The 11- and 12-point Higgs self-couplings coincide at the top edge

In the cubic field redefinition h → h + α P(h) of Alasfar et al., the Standard Model potential in the old field has degree 12 in h. At this top edge the 11- and 12-point contributions, with the normalization Nₙ = n! / 3! (so N₁₁ = 6652800, N₁₂ = 79833600), coincide, and the 12-point one has the closed form 246400 α⁴ / (1 + α)². These are algebraic identities in ℝ; the tree-level reparametrization is the hypothesis, not derived here.

Main results

• bhr_top_edge_collapse: the normalized 11- and 12-point top-edge contributions are equal.
• bhr_top_edge_closed_form: the 12-point coefficient equals 246400 α⁴ / (1 + α)².

Related

SMEFT.AllOrdersSelfCouplings.dk11_eq_dk12 states the same coincidence for the all-orders closed forms.

References

• L. Alasfar et al., arXiv:2304.01968 (field redefinition).

theorem SMEFT.TopEdgeCollapse.bhr_top_edge_collapse (prefactor v massSq top : ℝ) : 6652800 * (prefactor * (4 * (3 * v * top) * top ^ 3)) * (2 * v ^ 9) / massSq = 79833600 * (prefactor * top ^ 4) * (2 * v ^ 10) / massSq

At the degree-12 edge of the cubic reparametrization the normalized 11- and 12-point contributions coincide; massSq is an arbitrary common normalization.

theorem SMEFT.TopEdgeCollapse.bhr_top_edge_closed_form (alpha v massSq : ℝ) (hv : v ≠ 0) (hmassSq : massSq ≠ 0) : 79833600 * (massSq / (8 * v ^ 2)) * (alpha / (3 * v ^ 2)) ^ 4 * (2 * v ^ 10) / (massSq * (1 + alpha) ^ 2) = 246400 * alpha ^ 4 / (1 + alpha) ^ 2

The 12-point coefficient produced by the cubic top term is 246400 α⁴ / (1 + α)².