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Chiral anomaly

The chiral anomaly, also known as the axial anomaly, is a fundamental quantum mechanical effect in quantum field theory (QFT) where a classically conserved chiral (or axial) current—associated with transformations that distinguish left-handed and right-handed fermions—fails to remain conserved at the quantum level due to ultraviolet divergences that cannot be regularized in a chirally invariant manner.^[1]^[2] This anomaly arises in theories involving massless Dirac fermions coupled to gauge fields, such as quantum electrodynamics (QED) or quantum chromodynamics (QCD), and stems from the topological properties of the gauge field configurations, particularly instantons or configurations with nonzero Pontryagin index.^[1]^[2] Discovered independently in 1969 by Stephen Adler and by John Bell and Roman Jackiw through the analysis of the triangle Feynman diagram in the context of neutral pion decay (π⁰ → γγ), the chiral anomaly provided the first explicit example of how quantum corrections can break a classical symmetry, resolving long-standing puzzles in particle physics.^[1] In four spacetime dimensions, the anomaly is mathematically expressed by the non-conservation equation ∂μ j^μ_5 = \frac{g^2}{16\pi^2} \epsilon^{\mu\nu\rho\sigma} \mathrm{Tr}(F{\mu\nu} F_{\rho\sigma}) for a non-Abelian gauge theory, where j^μ_5 is the axial current, g is the coupling constant, F_{\mu\nu} is the field strength tensor, and the trace is over the gauge group representation; for Abelian QED, it simplifies to ∂μ j^μ_5 = \frac{e^2}{8\pi^2} F{\mu\nu} \tilde{F}^{\mu\nu}, with \tilde{F}^{\mu\nu} = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma} F_{\rho\sigma}.^[1]^[2] This topological term, proportional to the Chern-Simons density, reflects the index theorem relating the difference in the number of left- and right-handed zero modes of the Dirac operator to the winding number of the gauge field.^[1] The implications of the chiral anomaly extend across theoretical and experimental physics, prohibiting the gauging of anomalous chiral symmetries and thus constraining the structure of the Standard Model, where it explains the decay of the neutral pion into two photons and contributes to the large mass of the η' meson via the Witten-Veneziano mechanism.^[1]^[2] In non-perturbative contexts, it enables processes that violate baryon and lepton numbers, such as sphaleron-induced transitions in electroweak theory, which could have played a role in the early universe's baryogenesis.^[2] Beyond high-energy physics, the anomaly manifests in condensed matter systems like Weyl semimetals, where it predicts phenomena such as the chiral magnetic effect—negative magnetoresistance in the presence of parallel electric and magnetic fields—verified experimentally in materials like TaAs and NbAs.^[2] These diverse applications underscore the anomaly's role as a bridge between quantum field theory, topology, and observable phenomena.

Introduction

Definition and overview

The chiral anomaly refers to a quantum mechanical phenomenon in which a classical symmetry of the theory, specifically chiral symmetry distinguishing left-handed from right-handed massless fermions, is broken at the quantum level despite the invariance of the classical Lagrangian. This occurs in quantum field theories where these fermions couple to gauge fields, such as in quantum electrodynamics (QED) or quantum chromodynamics (QCD). The effect arises from the regularization and renormalization procedures required in quantum field theory calculations, leading to a mismatch between classical and quantum descriptions.^[3]^[4] A primary consequence of the chiral anomaly is the non-conservation of the axial current, which classically would be preserved due to the symmetry. In the quantum theory, this current acquires an anomalous divergence proportional to the gauge field strength, manifesting in physical processes like particle decays that violate the expected conservation laws. This non-conservation is exact and independent of perturbative expansions, highlighting the fundamental quantum nature of the breaking.^[4] In QCD, the chiral anomaly plays a crucial role in explaining the large mass of the η′ meson compared to other light pseudoscalar mesons. The U(1)_A axial symmetry, expected to yield a nearly massless Goldstone boson, is broken by the anomaly through non-perturbative effects like instantons, generating a substantial mass term for the η′ via the topological susceptibility of the QCD vacuum.^[5] Intuitively, the chiral anomaly can be compared to quantum effects in electromagnetism, where vacuum fluctuations alter classically conserved quantities; for instance, in QED, the infinite Dirac sea under a background field leads to a shift in fermion states, effectively changing the axial charge without classical counterparts.^[4]

Historical context

In the early 1960s, experimental observations of the neutral pion decay into two photons, \pi^0 \to \gamma\gamma, revealed inconsistencies with theoretical expectations based on partially conserved axial-vector currents (PCAC). Calculations assuming exact conservation of the axial current, such as those by Sutherland in 1967, indicated that this decay mode should be forbidden or extremely suppressed due to symmetry constraints, yet measurements showed a branching ratio of nearly 100% with a lifetime on the order of $10^{-16} seconds. These discrepancies hinted at underlying issues with the axial current's conservation in quantum field theories involving electromagnetism. The resolution came in 1969 with the seminal work of Stephen L. Adler and, independently, John S. Bell and Roman Jackiw, who demonstrated through perturbative quantum electrodynamics (QED) that the axial current suffers from an anomalous non-conservation proportional to the field strength tensor. Their calculation, involving the famous triangle Feynman diagram, provided the exact amplitude for \pi^0 \to \gamma\gamma, matching experimental rates and resolving the PCAC puzzle. This Adler-Bell-Jackiw (ABJ) anomaly marked a pivotal shift, revealing how quantum effects break classical symmetries. In the 1970s, the anomaly was extended to quantum chromodynamics (QCD), with Gerard 't Hooft showing in 1976 that non-perturbative instanton configurations generate multi-fermion interactions that violate the axial U(1) symmetry, connecting the phenomenon to the topology of gauge fields. This work addressed the U(1) problem in QCD, as articulated by Steven Weinberg in 1975, where the anomaly's effects explain the unexpectedly large mass of the \eta' meson compared to the light pseudoscalar nonet, preventing an additional Goldstone boson. By the 1980s, the anomaly gained recognition in electroweak theory, where its cancellation between quarks and leptons ensures consistency of the Standard Model. Influential contributions from Sidney Coleman, including proofs that only triangle diagrams contribute to the anomaly in certain limits, further solidified its theoretical foundations. Key milestones include precise experimental verifications of the \pi^0 \to \gamma\gamma decay rate, which align with anomaly predictions to within 1% accuracy as confirmed by measurements in the 1980s and beyond. Roman Jackiw and others highlighted the anomaly's topological origins via the Atiyah-Singer index theorem. Into the 2020s, modern reviews underscore these topological aspects, linking the anomaly to broader phenomena in gauge theories and condensed matter systems.

Theoretical foundations

Chiral symmetry in field theory

In quantum field theory, the foundation for chiral symmetry lies in the Dirac equation, which describes the relativistic dynamics of spin-1/2 fermions. For a free massless Dirac field ψ, the equation is given by

(i \gamma^\mu \partial_\mu) \psi = 0,

where the γ^μ are 4×4 Dirac matrices satisfying the anticommutation relations {γ^μ, γ^ν} = 2 g^{μν} I, with g^{μν} the Minkowski metric and I the identity matrix. These matrices, introduced in the context of combining quantum mechanics with special relativity, allow the Dirac spinor ψ to encode both particle and antiparticle degrees of freedom in a four-component form.^[6]^[7] The concept of chirality, or handedness, emerges from the massless limit of the Dirac theory through projection operators that separate left- and right-handed components. The pseudoscalar matrix γ^5 = i γ^0 γ^1 γ^2 γ^3 anticommutes with all γ^μ ({γ^5, γ^μ} = 0) and has eigenvalues ±1. The chiral projectors are defined as P_L = (1 - γ^5)/2 and P_R = (1 + γ^5)/2, satisfying P_L + P_R = 1 and P_L P_R = 0. These operators decompose the spinor as ψ = ψ_L + ψ_R, where ψ_L = P_L ψ and ψ_R = P_R ψ, corresponding to left- and right-handed Weyl spinors that decouple in the massless case.^[7]^[8] Chiral symmetry refers to the invariance of the classical Lagrangian for N massless Dirac fermions (flavors) under independent global unitary transformations on the left- and right-handed components: ψ_L → U_L ψ_L and ψ_R → U_R ψ_R, where U_L, U_R ∈ SU(N). The corresponding Lagrangian density is \mathcal{L} = \sum_f \bar{\psi}f i \gamma^\mu \partial\mu \psi_f = \sum_f (\bar{\psi}{fL} i \gamma^\mu \partial\mu \psi_{fL} + \bar{\psi}{fR} i \gamma^\mu \partial\mu \psi_{fR}), which remains unchanged since the kinetic terms for left- and right-handed fields are separately invariant. This structure, SU(N)_L × SU(N)_R, distinguishes chiral symmetry from the vector-like SU(N)_V symmetry, which acts identically on both chiralities.^[5] Associated with these symmetries are conserved Noether currents. The vector current is J^\mu = \sum_f \bar{\psi}f \gamma^\mu \psi_f, transforming under SU(N)V, while the axial current is J^\mu_5 = \sum_f \bar{\psi}f \gamma^\mu \gamma^5 \psi_f = \sum_f (\bar{\psi}{fR} \gamma^\mu \psi{fR} - \bar{\psi}{fL} \gamma^\mu \psi_{fL}), associated with the axial SU(N)_A subgroup. In the classical theory for massless fermions, both currents are conserved: ∂_μ J^\mu = 0 and ∂_μ J^\mu_5 = 0, as follows from the equations of motion and the antisymmetry of the massless Dirac operator.^[4]^[9] In quantum chromodynamics (QCD), the theory of strong interactions, chiral symmetry plays a central role for the light quarks (up, down, and strange, with masses much smaller than the QCD scale Λ_QCD ≈ 200 MeV). The QCD Lagrangian for N_f = 3 massless flavors exhibits an approximate SU(3)_L × SU(3)_R symmetry, extended by a U(1)_V baryon number but with U(1)_A broken by the quantum anomaly (though here we focus on the classical SU(3) structure). This symmetry explains patterns in hadron spectroscopy, such as the near-degeneracy of parity-doubled states in the quark model.^[10]^[9] The approximate chiral symmetry in QCD is spontaneously broken by the formation of a nonzero quark condensate in the vacuum, ⟨\bar{q} q⟩ ≠ 0, which is non-invariant under axial transformations. This dynamical breaking, akin to superconductivity in condensed matter, reduces the symmetry to the diagonal SU(3)_V, producing eight massless Goldstone bosons according to the Goldstone theorem—one for each broken axial generator. These bosons are identified with the pseudoscalar octet (π, K, η mesons), with the pions being the lightest and most prominent example for the SU(2) subgroup of up and down quarks. The pion mass arises primarily from explicit symmetry breaking rather than spoiling the Goldstone nature entirely.^[5]^[11] Finally, the chiral symmetry is explicitly broken by the small but nonzero current quark masses in the QCD Lagrangian term - \sum_f m_f \bar{\psi}_f \psi_f, which couples left- and right-handed components and violates SU(N)_L × SU(N)_R down to SU(N)_V. For light quarks, m_u, m_d, m_s ≪ Λ_QCD, this breaking is mild, making the symmetry approximate and the pseudoscalars pseudo-Goldstone bosons with small masses proportional to √m_q. This sets the stage for understanding quantum-level violations beyond the classical framework.^[5]^[9]

Axial anomaly in gauge theories

In gauge theories, the axial anomaly arises as a quantum effect that breaks the classical conservation of the axial current J^\mu_5 = \bar{\psi} \gamma^\mu \gamma_5 \psi, where \psi represents Dirac fermions transforming under a gauge group. This violation couples the divergence of the axial current to the topological structure of the gauge field strength tensor F_{\mu\nu}, ensuring that the anomaly is invariant under gauge transformations despite spoiling the axial symmetry.^[4] The general form of the anomaly in an Abelian gauge theory, such as QED, is given by the anomalous divergence

\partial_\mu J^\mu_5 = \frac{g^2}{16\pi^2} F_{\mu\nu} \tilde{F}^{\mu\nu},

where g is the coupling constant, F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu is the field strength, and \tilde{F}^{\mu\nu} = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma} F_{\rho\sigma} is its dual; this expression was first derived perturbatively for the axial-vector vertex. In non-Abelian gauge theories, the form generalizes to

\partial_\mu J^\mu_5 = \frac{g^2}{16\pi^2} \operatorname{Tr}_R \left[ F_{\mu\nu} \tilde{F}^{\mu\nu} \right],

where the trace is taken over the representation R of the fermions under the gauge group, and F_{\mu\nu} now includes the non-Abelian structure constants; for QCD with N_f massless quark flavors in the fundamental representation, the coefficient includes a factor of N_f.^[4] The topological origin of the anomaly stems from the Atiyah-Singer index theorem, which relates the difference in the number of left- and right-handed zero modes of the Dirac operator to the winding number of the gauge field configuration, given by the integral \nu = \frac{g^2}{32\pi^2} \int d^4x \, \operatorname{Tr} \left[ F_{\mu\nu} \tilde{F}^{\mu\nu} \right]; thus, the integrated anomaly equals $2 N_f \nu, reflecting chirality non-conservation in topologically non-trivial backgrounds like instantons. In chiral gauge theories with only left- or right-handed fermions, the anomaly coefficient must vanish for gauge invariance to hold at the quantum level, prohibiting unbalanced chiral fermion content unless anomalies cancel across representations, as required for consistency.^[4] The anomaly coefficient is determined exactly at one loop, with no higher-order corrections due to the Adler-Bardeen theorem, and its numerical factors depend on the fermion representation, such as the dimension or Casimir invariants entering the trace. Unlike the trace anomaly, which breaks scale invariance through terms involving the beta function and is encoded in the energy-momentum tensor trace \langle T^\mu_\mu \rangle, the axial anomaly is specifically tied to the pseudoscalar structure involving \gamma_5, affecting only chiral symmetries without altering conformal properties.^[4]

Derivations and calculations

Perturbative approach via triangle diagram

The perturbative approach to deriving the chiral anomaly focuses on the one-loop calculation of the axial-vector-vector (AVV) triangle Feynman diagram in quantum electrodynamics (QED) for a massless Dirac fermion. This diagram arises in the evaluation of the three-point correlation function involving one axial current J^\mu_5 = \bar{\psi} \gamma^\mu \gamma_5 \psi and two vector currents J^\mu = \bar{\psi} \gamma^\mu \psi, corresponding to the process where the axial current couples to two photons. The setup considers the amplitude \Delta_{\lambda \mu \nu}(k_1, k_2) in momentum space, with incoming momenta k_1 and k_2 for the vector vertices and outgoing momentum q = k_1 + k_2 for the axial vertex. The Feynman integral for the triangle loop is given by

i \Delta_{\lambda \mu \nu}(k_1, k_2) = (-ie)^3 \int \frac{d^4 p}{(2\pi)^4} \mathrm{Tr} \left[ \gamma_\lambda \gamma_5 \frac{i}{\slashed{p} - \slashed{q}} \gamma_\nu \frac{i}{\slashed{p} - \slashed{k}_1} \gamma_\mu \frac{i}{\slashed{p}} \right] + \text{(cyclic permutations)},

where the trace is over Dirac indices, and the propagators account for the massless fermion loop. This configuration captures the quantum violation of the classically conserved axial current due to the interplay of chiral symmetry and gauge invariance. The calculation proceeds by evaluating the Dirac trace and the momentum integral. The trace \mathrm{Tr}[\gamma_\lambda \gamma_5 \gamma^\alpha \gamma^\beta \gamma^\mu \gamma^\nu] simplifies using the properties of \gamma_5 and the Levi-Civita tensor, yielding terms proportional to \epsilon_{\lambda \mu \nu \alpha \beta} p^\alpha k^\beta after symmetrization over permutations to enforce Bose symmetry for the identical photons. Momentum routing is chosen to facilitate integration, often shifting the loop momentum p \to p + a with an offset a^\mu to manage divergences while preserving Ward identities for the vector currents. The integral is UV divergent, exhibiting both quadratic and logarithmic divergences, but the anomaly emerges as a finite, unambiguous contribution after regularization. Regularization is crucial to isolate the anomaly without introducing scheme-dependent artifacts. Common methods include Pauli-Villars regularization, introducing massive regulator fields to subtract divergences, or dimensional regularization in d = 4 - 2\epsilon dimensions, where the pole terms cancel for the anomalous part due to the evanescent nature of \gamma_5 in non-four dimensions. An alternative geometric approach involves shifting the integration contour to a surface term on the sphere at infinity in momentum space, revealing the anomaly as \int_{S^3} d\Omega \, \hat{k}^\mu f(k), where the finite result depends on the choice of shift that maintains vector gauge invariance (e.g., a = -\frac{1}{2} k_2). This yields the non-conservation of the axial current, with the divergence q^\lambda \Delta_{\lambda \mu \nu} = -\frac{i e^2}{2 \pi^2} \epsilon_{\mu \nu \alpha \beta} k_1^\alpha k_2^\beta. In position space, Fourier transforming back gives the local form of the anomaly. The key result from this calculation is the anomalous divergence of the axial current:

\partial_\mu J^\mu_5 = \frac{e^2}{16 \pi^2} \epsilon^{\mu \nu \rho \sigma} F_{\mu \nu} F_{\rho \sigma},

where F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu is the electromagnetic field strength, and the factor accounts for a single Dirac fermion in QED. This expression is equivalent to \frac{2 \alpha}{\pi} \vec{E} \cdot \vec{B} in the non-relativistic limit, with \alpha = e^2 / 4\pi. The anomaly term is gauge-invariant and antisymmetric under vector gauge transformations A_\mu \to A_\mu + \partial_\mu \Lambda, as the \epsilon^{\mu \nu \rho \sigma} F_{\mu \nu} F_{\rho \sigma} is a topological invariant unchanged by such shifts, ensuring consistency with the vector Ward identities while breaking the axial one. Historically, the explicit evaluation of this integral, resolving ambiguities in regularization and confirming the finite anomalous term, was first performed by Adler in 1969, building on earlier puzzles in pion decay processes and providing a rigorous diagrammatic proof of the quantum violation of chiral symmetry. This work, alongside the independent calculation by Bell and Jackiw, established the perturbative foundation for the chiral anomaly and its role in quantum field theory.

Non-perturbative methods

Non-perturbative methods for deriving the chiral anomaly rely on global properties of the path integral and topological features of the underlying gauge fields, providing insights that transcend perturbative expansions. One prominent approach is Fujikawa's method, which examines the transformation properties of the fermionic path integral measure under infinitesimal chiral rotations. In this framework, the classical action remains invariant under such transformations, but the quantum measure acquires a non-trivial Jacobian due to the regularization required for the functional integral. To handle the ultraviolet divergences, Fujikawa employs a heat kernel regularization, expanding the eigenmodes of the Dirac operator in a gauge-invariant manner. The resulting anomaly manifests as an effective change in the action, given by the local form

\partial_\mu j^\mu_5 = \frac{g^2}{16\pi^2} \operatorname{Tr} (F_{\mu\nu} \tilde{F}^{\mu\nu}),

where F_{\mu\nu} is the field strength tensor, \tilde{F}^{\mu\nu} its dual, and the trace is over gauge indices; this expression arises directly from the non-invariance of the regulated measure. This derivation captures the local form of the anomaly without relying on diagrammatic techniques. Another key non-perturbative perspective links the chiral anomaly to the topology of gauge field configurations via the Atiyah-Singer index theorem. This theorem relates the index of the Dirac operator—defined as the difference between the number of zero modes of positive and negative chirality—to a topological invariant of the background gauge field, specifically the integral of the Chern-Pontryagin density \frac{g^2}{32\pi^2} \operatorname{Tr} [F \wedge F]. In instanton backgrounds, the zero modes of the Dirac operator directly account for the anomalous violation of chiral charge, providing an exact, integer-valued measure of the anomaly that holds non-perturbatively. These methods offer significant advantages over perturbative approaches, as they sum all orders in the coupling constant and yield exact results for abelian anomalies. Moreover, Fujikawa's formalism extends naturally to lattice QCD, where it aids in understanding the implementation of chiral symmetry and the associated anomalies in discretized spacetime.^[12] By confirming the coefficient from the perturbative triangle diagram while uncovering global topological effects—such as the resolution of the U(1) problem through the generation of the \eta' meson mass via instanton contributions—these techniques reveal the anomaly's role in non-perturbative dynamics.

Physical manifestations

Neutral pion decay

The decay of the neutral pion (\pi^0) into two photons (\pi^0 \to \gamma \gamma) is a prominent manifestation of the chiral anomaly in quantum chromodynamics (QCD). Classically, this process is forbidden because the pion, as a pseudoscalar particle, couples to the axial current, while photons couple to the vector current, and the symmetry of the classical theory conserves the axial current. However, quantum effects via the anomaly violate this conservation, enabling the decay at the leading order through a quark triangle loop diagram involving up and down quarks. The amplitude arises from the anomalous axial-vector-vector (AVV) triangle diagram, where the anomaly coefficient is proportional to the number of colors N_c = 3 and the quark charges. The resulting decay rate is given by

\Gamma(\pi^0 \to \gamma \gamma) = \frac{\alpha^2 N_c m_{\pi}^3}{3 \pi^2 f_{\pi}^2} \cdot \frac{1}{64 \pi},

where \alpha is the fine-structure constant, m_{\pi} is the pion mass, and f_{\pi} is the pion decay constant (\approx 92 MeV). This leading-order prediction yields \Gamma \approx 7.76 eV, with higher-order chiral corrections increasing it by about 4.5% to \approx 8.10 eV.^[13] The chiral anomaly directly provides the Wess-Zumino-Witten (WZW) term in the effective low-energy Lagrangian of chiral perturbation theory, which encodes the non-trivial topology of the QCD vacuum and reproduces the \pi^0 \to \gamma \gamma amplitude without ultraviolet divergences. This term ensures consistency with the anomaly matching conditions and forms the foundation for describing Goldstone boson interactions at low energies.^[13] Experimental measurements of the decay, initiated in the 1970s using techniques like the Primakoff effect and direct lifetime determinations, have progressively refined the width to \Gamma = 7.826 \pm 0.117 eV (1.5% total uncertainty) as of 2020, in excellent agreement with the QCD anomaly prediction and confirming the coefficient's value tied to N_c = 3. These results validate the anomaly's role in hadronic physics and provide a benchmark for lattice QCD simulations of the process.^[14]

Baryon number violation

In the electroweak sector of the Standard Model, the chiral anomaly associated with the SU(2)_L gauge group leads to non-conservation of baryon number B, as the anomaly equation for the baryon current takes the form \partial_\mu J^\mu_B = N_f \frac{g^2}{32\pi^2} \operatorname{Tr}(W_{\mu\nu} \tilde{W}^{\mu\nu}), where N_f = 3 is the number of fermion generations, g is the SU(2)_L coupling constant, and W_{\mu\nu} is the weak field strength tensor.^[15] This implies that the change in baryon number is \Delta B = N_f \nu, where \nu is the topological winding number characterizing the gauge field configuration, and the left-handed fermion number changes by \Delta N_L = 2 N_f \nu due to the anomalous production or absorption of left-handed fermions. Similarly, lepton number L is violated with \Delta L = N_f \nu, while B - L remains conserved.^[16] This mechanism distinguishes the electroweak anomaly from the perturbative QCD case, as it induces global violations of B and L through non-perturbative effects. Instantons, which are classical solutions to the Euclidean Yang-Mills equations representing tunneling between topologically distinct vacua, mediate baryon number violation at zero temperature.^[17] These configurations change the Chern-Simons number by an integer \nu = \pm 1, enabling a multi-fermion process involving quarks and leptons from all generations that violates baryon number by \Delta B = N_f \nu = -3 for a single instanton (\nu = -1).^[15] However, the transition rate is exponentially suppressed by a factor of approximately e^{-4\pi / \alpha_w}, where \alpha_w = g^2 / 4\pi \approx 1/30 is the weak fine-structure constant, yielding rates on the order of $10^{-165} or smaller, far below observable thresholds. This suppression explains the absence of detected proton decay in experiments, despite theoretical predictions of such channels induced by instantons.^[17] At finite temperatures, such as those in the early universe, sphalerons provide a thermal activation mechanism for baryon number violation, acting as saddle-point configurations in the gauge field space with Chern-Simons number differing by 1/2 from the vacua.^[16] Unlike instantons, sphalerons do not require quantum tunneling but are overcome by thermal fluctuations when the temperature approaches the electroweak scale (T \sim 100 GeV), with the sphaleron energy barrier E_{\rm sph} \approx 9 TeV in the Standard Model. The associated rate \Gamma \sim \alpha_w^4 T^4 e^{-E_{\rm sph}/T} becomes unsuppressed above the critical temperature, facilitating rapid equilibration of B and L in the primordial plasma and playing a crucial role in electroweak baryogenesis.^[15] The topological nature of these processes is captured by the Pontryagin index \nu = \frac{g^2}{32\pi^2} \int d^4x \, \operatorname{Tr}(F_{\mu\nu} \tilde{F}^{\mu\nu}), which counts the difference in Chern-Simons numbers between asymptotic field configurations and directly links the anomaly to the non-trivial vacuum structure of the SU(2)_L gauge theory.^[17] This index \nu governs the multiplicity of fermion zero modes in the instanton background, enforcing the \Delta B = N_f \nu violation.^[15]

Applications and extensions

In the Standard Model

In the Standard Model (SM), the chiral anomaly plays a crucial role in ensuring the consistency of the theory by requiring the cancellation of gauge anomalies across the fermion content. The SM gauge group SU(3)_c × SU(2)_L × U(1)_Y demands that anomalies such as [SU(3)_c]^3, [SU(2)_L]^3, [U(1)_Y]^3, and mixed terms like [SU(3)_c]^2 U(1)_Y vanish for the theory to be renormalizable and unitary. This cancellation occurs automatically for each generation of quarks and leptons when assigning the left-handed quarks to SU(2)_L doublets with hypercharge Y = 1/6, right-handed up-quarks to singlets with Y = 2/3, right-handed down-quarks to singlets with Y = -1/3, left-handed leptons to doublets with Y = -1/2, and right-handed electrons to singlets with Y = -1, while right-handed neutrinos, if included, adjust accordingly.^[18] The three generations ensure cancellation of the mixed gravitational anomaly, but the gauge anomalies cancel per family, a non-trivial feature first recognized in the construction of the electroweak theory.^[18] Electroweak contributions from chiral anomalies influence precision tests of the SM, including those probing the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Mixed anomalies, such as [SU(2)_L]^2 U(1)_Y and gravity-SU(2)_L^2, are canceled by the specific hypercharge assignments of fermions, which underpin the consistency of electroweak radiative corrections entering CKM elements. Deviations in CKM unitarity, like the observed tension in the first-row sum |V_ud|^2 + |V_us|^2 + |V_ub|^2 = 0.9983 \pm 0.0010 (as of 2024) from superallowed beta decays and kaon decays, representing a ~2\sigma deviation from unitarity, can signal new physics if anomalies are not perfectly canceled in extensions, but in the SM, the exact cancellation ensures reliable predictions for unitarity tests at the percent level.^[19] These tests, performed at facilities like LHCb and Belle II, constrain beyond-SM contributions that could mimic anomalous effects in flavor-changing neutral currents.^[20] The chiral anomaly resolves the longstanding U(1)_A problem in quantum chromodynamics (QCD), explaining the unexpectedly large mass of the η' meson compared to the light pseudoscalar octet. In the absence of the anomaly, the U(1)_A symmetry would yield a ninth Goldstone boson, but instanton effects generate a topological term in the QCD action that breaks U(1)_A explicitly, contributing significantly to the η' mass via gluonic interactions. The Witten-Veneziano formula relates this mass to the topological susceptibility χ_top of pure Yang-Mills theory: m_{\eta'}^2 f_\pi^2 \approx 2 N_f \chi_{\rm top}, where N_f = 3 is the number of flavors and f_π ≈ 93 MeV is the pion decay constant, predicting m_η' ≈ 958 MeV in agreement with observation.^[21] This mechanism confirms the anomaly's non-perturbative origin in the QCD vacuum.^[22] Beyond the SM, the chiral anomaly informs extensions like grand unified theories (GUTs), where anomaly cancellation occurs automatically within unified representations (e.g., \bar{5} + 10 in SU(5)), while the unified gauge interactions lead to baryon number violation by ΔB = 1, predicting proton decay modes like p → e^+ π^0 with lifetime τ_p > 2.4 × 10^{34} years (90% CL, as of 2024) from Super-Kamiokande bounds.^[23] Similarly, axion models address the strong CP problem by introducing a Peccei-Quinn U(1)_{PQ} symmetry, spontaneously broken to yield a light axion a that dynamically relaxes the θ parameter via the anomaly: θ → θ + a/f_a, where f_a ∼ 10^{12} GeV suppresses CP violation to |θ| < 10^{-10}.^[21] These models, such as KSVZ or DFSZ, predict axion couplings testable in haloscope experiments like ADMX. Recent lattice QCD simulations in the 2020s have confirmed the chiral anomaly's manifestations in the SM, particularly its persistence across the chiral phase transition. Using overlap fermions to preserve exact chiral symmetry, calculations at temperatures T = 190–330 MeV demonstrate that the axial U(1)_A anomaly remains robust even near the crossover to quark-gluon plasma, with measures like the η'–η mass difference and topological susceptibility χ_top ≈ (175 MeV)^4 matching the Witten-Veneziano prediction. Further studies on the γ → 3π chiral anomaly via dispersion relations and lattice inputs yield decay widths Γ(π^0 → γγ) in precise agreement with experiment, underscoring the anomaly's role in low-energy hadron physics.^[24] These non-perturbative validations bolster SM simulations for beyond-threshold phenomena.

In condensed matter physics

In condensed matter physics, the chiral anomaly manifests in topological semimetals, particularly Weyl semimetals, where low-energy electronic excitations behave as massless Weyl fermions analogous to those in relativistic quantum field theory. These quasiparticles emerge at band-touching points in the Brillouin zone, known as Weyl nodes, which carry topological charges (monopole and antimonopole) that protect their stability against perturbations. The anomaly arises from the topological properties of the band structure, leading to non-conservation of chiral charge under parallel electric and magnetic fields, much like in high-energy physics but realized in non-relativistic solid-state systems.^[25] Theoretical descriptions map the quantum field theory anomaly to lattice models of Weyl semimetals using effective field theories that incorporate the Berry curvature of the Bloch bands. In these models, the Weyl nodes act as sources and sinks of Berry flux, translating the Adler-Bell-Jackiw anomaly into a topological pumping of electrons between nodes. Lattice realizations, such as tight-binding models on cubic or pyrochlore lattices, demonstrate how the anomaly persists despite ultraviolet regularization issues in discrete spaces, confirming its robustness through index theorems.^[26]^[27] A key manifestation is the chiral magnetic effect (CME), where an electric current flows along an applied magnetic field due to the imbalance of chiral quasiparticles induced by the anomaly. This effect, predicted in Weyl semimetals, generates a dissipationless current proportional to the magnetic field strength and chiral chemical potential, distinguishing it from classical magnetotransport. In materials like TaAs, the CME contributes to negative longitudinal magnetoresistance, where resistivity decreases quadratically with magnetic field under parallel electric and magnetic configurations.^[28]^[29] Experimental evidence emerged in the 2010s through angle-resolved photoemission spectroscopy (ARPES) and transport measurements in Dirac and Weyl semimetals. ARPES studies on TaAs confirmed the presence of Weyl nodes and surface Fermi arcs, providing direct visualization of the topological band structure essential for the anomaly. Transport experiments revealed anomalous Hall effects and negative magnetoresistance consistent with chiral pumping, particularly in TaAs and NbAs, where the observed responses matched predictions after accounting for finite Fermi level effects. These observations, starting with the 2015 discovery of Weyl fermions in TaAs, solidified the condensed-matter realization of the anomaly.^[30]^[31]^[32] Unlike in particle physics, where the anomaly occurs in vacuum at zero density, condensed-matter analogs operate at finite electron density near the Weyl nodes, introducing inter-node scattering and disorder that can mask but not eliminate the topological origin. Disorder effects, modeled via random potentials in effective theories, lead to diffusive broadening of Landau levels but preserve the anomaly's quantized pumping in clean limits. This finite-density regime enables tabletop probes of anomaly-driven phenomena, contrasting the high-energy accelerators needed for relativistic cases, while sharing the same underlying topological invariance.^[33]^[34]

References

[1]
[PDF] 20 Anomalies in Quantum Field Theory - Eduardo Fradkin
This term is the chiral (or axial) anomaly. The form of the anomaly term depends on the dimensionality, and turned out to have a topological meaning. Although ...
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[PDF] 2 Anomalies - University of Washington
One of the fascinating features of chiral symmetry is that sometimes it is not a sym- metry of the quantum field theory even when it is a symmetry of the ...
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[PDF] 5 The Dirac Equation and Spinors
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Experimental signatures of the chiral anomaly in Dirac–Weyl ...
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