Manuscript II@2 · II.tex · section · 50d64def

Species couplings and reference quartic

Locked kinematics, action, and contacts › Species couplings and reference quartic · lines 507–548

II@2#eq:T, II@2#eq:action

Let $d_{I J K}$ be real and completely symmetric, with

d_{111} = a, d_{112} = b, d_{122} = c, d_{222} = e .

Define

S_{3} (d) = I, J, K = 1 \sum^{2} d_{I J K} E_{3}^{[3]} (h^{I}, h^{J}, h^{K}),

T_{I J K L} (d) = \frac{1}{3} M = 1 \sum^{2} (d_{I J M} d_{K L M} + d_{I K M} d_{J L M} + d_{I L M} d_{J K M}),

and

S_{4, sym} (d) = I, J, K, L = 1 \sum^{2} T_{I J K L} (d) E_{4}^{[4]} (h^{I}, h^{J}, h^{K}, h^{L}) .

The action through order $g^{2}$ is

S (d, θ) = S_{PF} + g S_{3} (d) + g^{2} (S_{4, sym} (d) + ℓ \sum θ_{ℓ} O_{ℓ}) .

Every $θ_{ℓ}$ is constant across the external domain.

The ordered species sums and normalized polarizations ensure that a differentiated external vertex has coefficient $d_{J_{1} J_{2} J_{3}} v_{3}$ or $T_{J_{1} J_{2} J_{3} J_{4}} v_{4}$ . Repeated species do not introduce an additional factorial: all assignments to the distinct bookkeeping variables are already included.

Displayed formulas in this passage

II@2#eq:T · line 518
$T_{I J K L} (d) = \frac{1}{3} M = 1 \sum^{2} (d_{I J M} d_{K L M} + d_{I K M} d_{J L M} + d_{I L M} d_{J K M}),$
II@2#eq:action · line 534
$S (d, θ) = S_{PF} + g S_{3} (d) + g^{2} (S_{4, sym} (d) + ℓ \sum θ_{ℓ} O_{ℓ}) .$
unlabeled · line 510
$d_{111} = a, d_{112} = b, d_{122} = c, d_{222} = e .$
unlabeled · line 514
$S_{3} (d) = I, J, K = 1 \sum^{2} d_{I J K} E_{3}^{[3]} (h^{I}, h^{J}, h^{K}),$
unlabeled · line 528
$S_{4, sym} (d) = I, J, K, L = 1 \sum^{2} T_{I J K L} (d) E_{4}^{[4]} (h^{I}, h^{J}, h^{K}, h^{L}) .$