FCLoopRewriteOverdeterminedTopologies[expr , topos]
handles topologies with overdetermined propagator bases in the given
expression. The routine will automatically perform partial fraction
decomposition on the affected topologies, introduce new names for the
resulting topologies and return back the expression depending on those
new topologies together with a list of the corresponding topologies.
The input expression is expected to be of the form returned by
FCLoopFindTopologies, e.g. with numerators separated from
the denominators where the latter are written as GLIs.
The names of the automatically generated topology can be controlled
using the Names option.
Notice that the returned topologies can be related to each other,
while some of them may even have identical sets of propagators. This is
expected, because the output of this function usually gets passed to
FCLoopFindTopologyMappings.
Overview, FCTopology, FCLoopFindOverdeterminedTopologies, FCLoopFindTopologies, FCLoopFindTopologyMappings, SubtopologyMarker.
topos = {FCTopology[topo1, {SFAD[k1], SFAD[k1 + p], SFAD[k1 - p]}, {k1}, {p}, {}, {}]}\left\{\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{((\text{k1}+p)^2+i \eta )},\frac{1}{((\text{k1}-p)^2+i \eta )}\right\},\{\text{k1}\},\{p\},\{\},\{\}\right)\right\}
expr = FCGV["GLIProduct"][SPD[k1, p], GLI[topo1, {1, 1, 1}]]\text{FCGV}(\text{GLIProduct})\left(\text{k1}\cdot p,G^{\text{topo1}}(1,1,1)\right)
FCLoopRewriteOverdeterminedTopologies[expr, topos]\text{FCLoopFindTopologyMappings: }\;\text{Found }1\text{ overdetermined topologies.}
\text{FCLoopFindTopologyMappings: }\;\text{Generated }3\text{ new topologies through partial fractioning.}
\left\{\text{FCGV}(\text{GLIProduct})\left(-\frac{\text{k1}\cdot p}{p^2},G^{\text{fcPFRTopology1}}(1,1)\right)+\text{FCGV}(\text{GLIProduct})\left(\frac{\text{k1}\cdot p}{2 p^2},G^{\text{fcPFRTopology2}}(1,1)\right)+\text{FCGV}(\text{GLIProduct})\left(\frac{\text{k1}\cdot p}{2 p^2},G^{\text{fcPFRTopology3}}(1,1)\right),\left\{\text{FCTopology}\left(\text{fcPFRTopology1},\left\{\frac{1}{((\text{k1}+p)^2+i \eta )},\frac{1}{((\text{k1}-p)^2+i \eta )}\right\},\{\text{k1}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{fcPFRTopology2},\left\{\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{((\text{k1}-p)^2+i \eta )}\right\},\{\text{k1}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{fcPFRTopology3},\left\{\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{((\text{k1}+p)^2+i \eta )}\right\},\{\text{k1}\},\{p\},\{\},\{\}\right)\right\}\right\}