Complex Saddles, Quasi-Zero Modes, and Hidden Phases

Complex saddles and correlated-event integrals can supply phases that no dilute gas of isolated real saddles contains. The phase is meaningful only after the complexified action, observable, and integration contour are declared. For an instanton–anti-instanton molecule, the separation is a quasi-zero mode; its prescribed contour generates the logarithm and two-valued imaginary part required to cancel a Borel ambiguity.

Required background. Complex saddles, Lefschetz thimbles, and integration cycles supplies the complexified flow and cycle decomposition. Ambiguity cancellation and transseries consistency fixes the lateral convention and the cancellation target.

Helpful background. Small-circle Abelianization, bions, and adiabatic continuity gives a controlled gauge-theory regime in which correlated monopole events can be distinguished.

Quasi-zero modes are collective coordinates with a weak potential

For two well-separated events centered at $\tau_1$ and $\tau_2$, the relative separation

$$R=|\tau_1-\tau_2|$$

would be an exact modulus if the events did not interact. Their exponentially small overlap lifts it, so $R$ is a quasi-zero mode. The collective integral is not a Gaussian determinant:

$$\mathcal A_{I\bar I} \propto e^{-2\mathcal S_I/g} \int_{\Gamma_R}\mathrm dR\, \left[e^{-V_{\mathrm{int}}(R)/g}-1\right].$$

The subtraction removes the disconnected product already counted by two independent events. The contour $\Gamma_R$ and the range in which the asymptotic interaction is reliable are part of the definition. Treating the lifted mode as an exact zero mode would miss the logarithm; treating it as an ordinary positive Hessian eigenvalue would miss the large separation region.

In the symmetric double well, the instanton and anti-instanton attract. A schematic large-$R$ interaction is

$$V_{\mathrm{int}}(R)=-c e^{-\omega R}+\cdots, \qquad c,\omega>0.$$

The positive-$g$ real-$R$ integral is driven toward small separation, where the separated-event approximation fails. One must specify a continuation or determine the full complex thimble.

A prescribed quasi-zero-mode integral

Set $\omega=1$ temporarily and isolate the universal logarithmic model

$$I(g) =\int_0^\infty\mathrm dR\, \left[ \exp\!\left(\frac{c}{g}e^{-R}\right)-1 \right].$$

For $g=-|g|$, the interaction is repulsive and the real contour converges after the disconnected subtraction:

$$\begin{aligned} I(-|g|) &=\int_0^{c/|g|}\frac{\mathrm dt}{t}(e^{-t}-1)\\ &=-\gamma_E-\log\!\left(\frac{c}{|g|}\right) -E_1\!\left(\frac{c}{|g|}\right), \end{aligned}$$

where $t=(c/|g|)e^{-R}$. At weak coupling, $E_1(c/|g|)$ is exponentially small. The Bogomolny–Zinn-Justin prescription analytically continues this result to positive $g$ through the upper or lower half-plane. Since its analytic form contains $\log(-c/g)$,

$$I_\pm(g) =-\gamma_E-\log\!\left(\frac{c}{g}\right) \mp i\pi +O(e^{-c/g}), \qquad g>0.$$

Here $+$ means that $g$ approaches the positive axis through the upper half-plane; reversing that declaration reverses the signs. Restoring $\omega$ multiplies the result by $1/\omega$ after $r=\omega R$.

The complete molecule contains its determinant, Jacobian, and the factor $e^{-2\mathcal S_I/g}$. Thus the contour produces a paired ambiguity proportional to

$$\mp i\pi\,e^{-2\mathcal S_I/g}$$

times the independently computed prefactor. This is the phase traced on the previous page. Bogomolny’s original continuation is Bogomolny 1980, pp. 431–435; the necessity of complexifying the quasi-zero-mode contour in supersymmetric quantum-mechanical examples is shown in Behtash et al. 2015, §§2–4.

Complex saddles and phases

For a holomorphic complexified action,

$$S[z_\sigma]=S_{\sigma,R}+iS_{\sigma,I},$$

the saddle weight is

$$e^{-S[z_\sigma]/g} =e^{-S_{\sigma,R}/g}e^{-iS_{\sigma,I}/g}.$$

A phase sometimes called a hidden topological angle is the imaginary action, or a monodromy-related invariant phase, attached to a contributing complex saddle and its thimble. The phrase is justified only when:

1. the complex saddle solves the stated holomorphic equations;
2. its finite action and branch are specified;
3. its downward cycle has nonzero intersection with the original contour;
4. the fluctuation and quasi-zero-mode contours are fixed; and
5. the phase changes a named observable or enforces a checked cancellation.

Without these data, a phase inferred from a formal logarithm is not a saddle result. Exact complex bion solutions in specified quantum-mechanical models provide controlled examples; see Behtash et al. 2016, §§II–IV. The same paper is explicit that extending this construction to QFT is more difficult.

The shared thimble map shows why a complex critical point need not contribute to a chosen cycle. The Borel and transseries map connects the quasi-zero-mode phase to its ambiguity target. The exact and rigorous status comparison keeps an exact quantum-mechanical saddle distinct from a QFT conjecture.

Correlated events on a small circle

In center-stabilized or adjoint-matter gauge theories on $\mathbb R^3\times S^1$, explicit holonomy can make monopole-instanton constituents semiclassical. Magnetic bions and neutral bions are different composites:

• a magnetic bion carries magnetic charge and can generate a dual-photon potential and mass;
• a neutral bion has zero net magnetic charge, can affect the holonomy potential, and can possess a two-fold quasi-zero-mode ambiguity.

The allowed constituents depend on the gauge group’s global form, matter representation, fermion boundary conditions, and circle holonomy. A correlated-event amplitude in that controlled regime can realize a transseries cancellation. Identifying it with an infinite-volume renormalon requires a separate continuity argument and remains model dependent; Argyres and Ünsal 2012 states that proposed connection with its compactification assumptions.

Calculation boundary

This page establishes a controlled bridge in quantum mechanics and selected lower-dimensional or compactified theories. General steepest-descent geometry remains with the thimble chapter, and rigorous summability requires theorem-specific hypotheses. A claimed complex saddle in an unregulated four-dimensional path integral is not established by the finite-dimensional logarithmic model.

Common pitfalls

Integrating an attractive separation on the real contour without a prescription. The small-separation endpoint then dominates outside the asymptotic two-event description. Continue from the repulsive problem or determine the actual complex thimble.

Calling every lifted mode a quasi-zero mode. The eigenvalue must be parametrically small because of an approximate modulus, and its range must extend beyond a local Gaussian neighborhood.

Inferring a condensate from zero-mode saturation or a phase alone. A nonzero amplitude with the right quantum numbers is not by itself a vacuum expectation value; volume limits, symmetry, and observable insertions still matter.

Exercises

1. Derive the exact identity

$$\int_0^a\frac{e^{-t}-1}{t}\,\mathrm dt =-\gamma_E-\log a-E_1(a), \qquad a>0.$$

Solution

Differentiate both sides with respect to $a$. Since $E_1'(a)=-e^{-a}/a$, the derivative of the right side is $(e^{-a}-1)/a$. Both sides vanish as $a\to0^+$, using $E_1(a)=-\gamma_E-\log a+a+O(a^2)$.

2. If $V_{\mathrm{int}}(R)=-c e^{-\omega R}$, show how the ambiguity depends on $\omega$.

Solution

With $r=\omega R$, $\mathrm dR=\mathrm dr/\omega$. The entire logarithmic integral is multiplied by $1/\omega$, so

$$\operatorname{Im}I_\pm(g)=\mp\frac{\pi}{\omega}.$$

The molecule’s remaining prefactor and $e^{-2\mathcal S_I/g}$ are unchanged by this variable substitution.

References

• Argyres, Philip C., and Mithat Ünsal. “The Semi-Classical Expansion and Resurgence in Gauge Theories: New Perturbative, Instanton, Bion, and Renormalon Effects.” Journal of High Energy Physics 08 (2012): 063. arXiv:1206.1890; doi:10.1007/JHEP08(2012)063.
• Behtash, Alireza, Gerald V. Dunne, Thomas Schäfer, Tin Sulejmanpasic, and Mithat Ünsal. “Toward Picard–Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence.” Annals of Mathematical Sciences and Applications 2 (2017): 95–212. arXiv:1510.03435; doi:10.4310/AMSA.2017.v2.n1.a3.
• Behtash, Alireza, Erich Poppitz, Tin Sulejmanpasic, and Mithat Ünsal. “The Curious Incident of Multi-Instantons and the Necessity of Lefschetz Thimbles.” Journal of High Energy Physics 11 (2015): 175. arXiv:1507.04063; doi:10.1007/JHEP11(2015)175.
• Bogomolny, E. B. “Calculation of Instanton–Anti-Instanton Contributions in Quantum Mechanics.” Physics Letters B 91 (1980): 431–435. doi:10.1016/0370-2693(80)91014-X.