FCLoopFromGLI[exp, topologies] replaces
GLIs in exp with the corresponding loop
integrals in the FeynAmpDenominator notation according to
the information provided in topologies.
Overview, FCTopology, GLI, FCLoopValidTopologyQ.
topos = {
FCTopology["topoBox1L", {FAD[{q, m0}], FAD[{q + p1, m1}], FAD[{q + p2, m2}], FAD[{q + p3, m3}]},
{q}, {p1, p2, p3}, {}, {}],
FCTopology["topoTad2L", {FAD[{q1, m1}], FAD[{q2, m2}], FAD[{q1 - q2, 0}]}, {q1, q2}, {}, {}, {}]}\left\{\text{FCTopology}\left(\text{topoBox1L},\left\{\frac{1}{q^2-\text{m0}^2},\frac{1}{(\text{p1}+q)^2-\text{m1}^2},\frac{1}{(\text{p2}+q)^2-\text{m2}^2},\frac{1}{(\text{p3}+q)^2-\text{m3}^2}\right\},\{q\},\{\text{p1},\text{p2},\text{p3}\},\{\},\{\}\right),\text{FCTopology}\left(\text{topoTad2L},\left\{\frac{1}{\text{q1}^2-\text{m1}^2},\frac{1}{\text{q2}^2-\text{m2}^2},\frac{1}{(\text{q1}-\text{q2})^2}\right\},\{\text{q1},\text{q2}\},\{\},\{\},\{\}\right)\right\}
exp = a1 GLI["topoBox1L", {1, 1, 1, 1}] + a2 GLI["topoTad2L", {1, 2, 2}]\text{a1} G^{\text{topoBox1L}}(1,1,1,1)+\text{a2} G^{\text{topoTad2L}}(1,2,2)
FCLoopFromGLI[exp, topos]\frac{\text{a1}}{\left(q^2-\text{m0}^2\right) \left((\text{p1}+q)^2-\text{m1}^2\right) \left((\text{p2}+q)^2-\text{m2}^2\right) \left((\text{p3}+q)^2-\text{m3}^2\right)}+\frac{\text{a2}}{\left(\text{q1}^2-\text{m1}^2\right) \left(\text{q2}^2-\text{m2}^2\right)^2 (\text{q1}-\text{q2})^4}
Notice that it is necessary to specify all topologies present in
exp. The function will not accept GLIs defined
for unknown topologies
FCLoopFromGLI[GLI["topoXYZ", {1, 1, 1, 1, 1}], topos]
\text{\$Aborted}
FCLoopFromGLI can also handle products of
GLIs (currently only for standalone integrals or lists of
integrals but not for amplitudes). In this case it will automatically
introduce dummy names for the loop momenta.
FCLoopFromGLI[GLI["topoBox1L", {1, 0, 1, 0}] GLI["topoBox1L", {0, 1, 0, 1}], topos]\frac{1}{\left(\text{FCGV}(\text{lmom21})^2-\text{m0}^2\right) \left((\text{p1}+\text{FCGV}(\text{lmom11}))^2-\text{m1}^2\right) \left((\text{p3}+\text{FCGV}(\text{lmom11}))^2-\text{m3}^2\right) \left((\text{p2}+\text{FCGV}(\text{lmom21}))^2-\text{m2}^2\right)}
You can customize the naming scheme for the momenta via the
LoopMomentum option. The first argument gives the number of
the loop integral, while the second corresponds to a particular loop
momentum this integral depends on.
SelectNotFree[Options[FCLoopFromGLI], LoopMomenta]\{\text{LoopMomenta}\to (\{\text{FeynCalc$\grave{ }$FCLoopFromGLI$\grave{ }$Private$\grave{ }$x},\text{FeynCalc$\grave{ }$FCLoopFromGLI$\grave{ }$Private$\grave{ }$y}\}\to \;\text{FCGV}(\text{lmom}<>\text{ToString}[\text{FeynCalc$\grave{ }$FCLoopFromGLI$\grave{ }$Private$\grave{ }$x}]<>\text{ToString}[\text{FeynCalc$\grave{ }$FCLoopFromGLI$\grave{ }$Private$\grave{ }$y}]))\}
FCLoopFromGLI[GLI["topoBox1L", {1, 0, 1, 0}] GLI["topoBox1L", {0, 1, 0, 1}], topos,
LoopMomenta -> Function[{x, y}, "p" <> ToString[x] <> ToString[x]]]\frac{1}{\left(\text{p22}^2-\text{m0}^2\right) \left((\text{p11}+\text{p1})^2-\text{m1}^2\right) \left((\text{p22}+\text{p2})^2-\text{m2}^2\right) \left((\text{p11}+\text{p3})^2-\text{m3}^2\right)}
In general, FCLoopFromGLI can change the ordering of
propagators inside FeynAmpDenominator, as compared to the
their ordering inside FCTopology. This is because by
default it calls FeynAmpDenominatorCombine. Ordering may
also change when applying FeynAmpDenominatorSimplify. You
want the ordering to remain unchanged, the following should help
FCLoopFromGLI[exp, topos, FeynAmpDenominatorCombine -> False, List -> FeynAmpDenominator]\frac{\text{a1}}{\left(q^2-\text{m0}^2\right).\left((\text{p1}+q)^2-\text{m1}^2\right).\left((\text{p2}+q)^2-\text{m2}^2\right).\left((\text{p3}+q)^2-\text{m3}^2\right)}+\frac{\text{a2}}{\left(\text{q1}^2-\text{m1}^2\right).\left(\text{q2}^2-\text{m2}^2\right)^2.(\text{q1}-\text{q2})^4}