I am writing a code in Mathematica using the FeynCalc package to calculate an expression ( Y_1 ) that is required to evaluate the differential cross-section for the scattering of two massless fermions into two massive vector bosons.
The expected result for ( Y_1 ) is:
Y1 = -4 (6 a^2 b^2 + a^4 + b^4) MV^4 * (Pair[Momentum[p1], Momentum[p2]] * (12 MV^4 Pair[Momentum[p1], Momentum[p3]] - 12 MV^2 Pair[Momentum[p1], Momentum[p3]]^2 + MV^6) - 2 Pair[Momentum[p1], Momentum[p3]] Pair[Momentum[p2], Momentum[p3]] * (-4 MV^2 Pair[Momentum[p1], Momentum[p3]] + 4 Pair[Momentum[p1], Momentum[p3]]^2 + 5 MV^4) );However, when I run my code, I am getting a much longer and more complicated expression. Here is the code I am using:
(* Define the individual expressions *)expr1 = GS[p2] . GA[\[Mu]] . (a - b GA[5]) . (GS[p1] - GS[p3]);expr2 = GA[\[Mu]] . (a - b GA[5]) . GS[p1] . GA[\[Rho]] . (a - b GA[5]);expr3 = (GS[p1] - GS[p3]) . GA[\[Sigma]] . (a - b GA[5]);(* Compute the trace *)T1 = FullSimplify[TR[expr1 . expr2 . expr3]];(* Additional terms *)expr4 = -MT[\[Mu], \[Rho]] + Pair[FV[p3, \[Mu]], FV[p3, \[Rho]]]/MV^2;expr5 = -MT[\[Nu], \[Sigma]] + Pair[FV[p1 + p2 - p3, \[Nu]], FV[p1 + p2 - p3, \[Sigma]]]/MV^2;(* Combine the terms *)Y1 = T1 . expr4 . expr5;(* Momentum rules for simplification *)momentumRules = { Pair[Momentum[p1], Momentum[p1]] -> 0, Pair[Momentum[p2], Momentum[p2]] -> 0, Pair[Momentum[p3], Momentum[p3]] -> MV^2};(* Contract and simplify *)ContractedY1 = Contract[Y1];SimplifiedY1 = FullSimplify[ContractedY1 /. momentumRules];Despite applying Contract and FullSimplify with the momentum rules, the output is far more complicated than expected.
Question:
What could be causing the discrepancy? Am I missing additional simplification rules, or is there an issue with how I defined the expressions or applied the simplifications?
I would greatly appreciate any help or guidance. Thank you!