Topological Susceptibility in QCD

• Topological susceptibility is defined as the variance density of the global topological charge, representing the curvature of the vacuum free energy with respect to the QCD vacuum angle.
• It plays a critical role in mechanisms like the Witten–Veneziano explanation of the η′ mass, linking topological fluctuations to physical observables in QCD and Yang–Mills theory.
• Lattice implementations using flowed gluonic and chirally symmetric fermionic methods provide practical tests for sampling topology, especially in regimes affected by topology freezing or thermal transitions.

Topological susceptibility is the second cumulant of the topological-charge distribution, the zero-momentum two-point function of the topological charge density, and the curvature of the vacuum free energy with respect to the QCD vacuum angle $\theta$. In gauge theory notation,

$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$

with the proviso that the correlator representation is only formal until the contact singularity at $x=0$ is controlled. In QCD and Yang–Mills theory, $\chi_t$ quantifies topological fluctuations of the vacuum, enters the Witten–Veneziano mechanism for the $\eta'$ mass, and governs the $\theta$-dependence relevant for axion physics; on the lattice it is also a stringent diagnostic of whether topology is sampled correctly [(Lüscher et al., 2010); (Cè et al., 2016); (Dromard et al., 2016); (Petreczky et al., 2016)].

1. Formal definition and cumulant structure

The standard definition treats $\chi_t$ as the variance density of the global topological charge. In CP-symmetric ensembles one usually has $\langle Q\rangle=0$, so $\chi_t=\langle Q^2\rangle/V$. The same quantity is the second derivative of the vacuum energy density $e_{\mathrm{vac}}(\theta)$ at $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$0, $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$1, and therefore the first nontrivial even cumulant in the $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$2-expansion of the vacuum free energy [(Xiong et al., 2015); (Lüscher et al., 2010)].

Higher cumulants characterize departures from a Gaussian topological-charge distribution. The fourth cumulant is commonly written as

$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$3

or equivalently, in the notation used for fixed-topology analyses,

$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$4

A vanishing $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$5 indicates a Gaussian $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$6-distribution; nonzero $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$7 controls the leading nonquadratic $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$8-dependence and enters some fixed-sector expansions (Xiong et al., 2015, Bautista et al., 2015).

A recurrent subtlety is that the formal identity between $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$9 and the integrated density correlator is not automatic in the continuum. The local operator $x=0$0 generates non-integrable short-distance singularities in $x=0$1, so a valid definition of $x=0$2 requires a UV-safe prescription compatible with anomalous chiral Ward identities rather than a naive contact-term subtraction (Lüscher et al., 2010).

2. Renormalized definitions and lattice realizations

On the lattice, topological susceptibility can be defined through flowed gluonic observables, chirally symmetric fermionic constructions, or mixed density-chain representations. The core issue is universality: once a renormalized, short-distance-safe definition is adopted, distinct admissible definitions should converge to the same continuum limit. A prominent fermionic representation is the spectral-projector formula,

$x=0$3

where $x=0$4 approximates the projector onto the low-mode subspace of $x=0$5. In pure $x=0$6 gauge theory this definition showed negligible visible cutoff effects over $x=0$7–$x=0$8 fm and yielded $x=0$9, practically coincident with the earlier chiral-Dirac-operator value $\chi_t$0, thereby providing a direct universality test (Lüscher et al., 2010).

A related fermionic strategy uses twisted-mass spectral projectors. There the renormalized susceptibility is

$\chi_t$1

with $\chi_t$2 reconstructed from density chains or stochastic projector observables. An important technical result is that, at maximal twist, contact terms generated by integrated scalar and pseudoscalar density chains do not spoil automatic $\chi_t$3-improvement: the relevant $\chi_t$4 contributions are odd under the $\chi_t$5 symmetry and vanish. The same framework produced a quark-mass dependence compatible with LO ChPT and condensate determinations consistent with direct extractions (Cichy et al., 2013).

Gluonic definitions remain widely used, but their finite-$\chi_t$6 implementation depends on smoothing. In two-flavor lattice QCD with Wilson twisted mass fermions, the topological charge was measured from a field-strength definition after gradient flow, followed by a multiplicative renormalization and projection toward integer charge (Dromard et al., 2016). In the large asqtad study, by contrast, $\chi_t$7 was reconstructed from the integrated point-to-point correlator of a discretized $\chi_t$8 operator with HYP smearing and asymptotic-tail modeling; the continuum extrapolation agreed with lowest-order $\chi_t$9 ChPT at small quark mass, supporting the rooted staggered formulation (Collaboration et al., 2010).

3. Topology freezing and fixed-sector estimators

Modern fine-lattice simulations often encounter topological freezing: the Monte Carlo history remains trapped in a single integer-$\eta'$0 sector, especially for $\eta'$1 fm or for exactly or highly chirally symmetric fermions. In that regime, the naive estimator $\eta'$2 ceases to probe the underlying theory and instead reflects algorithmic trapping (Dromard et al., 2016).

One fixed-sector strategy is the AFHO method, based on the large-distance plateau of the charge-density correlator at fixed $\eta'$3. In the two-flavor QCD application,

$\eta'$4

valid when $\eta'$5 and $\eta'$6. In benchmark ensembles with $\eta'$7, the method reproduced the unfixed-topology result: $\eta'$8 from fixed sectors versus $\eta'$9 from the direct estimator (Dromard et al., 2016). Earlier numerical tests in the $\theta$0 model, the $\theta$1 model, $\theta$2 gauge theory, and $\theta$3 Yang–Mills employed the related asymptotic form

$\theta$4

so that the kurtosis correction $\theta$5 appears explicitly when one goes beyond the simplest Gaussian approximation (Bautista et al., 2015).

The second strategy is the slab method. Here the volume is partitioned into slabs $\theta$6 and $\theta$7, and one studies the topological charge $\theta$8 in a subvolume while the total charge $\theta$9 is fixed. Under a Gaussian assumption for the charge distribution,

$\chi_t$0

This estimator uses local subvolume fluctuations that can remain active even when the global integer charge is frozen. In sigma-model tests the method worked best in sectors $\chi_t$1 and $\chi_t$2, with robustness improving when $\chi_t$3; non-Gaussianity is monitored through $\chi_t$4 (Bietenholz et al., 2015). In two-flavor QCD the slab method yielded $\chi_t$5, again consistent with the unfixed benchmark (Dromard et al., 2016).

Gradient flow modifies the slab observable in a model-dependent way. In two-flavor QCD, the data after flow are well described by

$\chi_t$6

with an additive constant $\chi_t$7 that grows approximately like $\chi_t$8, whereas in the $\chi_t$9 model the same constant was not required over the explored flow range. The $\langle Q\rangle=0$0 model interpolates between these behaviors: short flow times preserve the pure parabola, long flow times generate a noticeable constant offset (Mejía-Díaz et al., 2017).

4. Thermal behavior from the crossover to the semiclassical regime

At finite temperature, $\langle Q\rangle=0$1 is a sensitive probe of deconfinement, chiral restoration, and the onset of dilute instanton physics. In quenched $\langle Q\rangle=0$2 on anisotropic lattices with over-improved stout-link smoothing, $\langle Q\rangle=0$3 remained nearly constant below $\langle Q\rangle=0$4 and then decreased rapidly across the transition, from $\langle Q\rangle=0$5 at $\langle Q\rangle=0$6 to $\langle Q\rangle=0$7 at $\langle Q\rangle=0$8. The fourth cumulant displayed a step-like change near $\langle Q\rangle=0$9, and the ratio $\chi_t=\langle Q^2\rangle/V$0 approached the dilute-instanton-gas value $\chi_t=\langle Q^2\rangle/V$1 at higher temperature (Xiong et al., 2015).

Dynamical-quark studies show a less universal pattern. In $\chi_t=\langle Q^2\rangle/V$2 Wilson twisted mass QCD with $\chi_t=\langle Q^2\rangle/V$3 MeV, the susceptibility over $\chi_t=\langle Q^2\rangle/V$4 MeV decreases only slowly above the crossover temperature $\chi_t=\langle Q^2\rangle/V$5 MeV, and Wilson flow, Wilson cooling, over-improved cooling, and stout smearing agree well when matched at the same smoothing scale (Trunin et al., 2015). In $\chi_t=\langle Q^2\rangle/V$6-flavor HISQ QCD near the physical point, continuum-extrapolated data exhibit two regimes: below about $\chi_t=\langle Q^2\rangle/V$7 MeV the falloff is milder than DIGA, while above $\chi_t=\langle Q^2\rangle/V$8 MeV the behavior is consistent with $\chi_t=\langle Q^2\rangle/V$9, with a fitted high-temperature exponent $e_{\mathrm{vac}}(\theta)$0 for $e_{\mathrm{vac}}(\theta)$1. Matching to the partial two-loop DIGA expression requires a temperature-independent $e_{\mathrm{vac}}(\theta)$2-factor $e_{\mathrm{vac}}(\theta)$3, and the resulting $e_{\mathrm{vac}}(\theta)$4 feeds directly into the axion bound $e_{\mathrm{vac}}(\theta)$5 in the post-inflation PQ scenario (Petreczky et al., 2016).

Very high temperatures pose a rare-event problem because nonzero-$e_{\mathrm{vac}}(\theta)$6 sectors become exponentially suppressed. Two complementary strategies have been used in pure gauge theory. The density-of-states method reconstructs the charge-sector histogram from constrained ensembles and reached $e_{\mathrm{vac}}(\theta)$7, finding $e_{\mathrm{vac}}(\theta)$8 at fixed $e_{\mathrm{vac}}(\theta)$9 and sector probabilities consistent with a Skellam distribution, as expected for a free instanton gas (Borsanyi et al., 2021). Reweighting methods, using a partially flowed proxy charge to enhance transitions through dislocation-like configurations, produced continuum-extrapolated values

$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$00

in agreement with the existing quenched literature (Jahn et al., 2018).

Recent physical-point $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$01 Möbius domain-wall simulations reinforce the extent of cutoff sensitivity. Preliminary $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$02 results gave $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$03 at $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$04 and $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$05 at $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$06, close to the HISQ continuum value $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$07, while topological sectors became nearly invisible by $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$08 MeV, where only $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$09 configurations were observed in the reported sample (Kanamori et al., 24 Mar 2026).

5. Universality, large-$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$10, and exceptional scaling failures

In pure Yang–Mills theory, the flowed definition of $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$11 behaves in the expected large-$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$12 manner. A combined continuum and large-$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$13 analysis of $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$14, $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$15, $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$16, and $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$17 gauge theory with open temporal boundary conditions gave

$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$18

with continuum values $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$19, $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$20, $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$21, and $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$22 for $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$23, respectively. The $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$24 corrections are therefore modest, quantitatively confirming the large-$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$25 picture behind the Witten–Veneziano mechanism (Cè et al., 2016).

The $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$26 model is exceptional. There the conventional scaling combination is $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$27, and long-standing evidence indicates that it diverges rather than approaching a finite continuum limit. Direct gradient-flow studies showed that GF strongly reduces $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$28, especially on fine lattices, but even when the smoothing radius reaches approximately $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$29 there is no sign that $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$30 converges; the growth with $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$31 persists and is well described by logarithmic or mild power-law ansätze (Bietenholz et al., 2018). Earlier GF/slab investigations had already found that $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$32 drops quickly at early flow time but had not yet reached flow radii large enough to settle the continuum question decisively (Mejía-Díaz et al., 2017). A common misconception is that GF automatically cures topological UV pathologies; the $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$33 evidence does not support that conclusion.

Infrared effective models can reproduce some, but not all, of the same physics. In the $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$34 random center-vortex world-surface ensemble, topological charge is generated by writhe and self-intersection of vortex surfaces, and the confined-phase susceptibility appears quantitatively consistent with Yang–Mills theory after accounting for systematic uncertainties from the hypercubic construction. In the deconfined phase, however, the susceptibility falls off significantly less than in lattice $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$35 Yang–Mills, indicating either hypercubic artefacts or incomplete effective vortex dynamics (Engelhardt, 2010).

6. External fields, dense matter, and alternative reformulations

Topological susceptibility admits nontrivial generalizations in external environments. In a weak uniform magnetic field, $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$36PT yields model-independent sum rules for the shifts of topological cumulants. For two flavors,

$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$37

while in three flavors,

$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$38

The shift of the fourth cumulant depends not only on condensate shifts but also on quark susceptibility shifts. In this framework the magnetic field enhances $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$39, and the same structure follows from Ward–Takahashi identities, not only from the low-energy expansion (Adhikari, 2021).

At finite density, WTIs can make the anomaly content of $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$40 explicit. In two-color QCD with two flavors and zero temperature,

$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$41

and in the vacuum low-energy regime,

$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$42

Within the linear sigma model used there, the susceptibility vanishes identically if the $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$43 anomaly term is absent; with anomaly effects present it is constant and nonzero in the hadronic phase and then decreases smoothly as $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$44 in the baryon-superfluid phase (Kawaguchi et al., 2023).

Dense three-color superconducting matter leads to a different structure. In a DSE/HDET/2PI treatment of CFL and 2SC phases, the curvature of the $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$45-dependent effective potential gives

$$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$46

where $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$47 is the asymptotic anomaly-induced gap component and $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$48 is the effective $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$49-breaking coupling. Within that framework, $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$50 in both CFL and 2SC remains of vacuum-like order, implying axion masses of the same order through $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$51 (Murgana et al., 26 Feb 2025).

A more formal reinterpretation identifies $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$52 with the quantum Fisher information density of the ground state with respect to $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$53. Under the assumptions made in that analysis, $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$54, and the axion field can be viewed as the maximally efficient quantum estimator of the $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$55-angle. This reformulation leaves the usual role of $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$56 intact—curvature of $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$57, integrated $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$58 correlator, and axion mass parameter—but recasts it as a metric notion of distinguishability in the space of $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$59-vacua (Gomez, 2020).

In aggregate, topological susceptibility is not a single computational object but a tightly constrained family of equivalent definitions whose equivalence is contingent on UV control, IR volume, and algorithmic ergodicity. In four-dimensional Yang–Mills and QCD, flowed and fermionic constructions now support a coherent continuum picture; in fixed-topology simulations, subvolume and correlator methods recover the same observable under controlled assumptions; in thermal, dense, and magnetized settings, $$\chi_t=\frac{\langle Q^2\rangle-\langle Q\rangle^2}{V},\qquad Q=\int d^4x\,q(x),\qquad \chi_t=\int d^4x\,\langle q(x)q(0)\rangle,$$60 remains the canonical measure of how anomalous topology reorganizes itself under external scales and symmetry breaking.