Zero Modes, Collective Coordinates, and Moduli Measures

A normalizable zero mode says that the quadratic approximation is constant along a direction in a family of saddles. The divergent Gaussian integral over that mode must be replaced by an integral over the corresponding modulus, with a Jacobian determined by the zero-mode norm. Gauge directions are different: they are redundancies to quotient, not physical moduli to integrate as distinct configurations.

Required background. Fluctuation operators and determinant ratios supplies the primed determinant and the spectral meaning of an omitted zero eigenvalue.

Helpful background. Changes of variables and regulated Jacobians supplies the finite-regulator definition of a functional measure change; the Faddeev–Popov construction supplies the quotient of gauge orbits and its determinant.

From a zero eigenfunction to a modulus

Let a $k$-parameter family of saddles be $\phi_\sigma(x;\gamma^1,\ldots,\gamma^k)$. Its tangent vectors are

$$u_a(x;\gamma)=\frac{\partial\phi_\sigma(x;\gamma)}{\partial\gamma^a}.$$

Differentiating the field equation with respect to $\gamma^a$ gives

$$M_\sigma u_a=0,$$

provided the differentiated configuration obeys the linearized boundary conditions. This last qualification matters: a formal symmetry variation need not be a zero mode in a finite box or in a fixed-boundary amplitude.

Define the moduli-space Gram matrix using the inner product appearing in the quadratic action,

$$G_{ab}(\gamma)=\langle u_a,u_b\rangle.$$

If $G$ is finite and nonsingular, the zero modes are normalizable and the local change of variables in the regulated functional measure is

$$\prod_{a=1}^{k}\frac{\mathrm dc_a}{\sqrt{2\pi g}} \longrightarrow \frac{\sqrt{\det G(\gamma)}}{(2\pi g)^{k/2}} \prod_{a=1}^{k}\mathrm d\gamma^a.$$

Here $c_a$ are coefficients along an orthonormal zero-mode basis. The one-saddle measure therefore contains

$$\mathrm d\mu_\sigma(\gamma) =\frac{\sqrt{\det G(\gamma)}}{(2\pi g)^{k/2}} \prod_a \mathrm d\gamma^a\, \left(\frac{\det{}'M_\sigma}{\det M_{\rm ref}}\right)^{-1/2}.$$

This formula fixes both the power of $g$ and the dimensions. It also shows why a primed determinant without its collective-coordinate measure is incomplete. Coleman 1985, ch. 7, §2.2, pp. 273–276 and Mariño 2015, §1.4, pp. 16–25 derive the replacement in the instanton path integral.

Shared calculation. The saddle-contribution anatomy displays this replacement beside the nonzero determinant and contour data. Shared comparison. The canonical comparison table gives the mode and measure checks for representative saddles.

The formula is local on moduli space. If several coordinate patches are required, $\sqrt{\det G}\,\mathrm d^k\gamma$ is the invariant volume element. Discrete identifications, stabilizers, and permutations of identical events must still be divided out. A noncompact modulus can produce a physical volume factor, such as total Euclidean time; a divergent integral over a size modulus is instead a signal that the semiclassical sector is infrared sensitive.

Translation modulus of the double-well instanton

For

$$x_I(\tau;\tau_0)=\tanh(\tau-\tau_0),$$

the modulus is the center $\tau_0$ and

$$u_{\tau_0}(\tau) =\frac{\partial x_I}{\partial\tau_0} =-\dot x_I =-\operatorname{sech}^2(\tau-\tau_0).$$

Its norm is

$$G_{\tau_0\tau_0} =\int_{-\infty}^{\infty}\mathrm d\tau\,\dot x_I^2 =\int_{-\infty}^{\infty}\mathrm d\tau\,\operatorname{sech}^4\tau =\frac43.$$

The same result follows from the first-order instanton equation:

$$\int \mathrm d\tau\,\dot x_I^2 =\int_{-1}^{1}(1-x^2)\,\mathrm dx =\mathcal S_I.$$

Thus the zero-mode integral becomes

$$\frac{\mathrm dc_0}{\sqrt{2\pi g}} \longrightarrow \left(\frac{\mathcal S_I}{2\pi g}\right)^{1/2}\mathrm d\tau_0 =\left(\frac{2}{3\pi g}\right)^{1/2}\mathrm d\tau_0.$$

On a long interval of length $T$, integration over an isolated center gives $T$ up to endpoint corrections. Dividing by $T$ converts the one-instanton amplitude to the event fugacity used in an energy or transition rate. If an $n$-event configuration is dilute, the ordered center integral produces $T^n/n!$; the factorial is a quotient by permutations, not another determinant.

In a finite interval, translation invariance is broken by the endpoints and the mode is lifted by an exponentially small eigenvalue. The correct large-$T$ calculation identifies the analytic translation vector and takes the projection and interval limits consistently. Treating the lifted eigenvalue as an ordinary Gaussian produces a spurious factor that diverges as $T\to\infty$.

Physical moduli versus gauge directions

Both a physical modulus and a gauge transformation can solve the linearized field equation, but their measures have different meanings.

For a physical modulus:

• changing $\gamma$ changes the location, size, or global orientation of the configuration relative to the boundary data;
• the tangent vector is normalizable in the declared inner product;
• the functional integral includes the invariant measure on the inequivalent family.

For a gauge direction:

• changing the gauge parameter does not change the physical configuration;
• the path integral must divide by the gauge-group volume;
• a gauge condition and Faddeev–Popov determinant remove the redundant direction.

Schematically, insert

$$1=\Delta_{\rm FP}[\phi]\int\mathcal D\alpha\, \delta\!\bigl(F[\phi^\alpha]\bigr)$$

and cancel the gauge-orbit volume. Residual transformations that change boundary data may become global symmetries and produce genuine orientation moduli; transformations in the stabilizer leave the saddle fixed and must be divided out. The answer therefore depends on the boundary conditions and on which gauge transformations are declared trivial.

A useful diagnostic is to ask whether a gauge-invariant observable changes when one moves along the proposed coordinate. If not, the direction is normally redundancy. This diagnostic does not replace the regulated Faddeev–Popov analysis, especially when stabilizers or Gribov copies are present.

Approximate zero modes

A coordinate describing widely separated instantons is not generally an exact modulus: interactions generate a shallow potential $V_{\rm eff}(\gamma)$. If its curvature is comparable to the terms omitted from the loop expansion, the Gaussian approximation along that direction is nonuniform. Retain the coordinate explicitly:

$$Z_\sigma \sim e^{-\mathcal S_\sigma/g} \int \mathrm d\gamma\,J(\gamma) \exp\!\left[-\frac{V_{\rm eff}(\gamma)}{g}\right] \left(\det{}'M_\sigma(\gamma)\right)^{-1/2}.$$

This treatment is also required for a size modulus whose integral explores scales where the running coupling becomes strong. A divergence of the moduli integral is not cured by assigning the zero eigenvalue a small arbitrary mass; it identifies missing infrared physics or a boundary of semiclassical control.

Common pitfalls

Integrating every symmetry parameter as a physical modulus. Local gauge transformations label redundant representatives. Gauge-fix and divide by the stabilizer before identifying any remaining global orientations.

Counting the zero mode twice. Once its eigenvalue is removed from $\det{}'M$, the corresponding integral appears exactly once in $\mathrm d\mu_\sigma$. Keeping both the Gaussian coefficient and the modulus overcounts the saddle family.

Assuming a formal zero mode is normalizable. Scale or gauge-orientation variations can have divergent norm. State the volume and boundary regulator, then test whether the regulated measure has a controlled limit.

Exercises

1. Verify $\int_{-\infty}^{\infty}\operatorname{sech}^4\tau\,\mathrm d\tau=4/3$.

Solution

Set $u=\tanh\tau$, so $\mathrm du=\operatorname{sech}^2\tau\,\mathrm d\tau$. Then

$$\int_{-\infty}^{\infty}\operatorname{sech}^4\tau\,\mathrm d\tau =\int_{-1}^{1}(1-u^2)\,\mathrm du =\left[u-\frac{u^3}{3}\right]_{-1}^{1} =\frac43.$$

2. Show that $\sqrt{\det G}\,\mathrm d^k\gamma$ is invariant under a change of moduli coordinates.

Solution

For $\gamma=\gamma(\widetilde\gamma)$, the metric transforms as

$$\widetilde G=J^{\mathsf T}GJ, \qquad J^a{}_b=\frac{\partial\gamma^a}{\partial\widetilde\gamma^b}.$$

Hence $\sqrt{\det\widetilde G}=|\det J|\sqrt{\det G}$. Since $\mathrm d^k\gamma=|\det J|\,\mathrm d^k\widetilde\gamma$, the coordinate Jacobian and metric transformation combine to give the same invariant volume element.

3. A two-instanton configuration has centers $\tau_1<\tau_2$ in an interval of length $T$. Neglecting endpoint and overlap corrections, compute the center integral.

Solution

The ordered region is half of the square:

$$\int_0^T \mathrm d\tau_2\int_0^{\tau_2}\mathrm d\tau_1 =\frac{T^2}{2}.$$

Equivalently, integrating two labeled centers over the full square gives $T^2$ and division by $2!$ removes their permutation. Interactions modify this result when $\tau_2-\tau_1$ is comparable to the core size.

References

• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 7, pp. 265–350. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, ch. 1, pp. 3–61. DOI.