Gauge and Renormalization-Scale Dependence of Decay Rates

A physical false-vacuum decay rate is gauge-fixing and renormalization-scale independent, although a background field, an effective potential, a bounce profile, and the conventional split between exponent and prefactor need not be. The cancellations follow from the Nielsen identity and the renormalization-group equation for the false-vacuum effective action. At finite order they are obtained only when the profile, derivative terms, determinants, zero-mode factors, and running parameters are treated with one consistent power counting.

Required background. Decay rates, the negative mode, and prefactors supplies the observable and the exponent–prefactor formula. The 1PI effective action and mean-field equations supplies functional stationarity and the derivative expansion. Large logarithms and RG improvement supplies running-coupling and residual-scale logic. Localized transformations and Ward–Takahashi identities supplies the change-of-variables reasoning behind gauge-parameter identities.

Helpful background. Standard Model running and vacuum-stability criteria develops the phenomenological application; this page keeps the argument at the general and Abelian toy-model level.

The false-vacuum effective action

The ordinary ground-state 1PI effective action is real and convex in its exact form, whereas decay is encoded by a false-vacuum persistence amplitude. The relevant functional, denoted $\Gamma_{\mathrm F}[\varphi;\xi,\mu]$, is defined with false-vacuum boundary conditions. It can be nonconvex and complex. Its stationary “quantum bounce” $\bar\varphi_b$ and stationary false vacuum $\bar\varphi_{\mathrm f}$ satisfy

$$\left.\frac{\delta\Gamma_{\mathrm F}}{\delta\varphi^i(x)} \right|_{\bar\varphi_b} =0, \qquad \left.\frac{\delta\Gamma_{\mathrm F}}{\delta\varphi^i(x)} \right|_{\bar\varphi_{\mathrm f}} =0.$$

The exact rate can be formulated from the false-vacuum functional evaluated at these stationary configurations. This formulation retains the contour and boundary data that are lost if one substitutes an exact convex ground-state potential into a classical bounce equation Plascencia and Tamarit 2016, §§2–4.

The notation $\varphi^i$ includes every background component needed by the gauge-fixed theory. Eliminating gauge, Goldstone, or ghost sectors before deriving the identities can remove terms required for the cancellation.

Nielsen identity on the bounce

For a gauge-fixing parameter $\xi$, the false-vacuum effective action obeys a functional Nielsen identity of the form

$$\frac{\partial\Gamma_{\mathrm F}}{\partial\xi} +\int\mathrm d^d x\, K^i[\varphi;\xi](x) \frac{\delta\Gamma_{\mathrm F}}{\delta\varphi^i(x)} =0.$$

The functional $K^i$ depends on the gauge-fixing convention. The important structure is that gauge-parameter variation is proportional to the equations of motion Nielsen 1975, §§2–3.

Differentiate the on-shell bounce value, including the implicit $\xi$ dependence of the profile:

$$\begin{aligned} \frac{\mathrm d}{\mathrm d\xi} \Gamma_{\mathrm F}[\bar\varphi_b(\xi);\xi] ={}& \left.\frac{\partial\Gamma_{\mathrm F}}{\partial\xi} \right|_{\bar\varphi_b}\\ &+ \int\mathrm d^d x\, \left. \frac{\delta\Gamma_{\mathrm F}}{\delta\varphi^i(x)} \right|_{\bar\varphi_b} \frac{\mathrm d\bar\varphi_b^i(x)}{\mathrm d\xi} =0. \end{aligned}$$

The same equation holds at $\bar\varphi_{\mathrm f}$. Hence their on-shell action difference is gauge independent. The profile itself can move in field space as $\xi$ changes; gauge independence never requires a gauge-independent profile.

A local derivative expansion illustrates why an effective-potential-only calculation is incomplete:

$$\Gamma_{\mathrm F}[\phi] = \int\mathrm d^d x \left[ V(\phi;\xi,\mu) +\frac12Z(\phi;\xi,\mu)(\partial_\mu\phi)^2 +\mathcal O(\partial^4) \right].$$

The potential-level identity has the schematic form

$$\partial_\xi V+C(\phi;\xi)V'=0.$$

It guarantees that $V$ at a homogeneous stationary point is gauge independent, while the coordinate $\phi$ of that point can be gauge dependent. A bounce is inhomogeneous and is not pointwise at $V'=0$. The identities for $Z$ and the higher-derivative coefficients supply the remaining terms. Solving a bounce using a gauge-dependent $V$ while fixing $Z=1$ at an inconsistent order generally leaves spurious $\xi$ dependence. This failure and its consistent derivative-expansion cure are explicit in radiatively generated barriers Metaxas and Weinberg 1996, §§II–IV.

Perturbative order and the exponent–prefactor split

Write schematically

$$\Gamma_{\mathrm F} = \Gamma_0+\hbar\Gamma_1+\hbar^2\Gamma_2+\cdots, \qquad \bar\varphi_b = \varphi_0+\hbar\varphi_1+\cdots.$$

If $\delta\Gamma_0/\delta\varphi|_{\varphi_0}=0$, then

$$\Gamma_{\mathrm F}[\bar\varphi_b] = \Gamma_0[\varphi_0] +\hbar\Gamma_1[\varphi_0] +\mathcal O(\hbar^2).$$

The term linear in $\varphi_1$ vanishes by leading stationarity, but $\varphi_1$ becomes necessary at the next order. This elementary expansion prevents two common mismatches: inserting a partially corrected profile into an uncorrected functional, and improving the potential while leaving derivative and determinant terms at a lower order.

For a tree-level barrier, the classical bounce usually supplies the leading exponent and the one-loop fluctuation determinant supplies the first prefactor. For a barrier generated by loops, couplings may themselves carry powers of $\hbar$ and the counting must be reorganized. Contributions that look like different loop orders in ordinary counting can then enter the same order in the decay rate. The separation

$$\frac{\Gamma}{V_{d-1}}=A\,e^{-B}$$

is useful, but finite pieces can move between $B$ and $A$ under a scheme change or reorganization. Gauge and scale independence apply to the consistently truncated $\ln(\Gamma/V_{d-1})$, not necessarily to each displayed factor separately. Precision calculations of scale-invariant and radiative bounces make this order mixing explicit Andreassen et al. 2017, §§2–5.

Renormalization-scale consistency

Let $g_a(\mu)$ denote all renormalized couplings and masses. The false-vacuum effective action obeys

$$\mathcal D_{\mathrm{RG}}\Gamma_{\mathrm F}=0,$$

with

$$\mathcal D_{\mathrm{RG}} = \mu\frac{\partial}{\partial\mu} +\sum_a\beta_a\frac{\partial}{\partial g_a} -\int\mathrm d^d x\, \gamma_i{}^j\varphi^i(x) \frac{\delta}{\delta\varphi^j(x)}.$$

At a stationary bounce and false vacuum, the field-redefinition term vanishes in their action difference. Explicit logarithms cancel the running of couplings and masses when both are retained to the same order. In the rate,

$$\mathcal D_{\mathrm{RG}} \left[ -B+\ln A \right] =0$$

to the computed order. A residual $\mu$ dependence of the same order as retained terms signals missing pieces; a residual of the first omitted order is expected and can help estimate truncation uncertainty. Choosing $\mu\sim R^{-1}$ may reduce logarithms, but it does not replace solving the RG equation when several disparate masses are present.

Changing renormalization scheme likewise redistributes finite terms among running parameters, the action, counterterms, and the determinant. A physical rate is scheme independent only through the order at which all those transformations have been applied.

First application: an Abelian gauge–scalar model

Consider a complex scalar $\Phi$ coupled to an Abelian gauge field,

$$\mathcal L_E = \frac14F_{\mu\nu}F_{\mu\nu} +\lvert D_\mu\Phi\rvert^2 +U(\lvert\Phi\rvert) +\mathcal L_{\mathrm{gf}}(\xi) +\mathcal L_{\mathrm{ghost}}.$$

Suppose the renormalized parameters produce a metastable scalar background. In a covariant gauge, the Goldstone and longitudinal-gauge contributions are gauge-parameter dependent. The ghost determinant can be background dependent or can reduce to a field-independent factor, depending on the gauge-fixing functional; either way, it must be treated in the same convention as the other fluctuations. A defensible fixed-order calculation proceeds as follows:

1. Define the false-vacuum persistence amplitude, regulator, subtraction scheme, and whether the barrier is tree-level or radiatively generated.
2. Derive one power counting for $V$, wave-function and higher-derivative terms, and the bounce profile.
3. Solve the stationary equation of that truncated false-vacuum effective action, not an isolated effective potential from a different order.
4. Evaluate the coupled gauge–Goldstone fluctuation operator, ghost determinant, physical scalar modes, collective-coordinate factors, and counterterms in the same gauge and scheme.
5. Combine exponent and prefactor before testing $\xi$ and $\mu$ dependence.
6. Vary $\xi$ only over a perturbatively regular range and vary $\mu$ around the physical bounce scales. The combined logarithmic rate should change first at the omitted order.

This is a consistency test, not a proof obtained by numerical flatness. A small variation can result from an accidental cancellation, whereas an exact Nielsen or RG identity identifies which terms must cancel. Conversely, singular gauges, infrared enhancements, or large logarithms can invalidate the nominal loop counting even when the formal identity remains true.

What a potential-only lifetime misses

An estimate based only on the depth, crossing point, or extremum of an effective potential omits at least the inhomogeneous profile, derivative terms, the negative-mode contour, translation Jacobians, determinants, and state normalization. It may be useful for locating candidate regimes, but it is not a decay rate. For a quantitative result, report:

• the false-vacuum observable and boundary conditions;
• the perturbative and derivative power counting;
• the gauge fixing and range used for the residual check;
• the renormalization scheme and scale prescription;
• the complete mode and counterterm content of the prefactor;
• the separate sizes of the first omitted loop, derivative, and large-log terms.

Shared calculation. The bounce control map locates the gauge, renormalization, determinant, and environmental checks in the full rate calculation.

Shared comparison. The instanton–bounce boundary and mode comparison keeps the false-vacuum decay observable distinct from level splitting or topological-sector tunneling.

Common pitfalls

Demanding a gauge-independent bounce profile. The Nielsen identity permits a gauge-dependent field coordinate and profile. It is the on-shell false-vacuum functional and the physical rate that are invariant.

RG-improving only the potential. Running couplings inside $V$ while leaving $Z$, determinants, zero-mode normalization, and matching fixed at another order does not constitute a consistent RG improvement.

Adding scale and gauge variations as if they were independent observables. Both probe missing terms in one truncated calculation and can be correlated. Report the variations and the power-counting estimate rather than treating their envelope as a theorem.

Exercises

1. Starting from the functional Nielsen identity, prove that the on-shell action difference between a stationary bounce and stationary false vacuum is gauge independent.

Solution

For either stationary configuration $\bar\varphi$,

$$\frac{\mathrm d}{\mathrm d\xi} \Gamma_{\mathrm F}[\bar\varphi(\xi);\xi] = \partial_\xi\Gamma_{\mathrm F} +\int\mathrm d^d x\, \frac{\delta\Gamma_{\mathrm F}}{\delta\varphi^i} \frac{\mathrm d\bar\varphi^i}{\mathrm d\xi}.$$

The second term vanishes by stationarity. The Nielsen identity makes the first term proportional to the same equations of motion, so it also vanishes. Subtracting the false-vacuum result from the bounce result preserves zero.

2. Show why the order-$\hbar$ shift of the bounce does not enter the on-shell action through order $\hbar$ when the leading bounce is stationary.

Solution

Taylor expansion gives

$$\begin{aligned} \Gamma[\varphi_0+\hbar\varphi_1] ={}& \Gamma_0[\varphi_0] +\hbar\int\mathrm d^d x\, \left.\frac{\delta\Gamma_0}{\delta\varphi}\right|_{\varphi_0} \varphi_1\\ &+\hbar\Gamma_1[\varphi_0] +\mathcal O(\hbar^2). \end{aligned}$$

The integral vanishes because $\varphi_0$ is stationary, leaving $\Gamma_0[\varphi_0]+\hbar\Gamma_1[\varphi_0]$ at this order.

3. Suppose a calculation finds

$$\frac{\mathrm dB}{\mathrm d\ln\mu}=c\,g^2, \qquad \frac{\mathrm d\ln A}{\mathrm d\ln\mu}=c\,g^2+\mathcal O(g^4).$$

What is the scale dependence of the logarithmic rate through order $g^2$?

Solution

Since $\ln(\Gamma/V)=-B+\ln A$,

$$\frac{\mathrm d}{\mathrm d\ln\mu} \ln\frac{\Gamma}{V} = -c\,g^2+c\,g^2+\mathcal O(g^4) = \mathcal O(g^4).$$

The exponent and prefactor are separately scale dependent, while their combination is stable through the retained order.

References

• Andreassen, A., Farhi, D., Frost, W., and Schwartz, M. D. (2017). “Precision Decay Rate Calculations in Quantum Field Theory.” Physical Review D 95, 085011. doi:10.1103/PhysRevD.95.085011. Open PDF.
• Metaxas, D., and Weinberg, E. J. (1996). “Gauge Independence of the Bubble Nucleation Rate in Theories with Radiative Symmetry Breaking.” Physical Review D 53, 836–843. doi:10.1103/PhysRevD.53.836.
• Nielsen, N. K. (1975). “On the Gauge Dependence of Spontaneous Symmetry Breaking in Gauge Theories.” Nuclear Physics B 101, 173–188. doi:10.1016/0550-3213(75)90301-6.
• Plascencia, A. D., and Tamarit, C. (2016). “Convexity, Gauge-Dependence and Tunneling Rates.” Journal of High Energy Physics 2016(10), 099. doi:10.1007/JHEP10(2016)099. Open PDF.