High-energy theory
Your amplitude grows and it should not
About 6 min if you have seen this material before, 25 min or so if it is new to you.
Background
Assumed background: polarization vectors of a massive vector boson, and that the longitudinal one behaves as at .
Finding the error in your own calculation is harder than finding it in someone else's. Every step is one you meant to take, so what is wrong is usually something you never wrote down at all.
You are computing in the Standard Model. Using the -channel neutrino exchange diagram, you find an amplitude growing as , which crosses the A limit on how large an amplitude can be, following from the requirement that probabilities never exceed one. around a TeV. The argument has the same structure as the one that predicted the W from A theory valid only below some energy scale, which can tell you from the inside where its own description stops working..
The Standard Model is tested well above a TeV, and pair production agrees with it at LEP and beyond.
Your calculation disagrees with a well-tested theory. Which is the most likely culprit? Answer with its number.
1. The high-energy form is an approximation, and using it has manufactured the growth.
2. A sign error in the neutrino propagator. With the correct sign the amplitude falls instead of growing.
3. The -channel photon and diagrams were never included. Each grows the same way, and only the sum of all three is bounded.
4. Nothing is wrong. The growth is physical and the Standard Model does need new physics near a TeV.
Past 45 min on this? That usually means a missing prerequisite rather than a lack of effort — worth reading up before pushing further, starting with Uses of effective Lagrangians in electroweak interactions (gauge cancellations).