FCLoopRewriteIncompleteTopologies[expr , topos] handles
topologies with incomplete propagator bases in the given expression. The
routine will automatically perform basis completions by adding missing
propagators, 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.
By default the basis completion approach (controlled by the
Method option) is set to Automatic. This means
that the function will use propagators already present in the list of
supplied topologies to find complete the bases. It is also possible to
specify the propagators explicitly as a list or use the option
ScalarProduct for automatically adding eikonal
propagators.
Overview, FCTopology, FCLoopFindIncompleteTopologies, FCLoopFindTopologies, FCLoopFindTopologyMappings, SubtopologyMarker.
topos = {FCTopology[topo1, {SFAD[k1], SFAD[k2]}, {k1, k2}, {}, {}, {}]}\left\{\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{(\text{k2}^2+i \eta )}\right\},\{\text{k1},\text{k2}\},\{\},\{\},\{\}\right)\right\}
expr = FCGV["GLIProduct"][SPD[k1, k2], GLI[topo1, {1, 1}]]\text{FCGV}(\text{GLIProduct})\left(\text{k1}\cdot \;\text{k2},G^{\text{topo1}}(1,1)\right)
FCLoopRewriteIncompleteTopologies[expr, topos, Method -> {SFAD[k1 - k2]}]\text{FCLoopRewriteIncompleteTopologies: }\;\text{Found }1\text{ incomplete topologies.}
\left\{\text{FCGV}(\text{GLIProduct})\left(\text{k1}\cdot \;\text{k2},G^{\text{topo1Aug}}(1,1,0)\right),\left\{\text{FCTopology}\left(\text{topo1Aug},\left\{\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{(\text{k2}^2+i \eta )},\frac{1}{((\text{k1}-\text{k2})^2+i \eta )}\right\},\{\text{k1},\text{k2}\},\{\},\{\},\{1\}\right)\right\}\right\}
```mathematica FCLoopRewriteIncompleteTopologies[expr, topos, Method -> ScalarProduct]
```mathematica
\text{FCLoopRewriteIncompleteTopologies: }\;\text{Found }1\text{ incomplete topologies.}
\left\{\text{FCGV}(\text{GLIProduct})\left(\text{k1}\cdot \;\text{k2},G^{\text{topo1Aug}}(1,1,0)\right),\left\{\text{FCTopology}\left(\text{topo1Aug},\left\{\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{(\text{k2}^2+i \eta )},\frac{1}{(\text{k1}\cdot \;\text{k2}+i \eta )}\right\},\{\text{k1},\text{k2}\},\{\},\{\},\{1\}\right)\right\}\right\}