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Nucleon Polarizabilities and Compton Scattering as a Playground for Chiral Perturbation Theory

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.


Introduction
The name Chiral Perturbation Theory (χPT) was first introduced in the seminal works of Pagels [1], who used it to describe a systematic expansion in the pion mass m π , 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 χPT acquired its present meaning by Gasser and Leutwyler [3,4] in this, more powerful, connotation. Since then, χPT 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 χPT 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. And, each new order brings more and more free parameters -the low-energy constants (LECs). This is why the cases where χPT 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 α E1 and β M1 . These quantities describe the size of the electric and magnetic dipole moments induced by an external electric E or magnetic H field: In loosely bound systems, such as atoms and molecules, these polarizabilities are roughly given by the volume of the system. In the nucleon, the sum of dipole polarizabilities is of the order of 10 −3 fm 3 . This is much smaller than the volume of the nucleon (∼ 1 fm 3  polarizabilities. These are more difficult to visualise in a classical picture. Nonetheless, χPT provides robust predictions for these quantities 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 χPT in the single-baryon sector. But they are not just a testing ground for χPT. The lattice QCD studies of nucleon polarizabilities are also closing in on the physical pion mass, see This mini-review is by no means comprehensive. A more proper review can be found in Ref. [7], 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 χPT and beyond can be found in Ref. [8]. Other commendable reviews include: Guichon and Vanderhaeghen [9] or Fonvieille et al. [10] (VCS and generalized polarizabilities), Drechsel et al. [11] or Pasquini and Vanderhaeghen [12] (dispersion relations for CS), Pascalutsa et al. [13] (∆(1232) resonance), Phillips [14] (neutron polarizabilities), Grießhammer et al. [15] (χEFT and RCS experiments), Holstein and Scherer [16] (pion, kaon, nucleon polarizabilities), Geng [17] (BχPT), Pascalutsa [18] (dispersion relations), Deur et al. [19] (nucleon spin structure). A textbook introduction to χPT can be found in Ref. [20].
The paper is organized as follows. In Sections 2 and 3, I briefly describe the χPT framework and the CS formalism. In Section 4, I summarize recent χPT results for the nucleon polarizabilities and compare to empirical and lattice QCD evaluations.

Baryon Chiral Perturbation Theory
The low-energy processes involving a nucleon, such as πN scattering or CS off the nucleon, can be described by SU(2) baryon chiral perturbation theory (BχPT), which is the manifestly Lorentz-invariant variant of χPT in the single-baryon sector [4,21,22]. 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 ∆(1232) resonance -into the χPT framework. In Section 2.3, I will show how the LECs can be fit to experimental data and discuss the predictive power of χPT for CS. In Section 2.4, I introduce the heavy-baryon chiral perturbation theory (HBχPT) and point out how its predictions differ from BχPT for certain polarizabilities. For more details on BχPT for CS, I refer to the following series of calculations: RCS [23][24][25], VCS [26] and forward VVCS [27][28][29].

BχPT with pions and nucleons
Consider the basic version of SU(2) BχPT including only pion and nucleon fields [4]: scalar iso-vector π a (x) and spinor iso-doublet N (x). Expanding the EFT Lagrangian [4] to leading orders in pion derivatives, mass and fields, one finds (see, e.g., Ref. [30]): with the covariant derivatives: the photon vector field A µ (x), and the charges: Here, τ a are the Pauli matrices, γ 5 = iγ 0 γ 1 γ 2 γ 3 are the Dirac matrices, and all other parameters are introduced in Table 1.
The key ingredient for the development of χPT 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, Λ χSB ∼ 4π f π ≈ 1 GeV. Therefore, one expects that χPT provides a systematic description of the strong interaction at energies well below 1 GeV. Considering only pion and nucleon fields, the chiral order O(p n ) of a Feynman diagram with L loops, N π (N N ) pion (nucleon) propagators, and V k vertices from k-th order Lagrangians [e.g., k = 1: Γ µ γNN interaction from Eq. (2a), k = 2: Γ µ γππ interaction from Eq. (2b)] is defined as [4]: In the case of CS, the low-energy scale p also includes the photon energy ν 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 ππ scattering the limiting scale of the perturbative expansion is set by the σ(600) and ρ(775) mesons [31,32]. In the single-nucleon sector, the breakdown scale is set by the excitation energy of the first nucleon resonance, the ∆(1232) isobar. That is unless the ∆(1232) is included explicitly in the effective Lagrangian.

Inclusion of the ∆(1232) and Power Counting
The ∆(1232) resonance as the lightest nucleon excitation has an excitation energy which is in the same ballpark as the pion mass. In the following, it will be included as an explicit degree of freedom: vector-spinor iso-quartet ∆ µ (x). The relevant Lagrangians read [30,33,34]: with the covariant derivative: and the charge: Here, h.c. stands for the hermitian conjugate, γ µν = − i 2 µναβ γ α γ β γ 5 and γ µνα = −i µναβ γ β γ 5 are Dirac matrices with 0123 = 1, and T a (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 Eqs. (2) and (7) denote their order as reflected by the number of comprised small quantities: pion mass, momentum and factors of e. Inclusion of the ∆(1232) introduces the excitation energy ∆ as another small scale, which has to be considered when defining a power-counting for the perturbative χPT expansion.
There are two prominent counting schemes for χPT with explicit inclusion of the ∆(1232). For simplicity, they both deduce a single expansion parameter from the two involved small mass scales: = m π /Λ χSB and δ = ∆/Λ χSB . In the -expansion (small-scale expansion) it is assumed that ∼ δ [35], while in the δ-expansion one assumes ∼ δ 2 with δ [36]. In this way, the δ-expansion defines a hierarchy between the two mass scales. Consequently, it defines two regimes where the ∆(1232) contributions need to be counted differently: • low-energy region: p ∼ m π ; • resonance region: p ∼ ∆.
This makes sense since the ∆(1232) is expected to be suppressed at low energies and dominating in the resonance region. The chiral order O(p n δ ) of a Feynman diagram with N 1∆R (N 1∆I ) one-∆-reducible (one-∆-irreducible) propagators is in the δ-expansion defined as: where An extensive review on the electromagnetic excitation of the ∆(1232)-resonance with more details on the formulation of the extended χPT framework and the chiral expansion in the resonance region can be found in Ref. [13]. As we will see in Section 4, BχPT calculations based on the [37] and the δ [27,29] counting scheme give significantly different predictions for the longitudinal-transverse polarizability of the proton shown in Figure 11.

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 χPT to QCD as the fundamental theory of the strong interaction, the renormalized LECs need to be fitted to experimental or lattice data. 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 ) [29]: The next-to-leading-order effect of the ∆(1232) 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 ∼ ∆/M N ∼ 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 BχPT calculation of low-energy CS in the δ-expansion are f π , g A , h A , g M , g E and g C . 1 They are listed in Table 1 together with the experiments used to constrain their values. As one can see, BχPT has "predictive power" for CS up to and including O(p 4 /∆) because all relevant LECs are matched to processes other than CS. 2 This makes χPT the perfect tool to study the low-energy structure of the nucleon as encoded in CS and the associated polarizabilities. Starting from O(p 4 ), LECs need to be fitted to the CS process as well, for instance through the Baldin sum rule, as done in Refs. [15,28,[38][39][40].

Heavy-Baryon Expansion
The theory of HBχPT was first introduced in Ref. [41], and later applied to CS and polarizabilities [42], including also the effect of the ∆(1232) [15,[43][44][45][46][47][48]. The results of HBχPT can be recovered from the BχPT results by expanding in powers of the inverse nucleon mass. HBχPT calculations tend to fail in describing the Q 2 evolution of the generalized nucleon polarizabilities [28,29]. Also for the static polarizabilities the 1 Note that the g E and g C couplings of the N-to-∆ transition would be strictly speaking of higher order. 2 Note that O(p 4 /∆) corresponds to O(p 7/2 ), cf. Eq. (10) with p 1/2 ∼ ∆ or p ∼ m π . Table 1. Low-energy constants and other parameters and the orders at which they appear in the chiral expansion when employing the low-energy δ-expansion counting scheme.

Order in chiral expansion
χPT parameters Values Sources  [34] heavy-baryon expansion can give significantly different predictions. Consider for instance the nucleon dipole polarizabilities. The BχPT prediction (in units of 10 −4 fm 3 ) [25]: at the expense of violating the naturalness requirement, see also Ref. [15]. This can be seen from the dimensionless LECs associated to δα and δβ [45], g 117 = 18.82 ± 0.79 and g 118 = −6.05 ∓ 0.66, that should be of O(1) to be consistent with estimates from power counting. This problem is discussed at length in Refs. [24,49].

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(q ) and nucleons p(p ): In general kinematics (p 2 = p 2 = M 2 N , q 2 = q 2 ), the CS amplitude can be described by 18 independent tensor structures. For VCS one needs 12 independent tensor structures; for RCS one needs 6 independent tensor structures [53,54]. In the forward limit, this reduces to 4 independent tensor structures for virtual photons and 2 independent tensor structures for real photons.
Splitting into spin-independent (symmetric) and spin-dependent (antisymmetric) parts, the forward VVCS decomposes into the following four scalar amplitudes: with with ν the photon lab-frame energy and terms which vanish upon contraction with the photon polarization vectors omitted. For real photons, the following two scalar amplitudes survive: Constraints relating the different kinematic regimes (RCS, VCS and forward VVCS) are discussed in Refs. [55] and [56,57] 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 [36]: with the overcomplete set of 8 tensors: and the incoming (outgoing) photon polarization vector ε ( ) and Dirac spinor u ( ) . Alternatively, one can choose the non-covariant decomposition with the minimal set of 6 tensors: with the incoming (outgoing) Pauli spinor χ ( ) and the scalar complex amplitudes: A(s, t) = A 1 , · · · , A 6 (s, t), O ij = δ ij , n i n j , i ijk σ k , δ ij i klm σ k n l n m , i klm σ k (δ il n m n j − δ jl n i n m ), where n ( ) is the direction of the incoming (outgoing) photon. The scalar amplitudes A 1,...,8 are related to the scalar amplitudes A 1,...,6 in the following way [38]: where are the nucleon and photon energies in the Breit frame ( p = − p ), and s, t, u are the usual Mandelstam variables. According to the low-energy theorem of Low [58], Gell-Mann and Goldberger [59], 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.,Ā 1,...,6 ) reads as: with z = cos θ B = 1 + t/2ω 2 B . The coefficients are given in terms of static nucleon polarizabilities: electric dipole (α E1 ), magnetic dipole (β M1 ), quadrupole (α E2 , β M2 ), dispersive (α E1ν , β M1ν ), and lowest-order spin polarizabilities (γ E1E1 , γ M1M1 , γ E1M2 and γ M1E2 ), see Figures 2, 3, 4, 5 and 7, respectively. The latter combine into the forward (see Figure 8) and backward spin polarizabilities: Studying the forward RCS and VVCS is of advantage because of their accessibility through sum rules. Based on the general principles of analyticity, causality 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 [11]: , with ν el = Q 2 /2M N the elastic threshold. Note that the structure functions f 1 , f 2 , g 1 and g 2 are functions of the Bjorken variable x = ν el /ν and the photon virtuality Q 2 . They are related to the photoabsorption cross sections σ T , σ L , σ TT and σ LT measured in electroproduction, defined here with the photon flux factor K(ν, Q 2 ) = ν 2 + Q 2 [60].
Performing low-energy expansions of the relativistic CS amplitudes [11,56,61] 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 [62], allowing for a precise data-driven evaluation of the sum of electric and magnetic dipole polarizabilities [63]: It follows from the ν 2 term in the low-energy expansion of the RCS amplitude f (ν): The extension of the Baldin sum rule to finite momentum-transfers [11], 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 β M1 (Q 2 ), which for VCS is defined in Eq. (B2b) of Ref. [55], and for forward VVCS could be defined either by generalizing the non-Born part of the subtraction function but is usually understood as part of the generalized Baldin sum rule (29). A recent measurement of the generalized α E1 (Q 2 ) and β M1 (Q 2 ) polarizabilities from VCS by the A1 Collaboration can be found in Ref. [64]. The generalized fourth-order Baldin sum rule is defined as: It differs from the Baldin sum rule (29) by the energy weighting of the cross section 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 ν 4 term in Eq. (28) [65,66]: see Figure 6. Similarly, the generalized forward spin polarizability is related to the helicity-difference cross section as [11]: while the fifth-order generalized forward spin polarizability sum rule is given by: see Figures 8 and 9, respectively. The polarizabilities involving longitudinal photon polarizations are absent from RCS. They are given as sum rule integrals over the longitudinal cross section, e.g., the longitudinal polarizability [27]: and the longitudinal-transverse cross section, e.g., the longitudinal-transverse polarizability [11]: see Figures 10 and 11, respectively.
The O(p 4 /∆) BχPT prediction [25] and the BχPT fit [39] of the proton dipole polarizabilities, see Figures 2 and 3, are in good agreement. A HBχPT fit, which also includes the lowest-order spin polarizabilities in Figures 7 and 8, agrees with the BχPT results [25,39] except for γ M1E2 . It is not possible to comment on the other HBχPT predictions for the dispersive, quadrupole and longitudinal polarizabilities [48,89] since they have no error bars. 3 The most studied polarizabilities are the electric and magnetic dipole polarizabilities, for which the Particle Data Group publishes recommended values [67]. They are needed as input for calculations of the proton-structure effects from two-photon exchange in the muonic-hydrogen Lamb shift. Of particular importance is β M1p . It enters the T 1 (0, Q 2 ) subtraction function (30), which has to be modeled [91] or predicted within χPT [28,55,92] because it cannot be measured in experiment or reconstructed from the unpolarized proton structure function f 1 in the dispersive approach. Recently, β M1p has therefore been extracted from the linear polarization beam asymmetry, measured for the proton by the A2 Collaboration [71] and LEGS [93]. Up to O(ν 2 ), the beam asymmetry Σ 3 provides access to β M1 independent of α E1 [94]: Presently, the extraction of β M1 from Σ 3 [71] 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 Refs. [54,95], with an update in Ref. [75]. Fixed-t subtracted dispersion relations are used in Ref. [66], and are applied together with a bootstrap-based fitting technique in the recent Ref. [76]. Unfortunately, the dispersive and χPT fits tend to disagree for certain polarizabilities, e.g., for α E1p and β M1p , cf. Figures 2 and 3. Recently, an independent partial-wave analysis of proton RCS data below pion-production threshold has 3 Note that the predictions for M    shown that the differences between dispersive approaches and BχPT extractions are due to inconsistent experimental data subsets and not a sign of "model-dependence" [81]. From this work, the result (Fit 0) of fitting the complete database with the dipole and lowest-order spin polarizabilities, cf. Figure 7, as free parameters and the customary constraints from the data-driven evaluations of the Baldin and forward spin polarizability sum rules [63,78] is shown in the summary figures. In Ref. [77], 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 5. Both Ref. [81] and Ref. [77] conclude that quantity and quality of the data has to increase for improved extractions of the nucleon polarizabilities. A trend is going towards the measurement of beam asymmetries, such as Σ 3 , and double-polarization observables: where dσ R(L) +x and dσ R(L) +z are the differential cross sections for right (left) circularly polarized photons scattering from a nucleon target polarized either in the transverse +x direction or in the incident beam direction +ẑ. 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 7. For the extraction of the polarizabilities from the MAMI data for Σ 2x [70,72], Σ 2z [69] and Σ 3 [71], as well as the older LEGS data for Σ 3 [93], one can use dispersive models [11,66,96] or χPT fits [24]. Besides experimental efforts, lattice QCD is making considerable progress. Most notably are the lattice QCD predictions for β M1 with chiral extrapolation to physical pion mass [82,97]. As well as the plentiful results for α E1n [83,[85][86][87]. By now, even direct lattice evaluations of the unpolarized forward VVCS amplitude T 1 became possible and lead to predictions of the generalized Baldin sum rule and its fourth-order variant in the region of Q 2 ∈ {2, 10} GeV 2 [98,99].

Summary and Conclusions
Chiral perturbation theory has predictive power for Compton scattering and the nucleon polarizabilities. Here, the chiral perturbation theory predictions have been compared to empirical determinations and lattice QCD predictions. While most predictions agree with the experimental values, a few rather small discrepancies remain. To pin down the nucleon polarizabilities and resolve the present discrepancies, more high-precision data are needed. Here, a trend is going towards measurements of beam asymmetries or double-polarization observables with improved systematic uncertainties. Chiral perturbation theory also provides a framework to fit low-energy Compton scattering data, and is used to design "optimal experiments" [102].
Knowledge of the proton polarizabilities is important as input for the proton-structure corrections to the muonic-hydrogen spectrum. These are not only relevant in the context of the proton-radius puzzle [103,104], but also for the planned measurements of the muonic-hydrogen ground-state hyperfine splitting [105][106][107].
Funding: Financial support from the Swiss National Science Foundation is gratefully acknowledged.