Definition: External domain

A label $\beta \in \mathcal{X}$ specifies:

1. four species $J_1,\dots ,J_4\in \{1,2\}$;

2. four nonzero real null momenta such that

   $$\sum _i{k}_i=0,s=(k_1+k_2{)}^2,t=(k_1+k_3{)}^2,u=(k_1+k_4{)}^2,stu\ne 0;$$

3. one replaced leg $j$ and a real gauge vector ${\xi }_j$;

4. for every other leg, a real symmetric tensor ${ϵ}_i$ and a timelike reference vector $t_i$ satisfying

   $${k_i}^{\mu }{ϵ}_{i\mu \nu }=0,{t_i}^{\mu }{ϵ}_{i\mu \nu }=0,{\eta }^{\mu \nu }{ϵ}_{i\mu \nu }=0.$$

The replaced tensor is $G(k_j,{\xi }_j)$. Polarization normalization is not imposed. Simultaneous permutations of all external data are admitted.