This post is part of a series on few-body dynamics in the forward-JWKB approximation.
In this post I will enumerate some one-loop contributions to the elastic process
$$ \Phi_{1}(p_{1}) + \Phi_{2}(p_{2}) \longrightarrow \Phi_{1}(p_{3}) + \Phi_{2}(p_{4}). $$
This process has an associated four-point scattering amplitude in the Scalars Toy Model. The one-loop contributions are either order \(g^{4}\), \(h^{4}\), or \(g^{2} h^{2}\) depending on the kind of intermedia that are being exchanged.
Square
There are four square contributions:
$$ \mathcal{S}_{gg}(x) \equiv g^{4} G_{A}(x_{1} \mid x_{2}) G_{2}(x_{4} \mid x_{2}) G_{A}(x_{4} \mid x_{3}) G_{1}(x_{3} \mid x_{1}), $$
$$ \mathcal{S}_{hh}(x) \equiv h^{4} G_{Y}(x_{1} \mid x_{2}) G_{2}(x_{4} \mid x_{2}) G_{Y}(x_{4} \mid x_{3}) G_{1}(x_{3} \mid x_{1}), $$
$$ \mathcal{S}_{gh}(x) \equiv g^{2} h^{2} G_{A}(x_{1} \mid x_{2}) G_{2}(x_{4} \mid x_{2}) G_{Y}(x_{4} \mid x_{3}) G_{1}(x_{3} \mid x_{1}), $$
$$ \mathcal{S}_{hg}(x) \equiv g^{2} h^{2} G_{Y}(x_{1} \mid x_{2}) G_{2}(x_{4} \mid x_{2}) G_{A}(x_{4} \mid x_{3}) G_{1}(x_{3} \mid x_{1}). $$
Twisted Square
There are four twisted square contributions:
$$ \mathcal{T}_{gg}(x) \equiv g^{4} G_{A}(x_{1} \mid x_{4}) G_{2}(x_{4} \mid x_{2}) G_{A}(x_{2} \mid x_{3}) G_{1}(x_{3} \mid x_{1}), $$
$$ \mathcal{T}_{hh}(x) \equiv h^{4} G_{Y}(x_{1} \mid x_{4}) G_{2}(x_{4} \mid x_{2}) G_{Y}(x_{2} \mid x_{3}) G_{1}(x_{3} \mid x_{1}), $$
$$ \mathcal{T}_{gh}(x) \equiv g^{2} h^{2} G_{A}(x_{1} \mid x_{4}) G_{2}(x_{4} \mid x_{2}) G_{Y}(x_{2} \mid x_{3}) G_{1}(x_{3} \mid x_{1}), $$
$$ \mathcal{T}_{hg}(x) \equiv g^{2} h^{2} G_{Y}(x_{1} \mid x_{4}) G_{2}(x_{4} \mid x_{2}) G_{A}(x_{2} \mid x_{3}) G_{1}(x_{3} \mid x_{1}). $$
Vertex Correction
There are two sets of four vertex correction contributions:
$$ \mathcal{U}_{gg}(x) \equiv g^{4} \delta( x_{4} - x_{2} ) G_{A}(x_{1} \mid x_{3}) \int \mathrm{d} y \left[ G_{1}(x_{1} \mid y) G_{A}(x_{2} \mid y) G_{1}(x_{3} \mid y) \right], $$
$$ \mathcal{U}_{hh}(x) \equiv h^{4} \delta( x_{4} - x_{2} ) G_{Y}(x_{1} \mid x_{3}) \int \mathrm{d} y \left[ G_{1}(x_{1} \mid y) G_{Y}(x_{2} \mid y) G_{1}(x_{3} \mid y) \right], $$
$$ \mathcal{U}_{gh}(x) \equiv g^{2} h^{2} \delta( x_{4} - x_{2} ) G_{A}(x_{1} \mid x_{3}) \int \mathrm{d} y \left[ G_{1}(x_{1} \mid y) G_{Y}(x_{2} \mid y) G_{1}(x_{3} \mid y) \right], $$
$$ \mathcal{U}_{hg}(x) \equiv g^{2} h^{2} \delta( x_{4} - x_{2} ) G_{Y}(x_{1} \mid x_{3}) \int \mathrm{d} y \left[ G_{1}(x_{1} \mid y) G_{A}(x_{2} \mid y) G_{1}(x_{3} \mid y) \right]; $$
and
$$ \mathcal{V}_{gg}(x) \equiv g^{4} \delta( x_{3} - x_{1} ) G_{A}(x_{2} \mid x_{4}) \int \mathrm{d} y \left[ G_{2}(x_{2} \mid y) G_{A}(x_{1} \mid y) G_{2}(x_{4} \mid y) \right], $$
$$ \mathcal{V}_{hh}(x) \equiv h^{4} \delta( x_{3} - x_{1} ) G_{Y}(x_{2} \mid x_{4}) \int \mathrm{d} y \left[ G_{2}(x_{2} \mid y) G_{Y}(x_{1} \mid y) G_{2}(x_{4} \mid y) \right], $$
$$ \mathcal{V}_{gh}(x) \equiv g^{2} h^{2} \delta( x_{3} - x_{1} ) G_{A}(x_{2} \mid x_{4}) \int \mathrm{d} y \left[ G_{2}(x_{2} \mid y) G_{Y}(x_{1} \mid y) G_{2}(x_{4} \mid y) \right], $$
$$ \mathcal{V}_{hg}(x) \equiv g^{2} h^{2} \delta( x_{3} - x_{1} ) G_{Y}(x_{2} \mid x_{4}) \int \mathrm{d} y \left[ G_{2}(x_{2} \mid y) G_{A}(x_{1} \mid y) G_{2}(x_{4} \mid y) \right]. $$
Vaccum Polarization
There are two sets of four vaccum polarization contributions:
$$ \mathcal{X}_{gg}(x) \equiv g^{4} \delta( x_{3} - x_{1} ) \delta( x_{4} - x_{2} ) \int \int \mathrm{d}y_{1} \mathrm{d}y_{2} \left[ G_{A}(x_{1} \mid y_{1}) G_{1}(y_{1} \mid y_{2}) G_{1}(y_{2} \mid y_{1}) G_{A}(y_{2} \mid x_{2}) \right], $$
$$ \mathcal{X}_{hh}(x) \equiv h^{4} \delta( x_{3} - x_{1} ) \delta( x_{4} - x_{2} ) \int \int \mathrm{d}y_{1} \mathrm{d}y_{2} \left[ G_{Y}(x_{1} \mid y_{1}) G_{1}(y_{1} \mid y_{2}) G_{1}(y_{2} \mid y_{1}) G_{Y}(y_{2} \mid x_{2}) \right], $$
$$ \mathcal{X}_{gh}(x) \equiv g^{2} h^{2} \delta( x_{3} - x_{1} ) \delta( x_{4} - x_{2} ) \int \int \mathrm{d}y_{1} \mathrm{d}y_{2} \left[ G_{A}(x_{1} \mid y_{1}) G_{1}(y_{1} \mid y_{2}) G_{1}(y_{2} \mid y_{1}) G_{Y}(y_{2} \mid x_{2}) \right], $$
$$ \mathcal{X}_{hg}(x) \equiv g^{2} h^{2} \delta( x_{3} - x_{1} ) \delta( x_{4} - x_{2} ) \int \int \mathrm{d}y_{1} \mathrm{d}y_{2} \left[ G_{Y}(x_{1} \mid y_{1}) G_{1}(y_{1} \mid y_{2}) G_{1}(y_{2} \mid y_{1}) G_{A}(y_{2} \mid x_{2}) \right]; $$
and
$$ \mathcal{Y}_{gg}(x) \equiv g^{4} \delta( x_{3} - x_{1} ) \delta( x_{4} - x_{2} ) \int \int \mathrm{d}y_{1} \mathrm{d}y_{2} \left[ G_{A}(x_{1} \mid y_{1}) G_{2}(y_{1} \mid y_{2}) G_{2}(y_{2} \mid y_{1}) G_{A}(y_{2} \mid x_{2}) \right], $$
$$ \mathcal{Y}_{hh}(x) \equiv h^{4} \delta( x_{3} - x_{1} ) \delta( x_{4} - x_{2} ) \int \int \mathrm{d}y_{1} \mathrm{d}y_{2} \left[ G_{Y}(x_{1} \mid y_{1}) G_{2}(y_{1} \mid y_{2}) G_{2}(y_{2} \mid y_{1}) G_{Y}(y_{2} \mid x_{2}) \right], $$
$$ \mathcal{Y}_{gh}(x) \equiv g^{2} h^{2} \delta( x_{3} - x_{1} ) \delta( x_{4} - x_{2} ) \int \int \mathrm{d}y_{1} \mathrm{d}y_{2} \left[ G_{A}(x_{1} \mid y_{1}) G_{2}(y_{1} \mid y_{2}) G_{2}(y_{2} \mid y_{1}) G_{Y}(y_{2} \mid x_{2}) \right], $$
$$ \mathcal{Y}_{hg}(x) \equiv g^{2} h^{2} \delta( x_{3} - x_{1} ) \delta( x_{4} - x_{2} ) \int \int \mathrm{d}y_{1} \mathrm{d}y_{2} \left[ G_{Y}(x_{1} \mid y_{1}) G_{2}(y_{1} \mid y_{2}) G_{2}(y_{2} \mid y_{1}) G_{A}(y_{2} \mid x_{2}) \right]. $$