With $\beta_h=0$ , the vacuum $H=\phi^0=\phi^\pm=0$ minimizes the potential

Proved

SMHiggsPotential.vacuum_minimizes_potential

by Lucas · Oct 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

higgsparticle-physicsstandard-model

Let $g,M,m_h$ be real and set $\beta_h=0$ . For all real $H,\phi^0$ and complex $\phi^+$ ,

\mathcal L_V(H,\phi^0,\phi^+)\le\mathcal L_V(0,0,0).

So the vacuum $\Phi=(0,v/\sqrt2)$ is a global minimum of the scalar potential $V=-\mathcal L_V$ of box 2. (The degenerate cases $M=0$ or $g=0$ are included; there $\mathcal L_V=-\frac12m_h^2H^2$ under the convention $x/0=0$ .)

Preamble

import Mathlib
import Definitions.Def_SMHiggsPotential_Defs
open Complex

Formal statement

namespace SMHiggsPotential

theorem vacuum_minimizes_potential (g M mh H φ0 : ℝ) (φp : ℂ) :
    scalarLagrangian g M mh 0 H φ0 φp ≤ scalarLagrangian g M mh 0 0 0 0 := by sorry

end SMHiggsPotential

Source

Standard Model Lagrangian in Veltman's conventions (Feynman-'t Hooft gauge), as printed in the five-box chart supplied by the proposer (after M. Veltman, *Diagrammatica: The Path to Feynman Diagrams*, Cambridge University Press, 1994). Box 2, the non-derivative scalar terms:

-\frac12 m_h^2H^2

, the

\beta_h[\dots]

term,

+\frac{2M^4}{g^2}\alpha_h

-g\alpha[H^3+H\phi^0\phi^0+2H\phi^+\phi^-]

and

-\frac18g^2\alpha_h[H^4+\dots]

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) - drafting agent, NON-BLIND (not an independent auditor)

NON-BLIND READ-BACK — NOT INDEPENDENT TESTIMONY. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statement, with full knowledge of the intended meaning and of the source image. It was not produced by a blind, independent auditor, and no reviewer should treat it as independent evidence that the formalization is faithful.

Binders. Real $g,M,m_h,H,\phi^0$ and complex $\phi^+$ ; no hypotheses. The tadpole parameter is fixed to $\beta_h=0$ on both sides.

Claim.

\mathcal L_V(g,M,m_h,0;H,\phi^0,\phi^+)\le\mathcal L_V(g,M,m_h,0;0,0,0).

This is a non-strict inequality, and $M=0$ or $g=0$ are allowed.

View graph

With βh=0\beta_h=0βh​=0, the vacuum H=ϕ0=ϕ±=0H=\phi^0=\phi^\pm=0H=ϕ0=ϕ±=0 minimizes the potential

With $\beta_h=0$ , the vacuum $H=\phi^0=\phi^\pm=0$ minimizes the potential