FCLoopTensorReduce[exp, topos] performs tensor reduction for the numerators of multi-loop integrals present in exp. Notice that exp is expected to be the output of FCLoopFindTopologies where all loop integrals have been written as fun[num, GLI[...]] with num being the numerator to be acted upon.
The reduction is done only for loop momenta contracted with Dirac matrices, polarization vectors or Levi-Civita tensors. Scalar products with external momenta are left untouched. The goal is to rewrite everything in terms of scalar products involving only loop momenta and external momenta appearing in the given topology. These quantities can be then rewritten in terms of inverse propagators (GLIs with negative indices), so that the complete dependence on loop momenta will go into the GLIs.
Unlike FCMultiLoopTID, this function does not perform any partial fractioning or shifts in the loop momenta.
The default value for fun is FCGV[“GLIProduct”] set by the option Head
Overview, FCLoopFindTopologies.
1-loop tadpole topology
topo1 = FCTopology["tad1l", {SFAD[{q, m^2}]}, {q}, {}, {}, {}]\text{FCTopology}\left(\text{tad1l},\left\{\frac{1}{(q^2-m^2+i \eta )}\right\},\{q\},\{\},\{\},\{\}\right)
amp1 = FCGV["GLIProduct"][GSD[q] . GAD[\[Mu]] . GSD[q], GLI["tad1l", {1}]]\text{FCGV}(\text{GLIProduct})\left((\gamma \cdot q).\gamma ^{\mu }.(\gamma \cdot q),G^{\text{tad1l}}(1)\right)
amp1Red = FCLoopTensorReduce[amp1, {topo1}]\text{FCGV}(\text{GLIProduct})\left(\frac{(2-D) q^2 \gamma ^{\mu }}{D},G^{\text{tad1l}}(1)\right)
topo2 = FCTopology[prop1l, {SFAD[{q, m^2}, {q - p, m^2}]}, {q}, {p}, {}, {}]\text{FCTopology}\left(\text{prop1l},\left\{\frac{1}{(q^2-m^2+i \eta ).((q-p)^2-m^2+i \eta )}\right\},\{q\},\{p\},\{\},\{\}\right)
1-loop self-energy topology
amp2 = gliProduct[GSD[q] . GAD[\[Mu]] . GSD[q], GLI[prop1l, {1, 2}]]\text{gliProduct}\left((\gamma \cdot q).\gamma ^{\mu }.(\gamma \cdot q),G^{\text{prop1l}}(1,2)\right)
amp2Red = FCLoopTensorReduce[amp2, {topo2}, Head -> gliProduct]\text{gliProduct}\left(-\frac{2 D p^{\mu } \gamma \cdot p (p\cdot q)^2-2 p^2 \gamma ^{\mu } (p\cdot q)^2-D p^4 q^2 \gamma ^{\mu }+3 p^4 q^2 \gamma ^{\mu }-2 p^2 q^2 p^{\mu } \gamma \cdot p}{(1-D) p^4},G^{\text{prop1l}}(1,2)\right)
If the loop momenta are contracted with some external momenta that do not appear in the given integral topologies, they should be listed via the option Uncontract
amp3 = gliProduct[SPD[q, x], GLI[prop1l, {1, 2}]]\text{gliProduct}\left(q\cdot x,G^{\text{prop1l}}(1,2)\right)
FCLoopTensorReduce[amp3, {topo2}, Uncontract -> {x}, Head -> gliProduct]\text{gliProduct}\left(\frac{(p\cdot q) (p\cdot x)}{p^2},G^{\text{prop1l}}(1,2)\right)