Chiral symmetry and effective field theories for
hadronic, nuclear and stellar matter

Historically, our understanding of the pion as a Nambu-Goldstone boson emerged [16, 17] in the pre-QCD era of the 1960’s and culminated in the current algebra approach [18] (combined with the PCAC relation for the pion). Inspired by the BCS theory of superconductivity, Nambu and Jona-Lasinio (NJL) [17] developed a model that helped clarify the dynamics that drives spontaneous chiral symmetry breaking and the formation of pions as pseudo-Goldstone bosons. The NJL model in the SM (standard model) context is known to be equivalent to the nonlinear sigma model. So one could just as well discuss what is given below in terms of a generalized nonlinear sigma model but we find it more transparent to follow the logic of the NJL which gives a hint at the link to QCD degrees of freedom. In the next section, we will address the connection between QCD variables and hadronic variables with the help of topology closely tied to quantum anomalies.

Starting from the color current of quarks, ${\bf J}_{\mu}^{a}=\bar{q}\gamma_{\mu}{\bf t}^{a}q$, where $q$ denotes the quark fields with $4N_{c}N_{f}$ components representing their spin, color and flavor degrees freedom $(N_{c}=3,N_{f}=2)$ and ${\bf t}^{a}$ ($a=1,...,8$) are the generators of the $SU(3)_{c}$ color gauge group, one can make the additional ansatz that the distance over which color propagates is restricted to a short correlation length $l_{c}$. Then the interaction between low-momentum quarks, mediated by the coupling of the quark color current to the gluon fields, can be schematically viewed as a local coupling between their color currents:

$${\cal L}_{\rm int}=-G_{c}\,{\bf J}_{\mu}^{a}(x)\,{\bf J}^{\mu}_{a}(x)\,\,,$$

(1)

where $G_{c}\sim\bar{g}^{2}\,l_{c}^{2}$ represents an effective coupling strength proportional to the square of the QCD coupling, $g$, averaged over the relevant distance scales, in combination with the correlation length, $l_{c}$, squared.

Given the local interaction in Eq. (1), the model Lagrangian for the quark fields

$${\cal L}=\bar{q}(x)(i\gamma^{\mu}\partial_{\mu}-m_{q})q(x)+{\cal L}_{\rm int}(\bar{q},q)$$

(2)

arises by “integrating out” the gluon degrees of freedom and absorbing them in the local four-fermion interaction ${\cal L}_{\rm int}$. In this way the local $SU(3)_{c}$ gauge symmetry of QCD is replaced by a global one. Confinement is lost but all symmetries of QCD are maintained. In Eq. (2) the mass matrix $m_{q}$ incorporates the small “bare” quark masses, and in the chiral limit ($m_{q}\rightarrow 0$) the Lagrangian in Eq. (2) has a chiral symmetry of left- and right-handed quarks, $SU(N_{f})_{L}\times SU(N_{f})_{R}$, analogous to that of the original QCD Lagrangian for $N_{f}$ massless quark flavors.

Fierz transforming the color current-current interaction in Eq. (1) produces a set of exchange terms acting in quark-antiquark channels. For the case of $N_{f}=2$:

$${\cal L}_{\rm int}\rightarrow{G\over 2}\left[(\bar{q}q)^{2}+(\bar{q}\,i\gamma_{5}\,\vec{\tau}\,q)^{2}\right]+...\,\,,$$

(3)

where $\vec{\tau}=(\tau_{1},\tau_{2},\tau_{3})$ represents the vector of isospin Pauli matrices. Included in Eq. (3) but not shown explicitly are a series of terms with vector and axial vector currents, both in color-singlet and color-octet channels. The new constant $G$ is proportional to the color coupling strength $G_{c}$, and their ratio is uniquely determined by the number of colors and flavors. This derivation can be viewed as a contemporary way of introducing the time-honored NJL model [17], which has been applied [19, 20] to a variety of problems in hadronic physics. The virtue of the model is its simplicity in illustrating the basic mechanism behind spontaneous chiral symmetry breaking, as we now outline.

In the mean-field approximation, the equation of motion associated with the Lagrangian in Eq. (2) leads to a gap equation

$$M_{q}=m_{q}-G\langle 0|\bar{q}q|0\rangle\,\,,$$

(4)

which connects the dynamically generated constituent quark mass $M_{q}$ to the appearance of the chiral quark condensate

$$\langle 0|\bar{q}q|0\rangle=-{\rm tr}\lim_{\,x\rightarrow 0}\langle 0|{\cal T}q(0)\bar{q}(x)|0\rangle=-2iN_{f}N_{c}\int{d^{4}p\over(2\pi)^{4}}{M_{q}\,\theta(\Lambda-|\vec{p}\,|)\over p^{2}-M_{q}^{2}+i\epsilon}\,\,.$$

(5)

The chiral condensate plays the role of an order parameter for the spontaneous breaking of chiral symmetry. In the chiral limit, where $m_{q}=0$, a non-zero constituent quark mass $M_{q}$ develops dynamically, together with a non-vanishing chiral condensate $\langle 0|\bar{q}q|0\rangle$, provided that $G$ exceeds a critical value on the order of $G_{\rm crit}\simeq 10$ GeV^-2. The integral in Eq. (5) requires a momentum-space cutoff $\Lambda\simeq 2M_{q}$ beyond which the interaction is “turned off”. The strong non-perturbative interactions polarize the vacuum and generate a condensate of quark-antiquark pairs, thereby turning an initially point-like quark with its small bare mass $m_{q}$ into a dressed quasi-particle with a size on the order of $(2M_{q})^{-1}$.

By solving the Bethe-Salpeter equations in the color-singlet quark-antiquark channels, the lightest mesons are generated as quark-antiquark excitations of the correlated QCD ground state (with its condensate structure). Several calculations have been performed previously in the three-flavor NJL model [19, 20, 21]. This model has an undesired $U(3)_{L}\times U(3)_{R}$ symmetry, but due to the axial $U(1)_{A}$ anomaly of QCD this symmetry³³3The axial $U(1)$ transformation acting on quark fields, $q\to\exp(i\alpha\gamma_{5})q$, constitutes a symmetry only of classical chromodynamics, not its quantum version. is reduced to $SU(3)_{L}\times SU(3)_{R}\times U(1)_{V}$. In the three-flavor NJL model, instanton-driven interactions are incorporated in a flavor determinant [22] $\det[\bar{q}_{i}(1\pm\gamma_{5})q_{j}]$. This interaction necessarily involves all three quark flavors $u,d,s$ simultaneously in a genuine three-body contact term.

In Fig. 1 we show the symmetry breaking pattern resulting from such a calculation of the pseudoscalar meson spectrum. Assuming massless $u,d$ and $s$ quarks, the pseudoscalar octet emerges as a set of massless Goldstone bosons of the spontaneously broken $SU(3)_{L}\times SU(3)_{R}$ symmetry, while the anomalously broken $U(1)_{A}$ symmetry drives the singlet $\eta_{0}$ away from the Goldstone boson sector. The inclusion of finite quark masses that explicitly break chiral symmetry shifts the pseudoscalar ($J^{\pi}=0^{-}$) nonet into its empirically observed hierarchy, including $\eta_{0}$-$\eta_{8}$ mixing.

The pion mass is related to the scalar quark condensate and bare quark masses through the famous Gell-Mann-Oakes-Renner relation [23]:

$$m^{2}_{\pi}\,f_{\pi}^{2}=-{1\over 2}(m_{u}+m_{d})\langle 0|\bar{q}q|0\rangle+{\cal O}(m^{2}_{u,d}\,{\rm ln}(m_{u,d})).$$

(6)

derived from current algebra and PCAC. It involves additionally the pion decay constant, $f_{\pi}=92.4$ MeV, which is defined by the matrix element connecting the pion state with the QCD vacuum via the isovector axial vector current, $A^{\mu}_{a}=\bar{q}\gamma^{\mu}\gamma_{5}{\tau_{a}\over 2}q$:

$$\langle 0|A^{\mu}_{a}(0)|\pi_{b}(p)\rangle=i\delta_{ab}\,p^{\mu}f_{\pi}\,.$$

(7)

The pion decay constant, like the chiral condensate $\langle 0|\bar{q}q|0\rangle$, is a measure of spontaneous chiral symmetry breaking associated with the scale $\Lambda_{\chi}\sim 4\pi f_{\pi}\sim 1$ GeV. The non-zero pion mass, $m_{\pi}\simeq 138$ MeV $\ll\Lambda_{\chi}$, is a reflection of the explicit chiral symmetry breaking by the small quark masses, with $m^{2}_{\pi}\sim m_{q}$. The quark masses $m_{u,d}$ as well as the scalar quark condensate $\langle 0|\bar{q}q|0\rangle$ are scale-dependent quantities, and only their product is renormalization group invariant. At the renormalization scale of 2 GeV, a typical average quark mass ${1\over 2}(m_{u}+m_{d})\simeq 3.5$ MeV leads to a condensate $\langle 0|\bar{q}q|0\rangle\simeq-(0.36$ GeV$)^{3}$.

The quark masses set the primary scales in QCD, and their classification into “light” and “heavy” quarks determines very different physics phenomena. The heavy quarks (e.g., the $t$, $b$ and – within limits – the $c$ quarks), whose reciprocal masses offer a natural “small parameter”, can be treated in non-relativistic approximations (that is, expansions of observables in powers of $1/m_{t,b,c}$). The sector of the light quarks (e.g., the $u$, $d$ quarks and – to some extent – the $s$ quark), however, is governed by quite different principles and rules. Evidently, the light quark masses themselves are now “small parameters” to be compared with a “large” scale of dynamical origin. This large scale is characterized by a mass gap of about 1 GeV separating the QCD vacuum from nearly all of its excitations, with the exception of the pseudoscalar meson octet shown in Fig. 1. This mass gap is comparable to $4\pi f_{\pi}$, the energy scale associated with the spontaneous breaking of chiral symmetry in QCD.

In this section we briefly summarize the steps [8, 9] for constructing the chiral Lagrangian in the pure meson sector (baryon number $B=0$). A chiral field is introduced as

$$U(x)=\exp\bigg{(}{i\over f_{\pi}}\,\vec{\tau}\cdot\vec{\pi}(x)\bigg{)}\in SU(2)~,$$

(8)

where $f_{\pi}$ is the the pion decay constant in the chiral limit ($m_{\pi}\rightarrow 0$). The physics of QCD is then phrased in terms of an effective Lagrangian involving the chiral field $U(x)$ and its derivatives:

$${\cal L}_{QCD}\to{\cal L}_{eff}(U,\partial^{\mu}U,...).$$

(9)

Since Goldstone bosons interact only when they carry non-zero four-momenta, the low-energy expansion of Eq. (9) allows for an ordering in powers of $\partial^{\mu}U$. From Lorentz invariance only even numbers of derivatives are permitted⁴⁴4Note that for $N_{f}=3$, there is the Wess-Zumino term with three derivatives.. One writes for the first few terms of the chiral Lagrangian:

$${\cal L}_{\rm eff}={\cal L}_{\pi\pi}^{(2)}+{\cal L}_{\pi\pi}^{(4)}+...\,,$$

(10)

where the leading term (encoding the current algebras, leading order in the non-linear sigma model) involves two derivatives:

$${\cal L}^{(2)}_{\pi\pi}={f_{\pi}^{2}\over 4}{\rm tr}(\partial^{\mu}U\partial_{\mu}U^{\dagger})\,.$$

(11)

At fourth order the additional terms permitted by symmetries are

$${\cal L}^{(4)}_{\pi\pi}={\ell_{1}\over 4}\Big{[}{\rm tr}(\partial^{\mu}U\partial_{\mu}U^{\dagger})\Big{]}^{2}+{\ell_{2}\over 4}{\rm tr}(\partial_{\mu}U\partial_{\nu}U^{\dagger})\,{\rm tr}(\partial^{\mu}U\partial^{\nu}U^{\dagger})+\dots\,,$$

(12)

where further contributions involving the light quark mass $m_{q}$ and external fields are not shown. The constants $\ell_{1},\ell_{2}$ absorb loop divergences and their finite scale-dependent parts must be fixed by matching to experiment.

The symmetry-breaking mass term is small, so that it too can be handled perturbatively. The leading contribution introduces a term that is linear in the quark mass matrix $m_{q}={\rm diag}(m_{u},m_{d})$:

$${\cal L}^{(2)}_{\pi\pi}={f_{\pi}^{2}\over 4}{\rm tr}(\partial_{\mu}U\partial^{\mu}U^{\dagger})+{f_{\pi}^{2}\over 4}m_{\pi}^{2}\,{\rm tr}(U+U^{\dagger})\,,$$

(13)

where $m_{\pi}^{2}\sim(m_{u}+m_{d})$. The fourth-order Lagrangian ${\cal L}^{(4)}_{\pi\pi}$ also receives explicit chiral symmetry breaking contributions (proportional to $m_{q}$ and $m_{q}^{2}$)⁵⁵5Higher-order quark mass terms that can be systematically brought in using the spurion field ${\cal M}$ that transforms like the chiral field $U$ with $\langle{\cal M}\rangle=m_{q}$ will figure in the skyrmion matter description of dense matter described below. with the introduction of additional low-energy constants $\ell_{3}$ and $\ell_{4}$.

Provided that the effective Lagrangian includes all terms allowed by the symmetries of QCD, chiral effective field theory is the low-energy equivalent [3, 24] of QCD. The framework for systematic perturbative calculations of (on-shell) $S$-matrix elements involving Nambu-Goldstone bosons, chiral perturbation theory (ChPT), is then defined by the following rules: Collect all Feynman diagrams generated by ${\cal L}_{eff}$. Classify individual terms according to powers of the small quantity $p/(4\pi f_{\pi})$, where $p$ stands generically for three-momenta of Goldstone bosons or for the pion mass $m_{\pi}$. Loops are evaluated in dimensional regularization and are renormalized by appropriate chiral counter terms.

How to incorporate massive mesons, i.e., the vector mesons $V$ and the dilaton $\chi$, as needed for certain processes in consistency with chiral symmetry, hidden local symmetry and spontaneously broken scale symmetry, will be described in detail below.

In a systematic large $N_{c}$ expansion to access QCD, the effective Lagrangian is given by a weakly coupled mesonic Lagrangian of the nonlinear sigma model form described above or hidden-local-symmetrized form to be described below. The baryon arises as a toplogical soliton with the topology lodged in the chiral field $U$ [25]. This is the famous skyrmion [26]. It will be seen in Section 3 that the skyrmion structure of the baryon characterized by topology can be “smoothly” transformed into the quark-bag structure of QCD. It thereby provides the “bridge” between the hadronic degrees of freedom and the QCD degrees of freedom. Furthermore when looked at in terms of holographic QCD that comes from string theory, the baryon appears as an instanton in 5D Yang-Mills Lagrangian, point-like in certain limits (large $N_{C}$ and large ’t Hooft $\lambda\equiv N_{c}g_{s}^{2}$ limit) [27]. It makes sense to take the baryon, the mass scale of which is much greater than that of fluctuating mesonic fields, $\pi$, $V$ and dilaton, as a point-like local field. Thus in the spirit of the Weinberg theorem, we may simply take the nucleon as a local field as in [27]. In nuclear physics a local nucleon field is simply introduced as a matter field subject to the proper symmetries involved. Certain subtle properties of the skyrmion structure in dense matter that is most likely inaccessible by the local baryon field description will be discussed.

The prominent role played by the pion as a pseudo-Goldstone boson of spontaneously broken chiral symmetry impacts as well the low-energy structure and dynamics of nucleons [28]. When probing the nucleon with a long-wavelength electroweak field, a substantial part of the response comes from the pion cloud, comprising the “soft” surface of the nucleon. The calculational framework for this, baryon chiral perturbation theory [29, 30] has been applied successfully to diverse low-energy processes (such as low-energy pion-nucleon scattering, threshold pion photo- and electro-production and Compton scattering on the nucleon).

We now consider the physics of the pion-nucleon system, the sector with baryon number $B=1$. The nucleon is represented by a local isospin-$\frac{1}{2}$ doublet field $\Psi=\left(\!\!\begin{array}[]{c}p\\ n\end{array}\!\!\right)$ of protons and neutrons with free Lagrangian

$${\cal L}_{N}^{\rm free}=\bar{\Psi}(i\gamma_{\mu}\partial^{\mu}-M_{0})\Psi$$

(14)

where $M_{0}$ is the nucleon mass in the chiral limit. Unlike the pion, the nucleon has a large mass of the same order as the chiral symmetry breaking scale $4\pi f_{\pi}$, even in the limit of vanishing quark masses. The additional term in the chiral Lagrangian involving the nucleon, denoted by ${\cal L}_{\pi N}$, is again dictated by chiral symmetry and expanded in powers of the quark masses and derivatives of the Goldstone boson field:

$${\cal L}_{\pi N}={\cal L}_{\pi N}^{(1)}+{\cal L}_{\pi N}^{(2)}+\dots$$

(15)

In the leading term, ${\cal L}_{\pi N}^{(1)}$, there is a vector current coupling between pions and the nucleon (arising from the replacement of $\partial^{\mu}$ by a chiral covariant derivative) as well as an axial vector coupling:

$${\cal L}_{\pi N}^{(1)}=\bar{\Psi}\Big{[}i\gamma_{\mu}(\partial^{\mu}+\Gamma^{\mu})-M_{0}+g_{A}\gamma_{\mu}\gamma_{5}\,u^{\mu}\Big{]}\Psi\,.$$

(16)

The vector and axial vector quantities involve the pion fields via $\xi=\sqrt{U}$ in the form

	$\displaystyle\Gamma^{\mu}$	$\displaystyle=$	$\displaystyle{1\over 2}[\xi^{\dagger},\partial^{\mu}\xi]={i\over 4f_{\pi}^{2}}\,\vec{\tau}\cdot(\vec{\pi}\times\partial^{\mu}\vec{\pi})+...~~,$		(17)
	$\displaystyle u^{\mu}$	$\displaystyle=$	$\displaystyle{i\over 2}\{\xi^{\dagger},\partial^{\mu}\xi\}=-{1\over 2f_{\pi}}\,\vec{\tau}\cdot\partial^{\mu}\vec{\pi}+...~~,$		(18)

where the last steps result from expanding $\Gamma^{\mu}$ and $u^{\mu}$ to leading order in the pion fields. Up to this point the only parameters that enter are the nucleon mass $M_{0}$, the pion decay constant $f_{\pi}$, and the nucleon axial vector coupling constant $g_{A}$, all three taken in the chiral limit.

The next-to-leading order pion-nucleon Lagrangian, ${\cal L}_{\pi N}^{(2)}$, contains the chiral symmetry breaking quark mass term, which shifts the nucleon mass to its physical value. The nucleon sigma term

$$\sigma_{N}=m_{q}\frac{\partial M_{N}}{\partial m_{q}}=\langle N|m_{q}(\bar{u}u+\bar{d}d)|N\rangle$$

(19)

measures the contribution of the non-vanishing quark mass to the nucleon mass $M_{N}$. Its empirical value, deduced [31] from low-energy pion-nucleon data, is in the range $\sigma_{N}\simeq(45\pm 8)$ MeV. Up to next-to-leading order, the $\pi N$ effective Lagrangian has the form

	$\displaystyle{\cal L}_{eff}^{N}$	$\displaystyle=$	$\displaystyle\bar{\Psi}(i\gamma_{\mu}\partial^{\mu}-M_{N})\Psi-{g_{A}\over 2f_{\pi}}\bar{\Psi}\gamma_{\mu}\gamma_{5}\vec{\tau}\,\Psi\cdot\partial^{\mu}\vec{\pi}$		(20)
			$\displaystyle-{1\over 4f_{\pi}^{2}}\bar{\Psi}\gamma_{\mu}\vec{\tau}\,\Psi\cdot(\vec{\pi}\times\partial^{\mu}\vec{\pi}\,)+{\sigma_{N}\over 2f_{\pi}^{2}}\,\bar{\Psi}\Psi\,\vec{\pi}^{\,2}+...~~,$

where we have not shown additional terms involving $(\partial^{\mu}\vec{\pi})^{2}$ that arise from the complete Lagrangian ${\cal L}_{\pi N}^{(2)}$. These terms come with further low-energy constants $c_{3}$ and $c_{4}$ that encode physics at smaller distance scales and that need to be fitted to experimental data, e.g., from pion-nucleon scattering.

The baryon in the MIT bag has its baryon charge entirely inside the bag. Classically the baryon charge is confined inside. It turns out that when the pions are coupled to the quarks in the chiral bag, this is no longer the case. This was noted a long time ago when the pion field outside the bag had a hedgehog configuration [35] at the “magic angle” [34] $\theta=\pi/2$,

$\displaystyle U=e^{i\tau\cdot\hat{r}\theta}.$

(22)

In analogy to a fermion coupled to a soliton (e.g., magnetic monopole) involving a zero mode for the fermion which turns out to have a 1/2 fermion charge [36], the bag containing 3 quarks was found to have a 1/2 baryon charge. The rediscovery of the skyrmion model for the nucleon in 1983 immediately led to the identification of the missing 1/2 baryon charge in the pion cloud [35, 34]. This was for the magic angle $\theta=\pi/2$. That the partition of one unit of baryon charge into the inside and outside of the bag takes place for any chiral angle was then shown by Goldstone and Jaffe [37] using the same chiral bag model. This meant that when the bag radius goes to $\infty$, the baryon charge is entirely inside the bag whereas when the bag radius shrinks to a point, the charge gets lodged entirely in the pion cloud.

Why and how this “leakage of baryon charge” occurs is a story of anomaly caused by infinities, often referred to as “infinite-hotel phenomenon.” [38, 39] A simple way of understanding this phenomenon is to recognize that the boundary condition resulting from imposing the axial-current conservation, i.e., chiral invariance, renders the $U(1)_{V}$ vector current to be violated at the surface and hence the baryon charge leaks out. This is because the chiral invariance generates at the boundary an abelian axial vector field and this leads to the vector anomaly. The chiral bag model is constructed in such a way that this leaking baryon charge is picked up precisely by the topology lodged in the pion field.

What this implies is that as far as the baryon charge is concerned, the bag size has no physical meaning. The statement that this phenomenon applies to all physical processes is the Cheshire Cat principle (CCP for short) [38], presaged in [34]. The boundary conditions connect the baryon, a fermion, to the pion, a boson. In (1+1) dimensions, the exact bosonization technique is available to establish the CCP. There is no exact bosonization in (3+1)D, hence the idea of CC can be at best approximate, and the chiral bag described by the action in Eq. (21) is expected to work at low energy, at best, approximately. As developed elsewhere, one can make the model applicable at higher-energy scale by incorporating hidden local fields.

The upshot of the reasoning developed here is that QCD dynamics can be traded in for hadron dynamics via topology. It is the boundary that does the trading.⁷⁷7Such a singular role of boundaries is fairly well recognized in condensed matter physics. Although this model has not been fully explored due to such subtleties as Casimir effects and associated $1/N_{c}$ corrections that are extremely difficult to treat reliably, what it suggests is that there is a hadron-quark duality in nuclear interactions. This feature will be an underlying principle of this article in a variety of nuclear processes at low energy here as well as in other sections.

The CCP for the baryon charge we saw above is exact thanks to the topological invariance, independently of the dimensions of the space-time and of specific dynamics involved. We now consider a case where the topology does not impose an obvious direct condition, and yet the CCP, nonetheless, holds, albeit approximately. A subtle nontrivial manifestation of CCP is in the role the color degree of freedom of QCD plays in hadron properties. A highly intriguing example is the flavor-singlet axial charge $g_{A}^{0}$ of the proton. This object has figured importantly in the so-called “proton spin problem.” The way $g_{A}^{0}$ enters into the proton spin is quite intricate and remains still controversial, and we won’t go into it.

The physically relevant object involved is the sum rule for the first moment of the proton, i.e.,

$$\Gamma^{P}_{1}(Q^{2})\equiv\int_{0}^{1}dxg_{1}^{p}(x,Q^{2})=\frac{1}{12}C_{1}^{NS}(\alpha_{s}(Q^{2}))\left(a^{3}+\frac{1}{3}a^{8}\right)$$

$\displaystyle+\frac{1}{9}C_{1}^{S}(\alpha_{S}(Q^{2}))a^{0}(Q^{2}).$

(23)

Here $C_{1}(\alpha_{s})$’s are first moments of the Wilson coefficients of the singlet ($S$) and nonsinglet ($NS$) axial currents and $\alpha_{s}$ the perturbatively running $QCD$ coupling constant. $a^{3}$, $a^{8}$ and $a^{0}(Q^{2})$ are the form factors in the forward proton matrix elements of the renormalized axial current, i.e.,

$$\langle p,s|A^{3}_{\mu}|p,s\rangle=s_{\mu}\frac{1}{2}a^{3},\;\;\;\;\;\langle p,s|A^{8}_{\mu}|p,s\rangle=s_{\mu}\frac{1}{2\sqrt{3}}a^{8},$$

and

$\displaystyle\langle p,s|A^{0}_{\mu}|p,s\rangle=s_{\mu}a^{0},$

(24)

where $p_{\mu}$ and $s_{\mu}$ are the momentum and the polarization vector of the proton. $a^{3}$ and $a^{8}$ can be chosen $Q^{2}$ independent and may be determined from the $\frac{G_{A}}{G_{B}}$ and $\frac{F}{D}$ ratios. $a^{0}(Q^{2})$ evolves due to the anomaly. Naive models or the $OZI$ approximation to $QCD$ lead, at low energies, to

$\displaystyle a^{0}\approx a^{8}\approx 0.69\pm 0.06.$

(25)

Experimentally [40]⁸⁸8This discrepancy between the constituent quark model and the experiment, with the FSAC interpreted in terms of the proton spin, led to the proton spin crisis.

$\displaystyle a^{0}=0.12(17)\ {\rm or}\ 0.29(6)\,,$

(26)

and lattice QCD gives $a_{0}=0.20(12)$.

As with other physical quantities considered above, we partition the axial current that we are interested in into a component inside and a component outside as

$\displaystyle A^{\mu}=A^{\mu}_{B}\Theta_{B}+A^{\mu}_{M}\Theta_{M}$

(27)

where $\Theta_{B}=\theta(R-r)$ and $\Theta_{M}=\theta(r-R)$ with $R$ being the radius of the bag which may be taken spherical. Since we are dealing only with the flavor-singlet axial current, we omit the flavor index in the current and understand in what follows that $A_{\mu}$ stands for the flavor-singlet axial current (FSAC for short).

Due to the $U(1)_{A}$ anomaly, the FSAC is not conserved. The divergence of the FSAC is taken to be given by⁹⁹9In what follows, $\eta$ will stand for $\eta^{\prime}$.

$\displaystyle\partial_{\mu}A^{\mu}=\frac{\alpha_{s}N_{F}}{2\pi}\sum_{a}\vec{E}^{a}\cdot\vec{B}^{a}\Theta_{B}+fm_{\eta}^{2}\eta\Theta_{M}.$

(28)

This formula should be subject to boundary conditions at the bag boundary

$\displaystyle n_{\mu}A^{\mu}_{B}=n_{\mu}A^{\mu}_{M}$

(29)

where $n^{\mu}$ is the outward normal to the bag surface with $n^{2}=-1$.

Due to the $U(1)_{A}$ anomaly, this boundary condition becomes a lot subtler than above. Since $\eta^{\prime}$ is present in addition to the pion field which accounts for spontaneously broken chiral symmetry, the vacuum fluctuations inside the bag that induce the baryon charge leakage into the skyrmion outside also induce a color leakage due to coupling to a pseudoscalar [41]. This is a quantum effect associated with the chiral anomaly. In order to preserve color invariance, this leakage of color charge has to be prevented and this can be done by putting into the CBM Lagrangian a counter term of the form (for $N_{c}=3$)

$\displaystyle{\cal L}_{CT}=i\frac{g_{s}^{2}}{32\pi^{2}}\oint_{\Sigma}d\beta K^{\mu}n_{\mu}({{\rm tr}}\,{{\rm ln}}\,U^{\dagger}-{{\rm tr}}\,{{\rm ln}}\,U)$

(30)

where $\beta$ is a point on a surface $\Sigma$, $U$ is the $U(N_{F})$ matrix-valued field written as $U=e^{i\pi/f}e^{i\eta/f}$ and $K^{\mu}$ the properly regularized Chern-Simons current $K^{\mu}=\epsilon^{\mu\nu\alpha\beta}(G_{\nu}^{a}G_{\alpha\beta}^{a}-\frac{2}{3}f^{abc}g_{s}G_{\nu}^{a}G_{\alpha}^{b}G_{\beta}^{c})$ given in terms of the color gauge field $G^{a}_{\mu}$. Note that Eq. (30) manifestly breaks color gauge invariance (both large and small, the latter due to the bag), so the action of the chiral bag model with this term is not gauge invariant at the classical level but when quantum fluctuations are calculated [41], there appears an induced anomaly term on the surface which is exactly canceled by Eq. (30). Gauge invariance is preserved, not at the classical level, i.e., in the Lagrangian, but at the quantum level.

For numerically evaluating the matrix elements, it is found to be convenient to rewrite the current in Eq. (27) as

$\displaystyle A^{\mu}=A_{B_{Q}}^{\mu}+A_{B_{G}}^{\mu}+A_{\eta}^{\mu}$

(31)

such that Eq. (28) is in the form

	$\displaystyle\partial_{\mu}(A_{B_{Q}}^{\mu}+A_{\eta}^{\mu})$	$\displaystyle=$	$\displaystyle fm_{\eta}^{2}\eta\Theta_{M},$		(32)
	$\displaystyle\partial_{\mu}A_{B_{G}}^{\mu}$	$\displaystyle=$	$\displaystyle\frac{\alpha_{s}N_{F}}{2\pi}\sum_{a}\vec{E}^{a}\cdot\vec{B}^{a}\Theta_{B}$		(33)

where the subindices Q and G imply that these currents are written in terms of quark and gluon fields respectively. It should be stressed that since one is dealing with an interacting theory, there is no unique way to separate the different contributions from the gluon, quark and $\eta$ components. In particular, the separation into Eqs. (32) and (33) is neither unique nor physically meaningful, although the sum has no ambiguity. Separately, they may not necessarily be gauge-invariant. The merit of this separation is a natural partition of the contributions in the framework of the bag description for the confinement mechanism that we are using here.

The flavor singlet matrix element $a^{0}\equiv g_{A}^{0}$ of Eq. (24) is comprised of two terms, “quarkish” ($Q$) and “gluonish” ($G$):

	$\displaystyle s^{\mu}a^{0}_{Q}$	$\displaystyle=$	$\displaystyle\langle p,s|(A_{B_{Q}}^{\mu}+A_{\eta}^{\mu})|p,s\rangle,$		(34)
	$\displaystyle s^{\mu}a^{0}_{G}$	$\displaystyle=$	$\displaystyle\langle p,s|A_{B_{G}}^{\mu}|p,s\rangle.$		(35)

The boundary condition Eq. (29) allows one to express the $\eta$ current contribution in terms of the quark-bag contribution,

$\displaystyle a^{0}_{Q}=(1+c(m_{\eta}R))a^{0}_{B_{Q}}$

(36)

with the coefficient $c$ that can be readily calculated with the (MIT) bag wavefunction. It is found that $c=1/2$ for $m_{\eta}R=0$ and $c=0$ for $m_{\eta}=\infty$. This shows that $a^{0}_{Q}$, zero for $R=0$, will increase as $R$ increases as one can see in Fig. 3. The $U(1)_{A}$ anomaly plays no role here.

On the contrary, the gluonish component of the FSAC is controlled strongly by the axial anomaly with the boundary term Eq. (30) playing a crucial role. The charge $a^{0}_{G}$ consists also of two terms. One, denoted $a^{0}_{G,stat}$, arises from the mixing of light quarks with the anomaly $\vec{E}^{a}\cdot\vec{B}^{a}$. For large bag size, this term contributes importantly with the sign opposite to the quarkish component. This is seen in Fig. 3. At $R=0$, both the quarkish charge and $a^{0}_{G,stat}$ vanish identically. This is what one would expect in the skyrmion result in the chiral limit. For $R\neq 0$, the two contributions largely cancel leaving a small positive value.

The other gluonish component is the Casimir effect due to vacuum fluctuations in the gluon sector. As in all Casimir problems, it is difficult to evaluate this term reliably. Fortunately, however, suppressed in the large $N_{c}$ limit, it is expected to be negligible except for $R\sim 0$. What is given in Fig. 3 has a large uncertainty but confirms this expectation. What is notable is that it is nonzero for $R\rightarrow 0$ and deviates from the pure skyrmion for $R=0$.

The message from this exercise is that although numerical values may change with more sophisticated treatments, the qualitative feature exhibiting the smallness of the FSAC is robust. While the baryon charge anomaly induced at the boundary gives rise to the baryon charge leakage from the bag assuring the exact CC for the baryon charge, here it is the color anomaly lodged at the boundary that assures the CC for the FSAC. Note that the principal action is at the boundary.

Another case where the Cheshire Cat principle is operative and shown to work in a surprising way is the vector dominance (VD) in the EM form factors of light-quark hadrons, in particular, the pion and the nucleon. Here the infinite tower of hidden gauge fields described in Section 6 play an essential role.

The key development on this matter is the holographic QCD modeled by Sakai and Sugimoto mentioned in Section 6. What transpires from the development of the gravity-gauge duality in hadron physics at low energy is that the EM form factors of the pion [43] and nucleon [44] are given entirely by the infinite tower of the isovector vector mesons $\rho,\rho^{\prime},\rho^{\prime\prime},...$ and the isoscalar vector mesons $\omega,\omega^{\prime},\omega^{\prime\prime},...$. That the pionic form factor is described by the infinite sum of the isosvector vector mesons is perhaps not surprising. Indeed, it has been known – as Sakurai vector dominance – for a long time that the dominance of the lowest-lying $\rho$ works well for the pion or more generally light-quark mesons. One can, therefore, think of the holographic result as an improvement over the Sakurai vector dominance. That the nucleon form factor is also given entirely by the sum of the infinite tower of vector mesons is highly non-trivial and significant.

It has been known since many years that in stark contrast to the pion, the Sakurai VD failed to work for the nucleon. One possible remedy for this defect was found in the hybrid structure of the nucleon with the quark bag and the cloud of the $\rho$ and $\omega$ mesons. In Ref. [45] was shown that a chiral bag in which the baryon number is partitioned equally into the interior and exterior of the bag, so-called chiral bag at magic angle $\theta=\pi/2$, can fairly well explain the nucleon form factors at low momentum transfers. What the holographic structure is indicating is that the physics of quark-bag with the explicit quark degree of freedom, identified pictorially as “Cheshire Cat smile,” can be entirely captured by the infinite tower of vector mesons. How the Cheshire Cat smile can “hide” in the infinite tower of vector meson cloud is explained by Zahed in the holographic picture of QCD [46]. The crucial difference in structure between mesons and baryons is the role of the Cheshire Cat: It is in the baryons but absent in the mesons. In this connection, an interesting possibility is that this difference in structure between the baryons and mesons, though not visible in other models such as constituent quark models, could be in QCD as suggested by Kaplan [47]. Another intriguing theoretical question is whether there is a process in which the Cheshire Cat smile cannot be completely “hidden.” A possible case was suggested in the proton decay [48] where the “smile” may provide a potentially important nonperturbative mechanism for suppressing the decay matrix element. This issue will be settled in future high-energy experiments.

In chiral effective field theory (ChEFT), the nuclear force is given in terms of explicit one-pion and multi-pion exchange process constrained by chiral symmetry as well as a complete set of contact-terms encoding unresolved short-distance dynamics (see Fig. 5). Diagrammatic contributions are organized in powers of small external momenta (or the pion mass) over the chiral symmetry breaking scale $\Lambda_{\chi}\sim 1$ GeV [13, 14, 15]. Two-body forces enter first at leading-order (LO), corresponding to $(q/\Lambda)^{0}$, and include the one-pion exchange contribution together with two contact terms acting in relative $S$-waves:

$$V_{NN}^{(\rm LO)}=-{g_{A}^{2}\over 4f_{\pi}^{2}}\,{\vec{\sigma}_{1}\cdot\vec{q}\,\,\vec{\sigma}_{2}\cdot\vec{q}\over m_{\pi}^{2}+q^{2}}\,\vec{\tau}_{1}\cdot\vec{\tau}_{2}+C_{S}+C_{T}\,\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\,.$$

(42)

Here, $\vec{\sigma}_{1,2}$ and $\vec{\tau}_{1,2}$ denote the spin- and isospin operators of the two nucleons, and $\vec{q}$ is the momentum transfer. Next-to-leading order (NLO) terms proportional to $(q/\Lambda)^{2}$ include two-pion exchange diagrams generated by the vertices of the chiral $\pi N$-Lagrangian ${\cal L}_{\pi N}^{(1)}$. All loop integrals can be performed analytically, and the resulting expression reads

	$\displaystyle V_{NN}^{(\rm 2\pi-NLO)}$	$\displaystyle=$	$\displaystyle{1\over 384\pi^{2}f_{\pi}^{4}}\bigg{\{}4m_{\pi}^{2}(1+4g_{A}^{2}-5g_{A}^{4})+q^{2}(1+10g_{A}^{2}-23g_{A}^{4})-{48g_{A}^{4}m_{\pi}^{4}\over 4m_{\pi}^{2}+q^{2}}\bigg{\}}L(q)\,\,\vec{\tau}_{1}\cdot\vec{\tau}_{2}$		(43)
			$\displaystyle+{3g_{A}^{4}\over 64\pi^{2}f_{\pi}^{4}}L(q)\,(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\,\vec{q}^{\,2}-\vec{\sigma}_{1}\cdot\vec{q}\,\,\vec{\sigma}_{2}\cdot\vec{q}\,)\,,$

which introduces a nontrivial logarithmic loop function

$$L(q)={\sqrt{4m_{\pi}^{2}+q^{2}}\over q}{\rm ln}{q+\sqrt{4m_{\pi}^{2}+q^{2}}\over 2m_{\pi}}\,.$$

(44)

The NLO contributions generate only isovector central, isoscalar spin-spin and isoscalar tensor forces. Additional polynomial pieces generated by the pion-loops have been left out in the above expression and can be absorbed into the most general contact-term at NLO:

	$\displaystyle V_{NN}^{(\rm ct-NLO)}$	$\displaystyle=$	$\displaystyle C_{1}\,\vec{q}^{\,2}+C_{2}\,\vec{k}^{\,2}+(C_{3}\,\vec{q}^{\,2}+C_{4}\,\vec{k}^{\,2})\,\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}+{i\over 2}C_{5}\,(\vec{\sigma}_{1}+\vec{\sigma}_{2})\cdot(\vec{q}\times\vec{k}\,)$		(45)
			$\displaystyle+C_{6}\,\vec{\sigma}_{1}\cdot\vec{q}\,\,\vec{\sigma}_{2}\cdot\vec{q}\,+C_{7}\,\vec{\sigma}_{1}\cdot\vec{k}\,\,\vec{\sigma}_{2}\cdot\vec{k}\,,$

with $\vec{k}=(\vec{p}+\vec{p}\,^{\prime})/2$ the half-sum of initial $\vec{p}$ and final $\vec{p}\,^{\prime}$ nucleon momenta. The low-energy constants $C_{1},\dots,C_{7}$ are adjusted to fit empirical NN scattering phase shifts. The coefficient $C_{5}$ parameterizes the strength of the short-distance spin-orbit interaction, which dominates over the finite-range contributions to the spin-orbit force arising at higher orders. Consequences for the nuclear energy density functional, where $C_{5}$ determines the self-consistent single-particle spin-orbit potential in finite nuclei proportional to the density gradient, will be discussed in Section 4.5.

The most important pieces from chiral two-pion exchange that generate attraction in the isoscalar central channel and reduce the strong $1\pi$-exchange isovector tensor force are still absent at NLO. These terms arise [70] from subleading $2\pi$-exchange through the chiral $\pi\pi NN$ contact couplings $c_{1,3,4}$ in ${\cal L}_{\pi N}^{(2)}$ (or from the inclusion of explicit $\Delta(1232)$-isobar degrees of freedom):

	$\displaystyle V_{NN}^{(\rm 2\pi-N^{2}LO)}$	$\displaystyle=$	$\displaystyle{3g_{A}^{2}\over 16\pi f_{\pi}^{4}}\Big{[}c_{3}\,q^{2}+2m_{\pi}^{2}(c_{3}-2c_{1})\Big{]}(2m_{\pi}^{2}+q^{2})A(q)$		(46)
			$\displaystyle+{g_{A}^{2}c_{4}\over 32\pi f_{\pi}^{4}}(4m_{\pi}^{2}+q^{2})A(q)\,(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\,\vec{q}^{\,2}-\vec{\sigma}_{1}\cdot\vec{q}\,\,\vec{\sigma}_{2}\cdot\vec{q}\,)\,\vec{\tau}_{1}\cdot\vec{\tau}_{2}\,,$

where the arctan loop function

$$A(q)={1\over 2q}\arctan{q\over 2m_{\pi}}\,.$$

(47)

In addition there are relativistic $1/M_{N}$-corrections to $2\pi$-exchange whose explicit form [70, 13] depends on the precise definition of the nucleon-nucleon potential $V_{NN}$, which itself is not an observable. Current state of the art NN potentials have been constructed up to order N³LO, $(q/\Lambda)^{4}$, which includes two-loop $2\pi$-exchange processes, $3\pi$-exchange terms and contact forces quartic in the momenta parameterized by 15 additional low-energy constants $D_{1},\dots,D_{15}$. When solving the Lippmann-Schwinger equation the chiral NN potential is multiplied by an exponential regulator function with a cutoff scale $\Lambda=400-700\,$MeV in order to restrict the potential to the low-momentum region where chiral effective field theory is applicable.

At order N³LO the chiral NN potential reaches the quality of a “high-precision” ($\chi^{2}/DOF\sim 1$) potential in reproducing empirical NN scattering phase shifts and deuteron properties. At the same time it provides the foundation for systematic nuclear structure studies of few- and many-body systems.

An advantage of the organization scheme provided by chiral effective field theory is that nuclear many-body forces are generated consistently with the nucleon-nucleon potential. Three-body forces arise first at N²LO and contain three different topologies. The two-pion exchange component contains terms proportional to the low-energy constants $c_{1}$, $c_{3}$, and $c_{4}$ and has the form

$$V_{3N}^{(2\pi)}=\sum_{i\neq j\neq k}\frac{g_{A}^{2}}{8f_{\pi}^{4}}\frac{\vec{\sigma}_{i}\cdot\vec{q}_{i}\,\vec{\sigma}_{j}\cdot\vec{q}_{j}}{(\vec{q_{i}}^{2}+m_{\pi}^{2})(\vec{q_{j}}^{2}+m_{\pi}^{2})}F_{ijk}^{\alpha\beta}\tau_{i}^{\alpha}\tau_{j}^{\beta},$$

(48)

where $\vec{q}_{i}$ is the difference between the final and initial momenta of nucleon $i$, and the isospin tensor $F_{ijk}^{\alpha\beta}$ is given by

$$F_{ijk}^{\alpha\beta}=\delta^{\alpha\beta}\left(-4c_{1}m_{\pi}^{2}+2c_{3}\,\vec{q}_{i}\cdot\vec{q}_{j}\right)+c_{4}\,\epsilon^{\alpha\beta\gamma}\tau_{k}^{\gamma}\,\vec{\sigma}_{k}\cdot\left(\vec{q}_{i}\times\vec{q}_{j}\right).$$

(49)

The low-energy constants $c_{1}$, $c_{3}$, and $c_{4}$ can be fit to empirical pion-nucleon [71] or nucleon-nucleon scattering [72, 73].

The one-pion exchange term proportional to the low-energy constant $c_{D}$ has the form

$$V_{3N}^{(1\pi)}=-\sum_{i\neq j\neq k}\frac{g_{A}c_{D}}{8f_{\pi}^{4}\,\Lambda}\,\frac{\vec{\sigma}_{j}\cdot\vec{q}_{j}}{\vec{q_{j}}^{2}+m_{\pi}^{2}}\,\vec{\sigma}_{i}\cdot\vec{q}_{j}\,{\vec{\tau}}_{i}\cdot{\vec{\tau}}_{j}\,,$$

(50)

and the three-nucleon contact force proportional to $c_{E}$ is written:

$$V_{3N}^{(\rm ct)}=\sum_{i\neq j\neq k}\frac{c_{E}}{2f_{\pi}^{4}\,\Lambda}\,{\vec{\tau}}_{i}\cdot{\vec{\tau}}_{j}\,,$$

(51)

where $\Lambda=700$ MeV sets a natural scale. Ideally the low-energy constants $c_{D}$ and $c_{E}$ are fit to properties of three-body systems only, and the triton binding energy and lifetime provide a pair of weakly-correlated observables for this purpose [74, 75].

At order N³LO in the chiral power counting, additional three- and four-nucleon forces arise without any additional undetermined low-energy constants. The N³LO three-body force is written schematically as

$\displaystyle V_{3N}^{(4)}=V_{2\pi}^{(4)}+V_{1\pi-\rm cont.}^{(4)}+V_{2\pi-1\pi}^{(4)}+V_{\rm ring}^{(4)}+V_{2\pi-\rm cont.}^{(4)}+V_{1/m}^{(4)},$

(52)

corresponding to the $2\pi$ and $1\pi-{\rm contact}$ topologies (present already at order N$2$LO) as well as the $2\pi-1\pi$, ring, $2\pi-{\rm contact}$ and $1/m$ topologies. All contributions have been worked out and presented in Refs. [76, 77, 78]. The N³LO four-nucleon force has been calculated [79] and the resulting expressions are considerably simpler than the N³LO three-nucleon force.

To facilitate the implementation of chiral three-nucleon forces in studies of finite nuclei and infinite nuclear matter, a strategy pursued in many recent works is to replace the full chiral three-body force with a normal-ordered two-body approximation [80, 81, 82, 83, 84, 85]. Then the three-body force in second-quantized notation

$\displaystyle V_{3N}=\frac{1}{36}\sum_{123456}\langle 123|\bar{V}|456\rangle\hat{a}^{\dagger}_{1}\hat{a}^{\dagger}_{2}\hat{a}^{\dagger}_{3}\hat{a}_{6}\hat{a}_{5}\hat{a}_{4}$

(53)

with antisymmetrized matrix elements $\langle 123|\bar{V}|456\rangle$, is written

	$\displaystyle V_{3N}$	$\displaystyle=$	$\displaystyle\frac{1}{6}\sum_{ijk}\langle ijk|\bar{V}_{3N}|ijk\rangle+\frac{1}{2}\sum_{ij14}\langle ij1|\bar{V}_{3N}|ij4\rangle:\!\hat{a}^{\dagger}_{1}\hat{a}_{4}\!:+\frac{1}{4}\sum_{i1245}\langle i12|\bar{V}_{3N}|i45\rangle:\!\hat{a}^{\dagger}_{1}\hat{a}^{\dagger}_{2}\hat{a}_{5}\hat{a}_{4}\!:$		(54)
			$\displaystyle+{1\over 36}\sum_{123456}\langle 123|\bar{V}_{3N}|456\rangle:\!\hat{a}^{\dagger}_{1}\hat{a}^{\dagger}_{2}\hat{a}^{\dagger}_{3}\hat{a}_{6}\hat{a}_{5}\hat{a}_{4}\!:,$

where the indices $i,j,k$ represent filled orbitals in the chosen reference state $|\Omega\rangle$, and $:\,\,:$ denotes the normal-ordered product of operators that satisfies

$\displaystyle:\!\hat{a}^{\dagger}_{1}\dots\hat{a}_{n}\!:|\Omega\rangle=0.$

(55)