The Effective Potential and Coleman–Weinberg

Goldstone and the Higgs mechanism analyze symmetry breaking at the classical level, reading the vacuum off the tree-level potential $V(\varphi )$. Quantum corrections modify the potential — and can cause symmetry breaking that is absent classically. The tool is the effective potential $V_{\text{eff}}$, the constant-field limit of the effective action, whose one-loop form is the Coleman–Weinberg potential.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

The effective potential as the true vacuum energy

The effective action $\Gamma [{\varphi }_{\text{cl}}]$ is the quantum-corrected action whose minimum gives the true vacuum. For a spacetime- constant field it reduces to the effective potential:

$$\Gamma [{\varphi }_0]=-(\text{vol})V_{\text{eff}}({\varphi }_0),\frac{d{V}_{\text{eff}}}{d{\varphi }_0}{|}_{⟨\varphi ⟩}=0.$$

The vacuum $⟨\varphi ⟩$ minimizes $V_{\text{eff}}$, not the classical $V$. At tree level $V_{\text{eff}}=V$; loops add corrections that can shift, create, or remove minima — and thereby change whether a symmetry is broken.

The one-loop Coleman–Weinberg potential

The one-loop correction is a functional determinant — a Gaussian integral over fluctuations around the constant background ${\varphi }_0$ — giving

$$\boxed{V_{\text{eff}}({\varphi }_0)=V({\varphi }_0)+\frac{1}{2}\int \frac{d^4{k}_E}{(2\pi {)}^4}\ln [k_E^2+V^{\prime \prime }({\varphi }_0)]+\cdots }$$

The loop integral is UV-divergent and must be renormalized: the divergences are absorbed into the tree-level couplings, and a renormalization condition fixes the finite part. The result, for a massless scalar, is the celebrated logarithmic form

$$V_{\text{eff}}({\varphi }_0)=\frac{\lambda }{4!}{\varphi }_0^4+\frac{{\lambda }^2{\varphi }_0^4}{256{\pi }^2}\left(\ln \frac{{\varphi }_0^2}{M^2}-\frac{25}{6}\right),$$

with $M$ the renormalization scale. The ${\varphi }_0^4\ln {\varphi }_0^2$ term is the quantum correction that classical analysis misses.

Radiative (dynamical) symmetry breaking

The Coleman–Weinberg discovery: even when the classical potential has its minimum at the symmetric point ${\varphi }_0=0$ (no classical breaking), the ${\varphi }_0^4\ln {\varphi }_0^2$ term can push the minimum of $V_{\text{eff}}$ to a nonzero ${\varphi }_0$, breaking the symmetry purely through radiative corrections. Symmetry breaking is thus not always a classical input — it can be generated by quantum fluctuations. This is dimensional transmutation in the scalar sector: a dimensionless coupling $\lambda$ trades for a dimensionful vev $⟨\varphi ⟩$, exactly as the running coupling generates ${\Lambda }_{\text{QCD}}$.

Convexity and the Maxwell construction

A subtlety: the true effective potential (as the Legendre transform of a convex generating functional $W[J]$) must be convex, yet the one-loop formula above is non-convex (it has the double-well shape) and even develops an imaginary part between the maxima. The resolution: the imaginary part signals an instability of the homogeneous field there, and the physical $V_{\text{eff}}$ is the convex hull — the Maxwell construction familiar from the liquid–gas transition. In the broken region the system phase-separates into domains of the two vacua rather than sitting at an unstable homogeneous configuration.

Applications

• Vacuum stability of the Standard Model. Running the Higgs quartic coupling with the RG and computing $V_{\text{eff}}$ at large field values tests whether the electroweak vacuum is absolutely stable, metastable, or unstable. With the measured top and Higgs masses, the Standard Model sits intriguingly near the metastability boundary — the electroweak vacuum may be a long-lived false vacuum.
• Finite-temperature breaking. Adding thermal corrections gives the temperature-dependent $V_{\text{eff}}({\varphi }_0,T)$, whose evolution drives symmetry restoration at high $T$ and the electroweak phase transition in the early universe — see finite-temperature QFT.
• Coleman–Weinberg inflation and beyond-SM model building, where radiative breaking fixes otherwise-free scales.

Summary

• The effective potential $V_{\text{eff}}$ (constant-field effective action) determines the true vacuum; loops correct the classical $V$.
• One-loop Coleman–Weinberg: a ${\varphi }^4\ln {\varphi }^2$ term that can induce radiative symmetry breaking even when $V$ is classically symmetric.
• The physical potential is convex (Maxwell construction); imaginary parts flag instability.
• Applications: SM vacuum (meta)stability, thermal symmetry restoration.

Where this leads

• The generating functional it descends from: the effective action.
• Thermal corrections and phase transitions: finite-temperature QFT.
• The tree-level mechanisms it refines: Goldstone, Higgs.

References

• Coleman & Weinberg, Phys. Rev. D 7, 1888 (1973).
• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 11.3–11.4.
• Weinberg, The Quantum Theory of Fields, Vol. 2, Ch. 16.
• Srednicki, Quantum Field Theory, Ch. 30, 34.