Prove2Me

Higgs mass: ∂H2LV∣0=−(mh2+βh)\partial_H^2\mathcal L_V|_{0}=-(m_h^2+\beta_h)

Proved
SMHiggsPotential.scalarLagrangian_higgs_mass

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

higgsparticle-physicsstandard-model

For all real g,M,mh,βhg,M,m_h,\beta_h, the second derivative of H↦LV(H,0,0)H\mapsto\mathcal L_V(H,0,0) at H=0H=0 is

d2dH2LV(H,0,0)∣H=0=−(mh2+βh).\frac{d^2}{dH^2}\mathcal L_V(H,0,0)\Big|_{H=0}=-(m_h^2+\beta_h).

In particular, for βh=0\beta_h=0 the Higgs field HH has mass mhm_h.

Preamble
import Mathlib
import Definitions.Def_SMHiggsPotential_Defs
open Complex
Formal statement
namespace SMHiggsPotential

theorem scalarLagrangian_higgs_mass (g M mh βh : ℝ) :
    deriv (deriv (fun H : ℝ => scalarLagrangian g M mh βh H 0 0)) 0 = -(mh ^ 2 + βh) := 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: −12mh2H2-\frac12 m_h^2H^2, the βh[… ]\beta_h[\dots] term, +2M4g2αh+\frac{2M^4}{g^2}\alpha_h, −gα[H3+Hϕ0ϕ0+2Hϕ+ϕ−]-g\alpha[H^3+H\phi^0\phi^0+2H\phi^+\phi^-] and −18g2αh[H4+… ]-\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,mh,βhg,M,m_h,\beta_h; no hypotheses.

Claim. Let f(H)=LV(g,M,mh,βh;H,0,0)f(H)=\mathcal L_V(g,M,m_h,\beta_h;H,0,0). Then the derivative (taken pointwise, with Lean's convention that a non-differentiable point gets derivative 00) of the derivative function f′f', evaluated at 00, equals −(mh2+βh)-(m_h^2+\beta_h).

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