Standard Model Running, Effective Potential, and Vacuum-Stability Criteria

Standard Model vacuum statements are conditional outputs of coupled running, threshold matching, an effective-potential prescription, and a tunneling calculation—not consequences of the sign of one running coupling alone. This page defines the durable workflow and its checks while leaving any verdict based on current masses or world averages to a dated evidence analysis.

Required background. Use the independent couplings in the Standard Model parameter map and the solution methods for coupled renormalization-group flows.

Helpful background. The one-particle-irreducible effective action explains why an effective potential is an off-shell slice of a larger functional and therefore carries gauge and field-coordinate subtleties.

A declared one-loop fixture

Take $V(H)=-m^2H^\dagger H+\lambda(H^\dagger H)^2$, retain only the top Yukawa among fermion Yukawas, use $Q=T_3+Y$, and define every beta function with respect to $\ln\mu$. In the $\overline{\mathrm{MS}}$ scheme above the stated Standard Model thresholds,

$$\begin{aligned} 16\pi^2\beta_{g'}&=\frac{41}{6}g'^3, &16\pi^2\beta_g&=-\frac{19}{6}g^3, &16\pi^2\beta_{g_s}&=-7g_s^3,\\ 16\pi^2\beta_{y_t} &=y_t\left(\frac92y_t^2-\frac{17}{12}g'^2 -\frac94g^2-8g_s^2\right), \end{aligned}$$

and

$$\begin{aligned} 16\pi^2\beta_\lambda={}&24\lambda^2 +\lambda\left(12y_t^2-9g^2-3g'^2\right)-6y_t^4\\ &+\frac38\left[2g^4+(g^2+g'^2)^2\right]. \end{aligned}$$

These equations are a fixture, not a precision prescription. A higher-order analysis must use a mutually consistent loop order for beta functions, threshold relations, and the effective potential; representative higher-order formulas and matching organization are collected in Buttazzo et al. 2013, appendices A–C.

The three gauge equations provide an analytic implementation check. If $16\pi^2\beta_{g_i}=b_i g_i^3$, then

$$\frac{1}{g_i^2(\mu)} =\frac{1}{g_i^2(\mu_0)} -\frac{b_i}{8\pi^2}\ln\frac{\mu}{\mu_0}.$$

A numerical coupled-flow solver must reproduce this expression when the Yukawa and scalar equations are switched off. It must also preserve positivity of $g_i^2$ within the interval before a perturbative pole and show decreasing $g_s$ toward higher scales.

Matching is part of the initial condition

Measured pole or threshold observables do not equal $\overline{\mathrm{MS}}$ parameters. A calculation first chooses a matching scale $\mu_0$, converts the input set to

$$\{g_s(\mu_0),g(\mu_0),g'(\mu_0),y_t(\mu_0),\lambda(\mu_0),m^2(\mu_0),\ldots\},$$

and attaches a perturbative order and mass/tadpole prescription to that conversion. In a mass-independent scheme, heavy fields do not decouple automatically. Crossing a threshold therefore requires an explicit matching relation and a declared effective field content on each side.

This distinction is numerically consequential even when no number is quoted: the top pole mass, a short-distance top mass, and a Monte-Carlo mass parameter are different objects. Feeding one into a matching equation for another invalidates the subsequent stability inference regardless of how accurately the RG equations are integrated.

Effective potential and gauge dependence

For a neutral background $H=(0,\phi/\sqrt2)^T$, an RG-improved finite-order potential has the schematic form

$$V_{\mathrm{eff}}(\phi;\mu,\xi) =-\frac12m^2(\mu)\phi^2 +\frac14\lambda(\mu)\phi^4 +\Delta V(\phi;\mu,\xi),$$

where $\xi$ denotes gauge-fixing parameters. Away from extrema, $V_{\mathrm{eff}}$ and the coordinate $\phi$ are gauge dependent. The Nielsen identity organizes that dependence,

$$\left(\xi\frac{\partial}{\partial\xi} +C(\phi,\xi)\frac{\partial}{\partial\phi}\right) V_{\mathrm{eff}}(\phi,\xi)=0.$$

At an exact stationary point, the second term vanishes and the potential value is gauge independent under the identity’s hypotheses, while the field location generally is not. At finite order, consistent power counting is required for this cancellation; minimizing an arbitrarily RG-improved expression and reading its field coordinate as an observable does not suffice. The identity originates in Nielsen 1975, pp. 173–188, and a practical order-by-order treatment is developed in Andreassen, Frost, and Schwartz 2015, §§2–4.

Choosing $\mu$ of order the background can reduce logarithms, but it cannot remove gauge dependence, matching uncertainty, or EFT sensitivity. Varying $\mu$ is one truncation diagnostic rather than an observable probability distribution.

Four different stability statements

Statement	Required calculation	Claim ceiling
Running-quartic diagnostic	solve $\lambda(\mu)$ in a stated scheme and truncation	locates where this coordinate changes behavior
Effective-potential comparison	construct extrema consistently and compare their action densities	conditional statement within the gauge/order prescription
Metastability	find the relevant bounce and fluctuation prefactor	decay rate within the EFT and semiclassical regime
Ultraviolet conclusion	control higher-dimensional operators and matching above the cutoff	cannot follow from Standard Model running alone

A zero of $\lambda(\mu)$ is therefore neither necessary nor sufficient, by itself, for a physical decay claim. The bounce probes a range of fields and scales, and higher-dimensional terms such as $c_6(H^\dagger H)^3/\Lambda^2$ can matter as the field approaches the EFT cutoff even if they are negligible near the electroweak vacuum.

Reproducible decision procedure

1. Freeze the theory definition: scheme, gauge, field content, input observables, mass definitions, and EFT cutoff.
2. Match inputs: record loop order, threshold scales, and covariance of the derived running parameters.
3. Run couplings: integrate the coupled system and reproduce the analytic gauge-coupling fixture.
4. Construct the potential: use compatible loop and resummation counting; verify the Nielsen-identity implications at stationary points.
5. Locate relevant configurations: distinguish a coordinate crossing, an extremum, and a bounce.
6. Propagate uncertainty: vary perturbative orders and scales, transform correlated inputs, and test higher-dimensional sensitivity.
7. State a bounded conclusion: name the parameter region, cutoff, approximation, and omitted effects.

No current mass value, confidence region, or “stable universe” verdict follows from this durable method without dated, versioned experimental and theoretical inputs. The reusable output is instead

$$\left\{\text{matched parameters and covariance},\ \beta_i,\ V_{\mathrm{eff}}\text{ prescription},\ \text{bounce criterion},\ \Lambda_{\mathrm{EFT}},\ \text{uncertainty decomposition}\right\}.$$

For deformations of observables at fixed EFT order, continue to SMEFT and HEFT. For a correlated conclusion using present data, combine this complete object with a dated, versioned dataset and likelihood.

References

• Andreassen, Anders, William Frost, and Matthew D. Schwartz. “Consistent Use of Effective Potentials.” Physical Review D 91, no. 1 (2015): 016009, §§2–4. DOI.
• Buttazzo, Dario, Giuseppe Degrassi, Pier Paolo Giardino, Gian F. Giudice, Filippo Sala, Alessandro Salvio, and Alessandro Strumia. “Investigating the Near-Criticality of the Higgs Boson.” Journal of High Energy Physics 2013, no. 12 (2013): 089, appendices A–C. DOI.
• Nielsen, N. K. “On the Gauge Dependence of Spontaneous Symmetry Breaking in Gauge Theories.” Nuclear Physics B 101, no. 1 (1975): 173–188. DOI.