# Failing to show $\xi$-gauge-independence in an abelian Spontaneously Broken Gauge Theories (SBGT)

I am studying the following paper:

Appelquist, Carazzone, Goldman & Quinn, Renormalization and Gauge Independence in Spontaneously Broken Gauge Theories, https://doi.org/10.1103/PhysRevD.8.1747

I am having trouble following its appendix calculations. More specifically following the feynman rules given below, I cannot include the external momenta factors that are presented in the paper's results. As a consequence I cannot show the $$\xi$$ gauge independence of the process.

The rules are:

The process is:

The appendix calculation:

My questions are the following:

1. How are the external momenta included (especially in A7 or A5), e.g $$q^2_1 + q^2_2 + q^2_3 - 6 h \lambda^2$$?

2. What is the general approach to this problem independent of the methodology used in the paper, should I just use brute force calculation and hope for the best, or is there any other way?

P.S The pencile drawed circles is what I am able to calculate using those rules. Be careful with the notation, the paper origins back to 1973.