feynlag.pheno.offshell

Off-shell (1→3) decay amplitudes: a scalar to one on-shell + one off-shell V.

Tier 2 of the decays roadmap. On-shell VVS gives zero below threshold (2m_V > M); the physical decay is S → V (V^* → f f̄') — a 1→3 process with an internal Breit–Wigner propagator. This module assembles the squared amplitude with the covariant engine (the internal q = p₂+p₃ flows into the trace exactly like an external momentum) and integrates it over the Dalitz region.

The current topology is SVV (scalar → vector + vector*→fermions); the assembler is written so other 1→3 topologies (FFFF, t→bW^*) can be added.

Two width routes, cross-checked against each other and the Keung–Marciano closed form: the numeric Dalitz integral (integrate) and, for massless fermions, an analytic SymPy integration.

Functions

offshell_scalar_vv_width(m_S, m_V, width_V, ...)	Γ(S → V f f̄') for one fermion channel.
scalar_offshell_vv_width(m_S, m_V, width_V, ...)	Γ(S → V V^* → V f f̄') summed over the V^* fermion channels.
scalar_vv_s12_integral(m_S, m_V)	The inner Dalitz integral ∫ |M|² ds₁₂ done analytically.
scalar_vv_squared(m_V, kin)	The reduced |M|² for S → V(p₁) [V^*→ f(p₂) f̄'(p₃)].