The chiral structure of the neutron as revealed in electron and photon scattering

In the limit of massless up and down quarks the QCD Lagrangian has an $SU(2)_{L}\times SU(2)_{R}$ symmetry. We know from lattice simulations that the vacuum of QCD in this limit does not share this symmetry of the Lagrangian, with the axial combination of $SU(2)$’s being spontaneously broken [1]. This confirms the key elements of the picture postulated by Nambu in talks and papers of 1960–1, work for which he was awarded the 2008 Nobel Prize in Physics [2, 3]. An immediate consequence of the spontaneous breaking of chiral symmetry is that there will be three Goldstone bosons: the $\pi^{+}$, $\pi^{0}$, and $\pi^{-}$. Since the up and down quarks are not exactly massless these pions are also not massless, but, to the extent that $m_{u}$ and $m_{d}$ are small compared to $\Lambda_{QCD}$, the pions will be much lighter than all other hadronic degrees of freedom.

This picture is systematized in an effective field theory (EFT) called ‘chiral perturbation theory’ ($\chi$PT) [4, 5, 6, 7]. Chiral perturbation theory encodes the $SU(2)_{L}\times SU(2)_{R}$ symmetry of QCD, and the pattern of its breaking, in a Lagrangian containing nucleon and pion fields. It includes all possible interactions consistent with the symmetry. These are organized in a derivative expansion, together with an expansion in powers of $m_{q}$ for the symmetry-breaking operators. The EFT can be used to calculate both tree and loop diagrams, which leads to a perturbation expansion for scattering amplitudes in powers of

$$P\equiv\frac{p,m_{\pi}}{\Lambda_{\chi}}.$$

(1)

Here $\Lambda_{\chi}\sim 4\pi f_{\pi}\approx 1200$ MeV (with $f_{\pi}\approx 93$ MeV the pion-decay constant) is the scale that emerges from loop calculations [8] as controlling the convergence of the expansion for meson-meson interactions. If we consider threshold processes, and so $p=0$, this is an expansion in powers of $m_{\pi}$, and so can be considered an expansion around the chiral limit of QCD, where $m_{\pi}=0$. In the meson sector the theory is mature, and calculations are routinely performed to two loops. For $\pi\pi$ scattering the results are extremely accurate, reaching the sub-1% level. (For a recent review see [9].)

Adding baryons to the theory as static sources facilitates a power counting in which each diagram has a definite order in the $P$ expansion [10]. The effects of baryon recoil constitute corrections to the static-source picture, and they can be systematically included in the Lagrangian via an expansion in powers of $p/M$ [10, 11]. The result is known as “heavy-baryon chiral perturbation theory” (HB$\chi$PT). HB$\chi$PT for nucleons and pions is summarized in Section 2. There has been a recent flurry of activity devoted to baryon $\chi$PT calculations that do not invoke this $p/M$ expansion [12, 13], but here I will be mainly using the heavy-baryon variant of the theory to describe neutron structure. Electromagnetic interactions are easily included in this theory, via minimal substitution and the addition of terms that are gauge invariant by themselves (e.g. an operator corresponding to magnetic fields coupling to the neutron anomalous magnetic moment). In the usual $\chi$PT counting the electron charge $e$ then counts as one power of $P$. For comprehensive reviews of $\chi$PT in the baryon sector (both heavy-baryon and “relativistic” formulations) see Refs. [14, 15, 16, 17].

The standard $\chi$PT Lagrangian includes only nucleons and pions, and so its breakdown scale is not $4\pi f_{\pi}$. Instead it is set by the lightest hadronic degree of freedom not explicitly included in the theory, which in most reactions is the Delta(1232). So, unless the $\chi$PT Lagrangian is extended to include Delta degrees of freedom, the scale in the denominator in Eq. (1) is not $\Lambda_{\chi}$, but is instead $M_{\Delta}-M_{N}\approx 290$ MeV. While there has been much progress in this direction [10, 18, 19, 20, 21, 22], for most of this article I will regard the Delta-nucleon mass difference as a high-energy scale.

My focus here is thus on the chiral structure of the neutron, as revealed by electromagnetic probes. I therefore report on computations of electron and photon scattering, with the main emphasis on results for these reactions obtained in HB$\chi$PT. I will be particularly concerned with the interplay between pion-cloud mechanisms and the role of higher-energy excitations. The ability to vary photon energy, $\omega$, in the case of Compton scattering, and three-momentum transfer, $\bf q$, in the case of electron scattering, allows us to map out this interplay over a significant kinematic domain. It also provides interesting data on the breakdown of $\chi$PT, since comparison of its predictions with experimental data over a range of $\omega$ and $|{\bf q}|$ can reveal where the theory is failing, and what additional degrees of freedom are needed in order to extend its radius of convergence.

But, practically speaking, these aspects of neutron structure cannot be examined separately from the chiral structure of nuclei. Neutrons cannot be collected in sufficient numbers to make electromagnetic scattering reactions on free neutrons an experimental reality. Hence everything we know about neutron electromagnetic structure is inferred from more complex processes, e.g. experiments on light nuclei. In order to extract information on neutron structure from such experiments an understanding of the multi-nucleon effects that are present in the data is also necessary.

Developments over the past 15 years have also allowed $\chi$PT to provide a systematic approach to multi-nucleon problems. Weinberg’s seminal papers on $\chi$PT expansions for the nuclear potential [23, 24] and the operators governing the interaction of external probes with the nucleus [25] facilitate a parallel and consistent $\chi$PT treatment of neutron structure and electromagnetic reactions in two- (and three- and …) nucleon systems. In Section 3 I will give a brief explanation of how $\chi$PT enables the consistent treatment of the operators governing processes in single- and multi-nucleon systems. Electron-neutron scattering and $\gamma n\rightarrow\gamma n$ will then be discussed in Sections 4 and 5 respectively. In this article I only have space to discuss those two processes in detail, so in the final section, Section 6, I briefly mention some other reactions that probe the same physics: virtual Compton scattering, as well as pion photo- and electro-production near its threshold. I also offer some concluding thoughts on the neutron’s chiral structure.

In this section I provide a brief summary of the aspects of chiral perturbation theory ($\chi$PT) that are relevant for the subsequent presentation. The discussion here follows the notation and conventions of Ref. [14]. In particular, I will use the ‘heavy-baryon’ formulation of $\chi$PT, in which a $p/M$ expansion is made alongside the expansion in powers of $p/\Lambda_{\chi}$. This expansion is not essential to the efficacy of $\chi$PT, but since $M\sim\Lambda_{\chi}$ it does not have a deleterious effect on its convergence—apart from a few specific cases that involve proximity to kinematic thresholds.

$\chi$PT is an effective field theory of QCD, that encodes the symmetries of the underlying theory which are pertinent to the regime where energies are of the order of Goldstone boson (here, pion) masses. As such the chiral Lagrangian is formulated in terms of objects with straightforward transformation properties under the chiral group $SU(2)_{L}\times SU(2)_{R}$. For details on the transformation properties of the building blocks of the chiral Lagrangian the reader is referred to Refs. [14, 26]. Here we choose a particular representation for the pion field, and summarize the building blocks as:

$$u=\exp\left(i\frac{\tau\cdot{\bf\pi}}{2f_{\pi}}\right);\quad U=u^{2};$$

(2)

where ${\bf\pi}$ is the pion iso-triplet. From these we construct the chiral covariant derivative of the nucleon field, $D_{\mu}$, and the axial-vector object with one derivative, $u_{\mu}$:

	$\displaystyle D_{\mu}\psi$	$\displaystyle=$	$\displaystyle\partial_{\mu}\psi+\Gamma_{\mu}\psi;$		(3)
	$\displaystyle\Gamma_{\mu}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left[u^{\dagger}\partial_{\mu}u+u\partial_{\mu}u^{\dagger}-ieA_{\mu}\left(u^{\dagger}{\cal Z}u+u{\cal Z}u^{\dagger}\right)\right];$		(4)
	$\displaystyle u_{\mu}$	$\displaystyle=$	$\displaystyle i\left[u^{\dagger}\partial_{\mu}u-u\partial_{\mu}u^{\dagger}-ieA_{\mu}\left(u^{\dagger}{\cal Z}u-u{\cal Z}u^{\dagger}\right)\right],$		(5)

with ${\cal Z}={1\over 2}(1+\tau_{3})$. Here we have specialized to the external-field case of interest for this paper: $A_{\mu}$ is an (external) photon field and there is no external axial field. Note that the vector field $\Gamma_{\mu}$ contains only even powers of the pion field, and $u_{\mu}$ contains only odd powers of it. The nucleon field $\psi$ is the usual iso-double. Both components of $\psi$ are Dirac fields.

The leading-order Lagrangian contains all possible terms with zero and one derivative and is:

$${\cal L}_{\pi N}^{(1)}=\bar{\psi}\left(i\not\!\!D-M+\frac{g_{A}}{2}\gamma^{\mu}u_{\mu}\gamma_{5}\right)\psi.$$

(6)

Strictly speaking $M$ and $g_{A}$ here are the mass and axial coupling of the nucleon in the chiral limit, although the distinction between their values at zero quark mass and the physical ones will not be important for any of the discussions here. This Lagrangian, supplemented by the standard leading-order meson-meson one:

$${\cal L}_{\pi\pi}^{(2)}=\frac{f_{\pi}^{2}}{4}[{\rm Tr}(D_{\mu}U^{\dagger}D^{\mu}U)-2B{\rm Tr}({\cal M}(U+U^{\dagger}))]$$

(7)

with $D_{\mu}U=\partial_{\mu}U-ieA_{\mu}[\frac{1}{2}\tau_{3},U]$, is all one needs to calculate the leading-order loop effects in chiral perturbation theory.

Such calculations are rendered more straightforward, and more transparent, if one removes the large energy scale associated with the nucleon mass. This energy is not available for particle production, because of baryon-number conservation. Hence it plays no role in the single-nucleon sector, and can be eliminated from consideration by what amounts to a different choice of the zero of energy. This is done explicitly by use of a field redefinition. As first pointed out by Jenkins and Manohar [10], the field $\psi$ can be re-expressed in terms of a “heavy-baryon” field $\psi_{HB}$, that has the straight-line propagation of a nucleon of mass $M$ removed from its dynamics:

$$\psi(x)\equiv\psi_{NR}(x)\exp(-iMv\cdot x).$$

(8)

If the nucleon is at rest then the choice $v=(1,\bf{0})$ is appropriate, although physical results should be independent of the choice of $v$. We also define the analogue of the non-relativistic spin

$$S_{\mu}=\frac{i}{2}\gamma_{5}\sigma_{\mu\nu}v^{\nu}.$$

(9)

If we choose $v=(1,\bf{0})$ and consider the case of four space-time dimensions then the spatial components of $S$ obey the algebra:

$$\{S^{i},S^{j}\}=\frac{1}{2}\delta_{ij};\quad[S^{i},S^{j}]=i\epsilon_{ijk}S^{k}.$$

(10)

Thus a representation for them is provided by the assignment $S^{i}=\Sigma^{i}/2$, with $\Sigma^{i}$ the block-diagonal Dirac matrix constructed out of the Pauli matrices $\sigma^{i}$.

Here we focus on the leading pieces of the heavy-baryon Lagrangian, i.e. the dominant terms that emerge from Eq. (6) in a $p/M$ expansion (with $p$ the three-momentum of the nucleon state). The key observation is that the operator structures in Eq. (6) do not couple the upper and lower components of $\psi_{B}$ to one another. Hence, the leading behavior of all Dirac bilinears in the $p/M$ expansion can be obtained by assuming that $\psi_{B}=H$ with $H$ obeying $\mbox{$\not\!v$}H=H$, i.e. assuming $\psi_{B}$ contains only two degrees of freedom. We identify these as the degrees of freedom associated with the usual non-relativistic spin operator.

When written in terms of bilinears of $H$, the operator structures in Eq. (6) involve only the four-vectors $v$ and $S$, since:

	$\displaystyle\bar{H}\gamma_{\mu}H$	$\displaystyle=$	$\displaystyle[v_{\mu}]\bar{H}H$		(11)
	$\displaystyle\bar{H}\gamma_{\mu}\gamma_{5}H$	$\displaystyle=$	$\displaystyle 2\bar{H}S_{\mu}H.$		(12)

These identities, when applied to the version of Eq. (6) with Eq. (8) inserted into it, produce the standard leading-order heavy-baryon Lagrangian:

$${\cal L}_{\pi N}^{(1)}=\bar{H}\left[i(v\cdot D)+g_{A}u_{\mu}S^{\mu}\right]H.$$

(13)

The structure of the propagator for the $H$ field can be understood by writing the $\psi$ field’s total four-momentum, $k$, as $k=Mv+p$. Then, in terms of the small residual (“off-shell”) momentum $p$, the Dirac propagator may be written:

$$S=\frac{i}{\not\!k-M}=i\frac{M\not\!v+\not\!p+M}{2Mv\cdot p+p^{2}}=\frac{i}{v\cdot p}\frac{1+\not\!v}{2}+\ldots.$$

(14)

The first term is this expansion is then the propagator for the $H$ field. In a typical $\chi$PT loop graph for a single-baryon process the baryon emits and absorbs a pion, and so it is easy to see that the loop graph will be dominated by contributions from momenta for which all four components of $p$ are of order $m_{\pi}$. The $\ldots$ terms in Eq. (14) are then suppressed by $m_{\pi}/M$. Exceptions to this rule occur in the vicinity of kinematic thresholds, but, as long as such thresholds are not nearby, the leading effects of the loop graphs in $\chi$PT can be obtained by replacing the full baryon propagator by the heavy-baryon one (14).

The corrections of higher order in $1/M$ can be systematically derived by writing

$$\psi_{B}=H+h$$

(15)

with

$$\not\!vH=H;\qquad\not\!vh=-h,$$

(16)

and then calculating the Lagrangian for the field $H$ that results from eliminating the degrees of freedom associated with $h$. This treatment of all $1/M$ corrections to the leading-order heavy-baryon Lagrangian (13) has been carried out by Bernard, Kaiser, Kambor, and Meißner to $O(P^{2})$ [11], Fettes, Meißner and Steininger to $O(P^{3})$ [27], and Fettes, Meißner, Mojžiš, and Steininger to $O(P^{4})$ [28]. At fourth order a large number of operators in the non-relativistic theory of which (13) is the leading Lagrangian have coefficients that are fixed by the requirement that S-matrix elements at low energies agree with the S-matrix elements of the relativistic theory.

Such considerations regarding the low-energy consequences of Lorentz invariance generate the first three terms in the second-order Lagrangian (for further details the reader is referred to Ref. [14]). But, since this is an effective theory, the second-order Lagrangian contains all operators that are consistent with the symmetries and have two derivatives or one power of the quark-mass matrix (since $m_{q}\sim m_{\pi}^{2}$ and $p\sim m_{\pi}$). There are six such independent operators in the isospin-symmetric limit $m_{u}=m_{d}$. The complete second-order Lagrangian is then [14]:

	$\displaystyle{\cal L}_{\pi N}^{(2)}$	$\displaystyle=$	$\displaystyle\bar{H}\left\{\frac{1}{2M}[(v\cdot D)^{2}-D\cdot D)]-\frac{ig_{A}}{2M}\{S\cdot D,v\cdot u\}+c_{1}{\rm Tr}\chi_{+}\right.$		(17)
			$\displaystyle+\left(c_{2}-\frac{g_{A}^{2}}{8M}\right)(v\cdot u)^{2}+c_{3}u\cdot u+\left(c_{4}+\frac{1}{4M}\right)[S_{\mu},S_{\nu}]u^{\mu}u^{\nu}$
			$\displaystyle-\left.\frac{i}{4M}[S_{\mu},S_{\nu}]\left[(1+\kappa_{v})f_{\mu\nu}^{+}+\frac{1}{2}(\kappa_{s}-\kappa_{v}){\rm Tr}(f_{\mu\nu}^{+})\right]\right\}H,$

with

$$f_{\mu\nu}^{+}=eF_{\mu\nu}\left(u^{\dagger}{\cal Z}u+u{\cal Z}u^{\dagger}\right),$$

(18)

the electromagnetic-field-strength tensor (containing both isovector and isoscalar parts) that transforms in the appropriate way under the chiral group. In this formulation of hadron dynamics the values of two of the low-energy constants (LECs) that appear in (17) have already been set to the measured isoscalar and isovector nucleon magnetic moments, $\kappa_{s}$ and $\kappa_{v}$ ¹¹1Strictly speaking, it is the chiral-limit magnetic moments that enter here, but the difference is a fourth-order effect.. The other LECs, $c_{1}$–$c_{4}$, encode the effects of more massive baryon and meson resonances ($\Delta(1232)$, Roper, $\rho$-meson, …) on low-energy dynamics.

The terms listed in Eqs. (7), (13), and (17) are all that is required for computation of complete one-loop effects in HB$\chi$PT. These loops are divergent, but the degree of divergence is mandated by the power of momentum carried by the vertices, and the behavior of the heavy-baryon and meson propagators. Dimensional analysis, combined with the usual topological identities for Feynman graphs, shows that a $\chi$PT graph with $L$ loops, $N_{d}^{\pi\pi}$ vertices from the $\pi\pi$ Lagrangian of dimension $d$, and $N_{d}^{\pi N}$ vertices from the $\pi$N Lagrangian of dimension $d$ has a total degree of divergence:

$$D=2L+2-A+\sum_{d}(d-2)N_{d}^{\pi\pi}+\sum_{d}(d-1)N_{d}^{\pi N},$$

(19)

with $A$ the number of nucleons.

This calculation of the degree of divergence of the graph is important, because, after renormalization is performed, the finite part of the graph must scale as $(p,m_{\pi})^{D}$. Therefore, provided $p\sim m_{\pi}$, the computation of $D$ tells us the order in $\chi$PT at which the graph contributes. If renormalization is performed using dimensional regularization and a mass-independent subtraction scheme then the effects of any particular HB$\chi$PT graph occur at one and only one chiral order $D$. This is an advantage of the heavy-baryon formulation: each graph scales precisely as $P^{D}$, and so has a definite order in the chiral counting. Specifically, Eq. (19) says that one-loop graphs with vertices from ${\cal L}_{\pi N}^{(1)}$ give pieces of the quantum-mechanical amplitude with $D=3$, while graphs with one insertion from ${\cal L}_{\pi N}^{(2)}$ give effects with $D=4$. Furthermore, since ${\cal L}_{\pi\pi}$ starts at $d=2$, and ${\cal L}_{\pi N}$ at $d=1$, the lowest order at which a two-loop graph can contribute is $D=5$. Thus one-loop graphs with vertices from ${\cal L}_{\pi N}^{(1)}$ and either zero or one insertion(s) from ${\cal L}_{\pi N}^{(2)}$—together with appropriate tree-level contributions—give the complete $\chi$PT result up to $O(P^{4})$.

Of course, the divergent loops at $O(P^{3})$ and $O(P^{4})$ need to be renormalized, and this is done with counterterms that occur in ${\cal L}_{\pi N}^{(3)}$ and ${\cal L}_{\pi N}^{(4)}$. Much effort has gone into finding a complete, minimal Lagrangian at these orders (see, e.g. Refs. [28, 29]). I will not attempt to do justice to that effort here, since only a few of the many contact interactions that occur in ${\cal L}_{\pi N}^{(3)}$ and ${\cal L}_{\pi N}^{(4)}$ enter the results presented below. I want to stress, though, that it is through such counterterms that hadronic states with excitation energies $>m_{\pi}$ affect $\chi$PT’s predictions. These states therefore produce effects that are slowly varying with respect to the external kinematic parameter $|{\bf q}|$ or $\omega$, while the rapid variation of measurable quantities with $|{\bf q}|$ and $\omega$ is provided by pion-cloud physics. This interplay of the chiral physics with the higher-energy hadron dynamics is captured in $\chi$PT, without the need for any assumptions about the details of that dynamics.

The previous section discussed chiral perturbation theory for the single-nucleon system. In subsequent sections we will examine the predictions of this theory for photon and electron scattering on the neutron. However,“experimental” data on these reactions can only be obtained using nuclear targets. In analyzing these data it is beneficial to treat the interactions that bind the nucleus within the same framework as that used to describe the chiral structure of the nucleon itself. This can be done in chiral perturbation theory, which not only provides a means to compute the low-energy interaction of photons and pions with nucleons, but also is used to calculate the forces between nucleons. This is the framework that will be used, e.g. in Section 5, to compute photon interactions with multi-nucleon systems, and we provide a brief summary of it here.

The first attempt in this direction was due to Weinberg [23, 24]. Weinberg pointed out that HB$\chi$PT could not be employed directly to compute the nucleon-nucleon amplitude, because a nucleon in a typical nuclear bound state has

$$p_{0}\sim{\bf p}^{2}/M;\quad|{\bf p}|\sim m_{\pi}.$$

(20)

Thus not all components of the (residual) nucleon four-momentum $p$ are order $m_{\pi}$, in contrast to the case inside a typical HB$\chi$PT loop graph. Because of the scaling (20) a different organization of the Lagrangian (13) plus (17) is required when treating NN interactions. The single-nucleon propagator becomes:

$$S(p)=\frac{i}{v\cdot p-\frac{(v\cdot p)^{2}-p^{2}}{2M}}\stackrel{{\scriptstyle v=(1,{\bf 0})}}{{\longrightarrow}}\frac{i}{p^{0}-\frac{{\bf p}^{2}}{2M}},$$

(21)

i.e. the usual non-relativistic propagator for a free particle of mass $M$. Weinberg further observed that in the regime (20) this non-relativistic propagator scales as $M/m_{\pi}^{2}$, which invalidates the degree-of-divergence formula (19).

However, if we consider only diagrams for $NN\rightarrow NN$ that do not contain an intermediate two-nucleon state—the so-called ‘two-particle irreducible’ (2PI) graphs—then all single-nucleon propagators have $p_{0}\sim m_{\pi}$, and Eq. (19) can be used. The goal thus becoomes to compute the 2PI NN amplitude up to a fixed chiral dimension $D$. In order to do this we must supplement ${\cal L}_{\pi N}$ and ${\cal L}_{\pi\pi}$ by a nucleon-nucleon Lagrangian that contains the effects of short-distance ($r\ll 1/m_{\pi}$) physics in the two-nucleon system. This physics appears as a string of NN contact interactions of increasing mass dimension, each of which has an LEC associated with it. We then identify the NN potential $V$ at order $D$ as the sum of all 2PI $NN$ graphs—loops plus contacts—up to that order. (Technically an extra step is desirable here, with the 2PI amplitude transformed into an energy-independent potential using e.g. the Okubo formalism [30].)

The full NN amplitude is then reconstructed from its 2PI parts via:

$$T({\bf p}^{\prime},{\bf p};E)=V({\bf p}^{\prime},{\bf p})+\int\frac{d^{3}p^{\prime\prime}}{(2\pi)^{3}}V({\bf p}^{\prime},{\bf p}^{\prime\prime})\frac{1}{E^{+}-{\bf p^{\prime\prime}}^{2}/M}T({\bf p}^{\prime\prime},{\bf p};E),$$

(22)

where an integral over $p_{0}$ has been performed to combine the two single-nucleon propagators into the propagator of an NN state of energy $E$. Since Eq. (22) is just the Lippmann-Schwinger equation, which is equivalent to the Schrödinger equation, we now have a non-relativistic quantum mechanics with the NN potential $V$. The resulting approach has been dubbed “chiral effective theory” ($\chi$ET) since it is a quantum mechanics at fixed particle number, rather than a quantum field theory. Within $\chi$ET not only $V$, but also quantum-mechanical operators for the interaction of external probes (pions, electrons, photons, …) can be computed up to a given chiral dimension $D$. If this is done to the same order for current operators and for $V$ then the relevant Ward identities are automatically satisfied. But, more generally, this $\chi$PT organization of quantum-mechanical operators is a systematically improvable way to compute few-nucleon structure and reactions at energies of order $m_{\pi}$.

In Ref. [31] the potential $V$ was computed at the one-loop level using ${\cal L}_{\pi N}^{(1)}$ and up to one insertion from ${\cal L}_{\pi N}^{(2)}$—reproducing the earlier result of Ref. [32]. The formula (19) tells us that when this is done we have the complete result for $V$ up to $O(P^{3})$. Representative diagrams at $O(P^{0})$ and $O(P^{2})$ are shown in Fig. 1. In Ref. [31] the potential was then used (in Born approximation) to obtain phase shifts in the higher partial waves of NN scattering. Because of the centrifugal barrier, these partial waves are sensitive mainly to the long-distance part of the force. A good description of a number of partial waves with $l\geq 2$ was found, with the relative contributions of the different chiral orders roughly in agreement with predictions of the chiral expansion. This suggests that $\chi$PT provides a useful organizing principle for the $r\sim 1/m_{\pi}$ part of the NN potential.

This is a key finding, since it opens the way for a treatment of nuclear physics with quantifiable uncertainties. If phase shifts can be obtained up to a fixed order $n$ in the $\chi$PT expansion parameter $P$, then that suggests that the remaining, omitted physics can impact the phase shift only at a level $\sim P^{n+1}$. But the crucial question of how to deal with low partial waves, where the NN interaction is non-perturbative, and in principle the nucleons probe arbitrarily short distances, was not tackled in Ref. [31].

In the seminal paper of Ordóñez et al. [32] the NN potential was computed up to $O(P^{3})$ and Eq. (22) solved for phase shifts in a number of NN partial waves. (See also the improved $O(P^{3})$ calculation of Ref. [33].) Such a non-perturbative treatment of potentials derived from $\chi$PT has also been advocated in Ref. [34, 35, 36, 37]. It has the advantage that the long-distance parts of $V$, and hence of $T$, are consistent with the pattern of chiral-symmetry breaking in QCD. It also means that for $r\sim 1/m_{\pi}$ ($\equiv|{\bf p}|\sim m_{\pi}$) the $\chi$PT counting provides a hierarchy of mechanisms within $V$: the leading-order one-pion exchange is more important than the $O(P^{2})$ two-pion exchange which in turn is more important than three-pion exchange, and so forth.

In practice, in order to solve Eq. (22) with the potentials derived from $\chi$PT one must introduce a cutoff, $\Lambda$, on the intermediate states in the integral equation, because the $\chi$PT potentials grow with the momenta ${\bf p}$ and ${\bf p}^{\prime}$. The contact interactions in ${\cal L}_{NN}$ should then absorb the dependence of the effective theory’s predictions on $\Lambda$ in all low-energy observables, or otherwise we conclude that $\chi$ET is unable to give reliable predictions. In Refs. [32, 33, 38, 39, 40] $V$ was computed to a fixed order, and then the NN LECs that appear in $V$ were fitted to NN data for a range of cutoffs between 500 and 800 MeV. The resulting predictions—especially the ones obtained with the $O(P^{4})$ potential derived in Refs. [39, 40]—contain very little residual cutoff dependence in this range of $\Lambda$’s, and describe NN data with considerable accuracy.

However, several recent papers showed that $\chi$ET—at least as presently formulated—does not yield stable predictions once cutoffs larger than $m_{\rho}$ ($\sim 800$ MeV) are considered [41, 42, 43, 44, 45, 46, 47, 48]. They argue that the theory described above is not properly renormalized, i.e. the impact of short-distance physics on the results is not under control. Thus the $\chi$ET described here cannot be used over a wide range of cutoffs. Meanwhile, in Ref. [49] it was argued that the cutoff $\Lambda$ should be varied only in the vicinity of $m_{\rho}$ in order to obtain an indication of the impact of omitted short-distance physics on the theory’s predictions. Since the short-distance physics of the effective theory for $p\gg m_{\rho}$ is (presumably) completely different to the short-distance physics of QCD itself, considering larger cutoffs that allow momentum integrals to probe $p>m_{\rho}$ does not yield any additional information regarding the true impact of short-distance physics on observables. Furthermore, Ref. [49] followed Ref. [34] in arguing that working with momenta in the LSE which are demonstrably within the domain of validity of $\chi$PT has the advantage that the $\chi$PT counting must then apply for all parts of $V$: including the contact operators. This justifies $\chi$ET as a systematic theory, but at the cost of limiting its use to cutoffs $m_{\pi}\ll\Lambda\ll\Lambda_{\chi{\rm SB}}$. Calculations of NN scattering using $\Lambda$’s in this window have been performed in Refs. [32, 33, 38, 39, 40] with, as noted above, considerable phenomenological success.

Consistent three-nucleon and four-nucleon forces have also been derived in $\chi$ET [50, 51, 52]. The three-nucleon force enters first at $O(P^{3})$, and the four-nucleon force at $O(P^{4})$. This provides an EFT-based explanation for the observed hierarchy of nuclear forces: two-nucleon forces provide the bulk of nuclear properties, with three-nucleon (3N) forces accounting for small but crucial corrections, and four-nucleon (4N) forces almost negligible. 3N and 4N forces have been implemented in calculations of the “low-cutoff” type described in the previous paragraph. For instance, the 3N force of Ref. [51], in conjunction with the two-nucleon force of Ref. [39], yields a good description of nuclides up to $A=13$ [53]. An $O(P^{3})$ calculation also explains a variety of neutron-deuteron scattering data for neutron beam energies up to about 100 MeV [51]. Similar success have been achieved for electron-deuteron scattering [54, 55, 56], Helium-4 photoabsorption [57], neutrino-deuteron scattering and breakup [58], pion-deuteron scattering at threshold [25, 59, 60], pion photoproduction [61, 62], and—as will be discussed extensively in Section 5—coherent Compton scattering from the NN system. From all of this evidence we conclude that structure and reactions in few-nucleon systems can be well understood using $\chi$ET—provided the cutoff is kept below about 800 MeV.

The neutron, as its name implies, is an object with zero nett charge. This does not, though, preclude the existence of charged substructure within the neutron. The distribution—and to some extent, motion—of charges within the neutron is encoded in its form factors. Since the neutron is a spin-half particle there are two of these: electric and magnetic in a non-relativistic formulation. These form factors carry the imprint of the chiral dynamics of QCD through its impact on the neutron’s internal structure.

The chiral structure of the neutron would be revealed in electron scattering from neutron targets—if such experiments were feasible. The gedankenscattering reaction is pictured in Fig. 2, where the one-photon-exchange approximation has been assumed. The amplitude is then written as

$${\cal M}=-\frac{e^{2}}{q^{2}}j^{lep}_{\mu}J^{\mu}.$$

(23)

where $q=k-k^{\prime}$ is the four-momentum of the exchanged photon (note $q^{2}<0$ for electron scattering) and $j^{lep}_{\mu}$ is the leptonic current:

$$j_{\mu}^{lep}=\bar{u}({\bf k}^{\prime})\gamma_{\mu}u({\bf k}).$$

(24)

The most general form of the neutron relativistic current is the usual:

$$J_{\mu}^{n}=\bar{u}({\bf p}^{\prime})\left[F_{1}^{n}(q^{2})\gamma_{\mu}+\frac{i}{2M}F_{2}^{n}(q^{2})\sigma_{\mu\nu}q^{\nu}\right]u({\bf p}).$$

(25)

While in the non-relativistic framework being employed here we have:

$$J_{\mu}^{n}=\xi_{m_{s}^{\prime}}^{\dagger}\left\{G_{E}^{n}(q^{2})v_{\mu}+\frac{1}{M}G_{M}^{n}(q^{2})[S_{\mu},S_{\nu}]q^{\nu}\right\}\xi_{m_{s}},$$

(26)

where $\xi$ is, in practice, a Pauli spinor ²²2It is a four-component spinor whose lower comonents are zero for the standard choice $v=(1,{\bf 0})$, and $G_{E}^{n}$ and $G_{M}^{n}$ are the neutron electric and magnetic form factors.

The following discussion is based on the HB$\chi$PT analysis of Refs. [11, 63]. For a parallel discussion within the context of dispersion relations and models on the role of the nucleon’s pion cloud in determining electromagnetic form factors see Refs. [64, 65].

For our purposes it is most straightforward to begin with the isoscalar and isovector form factors. These are related to the proton and neutron form factors by:

	$\displaystyle G_{E,M}^{s}$	$\displaystyle=$	$\displaystyle G_{E,M}^{p}+G_{E,M}^{n},$		(27)
	$\displaystyle G_{E,M}^{v}$	$\displaystyle=$	$\displaystyle G_{E,M}^{p}-G_{E,M}^{n}.$		(28)

This decomposition is useful because the leading pion-loop effects occur only in the isovector form factors. The quantum numbers of the pion guarantee that the first contribution of the pion cloud to the isoscalar electromagnetic form factors occurs only at the two-loop level. The one-loop calculation was first carried out in Ref. [11], and was extended and refined in Ref. [63]. Combining the $O(P^{3})$ diagrams shown in Fig. 3 with the relevant tree graphs yields the expression

	$\displaystyle G_{E}^{v}=1+\frac{q^{2}}{(4\pi f_{\pi})^{2}}\left\{\kappa_{v}\left(\frac{2\pi f_{\pi}}{M}\right)^{2}-\frac{2}{3}g_{A}^{2}-2B_{10}^{(r)}(\mu)-\left[\frac{5}{3}g_{A}^{2}+\frac{1}{3}\right]\log\left(\frac{m_{\pi}}{\mu}\right)\right\}$
	$\displaystyle+\frac{1}{(4\pi f_{\pi})^{2}}\int_{0}^{1}dx\,\left[(3g_{A}^{2}+1)m_{\pi}^{2}-q^{2}x(1-x)(5g_{A}^{2}+1)\right]\log\left[\frac{m_{\pi}^{2}-q^{2}x(1-x)}{m_{\pi}^{2}}\right]$
			(29)

Here we have worked in the Breit frame, and so $q^{2}=-|{\bf q}|^{2}\equiv Q^{2}$, with ${\bf q}={\bf p}^{\prime}-{\bf p}$ the three-momentum transfer to the struck neutron.

The first term, “1” is guaranteed by charge conservation. The second piece gives the isovector charge radius, according to the usual definition:

$$G_{E}^{v}=1-\frac{1}{6}\langle(r_{E}^{v})^{2}\rangle q^{2}+O(q^{4}).$$

(30)

The short-distance contribution to the isovector charge radius, $B_{10}^{(r)}$, renormalizes the divergent loop integrals. It arises from a term in ${\cal L}_{\pi N}^{(3)}$ of the form

$${\cal L}_{\pi N}^{(3)}\sim B_{10}\bar{H}D^{\alpha}f_{\alpha\nu}^{+}v^{\nu}H.$$

(31)

The formula (29) is obtained using dimensional regularization with the modified minimal subtraction of $\chi$PT for the calculation of the loop integrals. The physical result for $G_{E}^{v}$ is not dependent on the choice of regularization and renormalization scheme, but the finite parts of $B_{10}^{(r)}(\mu)$ will in general be so. The $\mu$ dependence is not, however, affected by the particular variant of minimal subtraction employed, since the $\mu$ dependence of $B_{10}^{(r)}(\mu)$ must cancel the $\mu$ dependence of the logarithimc term in the curly brackets that contributes to the isovector charge radius.

Note that $B_{10}$ only affects the charge radius, and all higher moments of the charge distribution are given—at $O(P^{3})$ accuracy—by the physics of the pion-loop graphs shown in Fig. 3. The presence of this short-distance contribution to $\langle(r_{E}^{v})^{2}\rangle$ means that the isovector charge radius cannot be predicted in $\chi$PT. In fact, the corresponding calculation for the isoscalar form factor gives purely short-distance effects:

$$G_{E}^{s}=1-\frac{q^{2}}{(4\pi f_{\pi})^{2}}\left[\kappa_{s}\left(\frac{2\pi f_{\pi}}{M}\right)^{2}-4\tilde{B_{1}}\right].$$

(32)

Here $\tilde{B}_{1}$ is the coefficient of the $\gamma NN$ contact interaction that is the isoscalar analog of Eq. (31). However, since there is no loop effect at this order $\tilde{B}_{1}$ is a finite LEC.

Thus the prediction for the neutron electric form factor, up to $O(P^{3})$, in HB$\chi$PT is:

	$\displaystyle G_{E}^{n}=-\frac{q^{2}}{(4\pi f_{\pi})^{2}}\tilde{B}_{e}^{n}-\frac{q^{2}\kappa^{n}}{4M^{2}}-\frac{1}{2(4\pi f_{\pi})^{2}}\int_{0}^{1}dx\,\left[(3g_{A}^{2}+1)m_{\pi}^{2}\right.$
	$\displaystyle\qquad\qquad\qquad\quad\left.-q^{2}x(1-x)(5g_{A}^{2}+1)\right]\log\left[\frac{m_{\pi}^{2}-q^{2}x(1-x)}{m_{\pi}^{2}}\right],$		(33)

with $\tilde{B}_{e}^{n}$ a cobination of $\tilde{B}_{1}$, $\bar{B}_{10}$, and finite terms with fixed coefficients. The measured value of the neutron charge radius squared is $\langle(r_{E}^{n})^{2}\rangle=-0.115(4)~{\rm fm}^{2}$ [66, 67]. Comparing this with the result of Eq. (33) reveals that, numerically, the second term on the the first line of Eq. (33)—the Foldy term—dominates the overall result, and $\tilde{B}_{e}^{n}$ is unnaturally small. The non-analytic behaviour beyond $O(q^{2})$ encoded in the integral in Eq. (33) is a prediction of $\chi$PT at this order, and arises because of the spontaneously broken chiral symmetry in QCD.

In the case of the magnetic form factor the only counterterm necessary is the one already written above in ${\cal L}_{\pi N}^{(2)}$. It renormalizes the anomalous magnetic moment, i.e. the value of the form factor at $q^{2}=0$. Consequently, for the neutron, we find:

$$G_{M}^{n}=\kappa^{n}+g_{A}^{2}\frac{2\pi M}{(4\pi f_{\pi})^{2}}\int_{0}^{1}dx\,\left(\sqrt{m_{\pi}^{2}-q^{2}x(1-x)}-m_{\pi}\right).$$

(34)

Expanding in powers of $q^{2}$ we find that the magnetic radius of the neutron blows up in the chiral limit as $m_{q}^{-1/2}$. This is to be compared to the $\log(m_{q})$ dependence of the electric radius seen in Eq. (29). These chiral-limit divergences of observables—together with the associated pre-factors—are model-independent predictions of $\chi$PT, since they result from the long-distance pieces of the relevant quantum loops.

In Eq. (34) short-distance physics contributes only to $\kappa^{n}$. All other moments of the magnetization distribution are dominated by pion-cloud physics, and have only small ($O(P^{4})$ and beyond) contributions from other baryons and mesons. Of course, this statement applies only to the regime $|{\bf q}|\sim m_{\pi}$. Once $|{\bf q}|\approx 600$ MeV a variety of other dynamics needs to be taken into consideration, since at that momentum scale the electromagnetic probe can excite shorter-degrees of freedom in the neutron’s electromagnetic structure.

The $O(P^{3})$ HB$\chi$PT predictions for $G_{E}^{n}$ and $G_{M}^{n}$ are plotted in Fig. 4, together with some “data” for both—on which, more below. Also shown is a recent dispersion-relation fit to a large database of electron-scattering experiments [67].

The upper panel of Fig. 4 shows that the $O(P^{3})$ result for $G_{E}^{n}$ has the same feature as this dispersion-relation description, namely a rise at low-$Q^{2}$, followed by a turn over. But the HB$\chi$PT prediction turns over too quickly. However, adjusting the input neutron charge radius to a slightly larger value produces a marked effect. And since $\langle(r_{E}^{n})^{2}\rangle$ results from a sizeable cancellation between isovector and isoscalar pieces of short-distance physics, this finding suggests that the neutron electric form-factor data can be accommodated within a “natural” scenario for that higher-energy dynamics. Indeed, the difficulty of describing $G_{E}^{n}$ accurately within HB$\chi$PT stems partly from these cancellations, which lead to a $G_{E}^{n}$ that is small throughout the entire realm of $\chi$PT’s applicability. This means that higher-order contributions will have a disproportionately large impact on the final answer.

The neutron magnetic form factor, which is order one in this domain, is more fertile ground for $\chi$PT. And indeed, the $O(P^{3})$ prediction for $G_{M}^{n}$ has the right general shape, but yields a magnetic radius of

$$\langle(r_{M}^{n})^{2}\rangle\equiv-\frac{6}{\kappa^{n}}\frac{dG_{M}^{n}}{dQ^{2}}=-\frac{g_{A}^{2}M}{16f_{\pi}^{2}\kappa_{n}\pi m_{\pi}}=0.51~{\rm fm}^{2},$$

(35)

which is a little smaller than the magnetic radius obtained from the dispersion-relation analysis. However, it must be remembered that there is an $O(P^{4})$ short-distance contribution to this quantity. Including that contribution in the calculation of $G_{M}^{n}$ yields the dotted curve shown in Fig. 4. This can be considered a partial $O(P^{4})$ calculation of $G_{M}^{n}$, since it does not include the additional loop mechanisms that are present at this order. But even this partial $O(P^{4})$ computation shows that the shift due to short-distance effects is about 20% at $Q^{2}=0.1$ GeV²—a magnitude that is consonant with this being a contribution of $O(Q^{2}/\Lambda_{\chi{\rm SB}}^{2})$ to $G_{M}^{n}/\kappa_{n}$. In fact, the $O(P^{4})$ contact interaction that shifts the magnetic radius to the empirical value is associated with a momentum scale of order 1 GeV. The LEC that is the coefficient of this contact interaction is certainly natural with respect to $\Lambda_{\chi{\rm SB}}$. But by $Q^{2}\approx 0.25$ GeV² the $O(P^{4})$ contribution is as large as the $O(P^{3})$ result—a clear sign that the $\chi$PT expansion is breaking down. The results for $G_{M}^{n}$ therefore confirm the $\chi$PT picture that the dominant effect in the neutron’s magnetization distribution is the pion cloud, while also showing that this picture is limited to momentum transfers below 500 MeV.

The results reported here are rigorous within HB$\chi$PT. Ref. [63] showed that the inclusion of Delta(1232) degrees of freedom in the effective field theory does not improve the agreement with data—the short-distance physics in these observables is associated with higher-mass states. The impact of corrections associated with nucleon recoil on neutron form-factor predictions is discussed in Refs. [73, 74]. These corrections are appreciable in the case of $G_{E}^{n}$, and there the $q/M$ series can be reorganized into a form that improves the agreement of the $\chi$PT predictions with data. The tree-level contribution of vector mesons is also considered in Refs. [73, 74] and it assists in the description of $G_{M}^{n}$.

The “data” points represented by circles and squares in Fig. 4 are obtained via electron-induced breakup reactions on deuterium targets. It needs to be emphasized that $\chi$PT calculation of these reactions would allow a completely consistent treatment of the pion dynamics that generates the higher-order terms in the neutron form factor and the pion-exchange currents which play, e.g. an important role in $\gamma d\rightarrow np$ at threshold [75]. These exchange currents occur at $O(P^{3})$ in the irreducible kernel for the interaction of real and virtual photons with the NN system, and so have the same chiral order as the single-nucleon loop effects that have been highlighted in this Section. But, no comprehensive analysis of $ed\rightarrow e^{\prime}np$ has been performed within $\chi$ET. Such an analysis, and its extension to the three-body system, would allow a consistent treatment of the chiral structure of the neutron and the target nucleus. It would also facilitate reliable estimates of the theoretical uncertainties in the neutron form-factor extraction.

The diamonds in the upper panel of Fig. 4 are obtained from an analysis of the deuteron’s quadrupole form factor, $G_{Q}$ [71]. In this case the AV18 potential [76] was used to generate the deuteron’s wave function, and the important correction to the deuteron’s charge operator [55, 77, 78] was included in the calculation. However, this calculation fails to reproduce the deuteron’s quadrupole moment, and this could have some consequences for the $Q^{2}$-dependence of the deuteron $G_{Q}$ [56]. $\chi$ET describes data on the ratio of deuteron charge to quadrupole form factors $G_{C}/G_{Q}$ well up to momentum transfers of about 0.3 GeV² [56]. A $\chi$ET extraction of $G_{E}^{(s)}$ from $G_{Q}$, and possibly from the $A(Q^{2})$ data of Ref. [79] and subsequent studies, could perhaps provide useful information on $G_{E}^{n}$ in this low-$Q^{2}$ regime.

The neutron has zero charge, but its composite structure means that it has a non-zero Compton scattering amplitude. The standard decomposition for the neutron Compton amplitude involves six non-relativistic invariants:

	$\displaystyle T_{\gamma N}=A_{1}{\bf\epsilon}^{\prime}\cdot{\bf\epsilon}+A_{2}{\bf\epsilon}\cdot\hat{k}\,{\bf\epsilon}\cdot\hat{k^{\prime}}+iA_{3}{\bf\sigma}\cdot({\bf\epsilon}^{\prime}\times{\bf\epsilon})+iA_{4}{\bf\sigma}\cdot(\hat{k^{\prime}}\times\hat{k})\,{\bf\epsilon^{\prime}}\cdot{\bf\epsilon}$
	$\displaystyle\!\!\!\!\!\!\!\!\!+iA_{5}{\bf\sigma}\cdot[({\bf\epsilon}^{\prime}\times\hat{k})\,{\bf\epsilon}\cdot\hat{k^{\prime}}-({\bf\epsilon}\times\hat{k^{\prime}})\,{\bf\epsilon}^{\prime}\cdot\hat{k}]+iA_{6}{\bf\sigma}\cdot[({\bf\epsilon}^{\prime}\times\hat{k^{\prime}})\,{\bf\epsilon}\cdot\hat{k^{\prime}}-({\bf\epsilon}\times\hat{k})\,{\bf\epsilon}^{\prime}\cdot\hat{k}],$
			(36)

where ${\bf k}$ and ${\bf\epsilon}$ (${\bf k}^{\prime}$ and ${\bf\epsilon}^{\prime}$) are the three-momentum and polarization vector of the incoming (outgoing) photon. The $A_{i}$’s, $i=1\ldots 6$, are scalar functions of photon energy and scattering angle.

At photon energies $\sim m_{\pi}$ Compton scattering from the neutron is driven by photon interactions with the neutron’s pion cloud. The dominant effects come from the $O(P^{3})$ mechanisms depicted in the diagrams (a)–(d) of Fig. 5 and the first line of Fig. 6. These loop diagrams individually contain divergences, but the divergences cancel in the sum. Indeed, they must do so, because no gauge-invariant $\gamma$n contact interaction can be constructed at $O(P^{3})$: at this level of accuracy there is no short-distance contribution to neutron Compton scattering. Consequently, up to $O(P^{3})$ there is a $\chi$PT prediction for this process. It, together with the associated proton result, is given by the following (Breit-frame) expressions for $A_{1}$–$A_{6}$ [80, 81]:

	$\displaystyle A_{1}$	$\displaystyle=$	$\displaystyle-\frac{{\mathcal{Z}}^{2}e^{2}}{M}+\frac{g_{A}^{2}m_{\pi}e^{2}}{8\pi f_{\pi}^{2}}\left\{1-\sqrt{1-\Upsilon^{2}}+\frac{2-t}{\sqrt{-t}}\left[\frac{1}{2}\arctan\frac{\sqrt{-t}}{2}-I_{1}(\Upsilon,t)\right]\right\},$
	$\displaystyle A_{2}$	$\displaystyle=$	$\displaystyle-\frac{e^{2}g_{A}^{2}\omega^{2}}{8\pi f_{\pi}^{2}m_{\pi}}\frac{2-t}{(-t)^{3/2}}\left[I_{1}(\Upsilon,t)-I_{2}(\Upsilon,t)\right],$
	$\displaystyle A_{3}$	$\displaystyle=$	$\displaystyle\frac{e^{2}\omega}{2M^{2}}[{\mathcal{Z}}({\mathcal{Z}}+2\kappa)-({\mathcal{Z}}+\kappa)^{2}\cos\theta]+\frac{{(2{\mathcal{Z}}-1)}e^{2}g_{A}m_{\pi}}{8\pi^{2}f_{\pi}^{2}}\frac{\Upsilon t}{1-t}$
			$\displaystyle+\frac{e^{2}g_{A}^{2}m_{\pi}}{8\pi^{2}f_{\pi}^{2}}\left[\frac{1}{\Upsilon}\arcsin^{2}\Upsilon-\Upsilon+2\Upsilon^{4}\sin^{2}\theta I_{3}(\Upsilon,t)\right],$
	$\displaystyle A_{4}$	$\displaystyle=$	$\displaystyle-\frac{({\mathcal{Z}}+\kappa)^{2}e^{2}\omega}{2M^{2}}+\frac{e^{2}g_{A}^{2}\omega^{2}}{4\pi^{2}f_{\pi}^{2}m_{\pi}}I_{4}(\Upsilon,t),$
	$\displaystyle A_{5}$	$\displaystyle=$	$\displaystyle\frac{({\mathcal{Z}}+\kappa)^{2}e^{2}\omega}{2M^{2}}-\frac{{(2{\mathcal{Z}}-1)}e^{2}g_{A}\omega^{2}}{8\pi^{2}f_{\pi}^{2}m_{\pi}}\frac{\Upsilon}{(1-t)}$
			$\displaystyle-\frac{e^{2}g_{A}^{2}\omega^{2}}{8\pi^{2}f_{\pi}^{2}m_{\pi}}[I_{5}(\Upsilon,t)-2\Upsilon^{2}\cos\theta I_{3}(\Upsilon,t)],$
	$\displaystyle A_{6}$	$\displaystyle=$	$\displaystyle-\frac{{\mathcal{Z}}({\mathcal{Z}}+\kappa)e^{2}\omega}{2M^{2}}+\frac{{(2{\mathcal{Z}}-1)}e^{2}g_{A}\omega^{2}}{8\pi^{2}f_{\pi}^{2}m_{\pi}}\frac{\Upsilon}{(1-t)}$		(37)
			$\displaystyle+\frac{e^{2}g_{A}^{2}\omega^{2}}{8\pi^{2}f_{\pi}^{2}m_{\pi}}[I_{5}(\Upsilon,t)-2\Upsilon^{2}I_{3}(\Upsilon,t)],$

where $\Upsilon=\omega/m_{\pi}$, $t=-2\Upsilon^{2}(1-\cos\theta)$, and

	$\displaystyle I_{1}(\Upsilon,t)$	$\displaystyle=$	$\displaystyle\int_{0}^{1}dz\,\arctan\frac{(1-z)\sqrt{-t}}{2\sqrt{1-\Upsilon^{2}z^{2}}},$
	$\displaystyle I_{2}(\Upsilon,t)$	$\displaystyle=$	$\displaystyle\int_{0}^{1}dz\,\frac{2(1-z)\sqrt{-t(1-\Upsilon^{2}z^{2})}}{4(1-\Upsilon^{2}z^{2})-t(1-z)^{2}},$
	$\displaystyle I_{3}(\Upsilon,t)$	$\displaystyle=$	$\displaystyle\int_{0}^{1}dx\,\int_{0}^{1}dz\,\frac{x(1-x)z(1-z)^{3}}{S^{3}}\left[\arcsin\frac{\Upsilon z}{R}+\frac{\Upsilon zS}{R^{2}}\right],$
	$\displaystyle I_{4}(\Upsilon,t)$	$\displaystyle=$	$\displaystyle\int_{0}^{1}dx\,\int_{0}^{1}dz\,\frac{z(1-z)}{S}\arcsin\frac{\Upsilon z}{R},$
	$\displaystyle I_{5}(\Upsilon,t)$	$\displaystyle=$	$\displaystyle\int_{0}^{1}dx\,\int_{0}^{1}dz\,\frac{(1-z)^{2}}{S}\arcsin\frac{\Upsilon z}{R},$		(38)

with

$$S=\sqrt{1-\Upsilon^{2}z^{2}-t(1-z)^{2}x(1-x)},\quad R=\sqrt{1-t(1-z)^{2}x(1-x)}.$$

(39)

Here and below formulae are given in Coulomb gauge, for real photons, so the polarization four-vectors and photon four-momenta satisfy:

$$k^{2}=0;\qquad k^{\prime 2}=0;\qquad v\cdot\epsilon=0;\qquad v\cdot\epsilon^{\prime}=0.$$

(40)

Note the impact of the Goldstone bosons of QCD—the pions—on the neutron Compton amplitude. The theory predicts that the Compton amplitude contains a cusp at $\omega=m_{\pi}$. That cusp is a consequence of the opening of the photoproduction channel at that energy ³³3In HB$\chi$PT the threshold is not at precisely the right photon energy, but this can be corrected through a resummation of higher-order terms that is motivated by a modified power counting in the vicinity of a threshold [80]., but its consequences are felt even for $\omega$ well below $m_{\pi}$. Thus Compton scattering—alone among reactions with real photons—probes chiral dynamics at photon energies well below the pion threshold.

The most famous manifestation of this chiral dynamics in neutron Compton scattering occurs in the neutron polarizabilities. The low-energy expansion of the $A_{i}$’s defines the electric and magnetic polarizabilities $\alpha$ and $\beta$ as well as the four “spin polarizabilities” $\gamma_{1}$–$\gamma_{4}$. The former occur as coefficients of the $\omega^{2}$ terms in $A_{1}$ and $A_{2}$, while the latter are coefficients of the $\omega^{3}$ terms in $A_{3}$–$A_{6}$—once the Born terms are accounted for. For the Breit-frame amplitude such an $\omega$-expansion gives (keeping terms up to $O(1/M^{3})$) [81, 82]:

	$\displaystyle A_{1}(\omega,\theta)\!$	$\displaystyle=$	$\displaystyle\!-{{\cal Z}^{2}e^{2}\over M}+{e^{2}\over 4M^{3}}\Bigl{(}({\cal Z}+\kappa)^{2}(1+\cos\theta)-{\cal Z}^{2}\Bigr{)}(1-\cos\theta)\,\omega^{2}$
			$\displaystyle\qquad\qquad\qquad\qquad+4\pi(\alpha+\beta\,\cos\theta)\omega^{2}+O(\omega^{4}),$
	$\displaystyle A_{2}(\omega,\theta)\!$	$\displaystyle=$	$\displaystyle\!{e^{2}\over 4M^{3}}\kappa(2{\cal Z}+\kappa)\omega^{2}\cos\theta-4\pi\beta\omega^{2}\,+O(\omega^{4}),$
	$\displaystyle A_{3}(\omega,\theta)\!$	$\displaystyle=$	$\displaystyle\!{e^{2}\omega\over 2M^{2}}\Bigl{(}{\cal Z}({\cal Z}+2\kappa)-({\cal Z}+\kappa)^{2}\cos\theta\Bigr{)}+A_{3}^{\pi^{0}}$
			$\displaystyle\qquad\qquad\qquad\qquad+4\pi\omega^{3}(\gamma_{1}-(\gamma_{2}+2\gamma_{4})\,\cos\theta)+O(\omega^{5}),$
	$\displaystyle A_{4}(\omega,\theta)\!$	$\displaystyle=$	$\displaystyle\!-{e^{2}\omega\over 2M^{2}}({\cal Z}+\kappa)^{2}+A_{4}^{\pi^{0}}+4\pi\omega^{3}\gamma_{2}+O(\omega^{5}),$
	$\displaystyle A_{5}(\omega,\theta)\!$	$\displaystyle=$	$\displaystyle\!{e^{2}\omega\over 2M^{2}}({\cal Z}+\kappa)^{2}+A_{5}^{\pi^{0}}+4\pi\omega^{3}\gamma_{4}+O(\omega^{5}),$
	$\displaystyle A_{6}(\omega,\theta)\!$	$\displaystyle=$	$\displaystyle\!-{e^{2}\omega\over 2M^{2}}{\cal Z}({\cal Z}+\kappa)+A_{6}^{\pi^{0}}+4\pi\omega^{3}\gamma_{3}+O(\omega^{5}),$		(41)

with $A_{3}^{\pi^{0}}$–$A_{6}^{\pi^{0}}$ the contributions from the pion-pole graph, Fig. 5(d).