Theta Dependence in Yang–Mills and QCD

A theta parameter does not change the local perturbative field equations on a closed spacetime, but it changes the quantum theory by assigning the phase $e^{i\theta\nu}$ to configurations of topological charge $\nu$. Integer charge gives $2\pi$ periodicity, CP maps $\theta$ to $-\theta$, and the vacuum energy probes how sectors interfere. Every part of that statement has qualifications: the global gauge group controls the allowed charges, fermion chiral rotations can move theta into mass phases, and a periodic vacuum energy can be assembled from individually nonperiodic branches.

Required background. Theta terms, periodicity, and vacuum sectors supplies the topological-term construction.

Helpful background. Regulated Jacobians and measure variation supplies the anomalous fermion-measure calculation used to shift theta.

Topological charge and declared normalization

Start in Euclidean signature, where the sector sum is cleanest. Keep the volume convention $D_\mu=\partial_\mu-igA_\mu$ and define

$$\widetilde F^a_{\mu\nu} \equiv\frac12\epsilon_{\mu\nu\rho\sigma}F^a_{\rho\sigma}, \qquad \nu[A] \equiv\frac{g^2}{32\pi^2} \int_M d^4x\,F^a_{\mu\nu}\widetilde F^a_{\mu\nu}.$$

The local convention table prevents the most common factor and periodicity errors.

Datum	Benchmark choice on this page	Consequence
Local gauge algebra	$\mathfrak{su}(N)$ with $\operatorname{tr}_F(T^aT^b)=\tfrac12\delta^{ab}$	Fixes the coefficient in $\nu$
Global group	Initially $SU(N)$	Smooth finite-action configurations on compactified $\mathbb R^4$ have $\nu\in\mathbb Z$
Orientation	Euclidean $\epsilon_{1234}=+1$	Fixes the sign of $\nu$
Sector weight	$e^{i\theta\nu}$	Fixes the sign used in the partition function and axial shifts
Spacetime	Closed oriented four-manifold unless stated	Avoids unaccounted boundary Chern–Simons terms
Matter masses	Declared separately	Determines whether theta is removable or combines with mass phases

With these choices the Euclidean action can be written

$$S_E(\theta)= \frac14\int d^4x\,F^a_{\mu\nu}F^a_{\mu\nu} -i\theta\nu[A]+S_{E,\mathrm{matter}},$$

so $e^{-S_E}$ contains the advertised phase. Many texts absorb $g$ into the connection, in which case the explicit $g^2$ in $\nu$ disappears. Formulas should be translated by matching $D_\mu$, not by copying the topological coefficient alone.

Locally, $F\widetilde F$ is a total derivative. Globally, its Chern–Simons current is not a single gauge-invariant function over all bundles, so the integral need not vanish. On a manifold with boundary, the surface contribution and its gauge variation must be included explicitly; boundary conditions, counterterms, or boundary degrees of freedom can then affect the allowed periodicity statement.

Sector sum and 2π periodicity

Decompose the path integral into fixed-charge sectors:

$$Z(\theta)=\sum_{\nu\in\mathbb Z}e^{i\theta\nu}Z_\nu, \qquad Z_\nu=\int_{\nu[A]=\nu}\!\mathcal DA\,\mathcal D\Phi\, e^{-S_E(0)}.$$

Then

$$Z(\theta+2\pi) =\sum_{\nu\in\mathbb Z}e^{i\theta\nu}e^{2\pi i\nu}Z_\nu =Z(\theta).$$

This is the complete periodicity derivation for the benchmark theory: it uses the integrality of every sector included in the sum. The construction of the Yang–Mills topological charge, sector weights, theta vacuum, and susceptibility is developed in Mariño 2015, §4.3, pp. 112–123.

In canonical language, classical zero-field configurations can be labeled by an integer winding number $n$, and a large gauge transformation shifts $n\mapsto n+1$. A theta vacuum transforms by a one-dimensional character and can be represented schematically as

$$|\theta\rangle=\sum_{n\in\mathbb Z}e^{in\theta}|n\rangle.$$

Euclidean configurations with nonzero $\nu$ provide transition amplitudes between these winding sectors. The semiclassical instanton calculation is one way to estimate those amplitudes, but the sector decomposition and periodicity do not depend on a dilute-instanton approximation.

The normalization has an independent check. Positivity of $(F\mp\widetilde F)^2$ gives

$$S_E(0)\ge\frac{8\pi^2}{g^2}|\nu|,$$

with equality for (anti-)self-dual fields. A different coefficient signals that $A$, $F$, or the generator trace has been normalized differently.

Vacuum energy, CP, and branches

For Euclidean four-volume $V_4$, define

$$E(\theta) =-\lim_{V_4\to\infty}\frac1{V_4}\log Z(\theta).$$

At $\theta=0$, assuming the usual positive measure conditions,

$$\chi_t \equiv E''(0) =\lim_{V_4\to\infty} \frac{\langle\nu^2\rangle_c}{V_4}.$$

The topological susceptibility $\chi_t$ is therefore an observable curvature of the vacuum energy. Higher connected moments determine the higher coefficients in

$$E(\theta)-E(0) =\frac12\chi_t\theta^2 \left(1+b_2\theta^2+b_4\theta^4+\cdots\right)$$

within the radius controlled by the nearest nonanalyticity.

Because $F\widetilde F$ is odd under parity and CP, CP sends $\theta\mapsto-\theta$. When the remaining couplings are CP invariant, $2\pi$ periodicity makes $\theta=0$ and $\theta=\pi$ the kinematically CP-invariant points. Kinematics alone does not decide the infrared realization at $\theta=\pi$: CP may remain unbroken, break spontaneously through degenerate vacua, or coexist with nontrivial gapless or topological degrees of freedom.

Periodicity also need not hold branch by branch. A common large-$N$ form is

$$E(\theta)=N^2\min_{k\in\mathbb Z} f\!\left(\frac{\theta+2\pi k}{N}\right).$$

The shift $\theta\mapsto\theta+2\pi$ permutes $k$, so the minimum is periodic even though one branch is not. A branch crossing can produce a cusp and degenerate vacua at $\theta=\pi$. This is a controlled organizing picture in appropriate large-$N$ limits, not a theorem for every finite-$N$ gauge theory; the large-$N$ branch structure is reviewed in Mariño 2015, §7.4, pp. 235–236.

Global form and fermion phases

The one-line proof of $2\pi$ periodicity fails if the allowed $\nu$ are not all integers. Quotient groups such as $SU(N)/\mathbb Z_N$ admit bundles whose instanton number can be fractional on suitable manifolds. In that case a $2\pi$ shift can permute theories distinguished by a discrete theta parameter, and the periodicity of one fixed theory may be larger. The answer can also depend on whether the manifold is spin and on which background higher-form fields are turned on. The relation among global form, genuine line choices, discrete theta data, and ordinary $2\pi$ shifts is discussed in Gaiotto et al. 2015, §4.2, pp. 17–19.

Fermions create a different identification. For QCD with mass matrix $M$, an anomalous axial redefinition shifts the measure contribution to theta while rotating the phase of $M$. In the convention used here, the invariant combination is

$$\bar\theta=\theta+\arg\det M.$$

If at least one quark is exactly massless and no other interaction fixes its axial phase, that field can be redefined to remove theta; observables cannot depend on $\bar\theta$. With nonzero masses, the same rotation merely moves the phase between the topological term and the mass matrix. The regulated Jacobian and the massless-quark qualification are derived in Mariño 2015, §5.4, pp. 183–188.

These facts yield a reliable hierarchy of claims.

Input established	Conclusion justified	Additional input still needed
Integer $\nu$ in every sector	$Z(\theta+2\pi)=Z(\theta)$	None for periodicity; dynamics for the shape of $E$
CP sends $\nu\to-\nu$	$0$ and $\pi$ are CP-invariant points	Infrared dynamics to decide symmetry realization
Exact massless fermion with an available axial rotation	Theta can be removed	Check anomalies and every interaction involving that fermion
Fractional sectors or discrete theta data	A $2\pi$ shift may permute distinct theories	Declare global form, manifold class, and backgrounds
Large-$N$ branch ansatz	Periodicity can arise by branch permutation	Control of corrections and finite-$N$ dynamics

Independent checks and failure diagnoses

• Normalization check: verify both $\nu\in\mathbb Z$ on a unit instanton and $S_E=8\pi^2/g^2$ for a self-dual unit-charge configuration.
• Reindexing check: prove the claimed theta period directly from the actual charge lattice, not from the local density alone.
• CP check: apply $\theta\mapsto-\theta$ and combine it with the established period before naming special points.
• Fermion check: track the phase of every mass and the anomalous Jacobian under the same axial rotation.
• Branch check: confirm that the full set of branches, not one selected branch, is periodic.
• Boundary check: if $\partial M\neq\varnothing$, include the induced boundary term and state what restores gauge invariance.

Common pitfalls

Omitting the coupling normalization. Whether $g^2$ appears in $\nu$ depends on whether it was absorbed into $A$. Declare $D_\mu$ and the generator trace first.

Inferring $2\pi$ periodicity from notation. The proof uses integer sectors. A quotient gauge group, background flux, or boundary can change the identification.

Claiming that a total derivative is irrelevant. It is locally a derivative but globally detects bundle topology and changes interference among sectors.

Treating a large-$N$ cusp as universal. Branch crossing at $\theta=\pi$ is dynamical. Separate kinematic CP invariance from the realized vacuum structure.

The explicit semiclassical saddle calculation continues in gauge instantons, charge, and moduli. Current nonperturbative questions and evidence standards are organized in Nonperturbative Gauge Dynamics.

References

• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, no. 2 (2015): 172, §4.2, pp. 17–19. DOI. Open PDF.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, §§4.3, 5.4, and 7.4, pp. 112–123, 183–188, and 235–236. DOI.