Nonperturbative Questions, Vacua, and Sector Dynamics

Nonperturbative vacuum physics begins where a local expansion is no longer the whole question. A candidate minimum can be mixed by tunneling; a path integral can decompose into topological sectors; a theta parameter can Fourier-transform those sectors; and the infinite-volume energy can develop branches constrained, but not uniquely determined, by symmetry and anomaly. This chapter supplies the distinctions, normalizations, and order-of-limits tests needed to move among those statements without conflating them.

Helpful background. Vacua, states, and representations supplies the state-space language, and theta terms, periodicity, and vacuum sectors supplies the general topological-response construction.

Check your preparation

You are ready to enter at the first page if you can answer three operational questions.

Can you name the approximation? Given a formal series, you should be able to say what tends to zero, which observable is expanded, and what a remainder would mean. If that is unfamiliar, repair it with asymptotic scales and remainders before reading the claim taxonomy.

Can you distinguish a state from a field configuration? A classical minimum, a wave packet localized near it, and an exact ground state are different objects. If those are still blending together, begin with vacua, states, and representations, then return to the double-well thread below.

Can you identify the global data of a gauge theory? The Lie algebra alone does not fix genuine line operators, allowed bundles, or the topological-charge lattice. If that statement is new, read global form and matter representations before making any periodicity claim.

Being unsure about anomaly matching does not block the first seven leaves. It only means that the final, research-level leaf should wait until you can couple a faithful symmetry to background fields and compare responses modulo local counterterms.

Scope and ownership

This chapter owns the durable logic connecting local expansions, global sectors, theta transforms, response cumulants, finite-volume mixing, and anomaly constraints. It does not compute instanton determinants or integration cycles, perform lattice continuum extrapolations, classify anomalies abstractly, or freeze a current verdict about confinement and phase selection. Those tasks hand off respectively to Semiclassical Expansions and Integration Cycles, Volume 8: Lattice QFT and Hamiltonian Methods, Symmetry, Gauge Structure, and Anomalies, and the dated Nonperturbative Gauge Dynamics research field.

The chapter’s examples are deliberately bounded. The double well and particle on a circle make the transform and limit logic exact; they do not reproduce four-dimensional bundle data. The Yang–Mills examples supply that global data under declared gauge-group, matter, manifold, and background-field assumptions; they do not turn anomaly matching into a solution of the dynamics.

Enter the chapter

Start from the question you can actually pose about an observable.

If your question is…	Read…	Result you should be able to use
What is the calculation nonperturbative relative to, and why should I trust it?	What Does Nonperturbative Mean?	State the expansion, observable, regime, control test, error, and falsifier
Can an expansion around one minimum determine the vacuum?	Local Expansions and Global Vacuum Structure	Separate local saddle data from the spectrum and global vacuum choice
What fixes the topological charge lattice?	Topological Sectors, Boundary Data, and Global Form	Derive sectors and periodicity from boundary, bundle, and global-form data
Is theta a coupling, a state label, or a Fourier variable?	Theta Parameters, Theta States, and Sector Sums	Translate consistently among the three roles
How can periodic physics have nonperiodic branches?	Theta Dependence, CP, and Model-Dependent Branches	Distinguish branch relabeling, the lower envelope, CP, and metastability
What do theta derivatives measure?	Topological Susceptibility and Vacuum Response	Derive charge cumulants and handle contact terms and volume limits
Why can finite-volume tunneling coexist with broken phases?	Tunneling, Superselection, and the Infinite-Volume Limit	Compare finite-size level splitting with infinite-volume pure sectors
What does an anomaly determine about the infrared?	Anomaly and Generalized-Symmetry Constraints on Infrared Phases	Exclude impossible phases while keeping all matching alternatives

The shortest conceptual route is

$$\begin{gathered} \text{theory and boundary data} \longrightarrow \text{charge sectors} \longrightarrow Z(\theta) \longrightarrow \text{response and branches},\\ \text{local minima} \longrightarrow \text{tunneling at finite volume} \longrightarrow \text{pure vacua at infinite volume},\\ \text{symmetry and backgrounds} \longrightarrow \text{anomaly class} \longrightarrow \text{allowed infrared realizations}. \end{gathered}$$

The arrows do not all reverse. For example, a susceptibility does not reconstruct the global theta dependence, and an anomaly-compatible phase is not thereby shown to occur.

Shared sector conventions

For an allowed charge set $\Lambda_Q$, the chapter uses the Euclidean Fourier sum

$$Z_{V_4}(\theta) =\sum_{Q\in\Lambda_Q}e^{i\theta Q}Z_{V_4,Q}, \qquad \mathcal E(\theta) =-\lim_{V_4\to\infty}\frac1{V_4} \log\frac{Z_{V_4}(\theta)}{Z_{V_4}(0)}.$$

The normalization of $Q$ and the set $\Lambda_Q$ must be declared before a theta period is inferred. If $\Lambda_Q=\mathbb Z$, the sum is $2\pi$ periodic. Fractional bundles, a different global gauge group, defects, or non-spin manifolds can alter the statement or make a $2\pi$ shift exchange theories with different discrete data; Aharony, Seiberg, and Tachikawa 2013, §§ 1–2 give explicit four-dimensional examples.

At finite volume,

$$\frac{\partial^n}{\partial\theta^n} \log Z_{V_4}(\theta) =i^n\langle Q^n\rangle_{\theta,c}.$$

Only after the regulator and contact prescription are fixed should this identity be converted into an integrated correlator of local topological densities. Only after the volume limit is controlled should a nonanalytic vacuum-energy branch or superselection claim be made. These are two different limits: removing a short-distance regulator defines the observable, whereas sending the volume to infinity can change the state space and analyticity.

When $V_4=\beta V_s$ comes from a positive thermal trace, the vacuum-energy shorthand means first taking $\beta\to\infty$ at fixed $V_s$ and then taking $V_s\to\infty$. On a generic Euclidean background where $Z$ may be complex, the same logarithm is an intensive generating functional with a declared branch, not automatically a physical energy.

One double-well thread, four lessons

The chapter temporarily displays $\hbar$ in this thread to make the semiclassical expansion visible; otherwise it follows the site-wide natural-unit conventions.

The symmetric potential

$$V(q)=\lambda(q^2-a^2)^2$$

provides the chapter’s simplest recurring example.

1. Expanding around $q=a$ gives a local perturbative series but cannot, at any finite order, produce the exponentially small splitting proportional to $e^{-S_0/\hbar}$.
2. The two minima are not two exact finite-volume ground states. Tunneling produces symmetric and antisymmetric eigenstates.
3. In a field theory, the tunneling action can grow with spatial volume, making the splitting vanish and leaving distinct pure phases after the thermodynamic limit.
4. This example teaches the branch and volume-limit logic, but it has no four-dimensional gauge-bundle data and no one-form-symmetry anomaly. Those conclusions require their own hypotheses.

For the one-dimensional instanton connecting $-a$ to $a$,

$$S_0 =\int_{-a}^{a}\mathrm dq\,\sqrt{2mV(q)} =\frac{4}{3}a^3\sqrt{2m\lambda}.$$

The exponential is a precise beyond-all-orders contribution relative to the local $\hbar$ expansion. The prefactor, multi-instanton corrections, and error estimate are further calculations, not consequences of the label “nonperturbative.” Coleman gives the classic semiclassical treatment in Coleman 1985, ch. 7, § 2.2, pp. 270–277, and Mariño develops the same quantum-mechanical setting in Mariño 2015, ch. 1, §§ 1.8–1.9, pp. 38–53.

Four checks before accepting a vacuum claim

Name the comparison. State the expansion variable or formulation relative to which the effect is nonperturbative.

Specify the theory globally. Include the gauge-group global form, allowed bundles and defects, boundary conditions, and spacetime structure relevant to the charge lattice.

Order the limits. Keep regulator removal, Euclidean-time projection, thermodynamic limits, source removal, large-$N$ limits, and mass limits in an explicit sequence.

Separate constraints from dynamics. Periodicity is a property of the complete theory; an anomaly excludes infrared responses; neither statement alone selects an energy branch or phase.

These checks are deliberately observable-centered. A method can be exact under its hypotheses and still answer the wrong observable or the wrong limiting problem.

What the chapter establishes

The eight leaves support one chain of reasoning, but its links have different logical strengths.

• Definitions specify the comparison expansion, charge sectors, theta transform, response coefficients, and infrared anomaly class.
• Exact identities relate fixed-charge and fixed-theta partition functions and relate derivatives of $\log Z$ to connected charge cumulants, once the regulator and conventions are fixed.
• Controlled calculations estimate quantities such as a double-well splitting or a dilute-gas energy only in a displayed small-parameter regime.
• Limit statements explain how a smooth finite-volume partition function can approach a cusped energy density and how tunneling-mixed states can approach disjoint pure phases.
• Constraints exclude anomaly-incompatible infrared phases but do not supply the missing dynamical phase selection.

The most important category error to avoid is moving upward in that list without new input. A sector decomposition is not an instanton approximation; periodicity is not a branch formula; a cusp is not automatically CP breaking; and an anomaly-compatible phase is not thereby the realized phase.

Review map

Use the prompts below as capability checks, not as completion bookkeeping.

Capability	A successful answer must show	Repair route
Identify a beyond-all-orders effect	Compare the effect with every power of the declared expansion variable and name the observable	What Does Nonperturbative Mean?
Derive a theta period	State the charge lattice, discrete theta data, manifold class, and whether a shift returns one theory or permutes theories	Topological Sectors, Boundary Data, and Global Form
Separate local minima from vacua	Include tunneling or exact sector separation and order the source, time, and volume limits	Local Expansions and Global Vacuum Structure and Tunneling, Superselection, and the Infinite-Volume Limit
Interpret theta derivatives	Use connected cumulants at a named theta value and retain the regulator/contact prescription	Topological Susceptibility and Vacuum Response
Use an anomaly without overclaiming	Specify faithful symmetry, backgrounds, counterterms, every matching option, and the extra dynamical discriminator	Anomaly and Generalized-Symmetry Constraints on Infrared Phases

Review the chapter

Why does the function $e^{-A/g}$ with $A>0$ escape every power series about $g=0^+$?

Solution

For every fixed $N$,

$$\lim_{g\to0^+}\frac{e^{-A/g}}{g^N}=0.$$

Repeated differentiation produces only powers of $1/g$ multiplying the same exponential, so every right derivative of its continuous extension at zero vanishes. Its Taylor series is therefore zero even though the function is nonzero for $g>0$.

If $Q\in\mathbb Z$, prove the $2\pi$ periodicity of the sector sum. Identify the premise that fails for a fixed theory with half-integral sectors.

Solution

For integer $Q$,

$$e^{i(\theta+2\pi)Q}=e^{i\theta Q}e^{2\pi iQ}=e^{i\theta Q},$$

term by term, hence $Z(\theta+2\pi)=Z(\theta)$. If half-integral charges are included, $e^{2\pi iQ}$ can equal $-1$, so the termwise argument fails. One must include the discrete theta data and derive whether the shift changes a parameter of the same theory or exchanges distinct theories.

Two models have the same susceptibility $\chi$ at theta zero:

$$\mathcal E_{\mathrm{quad}}(\theta) =\min_k\frac{\chi}{2}(\theta+2\pi k)^2, \qquad \mathcal E_{\mathrm{gas}}(\theta) =\chi(1-\cos\theta).$$

Give one observable distinction.

Solution

The quadratic envelope has a cusp at $\theta=\pi$, whereas the cosine is smooth there. Locally, their fourth derivatives also differ: $\mathcal E_{\mathrm{quad}}^{(4)}(0)=0$ but $\mathcal E_{\mathrm{gas}}^{(4)}(0)=-\chi$. Thus equal curvature at the origin does not determine the higher cumulants or global branch structure.

Why can a unique symmetric ground state in every finite box coexist with spontaneous symmetry breaking?

Solution

The splitting between symmetric and antisymmetric states can vanish as $e^{-cV_s/\hbar}$. An arbitrarily weak extensive source then dominates the splitting when $V_s\to\infty$. Taking the volume limit before sending the source to zero selects one clustering pure phase; reversing the limits retains the symmetric finite-volume expectation value.

A nonzero anomaly excludes a unique trivial symmetric gapped vacuum. Why does it not prove spontaneous symmetry breaking?

Solution

Gapless degrees of freedom, topological order, symmetry extension, or a genuine higher-dimensional bulk boundary can also reproduce the anomaly. Spontaneous breaking is established only after additional dynamics selects it and its vacua or walls match the same background response.

Continue from this chapter

• To calculate the exponential, determinant, zero-mode measure, and validity range that were only named here, continue to Semiclassical Expansions and Integration Cycles.
• To compute gauge-instanton measures and fermion selection rules, continue to Instantons, Fermion Zero Modes, and Tunneling.
• To turn topological charge and susceptibility into regulated numerical observables, use Topology and Lattice Index Diagnostics.
• To inspect dated evidence and unsettled phase-selection claims, use the Nonperturbative Gauge Dynamics research field. The present chapter deliberately supplies the durable definitions and constraints, not a frozen claim about the currently favored infrared mechanism.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 2013, no. 8 (2013): 115. arXiv. DOI.
• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 7, § 2.2, pp. 270–277. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, ch. 1, §§ 1.8–1.9, pp. 38–53. DOI.