LongitudinalVectorScatteringCert
The LongitudinalVectorScatteringCert packages the additive decomposition of the longitudinal vector-boson scattering amplitude into gauge and scalar pieces, its vanishing when the residues sum to zero, the implication from the Standard Model cancellation hypothesis, and the boundedness result under Recognition Science preservation of unitarity. High-energy physicists studying electroweak perturbative consistency would cite it when linking the Higgs mechanism to high-energy behavior. The structure is assembled as a direct wrapper over four prior
claimLet $A(a_g,a_s,v,s)$ be the longitudinal vector scattering amplitude. The certificate asserts $A(a_g,a_s,v,s)=A_g(a_g,v,s)+A_s(a_s,v,s)$ for $v≠0$, $A(a_g,a_s,v,s)=0$ whenever $a_g+a_s=0$, the hypothesis $a_s=-a_g$ implies $a_g+a_s=0$, and if the RS scalar residue equals the negative of the gauge residue then there exists $M$ such that $|A(a_g,a_{RS},v,s)|≤M$ for all $s$.
background
The module treats the high-energy limit of $W_L^+W_L^-$ scattering as the decisive test of the Higgs sector. Without cancellation the gauge diagrams produce an amplitude that grows as $s^2/v^4$ and violates unitarity above the TeV scale; the scalar exchange supplies the precise counter-term that cancels the leading piece, leaving a bounded remainder. This is the Lee-Quigg-Thacker identity formalized at the schema level inside the HiggsEFTBridge framework.
proof idea
The structure is populated by direct field assignment from four sibling lemmas: amplitude_decomposition supplies the additive split, amplitude_vanishes_under_cancellation gives vanishing under the condition, cancellation_of_SM_hypothesis shows the SM hypothesis implies the condition, and amplitude_bounded_under_SM_hypothesis (instantiated on the RS residue) yields the boundedness claim. It is a one-line wrapper with no additional tactics.
why it matters in Recognition Science
The certificate supplies the auditable surface for RS preservation of longitudinal unitarity and feeds directly into HiggsEFTLowEnergyLimitCert, which composes the full cost-geometry to SM-EFT bridge. It encodes the structural claim that the J-cost Taylor expansion fixes the scalar coupling to exactly the value required by the cancellation condition. The module notes that a fully kinematic four-point amplitude with Mandelstam variables remains open.
scope and limits
- Does not derive the explicit amplitude from the J-cost functional equation.
- Does not compute a numerical value for the bound M.
- Does not include loop corrections or non-perturbative effects.
- Does not formalize Mandelstam kinematics or helicity decomposition.
formal statement (Lean)
170structure LongitudinalVectorScatteringCert where
171 /-- THEOREM: amplitude decomposes additively into gauge + scalar pieces. -/
172 decomposition : ∀ a_g a_s v s, v ≠ 0 →
173 amplitudeS2 a_g a_s v s = amplitudeGaugeOnly a_g v s + amplitudeScalarOnly a_s v s
174 /-- THEOREM: the leading-order amplitude vanishes under the cancellation. -/
175 cancels_under_cond : ∀ a_g a_s v s,
176 CancellationCondition a_g a_s → amplitudeS2 a_g a_s v s = 0
177 /-- THEOREM: SM hypothesis implies cancellation. -/
178 sm_implies_cancel : ∀ a_g a_s,
179 SMCancellationHypothesis a_g a_s → CancellationCondition a_g a_s
180 /-- CONDITIONAL_THEOREM: RS preservation of unitarity implies the
181 cancellation holds (and hence the amplitude is bounded). -/
182 rs_implies_bounded : ∀ a_g a_RS v,
183 RSPreservesLongitudinalUnitarity a_g a_RS →
184 ∃ M, ∀ s, |amplitudeS2 a_g a_RS v s| ≤ M
185