Topological Susceptibility and Vacuum Response

Topological susceptibility is the curvature of the vacuum energy with respect to theta. In a finite Euclidean volume it is also the variance density of the total topological charge, and higher theta derivatives are higher connected charge cumulants. These identities are exact once the regulator, charge normalization, and contact-term prescription have been fixed; estimating the cumulants is a separate dynamical problem.

Required background. Theta dependence, CP, and model-dependent branches supplies the energy and branch conventions, while connected correlators and cumulants supplies the source-derivative identities.

Helpful background. Coincident products and contact terms explains why an integrated local-density correlator requires more than a formal continuum integral.

Shared comparison. The sector and periodicity comparison fixes charge normalization and global data, while the sector–theta–branch map locates finite-volume cumulants relative to the infinite-volume branch limit.

Theta derivatives generate charge cumulants

Work first at regulated finite four-volume $V_4$, where

$$Z_{V_4}(\theta) =\int\mathcal D\Phi\, e^{-S_E[\Phi]+i\theta Q[\Phi]}, \qquad K_{V_4}(\theta)=\log Z_{V_4}(\theta).$$

Differentiating the logarithm, rather than $Z$ itself, selects connected moments:

$$\frac{\partial K_{V_4}}{\partial\theta} =i\langle Q\rangle_\theta, \qquad \frac{\partial^2 K_{V_4}}{\partial\theta^2} =-\langle Q^2\rangle_{\theta,c},$$

and more generally

$$\frac{\partial^n K_{V_4}}{\partial\theta^n} =i^n\langle Q^n\rangle_{\theta,c}.$$

Define the finite-volume intensive theta response by

$$f_{V_4}(\theta) =-\frac1{V_4}\log\frac{Z_{V_4}(\theta)}{Z_{V_4}(0)}.$$

On a periodic cylinder $V_4=\beta V_s$ with a positive thermal trace, first taking $\beta\to\infty$ at fixed $V_s$ and then $V_s\to\infty$ turns this response into the vacuum-energy-density difference. On a generic Euclidean manifold or complex background it remains an intensive generating functional, and the logarithm branch must be declared.

At a CP-invariant point $\theta=0$, odd cumulants vanish when the vacuum and regulator preserve CP. For a nonnegative theta-zero Euclidean measure, the susceptibility is therefore

$$\chi_{V_4} =\left.\frac{\partial^2 f_{V_4}}{\partial\theta^2}\right|_{\theta=0} =\frac{\langle Q^2\rangle_{0,c}}{V_4}\geq0.$$

The inequality is the positivity of a variance under that measure assumption. Without positivity, the derivative identity still holds but the inequality needs an independent argument. It does not imply that the fourth or higher derivatives have a fixed sign. With

$$\mathcal E(\theta)-\mathcal E(0) =\frac{\chi}{2}\theta^2 \bigl(1+b_2\theta^2+b_4\theta^4+\cdots\bigr),$$

the fourth derivative gives

$$b_2 =-\frac{\langle Q^4\rangle_{0,c}} {12\langle Q^2\rangle_{0,c}}, \qquad \langle Q^4\rangle_{0,c} =\langle Q^4\rangle_0-3\langle Q^2\rangle_0^2,$$

after taking the same regulated large-volume limit in numerator and denominator. The displayed expansion assumes analyticity near zero and $\chi\neq0$.

Connectedness is what makes an energy density possible. In a phase with finite correlation length, divide a large volume into cells separated by more than that length. Charge fluctuations in distant cells are approximately independent, so every fixed connected cumulant is extensive:

$$\langle Q^{2n}\rangle_c=V_4\,c_{2n}+o(V_4).$$

Raw moments do not have this scaling—for example, $\langle Q^4\rangle$ contains the disconnected term $3\langle Q^2\rangle^2\propto V_4^2$. The logarithm of $Z$ removes precisely those disconnected pieces, leaving finite derivatives of $f$ and, after the physical limits above, of $\mathcal E$. Near a critical point or in a phase with long-range correlations, the cell argument fails and the volume scaling must be determined afresh.

From the total charge to a local correlator

If $Q=\int_{V_4}\mathrm d^4x\,q(x)$ and translation invariance is restored in the large-volume limit, then formally

$$\chi =\lim_{V_4\to\infty}\frac1{V_4} \int_{V_4}\mathrm d^4x\,\mathrm d^4y\, \langle q(x)q(y)\rangle_c =\int_{\mathbb R^4}\mathrm d^4x\, \langle q(x)q(0)\rangle_c.$$

The last equality is not a definition independent of regularization. The product $q(x)q(0)$ is singular at $x=0$, and local counterterms can contribute precisely where the two insertions coincide. A sound prescription defines the integrated correlator through a regulated $Z(\theta)$ or through Ward identities, and only then removes the regulator. Lüscher gives a regulator-independent QCD construction in terms of density correlators whose normalization is fixed by chiral Ward identities Lüscher 2004, §§ 2–5.

This distinction also explains why reflection-positivity arguments applied only at nonzero separation do not determine the sign of $\chi$. The contact contribution and the integrated definition are essential.

A controlled dilute-gas calculation

Suppose positive and negative unit-charge events are independent Poisson variables $N_+$ and $N_-$, each with mean $\zeta V_4$. Then $Q=N_+-N_-$ and

$$\begin{aligned} \frac{Z_{V_4}(\theta)}{Z_{V_4}(0)} &=\left\langle e^{i\theta(N_+-N_-)}\right\rangle\\ &=\exp\!\left[2\zeta V_4(\cos\theta-1)\right]. \end{aligned}$$

Consequently,

$$\mathcal E(\theta)-\mathcal E(0) =2\zeta(1-\cos\theta), \qquad \chi=2\zeta, \qquad b_2=-\frac1{12}.$$

The calculation is exact for the declared Poisson ensemble. In a field theory it is reliable only when a dilute, weakly interacting description of the relevant topological events is controlled. The result should not be transferred to strongly coupled zero-temperature Yang–Mills theory merely because both systems have integer charge sectors.

Chiral suppression and order of limits

In QCD with light dynamical quarks, anomalous chiral rotations relate theta to the phase of the quark-mass matrix. If at least one quark is exactly massless, theta can be removed and the vacuum energy is theta independent, so $\chi=0$. At small nonzero masses the chiral effective theory instead predicts a suppressed susceptibility whose value depends on the mass matrix and condensate; the finite-volume and chiral limits need not commute. Leutwyler and Smilga analyze this interplay through the QCD partition function and winding-number sectors in Leutwyler and Smilga 1992, §§ II–IV.

At a first-order point away from zero, such as a model-dependent cusp at $\theta=\pi$, derivatives of the infinite-volume energy can be discontinuous. Susceptibilities should then be labeled by the branch or one-sided limit. Taking derivatives at finite volume before the thermodynamic limit need not give the same answer as differentiating a nonanalytic limiting envelope.

Common pitfalls

Dropping the connected subscript. Derivatives of $\log Z$ generate cumulants. Raw moments agree only in special cases, such as $\langle Q\rangle=0$ for the second derivative.

Treating the density correlator as an ordinary function. Its coincident-point singularity is part of the observable’s definition. A contact prescription or Ward-identity construction is required.

Inferring global theta dependence from $\chi$. Susceptibility fixes only the local curvature at one point. The quadratic branch model and dilute gas can share the same $\chi$ while differing in every higher derivative and at $\theta=\pi$.

Exercises

Starting from $K(\theta)=\log Z(\theta)$, derive the signs relating the second and fourth derivatives of $\mathcal E$ to charge cumulants.

Solution

The cumulant identity gives $K''(0)=-\langle Q^2\rangle_c$ and $K^{(4)}(0)=\langle Q^4\rangle_c$. Since $\mathcal E=-K/V_4$ up to a theta-independent constant,

$$\mathcal E''(0)=\frac{\langle Q^2\rangle_c}{V_4}, \qquad \mathcal E^{(4)}(0)=-\frac{\langle Q^4\rangle_c}{V_4}.$$

Comparing the latter with $\mathcal E^{(4)}(0)=12\chi b_2$ and using $\chi=\langle Q^2\rangle_c/V_4$ yields the stated expression for $b_2$.

For the dilute gas, show directly that all even charge cumulants equal $2\zeta V_4$ and all odd cumulants vanish.

Solution

The cumulant generator for $Q$ is

$$\log\langle e^{tQ}\rangle =\zeta V_4(e^t-1)+\zeta V_4(e^{-t}-1) =2\zeta V_4(\cosh t-1).$$

Every even derivative at $t=0$ is $2\zeta V_4$, whereas every odd derivative vanishes. Substituting $t=i\theta$ reproduces the theta-dependent partition function.

Suppose $Q$ is exactly Gaussian with mean zero and variance $\sigma^2V_4$. Compute $\langle Q^4\rangle$, $\langle Q^4\rangle_c$, and $b_2$. Why would using the raw fourth moment give a nonsensical volume limit?

Solution

For a centered Gaussian, $\langle Q^4\rangle=3\langle Q^2\rangle^2=3\sigma^4V_4^2$. Therefore

$$\langle Q^4\rangle_c =\langle Q^4\rangle-3\langle Q^2\rangle^2=0, \qquad b_2=0.$$

Substituting the raw moment into the formula for $b_2$ would produce a quantity growing like $V_4$, even though the Gaussian cumulant generator is purely quadratic. The contradiction is the diagnostic that disconnected pieces were not removed.

Continue from the identity to a regulated measurement

The derivative and cumulant identities on this page define the target observable. Topology and Lattice Index Diagnostics supplies finite-spacing definitions of $Q$, diagnoses dislocations and topological freezing, and carries the susceptibility through a continuum and volume sequence. Current numerical values and axion phenomenology remain with their gauge-theory and Research owners.

References

• Leutwyler, Heinrich, and Andrei Smilga. “Spectrum of Dirac Operator and Role of Winding Number in QCD.” Physical Review D 46, no. 12 (1992): 5607–5632. DOI.
• Lüscher, Martin. “Topological Effects in QCD and the Problem of Short-Distance Singularities.” Physics Letters B 593, nos. 1–4 (2004): 296–301. arXiv. DOI.