--- title: Topological Susceptibility in QCD url: https://www.emergentmind.com/topics/topological-susceptibility type: topic --- # Topological Susceptibility in QCD 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 [1008.0732; 1607.05939; 1605.08637; 1606.03145]. ## 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 \(\theta=0\), \(c_2=\chi_t=d^2 e_{\mathrm{vac}}/d\theta^2|_{\theta=0}\), and therefore the first nontrivial even cumulant in the \(\theta\)-expansion of the vacuum free energy [1508.07704; 1008.0732]. Higher cumulants characterize departures from a Gaussian topological-charge distribution. The fourth cumulant is commonly written as $$ c_4=-\frac{\langle Q^4\rangle-3\langle Q^2\rangle^2}{V}, $$ or equivalently, in the notation used for fixed-topology analyses, $$ c_4=\frac{1}{V}\left(3\langle Q^2\rangle^2-\langle Q^4\rangle\right). $$ A vanishing \(c_4\) indicates a Gaussian \(Q\)-distribution; nonzero \(c_4\) controls the leading nonquadratic \(\theta\)-dependence and enters some fixed-sector expansions [1508.07704; 1503.06853]. A recurrent subtlety is that the formal identity between \(\chi_t\) and the integrated density correlator is not automatic in the continuum. The local operator \(q(x)\) generates non-integrable short-distance singularities in \(\langle q(x)q(0)\rangle\), so a valid definition of \(\chi_t\) requires a UV-safe prescription compatible with anomalous chiral Ward identities rather than a naive contact-term subtraction [1008.0732]. ## 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, $$ \chi_t= \frac{\langle \mathrm{Tr}\{R_M^4\}\rangle} {\langle \mathrm{Tr}\{\gamma_5 R_M^2 \gamma_5 R_M^2\}\rangle} \, \frac{\langle \mathrm{Tr}\{\gamma_5 R_M^2\}\,\mathrm{Tr}\{\gamma_5 R_M^2\}\rangle}{V} +O(a^2), $$ where \(R_M\) approximates the projector onto the low-mode subspace of \(D_m^\dagger D_m\). In pure \(SU(3)\) gauge theory this definition showed negligible visible cutoff effects over \(a\simeq0.05\)–\(0.10\) fm and yielded \(\chi_t^{1/4}=196.5(5.1)\,\mathrm{MeV}\), practically coincident with the earlier chiral-Dirac-operator value \(194.5(2.4)\,\mathrm{MeV}\), thereby providing a direct universality test [1008.0732]. A related fermionic strategy uses twisted-mass spectral projectors. There the renormalized susceptibility is $$ \chi_{\mathrm{top}}=\frac{Z_S^2}{Z_P^2}\frac{\langle Q^2\rangle}{V}, $$ with \(Q\) 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 \(O(a)\)-improvement: the relevant \(O(a)\) contributions are odd under the \(\mathcal R_5^{1,2}\) symmetry and vanish. The same framework produced a quark-mass dependence compatible with LO ChPT and condensate determinations consistent with direct extractions [1312.3535]. Gluonic definitions remain widely used, but their finite-\(a\) 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 [1605.08637]. In the large asqtad study, by contrast, \(\chi_t\) was reconstructed from the integrated point-to-point correlator of a discretized \(F\widetilde F\) operator with HYP smearing and asymptotic-tail modeling; the continuum extrapolation agreed with lowest-order \(SU(3)\) ChPT at small quark mass, supporting the rooted staggered formulation [1003.5695]. ## 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-\(Q\) sector, especially for \(a\lesssim0.05\) fm or for exactly or highly chirally symmetric fermions. In that regime, the naive estimator \(\chi_t=\langle Q^2\rangle/V\) ceases to probe the underlying theory and instead reflects algorithmic trapping [1605.08637]. One fixed-sector strategy is the AFHO method, based on the large-distance plateau of the charge-density correlator at fixed \(Q\). In the two-flavor QCD application, $$ \Big\langle q(t)q(0)\Big\rangle_{|Q|,V} \underset{t\to\infty}{=} -\frac{\chi_t}{V}\left(1-\frac{Q^2}{\chi_t V}\right) +\mathcal O\!\left(\frac{1}{\chi_t^2V^2},e^{-m_\eta t}\right), $$ valid when \(1/(\chi_t V)\ll1\) and \(Q^2/(\chi_t V)\ll1\). In benchmark ensembles with \(\chi_tV\approx10\), the method reproduced the unfixed-topology result: \(\chi_t a^4=7.69(22)\times10^{-5}\) from fixed sectors versus \(7.76(20)\times10^{-5}\) from the direct estimator [1605.08637]. Earlier numerical tests in the \(1d\ O(2)\) model, the \(2d\ O(3)\) model, \(2d\ U(1)\) gauge theory, and \(4d\ SU(2)\) Yang–Mills employed the related asymptotic form $$ \lim_{|x|\to\infty}\langle q_0 q_x\rangle_{|Q|} \approx -\frac{\chi_t}{V} +\frac{1}{V^2}\left(Q^2-\frac{c_4}{2\chi_t}\right), $$ so that the kurtosis correction \(c_4\) appears explicitly when one goes beyond the simplest Gaussian approximation [1503.06853]. The second strategy is the slab method. Here the volume is partitioned into slabs \(xV\) and \((1-x)V\), and one studies the topological charge \(q\) in a subvolume while the total charge \(Q\) is fixed. Under a Gaussian assumption for the charge distribution, $$ \langle(q-xQ)^2\rangle=\chi_t V\,x(1-x). $$ 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 \(Q=0\) and \(|Q|=1\), with robustness improving when \(L/\xi=O(10)\); non-Gaussianity is monitored through \(c_4\) [1509.06433]. In two-flavor QCD the slab method yielded \(\chi_t a^4=7.63(14)\times10^{-5}\), again consistent with the unfixed benchmark [1605.08637]. Gradient flow modifies the slab observable in a model-dependent way. In two-flavor QCD, the data after flow are well described by $$ \langle q'^2\rangle=\chi_t V\,x(1-x)-c, $$ with an additive constant \(c\) that grows approximately like \(0.38\sqrt t\), whereas in the \(2d\ O(3)\) model the same constant was not required over the explored flow range. The \(1d\ O(2)\) model interpolates between these behaviors: short flow times preserve the pure parabola, long flow times generate a noticeable constant offset [1712.01395]. ## 4. Thermal behavior from the crossover to the semiclassical regime At finite temperature, \(\chi_t(T)\) is a sensitive probe of deconfinement, chiral restoration, and the onset of dilute instanton physics. In quenched \(SU(3)\) on anisotropic lattices with over-improved stout-link smoothing, \(\chi_t\) remained nearly constant below \(T_c\) and then decreased rapidly across the transition, from \((188(1)\,\mathrm{MeV})^4\) at \(T=0\) to \((67(3)\,\mathrm{MeV})^4\) at \(1.9T_c\). The fourth cumulant displayed a step-like change near \(T_c\), and the ratio \(c_4/\chi_t\) approached the dilute-instanton-gas value \(-1\) at higher temperature [1508.07704]. Dynamical-quark studies show a less universal pattern. In \(N_f=2+1+1\) Wilson twisted mass QCD with \(m_{\pi^\pm}\simeq370\) MeV, the susceptibility over \(150