Renormalized Saddle Contributions and Validity Tests

A semiclassical calculation becomes a prediction only after its saddle action, determinant, collective measure, insertions, and counterterms are combined in one renormalization scheme and attached to a specified observable. Its error must include loop truncation, omitted saddles, dilute-ensemble corrections, regulator removal, and numerical uncertainty. This page gives that extraction procedure and sets out a direct spectral test of the double-well level splitting.

Required background. Multi-saddle sums and dilute ensembles supplies the relation between an event fugacity and a level splitting; fluctuation operators and determinant ratios supplies the one-loop prefactor and its zero-mode prescription.

Helpful background. Renormalization conditions, schemes, and finite parts supplies the distinction between a regulated determinant and a renormalized observable.

The renormalized saddle contribution

For an observable $\mathcal O$, a regulated saddle sector has the form

$$\mathcal Z_\sigma[\mathcal O] =n_\sigma e^{-S_{\sigma,\rm ren}/g} \int_{\mathcal M_\sigma}\mathrm d\mu_\sigma(\gamma;\mu) \mathcal P_{\sigma,\rm ren}(\mu) \mathcal I_\sigma[\mathcal O;\gamma,\mu] \left[1+\sum_{\ell=1}^{L}g^\ell c_{\sigma,\ell}(\mu)\right].$$

Here:

• $n_\sigma$ is fixed by the integration cycle;
• $S_{\sigma,\rm ren}$ is the saddle action relative to the reference sector;
• $\mathrm d\mu_\sigma$ includes physical collective coordinates and stabilizer quotients;
• $\mathcal P_{\sigma,\rm ren}$ is the regulated one-loop determinant prefactor, with only the identified zero modes omitted, combined with its local counterterms in one scheme;
• $\mathcal I_\sigma$ is the insertion evaluated with its fluctuation contractions;
• counterterms and operator renormalization use the same scale $\mu$ and scheme.

The normalized expectation value is

$$\langle\mathcal O\rangle =\frac{\sum_\sigma\mathcal Z_\sigma[\mathcal O]} {\sum_\sigma\mathcal Z_\sigma[1]}.$$

Disconnected vacuum factors cancel only after numerator and denominator have been expanded to compatible orders. A saddle-independent normalization can cancel; a sector-dependent determinant or counterterm cannot.

In QFT, the one-loop effective exponent is more transparently written

$$\mathcal W_\sigma =\frac{S_{\rm bare}[\phi_\sigma]-S_{\rm bare}[\phi_{\rm ref}]}{g_{\rm bare}} +\frac12\log\frac{\det{}'M_\sigma}{\det M_{\rm ref}} +S_{\rm ct}^{(1)}[\phi_\sigma]-S_{\rm ct}^{(1)}[\phi_{\rm ref}] +\cdots.$$

The heat-kernel coefficients of the determinant divergence are local and must match the available counterterms. After expressing the bare parameters through renormalized ones, a normalized renormalized observable—or a vector $\mathbf G_R$ of correlators whose insertions mix—obeys a Callan–Symanzik equation, not in general a partial derivative with every running parameter held fixed. A convention-neutral form is

$$\left( \bigl(\mathcal D_{\rm RG}+n\gamma_r\bigr)\mathbf 1 +\boldsymbol\gamma_{\mathcal O} \right)\mathbf G_R =\mathbf G_{R,\rm lead}\,O(g^{L+1}), \qquad \mathcal D_{\rm RG} =\mu\partial_\mu +\sum_i\beta_i\partial_{c_i} +\sum_a\beta_{m_a}\partial_{m_a}.$$

Here $c_i$ denotes renormalized dimensionless couplings, $\beta_{m_a}=\mu\,\mathrm d m_a/\mathrm d\mu$, $n\gamma_r$ accounts for the $n$ external renormalized fields in the site convention, and $\boldsymbol\gamma_{\mathcal O}$ acts on any additional mixing insertion vector, with its sign fixed by that definition. Set $n=0$ for a scalar observable with no external renormalized fields. The displayed remainder is relative to the leading retained result through $L$ loops. For an RG-invariant scalar observable, the mixing term is absent after the running parameters have been inserted. A separately defined saddle sector satisfies such an equation on its own only when its boundary conditions and operator basis are closed under RG evolution; in general the normalized physical sum is the object that must pass the test. Residual scale dependence estimates missing higher orders only after the running parameters, operator normalization, and all terms at the retained order have been included. Vassilevich 2003, §§2 and 4, pp. 285–317 gives the heat-kernel structure of one-loop divergences, while Dunne 2008, §§4–6, pp. 14–28 applies determinant methods to nontrivial backgrounds.

Extracting observables

Different observables use the same saddle ingredients but different boundary data.

Correlators. Insert the renormalized operators before the saddle expansion. The leading term evaluates the insertion on the saddle; higher terms contract fluctuation fields with the projected Green function $M_\sigma^{-1}$.

Energy shifts and splittings. Form the long-time transition matrix among perturbative vacua. The eigenvalues of its intensive logarithm give energies; off-diagonal one-event amplitudes generate exponentially small splittings.

Decay rates. Use the false-vacuum persistence amplitude and its prescribed lateral continuation. A rate requires the one-negative-mode contour and satisfies $\Gamma_{\rm decay}=-2\,\operatorname{Im}E_{\rm fv}$ in quantum mechanics. Callan and Coleman 1977, pp. 1762–1768 derives the renormalized one-bounce structure.

Densities and thermodynamic quantities. Take the logarithm before the large-volume limit so that connected clusters, rather than disconnected volume powers, define the intensive observable.

Shared comparison. The canonical saddle comparison records the action, mode count, determinant prescription, contour, renormalization, and principal failure boundary that must accompany each extraction.

Shared calculation. The saddle-contribution anatomy makes explicit which factors must be combined before the sector sum is interpreted as an observable.

Double-well splitting versus diagonalization

The Euclidean action used in this chapter corresponds to the Hamiltonian

$$H_g =-\frac g2\frac{\mathrm d^2}{\mathrm dx^2} +\frac{(x^2-1)^2}{2g}, \qquad g>0.$$

The two wells have small-oscillation frequency $2$. The leading instanton prediction derived from the action, translation Jacobian, and reduced determinant is

$$\Delta E_{\rm sc}(g) =8\sqrt{\frac{2}{\pi g}}\, \exp\!\left(-\frac{4}{3g}\right) \bigl[1+O(g)\bigr].$$

This can be tested without fitting the exponent:

1. solve the Schrödinger problem on $[-L,L]$ with parity-separated basis functions or a converged spectral grid;
2. increase $L$, basis size, and arithmetic precision until the two lowest parity eigenvalues are stable;
3. form $\Delta E_{\rm num}=E_{\rm odd}-E_{\rm even}$;
4. compare

$$R(g) =\frac{\Delta E_{\rm num}(g)} {8\sqrt{2/(\pi g)}\,e^{-4/(3g)}}.$$

In the controlled weak-coupling window,

$$R(g)=1+O(g),$$

and an exponent-only check gives

$$-g\log\Delta E_{\rm num} =\frac43+O(g\log g).$$

The ratio test is stronger because it checks the determinant and collective-coordinate normalization, not just the instanton action. At very small $g$, subtracting two nearly equal floating-point eigenvalues loses precision; diagonalizing even and odd sectors independently with extended precision avoids that failure. At larger $g$, $R(g)-1$ contains genuine loop and multi-event corrections, so it should not be labeled numerical error.

An independent parity-resolved diagonalization gives the following static benchmark. The calculation used a harmonic-oscillator basis with inverse length-squared scale $\Omega=2/g$, constructed $x$ and $p$ in a padded space of dimension $P=N+4$, projected to $N$ states, and diagonalized the exact even and odd blocks separately at 60-digit precision.

$g$	$\Delta E_{\rm num}$	$R(g)$
$0.20$	$0.0150877813243$	$0.830628592$
$0.15$	$0.00199735660138$	$0.878752523$
$0.10$	$3.01409101920\times10^{-5}$	$0.921975555$
$0.08$	$1.22349031017\times10^{-6}$	$0.938332685$

For these four rows, $N=120,160,200$ agree beyond the displayed digits; the $N=80\to120$ change in the gaps is below $3\times10^{-31}$. That basis and precision stability bounds the reported numerical error, but it does not turn $R(g)-1$ into numerical error: the monotone approach toward one is the expected weak-coupling trend, while the remaining deviation includes physical higher-loop and multi-event terms. A reusable interactive implementation remains a numerical-methods handoff; this table is an independently checked static benchmark, not a claim that the full semiclassical series converges.

Mariño 2015, §1.8, pp. 38–42 derives the analogous splitting in a different normalization and shows explicitly how rescaling changes the prefactor.

A quantitative error budget

For a leading saddle $\sigma$ with the nearest omitted saddle $\tau$, a useful relative-error model is

$$\delta_{\rm rel} \sim C_{\rm loop}g^{L+1} +C_{\tau\sigma} e^{-\operatorname{Re}(S_\tau-S_\sigma)/g} +C_{\rm overlap}\,\varrho_{\rm event}V_{\rm core} +C_{\rm FV}e^{-mL_{\rm box}} +\delta_{\rm ren} +\delta_{\rm num}.$$

For the same observable and compatible moduli domains, the omitted-sector coefficient contains the prefactor and measure ratio,

$$C_{\tau\sigma} \sim \left| \frac{n_\tau\int\mathrm d\mu_\tau\, \mathcal P_{\tau,\rm ren}\mathcal I_\tau} {n_\sigma\int\mathrm d\mu_\sigma\, \mathcal P_{\sigma,\rm ren}\mathcal I_\sigma} \right|.$$

The terms have different meanings:

• $C_{\rm loop}g^{L+1}$ is the next fluctuation order, provided no coefficient is anomalously large;
• the omitted-saddle term is meaningful only after its intersection number, action gap, and prefactor-and-measure ratio are known;
• $\varrho_{\rm event}V_{\rm core}$ measures event overlap and reduces to $\kappa\xi$ in one-dimensional quantum mechanics, with connected-cluster coefficients refining it;
• the finite-volume term assumes a mass gap $m$ and compatible boundaries;
• $\delta_{\rm ren}$ measures residual scale or regulator dependence after subtraction;
• $\delta_{\rm num}$ includes discretization, truncation, solver, and roundoff errors.

The model is not a universal bound, and its coefficients must be estimated in the specified theory and regime. It becomes an inequality only when each term has an independently established bound. Its purpose is to prevent one small number from concealing a different uncontrolled limit.

A claim should be qualified or withdrawn when any of the following occurs:

• a nonzero Hessian eigenvalue approaches the interaction scale;
• two contributing saddle actions become equal within the requested accuracy;
• a moduli integral reaches strong coupling or a volume endpoint;
• a negative-mode count changes under stable regulator refinement;
• counterterm or renormalization-scale dependence survives at the retained order;
• numerical and analytic normalizations cannot be matched dimensionally.

Common pitfalls

Comparing only the exponential slope. Agreement of $-g\log\Delta E$ with the classical action does not test the prefactor. Use a ratio such as $R(g)$ after fixing conventions.

Calling all disagreement numerical error. Loop truncation and multi-saddle overlap are physical approximation errors. Vary numerical controls separately from $g$ and volume to distinguish them.

Renormalizing the vacuum and saddle in different schemes. Finite parts then contaminate the claimed nonperturbative prefactor. Use the same renormalized parameters and operator normalization in both sectors.

Exercises

1. Derive the two-state splitting generated by an off-diagonal fugacity $\kappa$.

Solution

With

$$H_{\rm eff} =\begin{pmatrix}E_{\rm pert}&-\kappa\\-\kappa&E_{\rm pert}\end{pmatrix},$$

the symmetric and antisymmetric eigenvectors have eigenvalues $E_{\rm pert}-\kappa$ and $E_{\rm pert}+\kappa$. Their difference is $2\kappa$, which gives the displayed semiclassical prediction after substituting the one-loop fugacity.

2. Show that the prefactor changes the exponent-only diagnostic by $O(g\log g)$.

Solution

If $\Delta E=A g^{-1/2}e^{-S/g}[1+O(g)]$, then

$$-g\log\Delta E =S-g\log A+\frac g2\log g+O(g^2).$$

The logarithmic prefactor therefore produces an $O(g\log g)$ correction even when the classical action is exact.

3. Why must the logarithm be taken before the infinite-volume limit in a dilute gas?

Solution

Disconnected events generate powers of the volume and exponentiate. Taking $\log Z$ selects connected cluster coefficients, each proportional to one overall volume. Dividing $\log Z$ by that volume then has a finite intensive limit; dividing $Z$ itself does not.

4. A numerical splitting agrees with the one-loop exponential to high precision, but the prefactor ratio is $R=0.84$ and changes by less than $10^{-10}$ under every numerical refinement. Which error terms have been tested, which remain physical, and what claim is justified?

Solution

Grid or basis truncation, arithmetic precision, and solver stability have been tested to roughly $10^{-10}$ if the refinements are independent. The stable value $R-1=-0.16$ is not numerical error: it can contain higher-loop terms, multi-event corrections, and an omitted contributing saddle with its own prefactor. One may claim that the numerical eigenvalue and leading exponential scale are reproduced in the stated parameter window. One may not claim one-loop prefactor accuracy, convergence of the saddle series, or control outside that window until those physical terms and the relevant limits are bounded.

Continue from a saddle contribution to a specialized prediction

• For gauge-instanton measures, fermion zero-mode saturation, and size-modulus control, continue to Instantons, Fermion Zero Modes, and Tunneling.
• For a complete decay-rate derivation with false-vacuum boundary conditions, continue to Metastability and Vacuum Decay.
• For numerical continuum tests of a semiclassical prediction, use Volume 8: Lattice QFT and Hamiltonian Methods; numerical agreement validates the tested regime, not convergence of the saddle expansion.

References

• Callan, Curtis G., Jr., and Sidney Coleman. “Fate of the False Vacuum. II. First Quantum Corrections.” Physical Review D 16 (1977): 1762–1768. DOI.
• Dunne, Gerald V. “Functional Determinants in Quantum Field Theory.” Journal of Physics A: Mathematical and Theoretical 41 (2008): 304006. 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.
• Vassilevich, Dmitri V. “Heat Kernel Expansion: User’s Manual.” Physics Reports 388 (2003): 279–360. DOI.