A simple program to calculate the Feynman integral #262
iisiaccNewton
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I found that FCFeynmanParametrize can only handle some simple integrals, I don't know if there are other functions that can calculate integrals, but after reading the chapter LOOP in the manual, I think there shouldn't be any other functions.so I spent a few days writing a function to calculate the following integral:
here is the code:
{
spd[p_, q_] := FVD[p, a]FVD[q, a];
spd[p_] := spd[p, p];
rule2 = FVD[p, [Mu]_] -> FVD[p, [Mu]] + yFVD[q, [Mu]];
ASF[g_, {p_, q_}] := Expand[g[p, q] /. rule2];
extract[f_] := Module[
{factors1, factors2, product1, integrand},
factors1 = Cases[f, FVD[p, ], Infinity];
factors2 = Cases[f, spd[p], Infinity]; product1 = Times @@ factors1;
integrand = If[factors2 == {}, product1, FVD[p, a]*product1];
{
integrand,
f/integrand
}
];
FeynmanIntegral[{p, d_ : D}, 1, {p_, m_, n_}] := (-1)^n*
I/(4 [Pi])^(d/2)Gamma[n - d/2]/Gamma[n]1/(m^2)^(n - d/2);
FeynmanIntegral[{p_, d_ : D}, spd[p_], {p_, m_, n_}] := (-1)^n
I/(4 [Pi])^(d/2)Gamma[n - d/2 - 1]/
Gamma[n](-d/2)/(m^2)^(n - d/2 - 1);
FeynmanIntegral[{p_, d_ : D}, FVD[p_, a_], {p_, m_, n_}] := 0;
FeynmanIntegral[{p_, d_ : D},
FVD[p_, a_]FVD[p_, b_], {p_, m_, n_}] :=
MTD[a, b]/dFeynmanIntegral[{p, d}, spd[p], {p, m, n}];
FeynmanIntegral[{p_, d_ : D}, f_, {p_, m_, n_}] :=
If[FreeQ[f, Plus],
extract[f][[2]]
FeynmanIntegral[{p, d}, extract[f][[1]], {p, m, n}],
FeynmanIntegral[{p, d}, f, {p, m, n}]];
FeynmanIntegral[{p_, d_ : D}, f_ + g_, {p_, m_, n_}] :=
FeynmanIntegral[{p, d}, f, {p, m, n}] +
FeynmanIntegral[{p, d}, g, {p, m, n}];
FeynmanIntegral[{p_, d_ : D}, {f_, p_, q_}, {p_, s_, m_}, {p_ - q_,
r_, n_}] :=
With[
{INT = (n + m - 1)Binomial[n + m - 2, n - 1](1 - y)^(m - 1)*
y^(n - 1)*
Simplify[
FeynmanIntegral[{p, d},
ASF[f, {p, q}], {p,
Sqrt[s^2*(1 - y) + r^2y - y(1 - y)*spd[q]], n + m}]]},
HoldForm[Integrate[INT, {y, 0, 1}]]
];
}
there are some example;
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