feynlag.vertices.yangmills

Triple and quartic gauge self-couplings from group data.

Instead of expanding −¼ F_{μν}F^{μν} with explicit Lorentz indices, the self-interactions are computed group-theoretically (their Lorentz structures are universal and fixed by the catalog):

  • cubic: L ⊃ −g f^{abc} (∂_μ A^a_ν) A^{bμ} A^{cν} → vertex V(i,j,k) = −g F_ijk × [g^{μν}(p₁−p₂)^ρ + cyclic] with F_ijk = Σ_abc f^{abc} U[a,i] U[b,j] U[c,k],

  • quartic: pairwise tensors E_{(ij)(kl)} = Σ_e F^e_ij F^e_kl with F^e_ij = Σ_ab f^{abe} U[a,i] U[b,j], entering −g²/4 Σ_e (f^{abe}A^aA^b)(f^{cde}A^cA^d) structures.

U is the (possibly complex, unitary) matrix from adjoint components to physical bosons: A^a = Σ_i U[a,i] V_i (e.g. W¹,W²,W³,B → W⁺,W⁻,Z,γ; the identity if the bosons don’t mix). Overall sign/i conventions are pinned at UFO validation against the FeynRules SM model.

Functions

cubic_couplings(group[, physical, U])

Rotated cubic self-coupling tensor −g F_ijk.

quartic_couplings(group[, physical, U])

Rotated quartic tensor −g²/4 Σ_e F^e_ij F^e_kl per boson quadruple.

structure_constants(group)

f^{abc} of a gauge group, from the fundamental representation: f^{abc} = −2i Tr([T^a, T^b] T^c).