Axial U(1) Anomaly in QCD

• Axial U(1) anomaly is a quantum phenomenon where the classical conservation of the flavor-singlet axial current is broken due to non-invariance of the fermionic measure and topological gauge effects.
• It plays a critical role in QCD by inducing mass for the η' meson and affecting effective field theories, with implications for hadron spectra and phase transitions.
• Lattice studies using chirally symmetric operators, such as the overlap fermion formulation, confirm that topological fluctuations drive the anomaly, which effectively vanishes above the chiral restoration temperature.

The axial U(1) anomaly is a fundamental quantum field theoretic phenomenon that breaks the classical conservation of the flavor-singlet axial current in gauge theories with chiral fermions, such as quantum chromodynamics (QCD). While the Lagrangian of QCD with massless quarks is invariant under continuous axial U(1) transformations, quantum effects—specifically the non-invariance of the fermionic measure under chiral transformations—lead to an anomalous divergence of the axial current. The anomaly is tightly connected to the topology of the gauge field configurations and has profound implications for the hadron spectrum, the structure of effective field theories, the phase diagram of QCD at finite temperature and density, and the nature of global symmetries in quantum field theory.

1. Operator Origin and Field-Theoretic Structure

The central object of the axial U(1) anomaly is the flavor-singlet axial current,

$$J_5^\mu(x) = \sum_{f=1}^{N_f} \bar\psi_f(x) \gamma^\mu\gamma_5 \psi_f(x)$$

In classical massless QCD, this current is conserved, $\partial_\mu J_5^\mu = 0$. However, quantization of the theory spoils this conservation through the path-integral measure's non-invariance under U(1)$_A$ transformations. The anomaly is most transparently derived via Fujikawa's method, which shows that the Jacobian of the fermionic measure under a local chiral rotation is nontrivial: $$\partial_\mu J_5^\mu(x) = 2i\sum_{f=1}^{N_f} m_f\, \bar\psi_f(x)\gamma_5\psi_f(x) - \frac{N_f g^2}{16\pi^2} G^a_{\mu\nu} \tilde{G}^{a\mu\nu}$$ where $G^a_{\mu\nu}$ is the gluonic field strength, $$\tilde{G}^{a\mu\nu} = \tfrac12 \epsilon^{\mu\nu\rho\sigma} G^a_{\rho\sigma}$$, and $g$ is the gauge coupling (Karasik, 2021, Takeuchi, 10 Mar 2025). In the chiral limit ($m_f\to 0$), the divergence is entirely determined by the gauge field topology: $$\partial_\mu J_5^\mu(x) = -\frac{N_f g^2}{16\pi^2} G^a_{\mu\nu} \tilde{G}^{a\mu\nu}$$ The anomaly is directly related to the topological charge density $Q(x)$ and underpins physics associated with instantons and vacuum structure.

2. Topological Origin and Mathematical Formulations

The anomalous term in the divergence of $J_5^\mu$ is proportional to the second Chern-Pontryagin density, whose spacetime integral is an integer—the instanton number: $$Q = \int d^4x\, q(x)\,, \quad q(x) = \frac{g^2}{64\pi^2} \epsilon^{\mu\nu\rho\sigma} \operatorname{Tr} [F_{\mu\nu} F_{\rho\sigma}]$$ Gauge configurations with nonzero $Q$ interpolate between different topological vacua. The anomaly enforces that axial rotations are sensitive to the topological sector, leading to the breaking of the classical U(1)$_A$ symmetry down to a discrete subgroup $Z_{2N_f}$ (Azcoiti, 2019).

Lattice gauge theory, especially with Dirac operators satisfying the Ginsparg–Wilson relation (notably the overlap operator), provides a nonperturbative, regulator-independent realization of the anomaly. The index theorem relates zero modes of the Dirac operator to the topological charge, ensuring the correct quantization of the anomaly (Aoki et al., 2012, Cossu et al., 2013, collaboration et al., 2015).

3. Manifestations in Hadronic Physics

The most prominent physical consequence of the axial U(1) anomaly is the resolution of the so-called U(1) problem: the would-be ninth Goldstone boson, the $\eta'$, does not remain massless in the chiral limit due to explicit U(1)$_A$ breaking by the anomaly. The Witten–Veneziano relation ties the $\eta'$ mass to the topological susceptibility of pure Yang–Mills theory: $$m_{\eta'}^2 \sim \frac{2N_f}{f_\pi^2} \chi_\mathrm{top}$$ Effective hadronic models at zero temperature, e.g., linear sigma models including the 't Hooft determinant term or its variants, encode the anomaly through terms proportional to $\det\Phi$ (or variants like $(\det\Phi-\det\Phi^\dagger)^2$), which explicitly break U(1)$_A$ symmetry while preserving $SU(N_f)_V$. Phenomenological fits demonstrate that both the linear and quadratic determinant forms reproduce hadron spectra and decay phenomenology equally well at $T=0$; the anomaly term produces the correct $\eta$–$\eta'$ splitting and mixing (Kovacs et al., 2013).

4. The Anomaly at Finite Temperature and Density

Central to QCD thermodynamics is the fate of the U(1)$_A$ anomaly above the chiral restoration temperature $T_c$. Multiple high-precision lattice simulations employing chirally symmetric Dirac operators have revealed that:

• The near-zero eigenmodes of the Dirac operator (which saturate the anomaly) are suppressed at $T \gtrsim T_c$, leading to a gap in the Dirac spectrum (Cossu et al., 2013, Aoki et al., 2020, collaboration et al., 2015).
• Mesonic susceptibilities sensitive to U(1)$_A$ breaking, notably $\chi_{\pi}-\chi_\delta$, vanish in the chiral limit, reflecting the degeneracy of U(1)$_A$ partners (e.g., $\pi$ and $\delta$) (collaboration et al., 2015, Aoki et al., 2022).
• The topological susceptibility $\chi_\mathrm{top}$ falls rapidly with increasing temperature and vanishes with the same rate (in $m$) as $SU(N_f)_A$ restoration (Fukaya, 2017, Aoki et al., 2020, Aoki et al., 2024).

These findings demonstrate that, for $N_f=2$, the axial U(1) anomaly becomes "invisible"—i.e., it no longer leaves measurable imprints in two-point correlation functions or susceptibilities—immediately above $T_c$ (Aoki et al., 2012). The restoration is less abrupt but qualitatively similar for $N_f=2+1$ (Aoki et al., 2022). This "effective restoration" is a statement about observable correlators, not the absence of global quantum anomalies in the underlying path integral.

Implications include possible modifications of the chiral phase transition universality class (from O(4) to $U(2)_L \times U(2)_R$ or first order), with direct significance for heavy-ion phenomenology and axion cosmology (Aoki et al., 2024, Aoki et al., 2021).

5. Interactions with Generalized Symmetries and Non-Invertible Structures

Recent lattice and continuum work has illuminated new aspects of the anomaly in the context of generalized global (higher-form and non-invertible) symmetries. Explicitly, in lattice axion–QED models, the non-invertible symmetry operator implementing the discrete axial transformation acts trivially on suitably "dressed" 't Hooft line operators; the would-be anomalous phase possession is exactly canceled by the corresponding dressing factor, highlighting the necessity of higher-cup products and topological terms for consistency (Honda et al., 2024).

Furthermore, the "anomaly equation for large U(1)" transformations reveals an infinite set of anomalous conservation laws—one for each function $\epsilon(x)$ labeling the axial transformation—whose violation is universally controlled by $F \wedge F$, linking the anomaly with infrared soft theorems and memory effects (Takeuchi, 10 Mar 2025).

6. Alternative Mechanisms and Limitations of the Anomaly

There are proposals, based on lattice and analytic studies, suggesting alternative mechanisms for $\eta$–$\eta'$ mass generation without invoking an explicit chiral anomaly, instead leveraging the first-order disconnected contributions to meson correlators. These mechanisms can achieve 20%–level agreement with experiment for $m_\eta$, $m_{\eta'}$ by parameter tuning, and remain consistent with lattice results on U(1)$_A$ restoration at high $T$ (Yamanaka, 2024). Such approaches challenge the standard “instanton-density” paradigm, but the dynamical indistinguishability at the physical point and restoration at high T mitigate direct phenomenological contradictions.

7. Axial Anomaly in Other Contexts: Noncommutative Geometry and Gravity

The axial U(1) anomaly is not restricted to usual QCD-like settings. In noncommutative QED, the anomaly computation parallels the commutative case, provided the Moyal star-product and cyclicity of the trace are respected; the result remains exact at one loop (AlMasri, 2019). In the presence of gravity, the axial anomaly receives an additional contribution—the Kimura–Delbourgo–Salam–Eguchi–Toms gravitational anomaly—proportional to the Pontryagin density constructed from the Riemann tensor: $$D^\alpha\left\langle\bar\psi\gamma_\alpha\gamma_5\psi\right\rangle = \frac{1}{384\pi^2}\,\epsilon_{\mu\nu\rho\sigma}\,R^{\mu\nu\lambda\tau}R^{\rho\sigma}{}_{\lambda\tau}$$ Careful regularization and use of the universal energy–momentum tensor formulas derived via gradient flow reproduce this result (Morikawa et al., 2018).

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Summary Table: Key Lattice Results for U(1)$_A$ Anomaly near Chiral Restoration

Reference	Method/Setup	Anomaly Status at $T > T_c$	Observable(s)
(Cossu et al., 2013)	Overlap fermions, fixed topology	Effective restoration	Dirac gap, meson degeneracy
(collaboration et al., 2015)	Möbius DW, overlap determinant reweighting	Anomaly suppressed $\to$0 in chiral limit	$\chi_{\pi}-\chi_\delta$
(Aoki et al., 2020)	Möbius DW, overlap reweighting	Anomaly vanishes with $m\to0$	$\chi_\mathrm{top}$, spectra
(Aoki et al., 2022)	$2+1$ flavors, overlap reweighting	Anomaly vanishes at $T \gtrsim T_c$	$\Delta_{\pi-\delta}$, $\chi_t$
(Aoki et al., 2024, Aoki et al., 2021)	Overlap, mode decomposition	Anomaly dominates at $T \lesssim T_c$, suppressed at $T\gtrsim T_c$	Chiral susceptibility, eigenmodes
(Aoki et al., 2012)	Overlap, multi-point WTIs	Chiral restoration $\implies$ anomaly invisible	Susceptibilities

These findings collectively establish that the axial U(1) anomaly, while a foundational feature of QCD and related gauge theories, becomes dynamically "invisible" at high temperatures due to the suppression of topological fluctuations, with direct consequences for the properties of the QCD transition and effective field theory descriptions.