Auxiliary flat-space domain

All response functions in this section are defined on an auxiliary flat-space domain $\mathcal{X}$. Fourier waves are $e^{ik\cdot x}$, all momenta are incoming, and index operations use $\eta$.

A label specifies:

1. species $J_i\in \{1,2\}$, $i=1,\dots ,4$;

2. nonzero real null momenta satisfying

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

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

4. real symmetric physical tensors ${ϵ}_i$, $i\ne j$, and timelike references $t_i$, 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

No condition $k\cdot \xi =0$ is imposed. Simultaneous permutations of all external data are allowed.

These are formal vertex-evaluation labels. Their later assignment to local rods does not assert that they are compact wave packets, conical asymptotic states, or apparatus preparations.