,12,13,14,15], based on experimental cross sections of total photoabsorption on the nucleon, yield the most accurate information on proton [15] and neutron [13] dipole polarizabilities, we presently have

$$\begin{array}{ccc}\hfill {\alpha }_{E1p}+{\beta }_{M1p}& =& 14.0\left(2\right)\times {10}^{-4}{\mathrm{fm}}^3,\hfill \end{array}$$

(3a)

$$\begin{array}{ccc}\hfill {\alpha }_{E1n}+{\beta }_{M1n}& =& 15.2\left(4\right)\times {10}^{-4}{\mathrm{fm}}^3.\hfill \end{array}$$

(3b)

To disentangle ${\alpha }_{E1}$ and ${\beta }_{M1}$, one measures the angular distribution of low-energy CS. For example, the low-energy expansion of the unpolarized CS cross section is given by (to $\mathcal{O}\left({\nu }^2\right)$)

$$\frac{\mathrm{d}\sigma }{\mathrm{d}{\Omega }_L}-\frac{\mathrm{d}{\sigma }^{\mathrm{Born}}}{\mathrm{d}{\Omega }_L}=-\nu {\nu }^{\prime }{\left(\frac{{\nu }^{\prime }}{\nu }\right)}^2\frac{2\pi \alpha }{M_N}\left[\left({\alpha }_{E1}+{\beta }_{M1}\right){(1+cos{\theta }_L)}^2+\left({\alpha }_{E1}-{\beta }_{M1}\right){(1-cos{\theta }_L)}^2\right],$$

(4)

where ${\theta }_L$ is the scattering angle, $\mathrm{d}{\Omega }_L=2\pi \mathrm{d}cos{\theta }_L$, and $\nu \left({\nu }^{\prime }\right)$ is the energy of the incoming (scattered) photon, all in the lab frame. Here, in addition to the sum of dipole polarizabilities appearing in forward kinematics, one can measure their difference. Another interesting observable is the beam asymmetry ${\mathsf{\Sigma }}_3$ defined in Equation (39), which also provides access to ${\beta }_{M1}$ independent of ${\alpha }_{E1}$ at $\mathcal{O}\left({\nu }^2\right)$, cf. Equation (40).

In reality, the CS data are taken at finite energies (typically of ~100 MeV), rather than at infinitesimal energies required for a strict validation of the above low-energy expansion. For a model-independent empirical extraction of polarizabilities from the RCS data it is, therefore, important to have a systematic theoretical framework such as $\chi$PT or a partial-wave analysis (PWA).

There are other interesting polarizabilities, called the spin polarizabilities. These are more difficult to visualize in a classical picture, but they certainly characterize the spin structure of the nucleon. $\chi$PT provides robust predictions for the different nucleon polarizabilities at leading and next-to-leading order. Given the accurate empirical knowledge of the nucleon polarizabilities from dispersive sum rules and CS experiments, they become an important benchmark for $\chi$PT in the single-baryon sector, but not just for $\chi$PT—the lattice QCD studies of nucleon polarizabilities are also closing in on the physical pion mass, see Figure 2 and Figure 3.

It is worth mentioning that $\chi$PT can be used for calculating the proton-structure corrections to the muonic-hydrogen spectrum. These corrections are not only relevant in the context of the proton-radius puzzle [16,17], but also for the planned measurements of the muonic-hydrogen ground-state hyperfine splitting [18,19,20]. The $\chi$PT is thusfar the only theoretical framework which can reliably compute the polarizability effects in CS observables and, at the same time, in atomic spectroscopy. In this way, a calculation which is validated on experimental data of CS and photoabsorption (through sum rules) can be used to predict the effects in muonic hydrogen [21,22,23].

This mini-review is by no means comprehensive. A more proper review can be found in [24], whereas here I primarily provide an update on the nucleon polarizabilities. For the reader interested in the update only, I recommend to skip to Section 4 where a description of all summary plots is given. A recent theoretical discussion of nucleon polarizabilities in $\chi$PT and beyond can be found in [25]. Other commendable reviews include Guichon and Vanderhaeghen [26] or Fonvieille et al. [27] (VCS and generalized polarizabilities), Drechsel et al. [28] or Pasquini and Vanderhaeghen [29] (dispersion relations for CS), Pascalutsa et al. [30] ($\mathsf{\Delta }\left(1232\right)$ resonance), Phillips [31] (neutron polarizabilities), Griesshammer et al. [32] ($\chi$EFT and RCS experiments), Holstein and Scherer [33] (pion, kaon, nucleon polarizabilities), Geng [34] (B$\chi$PT), Pascalutsa [9] (dispersion relations), and Deur et al. [35] (nucleon spin structure). A textbook introduction to $\chi$PT can be found in [36].

The paper is organized as follows. In Section 2 and Section 3, I briefly describe the $\chi$PT framework and the CS formalism. In Section 4, I summarize recent $\chi$PT results for the nucleon polarizabilities and compare to empirical and lattice QCD evaluations.

2. Baryon Chiral Perturbation Theory

The low-energy processes involving a nucleon, such as $\pi N$ scattering or CS off the nucleon, can be described by SU(2) baryon chiral perturbation theory (B$\chi$PT), which is the manifestly Lorentz-invariant variant of $\chi$PT in the single-baryon sector [4,37,38]. To introduce it, I will start in Section 2.1 with the basic EFT including only pions and nucleons. Then, in Section 2.2, I will discuss different ways (counting schemes) for incorporation of the lowest nucleon excitation—the $\mathsf{\Delta }\left(1232\right)$ resonance—into the $\chi$PT framework. In Section 2.3, I will show how the LECs can be fit to experimental data and discuss the predictive power of $\chi$PT for CS. In Section 2.4, I introduce the heavy-baryon chiral perturbation theory (HB$\chi$PT) and point out how its predictions differ from B$\chi$PT for certain polarizabilities. For more details on B$\chi$PT for CS, I refer to the following series of calculations; RCS [39,40,41], VCS [42], and forward VVCS [43,44,45].

2.1. B$\chi$PT with Pions and Nucleons

Consider the basic version of SU(2) B$\chi$PT including only pion and nucleon fields [4]: scalar iso-vector ${\pi }^a\left(x\right)$ and spinor iso-doublet $\mathcal{N}\left(x\right)$. Expanding the EFT Lagrangian [4] to leading orders in pion derivatives, mass, and fields, one finds (see, e.g., in [46])

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mathcal{N}}^{\left(1\right)}& =& \overline{\mathcal{N}}\left(\cancel{D}-M_N\right)\mathcal{N}-\frac{g_A}{2f_{\pi }}\overline{\mathcal{N}}{\tau }^a\left({\cancel{D}}^{ab}{\pi }^b\right){\gamma }_5\mathcal{N},\hfill \end{array}$$

(5a)

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\pi }^{\left(2\right)}& =& \frac{1}{2}\left(D_{\mu }^{ab}{\pi }^b\right)\left(D_{ac}^{\mu }{\pi }^c\right)-\frac{1}{2}{m}_{\pi }^2{\pi }_a{\pi }^a,\hfill \end{array}$$

(5b)

with the covariant derivatives:

$$\begin{array}{cc}\hfill D_{\mu }^{ab}{\pi }^b=& {\delta }^{ab}{\partial }_{\mu }{\pi }^b+ie{Q}_{\pi }^{ab}{A}_{\mu }{\pi }^b,\hfill \end{array}$$

(6a)

$$\begin{array}{ccc}\hfill D_{\mu }\mathcal{N}& =& {\partial }_{\mu }\mathcal{N}+ie{Q}_N{A}_{\mu }\mathcal{N}+\frac{i}{4f_{\pi }^2}{\epsilon }^{abc}{\tau }^a{\pi }^b\left({\partial }_{\mu }{\pi }^c\right),\hfill \end{array}$$

(6b)

the photon vector field $A_{\mu }\left(x\right)$, and the charges:

$$\begin{array}{ccc}\hfill Q_{\pi }^{ab}& =& -i{\epsilon }^{ab3},\hfill \end{array}$$

(7a)

$$\begin{array}{ccc}\hfill Q_N& =& \frac{1}{2}(1+{\tau }^3).\hfill \end{array}$$

(7b)

Here, ${\tau }^a$ are the Pauli matrices, ${\gamma }_5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ are the Dirac matrices, ${\epsilon }^{ijk}$ is the Levi–Cevita symbol, and all other parameters are introduced in

Table 1. Low-energy constants (LECs) and other parameters and the orders at which they appear in the chiral expansion when employing the low-energy $\delta$-expansion counting scheme.

Order in Chiral Expansion	$\mathit{\chi }$PT Parameters		Values	Sources
$\mathcal{O}\left(p^2\right)$	fine-structure constant	$\alpha =MM{e}^2/4\pi$	$\simeq 1/137.04$
	nucleon mass	$M_N$	$938.27$ MeV
$\mathcal{O}\left(p^3\right)$	nucleon axial charge	$g_A$	$1.27$	neutron decay $n\to p{e}^{-}{\overline{\nu }}_e$ [47]
	pion decay constant	$f_{\pi }$	$92.21$ MeV	pion decay ${\pi }^{+}\to {\mu }^{+}{\nu }_{\mu }$ [47]
	pion mass	$m_{\pi }$	$139.57$ MeV
$\mathcal{O}(p^4/\Delta )$	$\mathcal{N}$-to-$\Delta$ axial coupling	$h_A$	$2.85$	$P_{33}$ partial wave in $\pi N$ scattering and $\Delta \left(1232\right)$ decay width [30,48,49]
	$\Delta \left(1232\right)$ mass	$M_{\Delta }$	1232 MeV
	magnetic (M1) coupling	$g_M$	$2.97$	pion electroproduction $e^{-}N\to e^{-}N\pi$ [50]
	electric (E2) coupling	$g_E$	$-1.0$
	Coulomb (C2) coupling	$g_{\mathrm{C}}$	$-2.6$

.

The key ingredient for the development of $\chi$PT as a low-energy EFT of QCD was the observation that the pion couplings are proportional to their four-momenta [2,3,4]. Therefore, at low momenta the couplings are weak and a perturbative expansion is possible. This chiral expansion is done in powers of pion momentum and mass, commonly denoted as p, over the scale of spontaneous chiral symmetry breaking, ${\mathsf{\Lambda }}_{\chi \mathrm{SB}}\sim 4\pi f_{\pi }\approx 1$ GeV. Therefore, one expects that $\chi$PT provides a systematic description of the strong interaction at energies well below 1 GeV. Considering only pion and nucleon fields, the chiral order $\mathcal{O}\left(p^n\right)$ of a Feynman diagram with L loops, $N_{\pi }$ ($N_N$) pion (nucleon) propagators, and $V_k$ vertices from k-th order Lagrangians (e.g., $k=1$: $\gamma NN$ interaction from Equation (5a) and $k=2$: $\gamma \pi \pi$ interaction from Equation (5b)) is defined as [4]

$$n=4L-2N_{\pi }-N_N+\sum _{k}k{V}_k.$$

(8)

In the case of CS, the low-energy scale p also includes the photon energy $\nu$ and virtuality Q, which therefore should be much smaller than 1 GeV. However, the presence of bound states or low-lying resonances may lead to a breakdown of this perturbative expansion. For example, in $\pi$–$\pi$ scattering the limiting scale of the perturbative expansion is set by the $\sigma \left(600\right)$ and $\rho \left(775\right)$ mesons [51,52]. In the single-nucleon sector, the breakdown scale is set by the excitation energy of the first nucleon resonance, the $\mathsf{\Delta }\left(1232\right)$ isobar. That is unless the $\mathsf{\Delta }\left(1232\right)$ is included explicitly in the effective Lagrangian.

2.2. Inclusion of the $\mathsf{\Delta }\left(1232\right)$ and Power Counting

The $\mathsf{\Delta }\left(1232\right)$ resonance as the lightest nucleon excitation has an excitation energy

$$\mathsf{\Delta }=M_{\mathsf{\Delta }}-M_N\simeq 293\mathrm{MeV},$$

(9)

which is of the same order of magnitude as the pion mass. In the following, it will be included as an explicit degree of freedom: vector-spinor iso-quartet ${\mathsf{\Delta }}_{\mu }\left(x\right)$. The relevant Lagrangians read [46,50,53]:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\pi \mathsf{\Delta }\mathcal{N}}^{\left(1\right)}& =& \frac{i{h}_A}{2f_{\pi }{M}_{\mathsf{\Delta }}}\overline{\mathcal{N}}{T}^a{\gamma }^{\mu \nu \lambda }\left(D_{\mu }{\mathsf{\Delta }}_{\nu }\right)\left(D_{\lambda }^{ab}{\pi }^b\right)+\mathrm{h}.\mathrm{c}.,\hfill \\ \hfill {\mathcal{L}}_{\gamma N\mathsf{\Delta }}^{\left(2\right)\mathrm{non}-\mathrm{minimal}}& =& \frac{3e}{2M_N(M_N+M_{\mathsf{\Delta }})}\left[\overline{\mathcal{N}}{T}_3\right\{i{g}_M\left({\partial }_{\mu }{\mathsf{\Delta }}_{\nu }\right){\tilde{F}}^{\mu \nu }-g_E{\gamma }_5\left({\partial }_{\mu }{\mathsf{\Delta }}_{\nu }\right)F^{\mu \nu }\hfill \end{array}$$

(10a)

$$\begin{array}{ccc}& & +i\frac{g_{\mathrm{C}}}{M_{\mathsf{\Delta }}}{\gamma }_5{\gamma }^{\alpha }({\partial }_{\alpha }{\mathsf{\Delta }}_{\nu }-{\partial }_{\nu }{\mathsf{\Delta }}_{\alpha }){\partial }_{\mu }{F}^{\mu \nu }\}+\mathrm{h}.\mathrm{c}.],\hfill \end{array}$$

(10b)

with the covariant derivative

$$D_{\mu }{\mathsf{\Delta }}_{\nu }={\partial }_{\mu }{\mathsf{\Delta }}_{\nu }+ie{Q}_{\mathsf{\Delta }}{A}_{\mu }{\mathsf{\Delta }}_{\nu }+\frac{i}{2f_{\pi }^2}{\epsilon }^{abc}{\mathcal{T}}^a{\pi }^b\left({\partial }_{\mu }{\pi }^c\right),$$

(11)

and the charge

$$Q_{\mathsf{\Delta }}=\frac{1}{2}(1+3{\mathcal{T}}^3).$$

(12)

Here, h.c. stands for the hermitian conjugate, ${\gamma }^{\mu \nu }=-\frac{i}{2}{ϵ}^{\mu \nu \alpha \beta }{\gamma }_{\alpha }{\gamma }_{\beta }{\gamma }^5$ and ${\gamma }^{\mu \nu \alpha }=-i{ϵ}^{\mu \nu \alpha \beta }{\gamma }_{\beta }{\gamma }^5$ are Dirac matrices with ${ϵ}_{0123}=1$, $F^{\mu \nu }={\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu }$ is the electromagnetic field strength tensor, ${\tilde{F}}^{\mu \nu }={ϵ}^{\mu \nu \rho \lambda }{\partial }_{\rho }{A}_{\lambda }$ is its dual, and $T^a$ (${\mathcal{T}}^a$) are the isospin $1/2$ ($3/2$) to $3/2$ transition matrices. The latter commute with the Dirac matrices. The superscripts of the Lagrangians in Equations (5) and (10) denote their order as reflected by the number of comprised small quantities: pion mass, momentum, and factors of e. Inclusion of the $\mathsf{\Delta }\left(1232\right)$ introduces the excitation energy $\mathsf{\Delta }$ as another small scale, which has to be considered when defining a power-counting for the perturbative $\chi$PT expansion.

There are two prominent counting schemes for $\chi$PT with explicit inclusion of the $\mathsf{\Delta }\left(1232\right)$. For simplicity, they both deduce a single expansion parameter from the two involved small mass scales: $ϵ=m_{\pi }/{\mathsf{\Lambda }}_{\chi \mathrm{SB}}$ and $\delta =\mathsf{\Delta }/{\mathsf{\Lambda }}_{\chi \mathrm{SB}}$. In the $ϵ$-expansion (small-scale expansion), it is assumed that $ϵ\sim \delta$ [54], while in the $\delta$-expansion one assumes $ϵ\sim {\delta }^2$ with $ϵ\ll \delta$ [55]. In this way, the $\delta$-expansion defines a hierarchy between the two mass scales. Consequently, it defines two regimes where the $\mathsf{\Delta }\left(1232\right)$ contributions need to be counted differently:

• low-energy region: $p\sim m_{\pi }$;
• resonance region: $p\sim \mathsf{\Delta }$.

This makes sense as the $\mathsf{\Delta }\left(1232\right)$ is expected to be suppressed at low energies and dominating in the resonance region. The chiral order $\mathcal{O}\left(p^{n_{\delta }}\right)$ of a Feynman diagram with $N_1\mathsf{\Delta }\mathrm{R}$ ($N_1\mathsf{\Delta }\mathrm{I}$) one-$\mathsf{\Delta }$-reducible (one-$\mathsf{\Delta }$-irreducible) propagators is in the $\delta$-expansion defined as

$$n_{\delta }=\left\{\begin{array}{cc}n-MM1/2N_{\mathsf{\Delta }},\hfill & p\sim m_{\pi },\hfill \\ n-3N_1\mathsf{\Delta }\mathrm{R}-N_1\mathsf{\Delta }\mathrm{I},\hfill & p\sim \mathsf{\Delta },\hfill \end{array}\right.$$

(13)

where

$$N_{\mathsf{\Delta }}=N_1\mathsf{\Delta }\mathrm{R}+N_1\mathsf{\Delta }\mathrm{I}.$$

(14)

An extensive review on the electromagnetic excitation of the $\mathsf{\Delta }$(1232) resonance with more details on the formulation of the extended $\chi$PT framework and the chiral expansion in the resonance region can be found in [30]. As we will see in Section 4, B$\chi$PT calculations based on the $ϵ$ [56] and the $\delta$ [43,45] counting schemes give significantly different predictions for the longitudinal-transverse polarizability of the proton shown in Figure 4 (upper panel) and Figure 5.

2.3. Low-Energy Constants and Predictive Orders

At any given order in the chiral expansion, the divergencies of the EFT are absorbed by renormalization of a finite number of LECs. To match $\chi$PT to QCD as the fundamental theory of the strong interaction, the renormalized LECs need to be fitted to experimental or lattice data. Thereby, it is important that the LECs are constrained to be of natural size. Take, for instance, the fifth-order forward spin polarizability (in units of ${10}^{-4}$ fm${}^6$) [45]:

$$\begin{array}{cc}\hfill & {\overline{\gamma }}_{0p}=1.12\left(30\right)\approx 2.08(\pi N\mathrm{loop})-0.96(\Delta \mathrm{exchange})-0.01(\pi \Delta \mathrm{loop}),\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill & {\overline{\gamma }}_{0n}=1.95\left(30\right)\approx 2.92(\pi N\mathrm{loop})-0.96(\Delta \mathrm{exchange})-0.01(\pi \Delta \mathrm{loop}),\hfill \end{array}$$

(15b)

also shown in Figure 6. The next-to-leading-order effect of the $\mathsf{\Delta }\left(1232\right)$ is two to three times smaller than the leading-order effect of the pion cloud. This is consistent with estimates from power counting, according to which each subleading order is expected to be suppressed with respect to the previous one by a factor of $\sim \mathsf{\Delta }/M_N\sim 1/3$. Therefore, implementing this naturalness allows to estimate the uncertainty due to neglect of higher-order effects.

The LECs entering a next-to-next-to-leading-order B$\chi$PT calculation of low-energy CS in the $\delta$-expansion are $f_{\pi }$, $g_A$, $h_A$, $g_M$, $g_E$, and $g_C$. They are listed in Table 1 together with the experiments used to constrain their values. As one can see, B$\chi$PT has “predictive power” for low-energy CS up to and including $\mathcal{O}(p^4/\mathsf{\Delta })$ because all relevant LECs are matched to processes other than CS. This makes $\chi$PT the perfect tool to study the low-energy structure of the nucleon as encoded in CS and the associated polarizabilities. Starting from $\mathcal{O}\left(p^4\right)$, LECs need to be fitted to the CS process as well, for instance through the Baldin sum rule, as done in [32,44,62,63,64,65,66].

2.4. Heavy-Baryon Expansion

The theory of HB$\chi$PT was first introduced in [67], and later applied to CS and polarizabilities [68], including also the effect of the $\mathsf{\Delta }\left(1232\right)$[32,60,69,70,71,72,73]. The results of HB$\chi$PT can be recovered from the B$\chi$PT results by expanding in powers of the inverse nucleon mass. HB$\chi$PT calculations tend to fail in describing the $Q^2$ evolution of the generalized nucleon polarizabilities [44,45]. Moreover, for the polarizabilities at the real-photon point ($Q^2=0$), the heavy-baryon expansion can give significantly different predictions. Consider, for instance, the nucleon dipole polarizabilities. The B$\chi$PT prediction (in units of ${10}^{-4}$fm${}^3$) [41]:

$$\begin{array}{ccc}\hfill {\alpha }_{E1p}& =& 6.9(\pi N\mathrm{loop})-0.1(\Delta \mathrm{exchange})+4.4(\pi \Delta \mathrm{loop})=11.2\pm 0.7,\hfill \end{array}$$

(16a)

$$\begin{array}{ccc}\hfill {\beta }_{M1p}& =& -1.8(\pi N\mathrm{loop})+7.1(\Delta \mathrm{exchange})-1.4(\pi \Delta \mathrm{loop})=3.9\pm 0.7,\hfill \end{array}$$

(16b)

is in good agreement with empirical evaluations, see Figure 2 and Figure 3. In HB$\chi$PT, however, the $\mathsf{\Delta }\left(1232\right)$ contributions to the nucleon polarizabilities turn out to be large [70] and need to be canceled by promoting the higher-order [$\mathcal{O}\left(p^4\right)$] counterterms $\delta \alpha$ and $\delta \beta$ (in units of ${10}^{-4}$fm${}^3$) [71]:

$$\begin{array}{ccc}{\alpha }_{E1p}\left(HB\right)\hfill & =\hfill & 11.87(\pi N\mathrm{loop})+0(\Delta \mathrm{exch}.)+5.09(\pi \Delta \mathrm{loop})-(5.92\pm 1.36)\left(\delta \alpha \right)\hfill \\ & =\hfill & 11.04\pm 1.36,\hfill \end{array}$$

(17a)

$$\begin{array}{ccc}{\beta }_{M1p}\left(HB\right)\hfill & =\hfill & 1.25(\pi N\mathrm{loop})+(11.33\pm 0.70)(\Delta \mathrm{exch}.)+0.86(\pi \Delta \mathrm{loop})\hfill \\ & & -(10.68\pm 1.17)\left(\delta \beta \right)\hfill \\ & =\hfill & 2.76\mp 1.36,\hfill \end{array}$$

(17b)

at the expense of violating the naturalness requirement, see also in [32]. This can be seen from the dimensionless LECs associated to $\delta \alpha$ and $\delta \beta$, $g_{117}=18.82\pm 0.79$ and $g_{118}=-6.05\mp 0.66$[71], that should be of $\mathcal{O}\left(1\right)$ to be consistent with estimates from power counting. This problem is discussed at length in [40,74].

3. Compton Scattering Formalism

The CS process, shown in Figure 1, gives the most direct access to the nucleon polarizabilities. Of interest are the following kinematic regimes, described by the four-momenta of incoming (outgoing) photons $q\left(q^{\prime }\right)$ and nucleons $p\left(p^{\prime }\right)$.

• Real Compton scattering (RCS): $q^2=q^{\prime 2}=0$;
• Virtual Compton scattering (VCS): $q^2=-Q^2<0$ and $q^{\prime 2}=0$;
• Forward doubly-virtual Compton scattering (VVCS): $q=q^{\prime }$ (thus $p=p^{\prime }$) and $q^2=-Q^2<0$.

In general kinematics ($p^2=p^{\prime 2}=M_N^2$, $q^2\ne q^{\prime 2}$), the CS amplitude can be described by 18 independent tensor structures. For VCS one needs 12 independent tensor structures; for RCS one needs six independent tensor structures [75,76]. In the forward limit, this reduces to four independent tensor structures for virtual photons and two independent tensor structures for real photons.

Splitting into spin-independent (symmetric) and spin-dependent (antisymmetric) parts, the forward VVCS decomposes into four scalar amplitudes $T_i(\nu ,Q^2)$ and $S_i(\nu ,Q^2)$:

$$T^{\mu \nu }(q,p)=\left[T_S^{\mu \nu }+T_A^{\mu \nu }\right](q,p),$$

(18a)

with

$$\begin{array}{ccc}\hfill T_S^{\mu \nu }(q,p)& =& -g^{\mu \nu }{T}_1(\nu ,Q^2)+\frac{p^{\mu }{p}^{\nu }}{M_N^2}{T}_2(\nu ,Q^2),\hfill \end{array}$$

(18b)

$$\begin{array}{ccc}\hfill T_A^{\mu \nu }(q,p)& =& -\frac{1}{M_N}{\gamma }^{\mu \nu \alpha }{q}_{\alpha }{S}_1(\nu ,Q^2)+\frac{Q^2}{M_N^2}{\gamma }^{\mu \nu }{S}_2(\nu ,Q^2),\hfill \end{array}$$

(18c)

with $\nu$ the photon lab-frame energy, $Q^2$ the photon virtuality, and terms which vanish upon contraction with the photon polarization vectors omitted. For real photons, the following two scalar amplitudes survive,

$$f\left(\nu \right)=\frac{1}{4\pi }{T}_1(\nu ,0),g\left(\nu \right)=\frac{\nu }{4\pi M_N}{S}_1(\nu ,0).$$

(19)

Constraints relating the different kinematic regimes (RCS, VCS, and forward VVCS) are discussed in [77,78,79] for the unpolarized and polarized CS, respectively. Here, the focus is on RCS and forward VVCS.

The off-forward RCS is conveniently described by the covariant decomposition [55]:

$${\overline{u}}^{\prime }({\epsilon }^{\prime }·T·\epsilon )u=4\pi \alpha {\widehat{\mathcal{A}}}^T(s,t){\overline{u}}^{\prime }{\widehat{\mathcal{O}}}^{\mu \nu }u{\mathcal{E}}_{\mu }^{\prime }{\mathcal{E}}_{\nu },$$

(20a)

with the overcomplete set of eight tensors:

$$\begin{array}{ccc}& & \widehat{\mathcal{A}}(s,t)=\left\{{\mathcal{A}}_1,\cdots ,{\mathcal{A}}_8\right\}(s,t),\hfill \\ & & {\widehat{\mathcal{O}}}^{\mu \nu }=\{-g^{\mu \nu },q^{\mu }{q}^{\prime \nu },-{\gamma }^{\mu \nu },g^{\mu \nu }(q^{\prime }·\gamma ·q),q^{\mu }{q}_{\alpha }^{\prime }{\gamma }^{\alpha \nu }-{\gamma }^{\alpha \mu }{q}_{\alpha }{q}^{\prime \nu },q^{\mu }{q}_{\alpha }{\gamma }^{\alpha \nu }-{\gamma }^{\alpha \mu }{q}_{\alpha }^{\prime }{q}^{\prime \nu },\hfill \end{array}$$

(20b)

$$\begin{array}{ccc}& & q^{\mu }{q}^{\prime \nu }(q^{\prime }·\gamma ·q),-i{\gamma }_5{ϵ}^{\mu \nu \alpha \beta }{q}_{\alpha }^{\prime }{q}_{\beta }\},\hfill \end{array}$$

(20c)

$$\begin{array}{ccc}& & {\mathcal{E}}_{\mu }={\epsilon }_{\mu }-\frac{P·\epsilon }{P·q}{q}_{\mu },{\mathcal{E}}_{\mu }^{\prime }={\epsilon }_{\mu }^{\prime }-\frac{P·{\epsilon }^{\prime }}{P·q}{q}_{\mu }^{\prime },P_{\mu }=\frac{1}{2}{(p+p^{\prime })}_{\mu },P·q=P·q^{\prime },\hfill \end{array}$$

(20d)

and the incoming (outgoing) photon polarization vector ${\epsilon }^{(\prime )}$ and Dirac spinor $u^{(\prime )}$. Alternatively, one can choose the non-covariant decomposition with the minimal set of six tensors:

$${\overline{u}}^{\prime }({\epsilon }^{\prime }·T·\epsilon )u=8\pi \alpha M_N{\widehat{A}}^T(s,t){\chi }^{\prime }{\epsilon }_i^{\prime }{\widehat{O}}_{ij}{\epsilon }_j\chi ,$$

(21a)

with the incoming (outgoing) Pauli spinor ${\chi }^{(\prime )}$ and the scalar complex amplitudes [80]:

$$\begin{array}{ccc}& & \widehat{A}(s,t)=\left\{A_1,\cdots ,A_6\right\}(s,t),\hfill \\ & & {\widehat{O}}_{ij}=\{{\delta }_{ij},n_i{n}_j^{\prime },i{ϵ}_{ijk}{\sigma }_k,{\delta }_{ij}i{ϵ}_{klm}{\sigma }_k{n}_l^{\prime }{n}_m,i{ϵ}_{klm}{\sigma }_k({\delta }_{il}{n}_m{n}_j^{\prime }-{\delta }_{jl}{n}_i{n}_m^{\prime }),\hfill \end{array}$$

(21b)

$$\begin{array}{ccc}& & i{ϵ}_{klm}{\sigma }_k({\delta }_{il}{n}_m^{\prime }{n}_j^{\prime }-{\delta }_{jl}{n}_i{n}_m)\},\hfill \end{array}$$

(21c)

where ${\overrightarrow{n}}^{(\prime )}$ is the direction of the incoming (outgoing) photon, ${\sigma }_k$ are the Pauli matrices and ${\delta }_{ij}$ is the Kronecker delta. The scalar amplitudes ${\mathcal{A}}_{1,\cdots ,8}$ are related to the scalar amplitudes $A_{1,\cdots ,6}$ in the following way [62]:

$$\begin{array}{ccc}\hfill A_1& =& \frac{{ϵ}_B}{M_N}{\mathcal{A}}_1+\frac{{\omega }_{B}t}{2M_N}{\mathcal{A}}_4,\hfill \end{array}$$

(22a)

$$\begin{array}{ccc}\hfill A_2& =& \frac{{ϵ}_B{\omega }_B^2}{M_N}{\mathcal{A}}_2+\frac{{\omega }_B^3}{M_N}\left({\mathcal{A}}_5+{\mathcal{A}}_6-\frac{1}{2}t{\mathcal{A}}_7\right),\hfill \end{array}$$

(22b)

$$\begin{array}{ccc}\hfill A_3& =& \frac{{ϵ}_B}{M_N}{\mathcal{A}}_3-\frac{M_N^2\eta t}{4M_N^2-t}\left(\frac{{\mathcal{A}}_5+{\mathcal{A}}_6}{2M_N({ϵ}_B+M_N)}-{\mathcal{A}}_7\right)-\frac{{\omega }_{B}t}{2M_N}{\mathcal{A}}_8,\hfill \end{array}$$

(22c)

$$\begin{array}{ccc}\hfill A_4& =& {\omega }_B^2{\mathcal{A}}_4,\hfill \end{array}$$

(22d)

$$\begin{array}{ccc}\hfill A_5& =& {\omega }_B^2{\mathcal{A}}_5+\frac{{\omega }_B^2}{2M_N({ϵ}_B+M_N)}\left[\frac{1}{2}{\mathcal{A}}_3+\frac{M_N^2\eta }{4M_N^2-t}\left({\mathcal{A}}_5+{\mathcal{A}}_6\right)\right]-{\omega }_B^2({\omega }_B^2+\frac{1}{2}t){\mathcal{A}}_7+\frac{{\omega }_B^3}{2M_N}{\mathcal{A}}_8,\hfill \end{array}$$

(22e)

$$\begin{array}{ccc}\hfill A_6& =& {\omega }_B^2{\mathcal{A}}_6-\frac{{\omega }_B^2}{2M_N({ϵ}_B+M_N)}\left[\frac{1}{2}{\mathcal{A}}_3+\frac{M_N^2\eta }{4M_N^2-t}\left({\mathcal{A}}_5+{\mathcal{A}}_6\right)\right]+{\omega }_B^4{\mathcal{A}}_7-\frac{{\omega }_B^3}{2M_N}{\mathcal{A}}_8,\hfill \end{array}$$

(22f)

where

$$\begin{array}{ccc}\hfill {\omega }_B& =& \frac{s-u}{2\sqrt{4M_N^2-t}},\hfill \end{array}$$

(23a)

$$\begin{array}{ccc}\hfill {ϵ}_B& =& \frac{1}{2}\sqrt{4M_N^2-t}.\hfill \end{array}$$

(23b)

are the nucleon and photon energies in the Breit frame (${\overrightarrow{p}}^{\prime }=-\overrightarrow{p}$),

$$\eta =\frac{M_N^4-su}{M_N^2},$$

(24)

and s, t, and u are the usual Mandelstam variables.

According to the low-energy theorem of Low [81], Gell-Mann, and Goldberger [82], the leading terms in a low-energy expansion of the RCS amplitudes are determined by charge, mass, and anomalous magnetic moment of the nucleon. At higher orders in the low-energy expansion, various polarizabilities emerge. The low-energy expansion of the non-Born RCS amplitudes (denoted by an overline, e.g., ${\overline{A}}_{1,\cdots ,6}$) reads as

$$\begin{array}{ccc}\hfill \alpha {\overline{A}}_1({\omega }_B,t)& =& {\omega }_B^2\left[{\alpha }_{E1}+{\beta }_{M1}+{\omega }_B^2({\alpha }_{E1\nu }+{\beta }_{M1\nu })\right]+\frac{1}{2}t\left({\beta }_{M1}+{\omega }_B^2{\beta }_{M1\nu }\right)\hfill \end{array}$$

(25a)

$$\begin{array}{ccc}& & +{\omega }_B^4\frac{1}{12}({\alpha }_{E2}+{\beta }_{M2})+\frac{1}{2}t(4{\omega }_B^2+t)\frac{1}{12}{\beta }_{M2}+\mathcal{O}\left({\omega }_B^6\right),\hfill \end{array}$$

(25b)

$$\begin{array}{ccc}\hfill \alpha {\overline{A}}_2({\omega }_B,t)& =& -{\omega }_B^2\left({\beta }_{M1}+{\omega }_B^2{\beta }_{M1\nu }\right)+{\omega }_B^4\frac{1}{12}({\alpha }_{E2}-{\beta }_{M2})-t{\omega }_B^2\frac{1}{12}{\beta }_{M2}+\mathcal{O}\left({\omega }_B^6\right),\hfill \end{array}$$

(25c)

$$\begin{array}{ccc}\hfill \alpha {\overline{A}}_3({\omega }_B,t)& =& -{\omega }_B^3\left[{\gamma }_{E1E1}+{\gamma }_{E1M2}+z({\gamma }_{M1E2}+{\gamma }_{M1M1})\right]+\mathcal{O}\left({\omega }_B^5\right),\hfill \end{array}$$

(25d)

$$\begin{array}{ccc}\hfill \alpha {\overline{A}}_4({\omega }_B,t)& =& {\omega }_B^3({\gamma }_{M1E2}-{\gamma }_{M1M1})+\mathcal{O}\left({\omega }_B^5\right),\hfill \end{array}$$

(25e)

$$\begin{array}{ccc}\hfill \alpha {\overline{A}}_5({\omega }_B,t)& =& {\omega }_B^3{\gamma }_{M1M1}+\mathcal{O}\left({\omega }_B^5\right),\hfill \end{array}$$

(25f)

$$\begin{array}{ccc}\hfill \alpha {\overline{A}}_6({\omega }_B,t)& =& {\omega }_B^3{\gamma }_{E1M2}+\mathcal{O}\left({\omega }_B^5\right),\hfill \end{array}$$

(25g)

with $z=cos{\theta }_B=1+t/2{\omega }_B^2$ and ${\theta }_B$ the scattering angle in the Breit frame. The coefficients are given in terms of static nucleon polarizabilities: electric dipole (${\alpha }_{E1}$), magnetic dipole (${\beta }_{M1}$), quadrupole (${\alpha }_{E2}$, ${\beta }_{M2}$), dispersive (${\alpha }_{E1\nu }$, ${\beta }_{M1\nu }$), and lowest-order spin polarizabilities (${\gamma }_{E1E1}$, ${\gamma }_{M1M1}$, ${\gamma }_{E1M2}$, and ${\gamma }_{M1E2}$), see Figure 2, Figure 3, and Figure 7, Figure 8 and Figure 9, respectively. The latter combine into the forward (see Figure 10) and backward spin polarizabilities:

$$\begin{array}{ccc}\hfill {\gamma }_0& =& -{\gamma }_{E1E1}-{\gamma }_{M1M1}-{\gamma }_{E1M2}-{\gamma }_{M1E2},\hfill \end{array}$$

(26a)

$$\begin{array}{ccc}\hfill {\gamma }_{\pi }& =& -{\gamma }_{E1E1}+{\gamma }_{M1M1}-{\gamma }_{E1M2}+{\gamma }_{M1E2}.\hfill \end{array}$$

(26b)

Studying the forward RCS and VVCS is of advantage because of their accessibility through sum rules. Based on the general principles of causality, unitarity and crossing symmetry, the forward VVCS amplitudes can be expressed in terms of the nucleon structure functions by means of dispersion relations and the optical theorem [28]:

$$\begin{array}{ccc}\hfill T_1(\nu ,Q^2)& =& T_1(0,Q^2)+\frac{32\pi \alpha M_N{\nu }^2}{Q^4}{\int }_0^1\mathrm{d}x\frac{x{f}_1(x,Q^2)}{1-x^2{(\nu /{\nu }_{\mathrm{el}})}^2-i{0}^{+}}\hfill \\ & =& T_1(0,Q^2)+\frac{2{\nu }^2}{\pi }{\int }_{{\nu }_{\mathrm{el}}}^{\infty }\frac{\mathrm{d}{\nu }^{\prime }}{{\nu }^{\prime }}\frac{\sqrt{{\nu }^{\prime 2}+Q^2}{\sigma }_T({\nu }^{\prime },Q^2)}{{\nu }^{\prime 2}-{\nu }^2-i{0}^{+}},\hfill \end{array}$$

(27a)

$$\begin{array}{ccc}\hfill T_2(\nu ,Q^2)& =& \frac{16\pi \alpha M_N}{Q^2}{\int }_0^1\mathrm{d}x\frac{f_2(x,Q^2)}{1-x^2{(\nu /{\nu }_{\mathrm{el}})}^2-i{0}^{+}}\hfill \\ & =& \frac{2Q^2}{\pi }{\int }_{{\nu }_{\mathrm{el}}}^{\infty }\mathrm{d}{\nu }^{\prime }\frac{{\nu }^{\prime }[{\sigma }_T+{\sigma }_L]({\nu }^{\prime },Q^2)}{\sqrt{{\nu }^{\prime 2}+Q^2}({\nu }^{\prime 2}-{\nu }^2-i{0}^{+})},\hfill \end{array}$$

(27b)

$$\begin{array}{ccc}\hfill S_1(\nu ,Q^2)& =& \frac{16\pi \alpha M_N}{Q^2}{\int }_0^1\mathrm{d}x\frac{g_1(x,Q^2)}{1-x^2{(\nu /{\nu }_{\mathrm{el}})}^2-i{0}^{+}}\hfill \\ & =& \frac{2M_N}{\pi }{\int }_{{\nu }_{\mathrm{el}}}^{\infty }\mathrm{d}{\nu }^{\prime }\frac{{\nu }^{\prime 2}\left[\frac{Q}{{\nu }^{\prime }}{\sigma }_{LT}+{\sigma }_{TT}\right]({\nu }^{\prime },Q^2)}{\sqrt{{\nu }^{\prime 2}+Q^2}({\nu }^{\prime 2}-{\nu }^2-i{0}^{+})},\hfill \end{array}$$

(27c)

$$\begin{array}{ccc}\hfill \nu S_2(\nu ,Q^2)& =& \frac{16\pi \alpha M_N^2}{Q^2}{\int }_0^1\mathrm{d}x\frac{g_2(x,Q^2)}{1-x^2{(\nu /{\nu }_{\mathrm{el}})}^2-i{0}^{+}}\hfill \\ & =& \frac{2M_N^2}{\pi }{\int }_{{\nu }_{\mathrm{el}}}^{\infty }\mathrm{d}{\nu }^{\prime }\frac{{\nu }^{\prime 2}\left[\frac{{\nu }^{\prime }}{Q}{\sigma }_{LT}-{\sigma }_{TT}\right]({\nu }^{\prime },Q^2)}{\sqrt{{\nu }^{\prime 2}+Q^2}({\nu }^{\prime 2}-{\nu }^2-i{0}^{+})},\hfill \end{array}$$

(27d)

with ${\nu }_{\mathrm{el}}=Q^2/2M_N$ being the elastic threshold. Note that the structure functions $f_1$, $f_2$, $g_1$, and $g_2$ are functions of the Bjorken variable $x={\nu }_{\mathrm{el}}/\nu$ and the photon virtuality $Q^2$. They are related to the photoabsorption cross sections ${\sigma }_T$, ${\sigma }_L$, ${\sigma }_{TT}$, and ${\sigma }_{LT}$ measured in electroproduction, defined here with the photon flux factor $K(\nu ,Q^2)=\sqrt{{\nu }^2+Q^2}$ [84].

Performing low-energy expansions of the relativistic CS amplitudes [28,78,85] and combining these with dispersion relations and the optical theorem leads to various sum rules for the polarizabilities. A famous sum-rule example is the Baldin sum rule [7], allowing for a precise data-driven evaluation of the sum of electric and magnetic dipole polarizabilities, cf. Equations (2) and (3). It follows from the ${\nu }^2$ term in the low-energy expansion of the RCS amplitude $f\left(\nu \right)$:

$$f\left(\nu \right)=-\frac{Z^2\alpha }{M_N}+\left[{\alpha }_{E1}+{\beta }_{M1}\right]{\nu }^2+\left[{\alpha }_{E1\nu }+{\beta }_{M1\nu }+MM1/12({\alpha }_{E2}+{\beta }_{M2})\right]{\nu }^4+\mathcal{O}\left({\nu }^6\right).$$

(28)

The extension of the Baldin sum rule to finite momentum-transfers [28],

$$\left[{\alpha }_{E1}+{\beta }_{M1}\right]\left(Q^2\right)=\frac{1}{2{\pi }^2}{\int }_{{\nu }_0}^{\infty }\mathrm{d}\nu \sqrt{1+\frac{Q^2}{{\nu }^2}}\frac{{\sigma }_T(\nu ,Q^2)}{{\nu }^2},$$

(29)

defines the $Q^2$ dependent sum of generalized dipole polarizabilities. Be aware that while the definitions of the polarizabilities in the real-photon limit are unambiguous, the generalized polarizabilities defined in VCS and forward VVCS can differ. As an example, one can consider the magnetic dipole polarizability ${\beta }_{M1}\left(Q^2\right)$, which for VCS is defined in Equation (B2b) of the work in [77], and for forward VVCS could be defined either by generalizing the non-Born part of the subtraction function

$$\frac{{\overline{T}}_1(0,Q^2)}{4\pi }={\beta }_{M1}{Q}^2+\mathcal{O}\left(Q^4\right),$$

(30)

but is usually understood as part of the generalized Baldin sum rule (29). A recent measurement of the generalized ${\alpha }_{E1}\left(Q^2\right)$ and ${\beta }_{M1}\left(Q^2\right)$ polarizabilities from VCS by the A1 Collaboration can be found in [92].

The generalized fourth-order Baldin sum rule is defined as

$$M_1^{\left(4\right)}\left(Q^2\right)=\frac{1}{2{\pi }^2}{\int }_{{\nu }_0}^{\infty }\mathrm{d}\nu \sqrt{1+\frac{Q^2}{{\nu }^2}}\frac{{\sigma }_T(\nu ,Q^2)}{{\nu }^4}.$$

(31)

It differs from the generalized Baldin sum rule (29) by the energy weighting of the total photoabsorption cross section ${\sigma }_T$ in the sum rule integral. In the real-photon limit, it is related to a linear combination of the dispersive and quadrupole polarizabilities given by the ${\nu }^4$ term in Equation (28) [93,94]:

$$M_1^{\left(4\right)}\left(0\right)={\alpha }_{E1\nu }+{\beta }_{M1\nu }+\frac{1}{12}({\alpha }_{E2}+{\beta }_{M2}),$$

(32)

see Figure 11.

Similarly, the low-energy expansion of the RCS amplitude $g\left(\nu \right)$:

$$g\left(\nu \right)=-\frac{\alpha {\kappa }_N^2}{2M_N^2}\nu +{\gamma }_0{\nu }^3+{\overline{\gamma }}_0{\nu }^5+\mathcal{O}\left({\nu }^6\right),$$

(33)

allows to express the anomalous magnetic moment of the nucleon (${\kappa }_p\sim 1.79,{\kappa }_n\sim -1.91$) and the forward spin polarizabilities as sum rule integrals over the helicity-difference photoabsorption cross section ${\sigma }_{TT}$, cf. Equation (27c). The Gerasimov–Drell–Hearn sum rule [95,96],

$$-\frac{\alpha }{2M_N^2{\kappa }_N^2}=\frac{1}{2{\pi }^2}{\int }_{{\nu }_0}^{\infty }\mathrm{d}\nu \frac{{\sigma }_{TT}\left(\nu \right)}{\nu },$$

(34)

has been experimentally verified for the nucleon by MAMI (Mainz) and ELSA (Bonn) [97,98]. The same cross section input can be used to evaluate the forward spin polarizabilities at the real-photon point, cf. Figure 6 and Figure 10. Considering the extension to finite momentum-transfers, the generalized forward spin polarizability reads [28]

$${\gamma }_0\left(Q^2\right)=\frac{1}{2{\pi }^2}{\int }_{{\nu }_0}^{\infty }\mathrm{d}\nu \sqrt{1+\frac{Q^2}{{\nu }^2}}\frac{{\sigma }_{TT}(\nu ,Q^2)}{{\nu }^3},$$

(35)

while the fifth-order generalized forward spin polarizability sum rule is given by

$${\overline{\gamma }}_0\left(Q^2\right)=\frac{1}{2{\pi }^2}{\int }_{{\nu }_0}^{\infty }\mathrm{d}\nu \sqrt{1+\frac{Q^2}{{\nu }^2}}\frac{{\sigma }_{TT}(\nu ,Q^2)}{{\nu }^5},$$

(36)

see Figure 12 upper and lower panel, respectively.

The polarizabilities involving longitudinal photon polarizations are absent from RCS. They are given as sum rule integrals over the longitudinal photoabsorption cross section ${\sigma }_L$, e.g., the longitudinal polarizability [43]:

$${\alpha }_L\left(Q^2\right)=\frac{1}{2{\pi }^2}{\int }_{{\nu }_0}^{\infty }\mathrm{d}\nu \sqrt{1+\frac{Q^2}{{\nu }^2}}\frac{{\sigma }_L(\nu ,Q^2)}{Q^2{\nu }^2},$$

(37)

cf. Figure 13, and the longitudinal-transverse cross section ${\sigma }_{LT}$, e.g., the longitudinal-transverse polarizability [28]:

$${\delta }_{LT}\left(Q^2\right)=\frac{1}{2{\pi }^2}{\int }_{{\nu }_0}^{\infty }\mathrm{d}\nu \sqrt{1+\frac{Q^2}{{\nu }^2}}\frac{{\sigma }_{LT}(\nu ,Q^2)}{Q{\nu }^2},$$

(38)

see Figure 4 and Figure 5.

4. Nucleon Polarizabilities

In the following, I want to discuss the nucleon polarizabilities, focusing on new empirical results from the last five years and comparisons to $\chi$PT predictions. References quoted in the summary figures are: PDG [106], MAID [104], experiments [86,87,88,89,107,108], dispersion relations [11,76,83,94,101,105,109,110], PWA [91], lattice QCD [111,112,113,114,115,116], HB$\chi$PT fit [62,64,66], B$\chi$PT fit [63], HB$\chi$PT [73,94,117], B$\chi$PT $\delta$-expansion [41,43,44,45] and B$\chi$PT $ϵ$-expansion [56,59].

Most recent HB$\chi$PT [32,62,64,65,66] and B$\chi$PT [39,40,41,42,43,44,45,63] calculations and fits of CS observables employ the $\delta$-expansion power counting. An exception are the works of Bernard et al. [56] and Thürmann et al. [59]. As one can see from Figure 4 (upper panel), B$\chi$PT predictions for ${\delta }_{LTp}$ within the $\delta$-expansion [43,45] or the $ϵ$-expansion [56,59] deviate substantially, since they include the $\mathsf{\Delta }\left(1232\right)$ in different ways. In the $ϵ$-expansion, the longitudinal-transverse polarizability receives a large contribution from diagrams where the photons couple directly to the $\mathsf{\Delta }\left(1232\right)$ inside a loop. These diagrams are absent in the $\delta$-expansion at $\mathcal{O}(p^4/\mathsf{\Delta })$, thus, there the effect of the $\mathsf{\Delta }\left(1232\right)$ is small and agrees with the MAID model [104]. For the generalized longitudinal-transverse polarizability ${\delta }_{LTp}\left(Q^2\right)$ a similar $Q^2$ evolution is found in both power-counting schemes, see Figure 5 (left panel). Therefore, the discrepancy found for the polarizability ${\delta }_{LTp}$ at the real-photon point continues as a constant shift for all $Q^2$[45]. Another difference between the B$\chi$PT calculations [43,45,56,59] is the implementation of the magnetic-dipole $\mathcal{N}$-to-$\mathsf{\Delta }$ transition and the coupling $g_M$[118]. This “${\delta }_{LT}$ puzzle” could soon be resolved by an empirical evaluation based on new data for the proton spin structure function $g_2$ from the Jefferson Lab “Spin Physics Program”. A preliminary analysis [119] favored the $\delta$-expansion power counting [45], just like the MAID model does, cf. Figure 4 and Figure 5. Note that the $\delta$-expansion results in Refs. [43,45] are both $\mathcal{O}(p^4/\mathsf{\Delta })$. They differ by an improved error estimate and inclusion of the Coulomb coupling $g_C$[45]. The $ϵ$-expansion results in Refs. [56,59] are $\mathcal{O}\left({ϵ}^3\right)$ and $\mathcal{O}({ϵ}^3+p^4)$, respectively.

Similarly, we observe that the extensive set of empirical evaluations of the generalized forward spin polarizability ${\gamma }_0\left(Q^2\right)$ at $Q^2<0.3$ GeV${}^2$ agrees perfectly with the $\delta$-expansion prediction [45], but differs from the $ϵ$-expansion prediction [56,59], cf. Figure 10 and Figure 12 (upper panel). For the higher-order analogue ${\overline{\gamma }}_0\left(Q^2\right)$, shown in Figure 12 (lower panel), the situation is less obvious. Only the dispersive evaluations of ${\overline{\gamma }}_{0p}$ at the real-photon point, cf. Figure 6, are in slight disagreement with the $\mathcal{O}(p^4/\mathsf{\Delta })$ prediction [45], while conform with the $\mathcal{O}({ϵ}^3+p^4)$ prediction [59].

The most studied polarizabilities are the electric and magnetic dipole polarizabilities, for which the Particle Data Group publishes recommended values [106]. They are needed as input for calculations of the proton-structure effects in the muonic-hydrogen Lamb shift from two-photon exchange. Of particular importance is ${\beta }_{M1p}$. It enters the ${\overline{T}}_1(0,Q^2)$ subtraction function (30), which has to be modeled [120] or predicted within $\chi$PT [44,77,121] because it cannot be measured in experiment or reconstructed from the unpolarized proton structure function $f_1$ in the dispersive approach, cf. Equation (27a). Recently, ${\beta }_{M1p}$ has therefore been extracted from the linear polarization beam asymmetry,

$${\mathsf{\Sigma }}_3=\frac{\mathrm{d}{\sigma }_{\left|\right|}-\mathrm{d}{\sigma }_{\perp }}{\mathrm{d}{\sigma }_{\left|\right|}+\mathrm{d}{\sigma }_{\perp }},$$

(39)

measured for the proton by the A2 Collaboration [89] and LEGS [90]. Up to $\mathcal{O}({\nu }^2$), the beam asymmetry ${\mathsf{\Sigma }}_3$ provides access to ${\beta }_{M1}$ independent of ${\alpha }_{E1}$ [122]:

$$\begin{array}{c}\hfill {\overline{\mathsf{\Sigma }}}_3=-\frac{4M_N{\omega }_B^{2}cos{\theta }_B{sin}^2{\theta }_B}{{(1+{cos}^2{\theta }_B)}^2}{\alpha }^{-1}{\beta }_{M1}.\end{array}$$

(40)

Presently, the extraction of ${\beta }_{M1p}$ from ${\mathsf{\Sigma }}_3$ [89] is not competitive with the standard dispersive analyses of unpolarized CS cross sections. New high-precision measurements with significantly higher statistics should change this.

Analyses of CS data with fixed-t unsubtracted dispersion relations can be found in [76,123], with an update in [109]. Fixed-t subtracted dispersion relations are used in [94], and are applied together with a bootstrap-based fitting technique in the recent work in [110]. Unfortunately, the dispersive and $\chi$PT fits tend to disagree for certain polarizabilities, e.g., for ${\alpha }_{E1p}$ and ${\beta }_{M1p}$, cf. Figure 2 and Figure 3 (upper panels). The $\mathcal{O}(p^4/\mathsf{\Delta })$ B$\chi$PT prediction [41] and the B$\chi$PT fit [63] of the proton dipole polarizabilities, see Figure 2 and Figure 3 (upper panels), are in good agreement. A HB$\chi$PT fit, which also includes the lowest-order spin polarizabilities in Figure 9 and Figure 10, agrees with the B$\chi$PT results [41,63] except for ${\gamma }_{M1E2p}$. Recently, a model-independent PWA of proton RCS data below pion-production threshold has shown [91] that the differences between dispersive approaches and B$\chi$PT results are due to inconsistent experimental data subsets, rather than the “model-dependence” of the theoretical frameworks. In the summary figures for the dipole and lowest-order spin polarizabilities, cf. Figure 2, Figure 3, and Figure 9 (upper panels), I show the spread of results from their PWA fits of different data subsets [91]. Note that all fits use the data-driven evaluations of the Baldin and forward spin polarizability sum rules from in [15,101] as input. Their analysis shows that the difference of proton scalar polarizabilities is constrained to a rather broad interval [91]: ${\alpha }_{E1p}-{\beta }_{M1p}=(6.9\cdots 10.9)\times {10}^{-4}$fm${}^3$. In [83], the dipole dynamical polarizabilities entering the multipole decomposition of the scattering amplitudes were for the first time extracted from proton RCS data below pion-production threshold. At lowest order, they are related to the static dipole and dispersive polarizabilities, see Figure 8 (upper panel).

Both the partial-wave and the dispersive analysis in [83,91] come to the conclusion that quantity and quality of the CS data has to increase for improved extractions of the nucleon polarizabilities. A trend is going towards the measurement of beam asymmetries, such as ${\mathsf{\Sigma }}_3$, and double-polarization observables:

$$\begin{array}{ccc}\hfill {\mathsf{\Sigma }}_{2x}& =& \frac{\mathrm{d}{\sigma }_{+x}^R-\mathrm{d}{\sigma }_{+x}^L}{\mathrm{d}{\sigma }_{+x}^R+\mathrm{d}{\sigma }_{+x}^L},\hfill \end{array}$$

(41a)

$$\begin{array}{ccc}\hfill {\mathsf{\Sigma }}_{2z}& =& \frac{\mathrm{d}{\sigma }_{+z}^R-\mathrm{d}{\sigma }_{+z}^L}{\mathrm{d}{\sigma }_{+z}^R+\mathrm{d}{\sigma }_{+z}^L},\hfill \end{array}$$

(41b)

where $\mathrm{d}{\sigma }_{+x}^{R\left(L\right)}$ and $\mathrm{d}{\sigma }_{+z}^{R\left(L\right)}$ are the differential cross sections for right (left) circularly polarized photons scattering from a nucleon target polarized either in the transverse $+\widehat{x}$ direction or in the incident beam direction $+\widehat{z}$. Here, the advantage is that systematic uncertainties, e.g., variations in photon flux or uncertainties in target thickness, are canceling out. Combining double-polarization observable and beam-asymmetry measurements, one is sensitive to the lowest-order spin polarizabilities, see Figure 9. For the extraction of the polarizabilities from the MAMI data for ${\mathsf{\Sigma }}_{2x}$[86,87], ${\mathsf{\Sigma }}_{2z}$[88] and ${\mathsf{\Sigma }}_3$[89], as well as the older LEGS data for ${\mathsf{\Sigma }}_3$[90], one can use dispersive models [28,94,124] or $\chi$PT fits [40].

Besides experimental efforts, lattice QCD is making considerable progress. Most notably are the lattice QCD predictions for ${\beta }_{M1}$ with chiral extrapolation to physical pion mass [111,125], as well as the plentiful results for ${\alpha }_{E1n}$[112,114,115,116]. By now, even direct lattice evaluations of the unpolarized forward VVCS amplitudes became possible and lead to predictions of, e.g, the generalized Baldin sum rule and its fourth-order variant in the region of $Q^2\in \{2,10\}$ GeV${}^2$ [126,127,128].

In Figure 4 and Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, one can see updated results from the recent $\mathcal{O}(p^4/\mathsf{\Delta })$ B$\chi$PT prediction of unpolarized VVCS [44], related to ${\alpha }_L$ and $M_1^{\left(4\right)}$, and polarized VVCS [45], related to ${\delta }_{LT}$, ${\gamma }_0$, and ${\overline{\gamma }}_0$. The latter could be compared to new results from the Jefferson Lab “Spin Physics Program” for the proton spin structure functions $g_1$ and $g_2$, see for instance the E08-027 experiment [102] and the E97-110 experiment [129]. Note that the HB$\chi$PT predictions for $M_1^{\left(4\right)}$ and ${\alpha }_L$ shown in Figure 11 and Figure 13 were extracted from the VVCS amplitudes presented in Ref. [73], but are not quoted in the original work.

5. Conclusions and Outlook

The chiral EFT expansion for nucleon polarizabilities begins with inverse powers of pion mass and other light scales, such as the nucleon-$\mathsf{\Delta }$ mass difference. These inverse powers ($1/m_{\pi }$, $1/\mathsf{\Delta }$, etc.) along with the chiral logs constitute predictions of $\chi$PT. As such, the polarizabilities, and, in fact, the entire process of CS at low energies, provide a testing ground for $\chi$PT.

Moreover, the interpretation of low-energy CS data and the extraction of nucleon polarizabilities therefrom should rely on a systematic theoretical framework such as $\chi$PT. In what we have seen thus far, $\chi$PT is quite successful in the prediction of nucleon polarizabilities. It can as well be used to design “optimal” future experiments for improving the empirical determinations of nucleon polarizabilities [130].

An alternative to $\chi$PT, in the field of nucleon CS, is provided by models based on fixed-t dispersion relations [131,132]. The theoretical uncertainties of the dispersive approach are harder to understand, but, at least within the quoted uncertainties, the extracted values of polarizabilities are overall comparable to those found in $\chi$PT. However, a few discrepancies remain. For example, the tension in the value of the proton magnetic dipole polarizability still persists, cf. “Disp. Rel.” vs. $\chi$PT results in Figure 3 (upper panel). A model-independent PWA shows [91] that this discrepancy is likely to be caused by the experimental CS database, rather than the differences between the theoretical frameworks. With MAMI [133] and HIGS [134] experiments underway, the database will soon be greatly improved. It is worth mentioning that MAMI is also finalising a program to measure the CS double-polarization observables (${\mathsf{\Sigma }}_{2x}$, ${\mathsf{\Sigma }}_{2z}$) which will lead to an improved extraction of proton spin polarizabilities [86,87,88].

Even among the various $\chi$PT calculations there are significant discrepancies that need to be understood. The differences between the heavy-baryon (HB$\chi$PT) and the Lorentz-invariant covariant (B$\chi$PT) results are not difficult to track. However, differences among various B$\chi$PT calculations are more troublesome. A prominent example is the longitudinal-transverse polarizability of the proton (upper panel of Figure 4 and left panel of Figure 5), where the $\delta$- and $ϵ$-expansion B$\chi$PT calculations are different by about a factor of 2. This “${\delta }_{LT}$ puzzle” could soon receive an experimental resolution, when the long-promised data from Jefferson Lab “Spin Physics Program” [102,129,135] on the proton spin structure function $g_2$ will be published [119]. Besides the polarizabilities, the Gerasimov–Drell–Hearn sum rule for the neutron will be verified by the E97-110 experiment using a helium-3 target [136].

In the mean time, lattice QCD calculations of nucleon polarizabilities are advancing towards the physical pion mass. Until now, however, $\chi$PT has been used to extrapolate the lattice results to the physical mass [111,113]. A significant progress has recently been achieved in calculating the proton polarizabilities [111,114], and in direct calculations of the spin-independent forward VVCS [126,127,128].

In the next few years, one can expect a lot of progress in this field, mainly due to the upcoming data from MAMI, HIGS, and Jefferson Lab. New $\chi$PT and lattice QCD calculations will certainly continue to advance and will, hopefully, bring some clarity on the aforementioned discrepancies.