Chiral Symmetry in QCD

For $N_f$ massless quark flavors, the QCD Lagrangian has independent left- and right-handed flavor rotations, customarily written $SU(N_f)_L\times SU(N_f)_R\times U(1)_V\times U(1)_A$ up to finite identifications. Quark masses explicitly reduce this symmetry, the quantum measure anomalously removes the continuous singlet axial factor, and the vacuum is expected to realize the remaining non-Abelian chiral symmetry spontaneously as $SU(N_f)_L\times SU(N_f)_R\to SU(N_f)_V$.

Required background. QCD fields, scales, and the perturbative domain supplies the quark representations and covariant derivative; Quantum Currents, Improvements, and Conservation supplies the regulated-current and Ward-identity logic.

Helpful background. Regulated Jacobians and Measure Variation derives the anomaly from the fermion measure.

Left- and right-handed flavor symmetry

Let

$$P_{L,R}=\frac{1\mp\gamma_5}{2}, \qquad q_{L,R}=P_{L,R}q,$$

and collect $N_f$ quark flavors into $q$. Suppressing gauge and flavor indices, the quark terms are

$$\mathcal L_q= i\bar q_L\gamma^\mu D_\mu q_L i\bar q_R\gamma^\mu D_\mu q_R -\bar q_L M q_R-\bar q_R M^\dagger q_L.$$

At $M=0$, the classical action is invariant under

$$q_L\mapsto Lq_L, \qquad q_R\mapsto Rq_R, \qquad L,R\in U(N_f).$$

Decomposing the two unitary factors gives the familiar local Lie-algebra description

$$SU(N_f)_L\times SU(N_f)_R\times U(1)_V\times U(1)_A.$$

The true global group is a quotient by overlapping finite centers; that quotient matters for global anomalies and extended operators, but not for the local current algebra derived here. $U(1)_V$ is quark number, with baryon number obtained by assigning charge $1/N_c$ to each quark. The singlet axial rotation is $q\mapsto e^{i\alpha\gamma_5}q$.

For Hermitian flavor generators $T^a$ normalized by $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$, define

$$V_a^\mu=\bar q\gamma^\mu T^a q, \qquad A_a^\mu=\bar q\gamma^\mu\gamma_5T^a q,$$

and singlet currents $V^mu=\bar q\gamma^\mu q$, $J_5^\mu=\bar q\gamma^\mu\gamma_5q$. The non-singlet currents generate vector rotations $L=R$ and axial rotations $L=R^\dagger$, respectively.

Mass spurions and explicit breaking

The mass term is formally invariant if the mass matrix is treated as a spurion transforming as

$$M\mapsto LMR^\dagger.$$

Freezing $M$ to its physical value then exposes the surviving subgroup. After choosing the ordinary real, Hermitian mass basis (and treating the physical vacuum angle separately), the equations of motion give the non-singlet Ward identities

$$\partial_\mu V_a^\mu=i\bar q[M,T^a]q, \qquad \partial_\mu A_a^\mu=i\bar q\gamma_5\{M,T^a\}q.$$

These equations provide immediate checks:

• if $M=m\mathbf1$, every vector current is conserved, while all non-singlet axial currents are explicitly broken for $m\neq0$;
• if $M$ is diagonal with unequal masses, only vector generators commuting with $M$ remain exact;
• if a subset of masses vanishes and is degenerate, that subset retains its corresponding chiral symmetry before spontaneous breaking.

The spurion transformation is more than notation: it determines which mass-dependent operators may occur in the chiral effective theory. It also makes clear that a statement such as “isospin is exact” always includes an approximation about $m_u-m_d$ and electromagnetic interactions.

The singlet anomaly

The non-singlet axial currents have traceless flavor generators, so their potential gluonic anomalies cancel. The singlet current does not. With

$$q_{\rm top}(x)=\frac{g_s^2}{32\pi^2} G_{\mu\nu}^a\widetilde G^{a\mu\nu}, \qquad \widetilde G^{a\mu\nu}=\frac12\epsilon^{\mu\nu\rho\sigma}G^a_{\rho\sigma},$$

the renormalized anomalous Ward identity is

$$\partial_\mu J_5^\mu =2i\bar qM\gamma_5q+2N_f q_{\rm top}(x).$$

The coefficient is fixed by the regulated fermion measure; the path-integral derivation is given by Fujikawa 1979, pp. 1195–1198. Operator mixing and contact terms require a consistent renormalization prescription, but they do not restore a conserved continuous $U(1)_A$ current.

An axial rotation changes the measure by a phase proportional to $2N_f\alpha Q$, where $Q=\int d^4x\,q_{\rm top}$. For integer $Q$, a discrete axial subgroup survives, conventionally denoted $\mathbb Z_{2N_f}$ before quotienting by transformations already contained in vector centers and fermion parity. Thus the anomaly is explicit quantum breaking of the continuous $U(1)_A$, not spontaneous breaking of an exact continuous symmetry.

Vacuum realization and order parameters

The standard chiral realization is diagnosed by the bifundamental bilinear

$$\Phi^{ij}=\bar q_R^{,j}q_L^{,i}, \qquad \Phi\mapsto L\Phi R^\dagger.$$

In the massless infinite-volume limit, a flavor-symmetric expectation value

$$\langle\Phi^{ij}\rangle=C\,\delta^{ij}, \qquad C\neq0,$$

is invariant precisely under $L=R$. It therefore realizes

$$SU(N_f)_L\times SU(N_f)_R \longrightarrow SU(N_f)_V$$

and breaks $N_f^2-1$ generators. Goldstone’s theorem then requires that many massless modes, subject to its infinite-volume and locality hypotheses. For $N_f=2$ these are the three pions; for $N_f=3$ they form the pseudoscalar octet in the chiral limit. The current-algebra construction is developed on Chiral Order Parameters, Current Algebra, and Pions.

The expectation value is not a finite-volume invariant without a symmetry-breaking source. The operational order is

$$\lim_{m\to0^+}\lim_{V\to\infty}\langle\bar q q\rangle_{m,V},$$

not the reverse. Moreover, the renormalized scalar density is scheme and scale dependent; its products with quark masses and the Ward identities are the safer invariant statements.

Symmetry classification

Effect	Symmetry affected	Diagnostic	Low-energy consequence
Nondegenerate quark masses	Axial flavor and part of vector flavor	$[M,T^a]$ and $\{M,T^a\}$ in the current divergences	Pseudo-Goldstone masses and isospin or flavor breaking
Gluonic anomaly	Continuous $U(1)_A$	$2N_fq_{\rm top}$ in $\partial_\mu J_5^\mu$	No Goldstone theorem for the singlet axial generator
Vacuum alignment	$SU(N_f)_L\times SU(N_f)_R$	Nonzero bifundamental order parameter or equivalent spectral diagnostics	$N_f^2-1$ Goldstone modes at zero quark mass
Degenerate vector subgroup	$SU(N_f)_V$	$[M,T^a]=0$ and an aligned vacuum	Multiplet classification of hadrons

The entries are logically distinct. A quark mass does not “cause the anomaly,” and the anomaly does not explicitly break the non-singlet chiral group. Likewise, the condensate describes vacuum realization; it is not the origin of the anomalous divergence.

Common pitfalls

Calling $U(1)_A$ spontaneously broken. The continuous singlet axial current is already anomalous in massless quantum QCD. Topological dynamics determine its spectral consequences, but there is no exact continuous generator to which the ordinary Goldstone theorem applies.

Forgetting the mass matrix in a symmetry claim. Equal nonzero masses preserve vector flavor but explicitly break axial flavor; unequal masses preserve only the commuting vector subgroup. Write $M$ before naming the exact symmetry.

Taking a condensate at finite volume as an order parameter. At zero source, symmetry averaging makes the finite-volume expectation vanish. Take the thermodynamic limit before removing the source or use finite-volume spectral and Ward-identity diagnostics.

References

• Fujikawa, Kazuo. “Path-Integral Measure for Gauge-Invariant Fermion Theories.” Physical Review Letters 42 (1979): 1195–1198. DOI.
• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007, §83. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge: Cambridge University Press, 1996, §§19.4 and 22.2. DOI.