Fujikawa Model: BRST Symmetry & Infrared QCD

• The Fujikawa model is a BRST-invariant scalar framework that organizes four fields into doublets, enabling spontaneous symmetry breaking with two massive scalar and two massless fermionic modes.
• It is embedded in infrared QCD where effective composite operators generate gluon and ghost masses and reproduce the Curci–Ferrari model under specific parameter choices.
• Distinct from Fujikawa’s measure formalism for anomalies, the model employs extended BRST transformations to preserve nilpotency in its sophisticated symmetry-breaking construction.

In one specific usage within quantum field theory, the Fujikawa model denotes a BRST-invariant scalar model built from a quartet of fields arranged into BRST doublets and designed to realize spontaneous BRST symmetry breaking. In recent infrared QCD constructions, that quartet is reinterpreted as an effective composite sector coupled to elementary Yang–Mills fields, with the condensates of the Fujikawa fields generating effective gluon and ghost masses and reproducing the Curci–Ferrari model as a special case (Fazio et al., 31 Mar 2026). In the wider literature, the surname Fujikawa also labels distinct but related constructions, notably the path-integral Jacobian formalism for anomalies, the Fujikawa–Takata recoil formalism in hard-x-ray photoemission, and Fujikawa’s higher-order Ginsparg–Wilson relation, so the term requires contextual disambiguation (Smith et al., 2 May 2025, Ritarossi et al., 12 May 2026, Singh, 26 May 2025).

1. Original quartet construction

Fujikawa’s original non-gauge model is a BRST-invariant scalar model with four fields $\{\varphi,\eta,\xi,B\}$ forming two BRST doublets (Fazio et al., 31 Mar 2026). The BRST transformations are

$$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$

Its BRST-invariant Lagrangian is

$$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$

For suitable parameters, this model develops a nontrivial vacuum with

$$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$

and the resulting spectrum contains two massive scalar modes and two massless fermionic modes $\xi,\eta$, interpreted as Nambu–Goldstone modes of spontaneously broken BRST symmetry (Fazio et al., 31 Mar 2026).

The defining structural feature is therefore not merely the presence of BRST symmetry, but the organization of the field content into BRST doublets together with a potential that permits spontaneous breaking. This distinguishes the Fujikawa model from the much broader Fujikawa measure formalism used in anomaly calculations.

2. Symmetry-breaking pattern and Nambu–Goldstone content

In the infrared QCD realization, the Fujikawa sector is recast in terms of fields $\varphi,\pi,\bar\pi,\bar\phi$, and the symmetry analysis is extended from BRST alone to simultaneous BRST and anti-BRST invariance (Fazio et al., 31 Mar 2026). The relevant BRST transformations include

$$\delta A^\mu_a=\theta(D^\mu c)_a,\qquad \delta c_a=-\frac{g}{2}\theta(c\times c)_a,\qquad \delta\bar c_a=\theta b_a,\qquad \delta b_a=0,$$

together with

$$\delta\varphi=\theta\pi,\qquad \delta\pi=0,\qquad \delta\bar\pi=\theta\bar\phi,\qquad \delta\bar\phi=0.$$

The anti-BRST transformations are defined analogously, with

$$\bar\delta A^\mu_a=\bar\theta(D^\mu\bar c)_a,\qquad \bar\delta\bar c_a=-\frac{g}{2}\bar\theta(\bar c\times\bar c)_a,\qquad \bar\delta c_a=\bar\theta\bar b_a,$$

and

$$\bar\delta\varphi=\bar\theta\bar\pi,\qquad \bar\delta\pi=\bar\theta\phi=-\bar\theta\bar\phi.$$

The nilpotency relations are

$$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$0

A central claim of the effective construction is that BRST breaking alone is not enough if one wants the Fujikawa sector to contain two massless Nambu–Goldstone modes. The two order-parameter conditions are

$$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$1

and

$$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$2

Accordingly, the model contains two massless fermionic Nambu–Goldstone modes, $$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$3 and $$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$4, one associated with broken BRST and one with broken anti-BRST symmetry (Fazio et al., 31 Mar 2026).

This places the Fujikawa model in a class of spontaneous fermionic-symmetry-breaking constructions whose massless sector is fixed by nilpotent charges rather than by ordinary internal Lie-algebra generators.

3. Infrared QCD embedding

The effective low-energy QCD construction is organized as

$$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$5

where $$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$6 is the Yang–Mills term, $$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$7 is the gauge-fixing/ghost sector, $$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$8 is the generalized Fujikawa sector, and $$\delta\varphi=\theta\eta,\qquad \delta\eta=0,\qquad \delta\xi=\theta B,\qquad \delta B=0.$$9 couples the Fujikawa fields to the elementary sector (Fazio et al., 31 Mar 2026).

The paper interprets the Fujikawa fields as effective composite fields built from the elementary gluon and ghost fields. The lowest-dimensional operators with the right quantum numbers are taken from the Yang–Mills sector, e.g.

$$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$0

$$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$1

and

$$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$2

The generalized Fujikawa sector is written as

$$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$3

Its expanded renormalizable form includes all allowed $$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$4, $$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$5, $$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$6, and mixed terms with coefficients $$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$7, and the original Fujikawa model is recovered for the particular choice

$$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$8

with the rest zero (Fazio et al., 31 Mar 2026).

Once the Fujikawa fields condense, the elementary Yang–Mills fields acquire effective masses through the portal couplings: $$\mathcal{L}_\mathrm{F}^0 =\tfrac{1}{2}\partial_\mu B\partial^\mu B+\partial_\mu B\partial^\mu\varphi-\partial_\mu\xi\partial^\mu\eta -\tfrac{1}{2}M^2B^2-m_0^2(B\varphi-\xi\eta) -g(B\varphi-\xi\eta)^2-g'B^4-g''B^2(B\varphi-\xi\eta).$$9 In the more general field-dependent version,

$$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$0

The mass generation is therefore tied to the condensation of the composite Fujikawa sector rather than to explicit mass insertions (Fazio et al., 31 Mar 2026).

4. Curci–Ferrari limit and extended-BRST completion

A major structural result is that the Curci–Ferrari model is reproduced as a special case of the effective theory after spontaneous BRST symmetry breaking (Fazio et al., 31 Mar 2026). The Curci–Ferrari Lagrangian is written as

$$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$1

The effective construction reproduces this elementary-field sector for

$$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$2

together with suitable $$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$3 (Fazio et al., 31 Mar 2026).

The same paper emphasizes that the modified BRST symmetry characteristic of the Curci–Ferrari model is non-nilpotent: $$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$4 To recover this structure while retaining an underlying nilpotent symmetry, it introduces an extended BRST transformation mixing the elementary and Fujikawa sectors: $$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$5

$$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$6

These transformations satisfy

$$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$7

$$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$8

so the extension is nilpotent (Fazio et al., 31 Mar 2026).

The construction is implemented by the shift

$$\langle B\rangle=B_0,\qquad \langle\varphi\rangle=\varphi_0,$$9

and a convenient choice of field-dependent couplings is

$\xi,\eta$0

implying

$\xi,\eta$1

After the Fujikawa shifts, the generalized theory splits as

$\xi,\eta$2

with the elementary sector exactly Curci–Ferrari and the remaining terms preserving the hidden nilpotent extended-BRST symmetry (Fazio et al., 31 Mar 2026).

5. Relation to Fujikawa’s measure formalism

A recurrent terminological confusion identifies the Fujikawa model with the Fujikawa method. The latter is a measure-theoretic formalism for anomalies, not the BRST quartet model. In Fujikawa’s anomaly framework, fermions are expanded in a Dirac eigenbasis,

$\xi,\eta$3

with measure

$\xi,\eta$4

Under a local chiral transformation, the measure acquires the Jacobian

$\xi,\eta$5

and the anomaly is encoded in a regulated spectral trace (Smith et al., 2 May 2025).

Recent work generalizes this by defining the regularized measure directly through an operator-valued $\xi,\eta$6-regularization,

$\xi,\eta$7

typically with

$\xi,\eta$8

thereby making the connection between spectral asymmetry, Atiyah–Singer index theory, and the regularized measure explicit (Smith et al., 2 May 2025). A complementary effective-field-theory formulation rewrites the Jacobian as a ratio of determinants and evaluates it with the Covariant Derivative Expansion, deriving covariant, consistent, gravitational, and scale anomalies within one framework (Filoche et al., 2022).

This wider measure formalism is conceptually adjacent to the Fujikawa model only in surname and in its concern with BRST- and chiral-symmetry structures. The two constructions solve different problems: one is an effective model of spontaneous fermionic symmetry breaking, the other a formalism for anomalous Jacobians.

6. Specialized extensions, limits, and contested usages

Several recent works delimit the range of validity of Fujikawa-type constructions. In radiative strong-$\xi,\eta$9 studies, the standard Fujikawa formula

$\varphi,\pi,\bar\pi,\bar\phi$0

is found not to cover all contributions at two loops. It captures the one-loop mass phase, but misses contributions from CEDM-induced effects and genuine two-loop threshold pieces. When there is a strong hierarchy in the $\varphi,\pi,\bar\pi,\bar\phi$1-violating sector, the Fujikawa evaluation is numerically sufficient; when the masses are comparable, the full two-loop calculation is required (Banno et al., 2023).

In gravitational path integrals, the paper "Diffeomorphism invariance of the effective gravitational action" argues that the Fujikawa measure

$\varphi,\pi,\bar\pi,\bar\phi$2

is not diffeomorphism invariant, whereas the Fradkin–Vilkovisky measure

$\varphi,\pi,\bar\pi,\bar\phi$3

is diffeomorphism invariant. In that analysis, the $\varphi,\pi,\bar\pi,\bar\phi$4 factor is necessary to cancel the nontrivial contribution arising from the change of lattice and time-ordering structure under diffeomorphisms (Branchina et al., 5 Jun 2025).

Outside anomaly theory proper, the name also appears in formally distinct constructions. In graphene hard-x-ray C 1s photoemission, a graphene-specific implementation of the Fujikawa–Takata cumulant formalism models phonon recoil through

$\varphi,\pi,\bar\pi,\bar\phi$5

with anisotropic mode-resolved spectral densities, and captures recoil scaling with photon energy and emission geometry. The baseline recoil model, however, fails to reproduce the pronounced asymmetric tails of the measured spectra, which require explicit convolution with an intrinsic asymmetric electronic line shape (Ritarossi et al., 12 May 2026). In Hamiltonian lattice theory, Fujikawa’s higher-order Ginsparg–Wilson relation

$\varphi,\pi,\bar\pi,\bar\phi$6

yields order-$\varphi,\pi,\bar\pi,\bar\phi$7 overlap Hamiltonians with an exactly conserved but nonquantized chiral charge that becomes quantized as $\varphi,\pi,\bar\pi,\bar\phi$8, at the price of worsening locality (Singh, 26 May 2025).

Taken together, these usages show that Fujikawa model is not a single universal object. In its most precise field-theoretic sense, it is the BRST quartet model and its infrared QCD descendants (Fazio et al., 31 Mar 2026). In adjacent literatures, the same surname denotes a measure formalism, recoil kernels, higher-order lattice chiral constructions, and other context-specific frameworks whose mathematical content is distinct even when the underlying theme remains symmetry, regularization, or anomalous response.