FeynCalc manual (development version)

FCLoopFindTopologies[exp, {q1, q2, ...}] attempts to identify the loop integral topologies present in exp by looking at the propagator denominators that depend on the loop momenta q1, q2, ... . It returns a list of two entries, where the first one is the original expression with the denominators rewritten as GLIs, and the second one is the set of the identified topologies.

Each of the identified topologies must contain linearly independent propagators (unless the option FCLoopBasisOverdeterminedQ is set to True), but may lack propagators needed to form a complete basis.

Scaleless topologies are automatically removed, but this can be disabled by setting the option FCLoopScalelessQ to True.

See also

Examples

amp = Get[FileNameJoin[{$FeynCalcDirectory, "Documentation", "Examples", 
      "Amplitudes", "Gh-Gh-2L.m"}]];

res = FCLoopFindTopologies[amp, {q1, q2}];

\text{FCLoopFindTopologies: Number of the preferred topologies among the unique topologies: }0

\text{FCLoopFindTopologies: }\;\text{Your topologies depend on the follwing kinematic invariants that are not all entirely lowercase: }\{\text{Pair[Momentum[p, D], Momentum[p, D]]}\}

\text{FCLoopFindTopologies: }\;\text{This may lead to issues if these topologies are meant to be processed using tools such as FIRE, KIRA or Fermat.}

res // Last

\left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((\text{q1}+\text{q2})^2+i \eta )},\frac{1}{((p+\text{q1})^2+i \eta )},\frac{1}{((p-\text{q2})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology2},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p+\text{q2})^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta )},\frac{1}{((p-\text{q1}+\text{q2})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right)\right\}

topos = Get[FileNameJoin[{$FeynCalcDirectory, "Documentation", "Examples", 
      "Topologies", "BMixingTopos2L.m"}]];

topos // Length

res = FCLoopFindTopologies[topos, {k1, k2}];

\text{FCLoopFindTopologies: Number of the preferred topologies among the unique topologies: }0

\text{FCLoopFindTopologies: }\;\text{Your topologies depend on the follwing kinematic invariants that are not all entirely lowercase: }\{\text{mb},\text{mc},\text{Pair[Momentum[p1, D], Momentum[p1, D]]}\}

\text{FCLoopFindTopologies: }\;\text{This may lead to issues if these topologies are meant to be processed using tools such as FIRE, KIRA or Fermat.}

(res // Last)[[1 ;; 2]]

\left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{((\text{k2}+\text{p1})^2+i \eta )},\frac{1}{(\text{k2}^2-\text{mb}^2+i \eta )},\frac{1}{(\text{k1}^2-\text{mc}^2+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2-\text{mc}^2+i \eta )},\frac{1}{((\text{k1}-\text{p1})^2-\text{mc}^2+i \eta )}\right\},\{\text{k1},\text{k2}\},\{\text{p1}\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology2},\left\{\frac{1}{((\text{k2}-\text{p1})^2+i \eta )},\frac{1}{(\text{k2}^2-\text{mb}^2+i \eta )},\frac{1}{(\text{k1}^2-\text{mc}^2+i \eta )},\frac{1}{((\text{k1}+\text{p1})^2-\text{mc}^2+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2-\text{mc}^2+i \eta )}\right\},\{\text{k1},\text{k2}\},\{\text{p1}\},\{\},\{\}\right)\right\}

In practical calculations even the simple extraction of topologies from the given list of diagrams can take considerable amount of time. This is why it is better to parallelize this process as much as possible.

We can split this part into two steps, were we first apply FCLoopIsolate to the resulting amplitudes and then pass the output to FCLoopFindTopologies. To avoid the unnecessary application of FCLoopIsolate during this step, we should specify the head with which the topologies have been wrapped via the option FCLoopIsolate.

isolatedTopos = FCLoopIsolate[topos[[44 ;; 48]], {k1, k2},  Collecting -> False, Factoring -> False, Numerator -> False, Head -> loopDen];

res = FCLoopFindTopologies[isolatedTopos, {k1, k2}, FCLoopIsolate -> loopDen, Head -> ampDen, Collecting -> False];

\text{FCLoopFindTopologies: Number of the preferred topologies among the unique topologies: }0

\text{FCLoopFindTopologies: }\;\text{This may lead to issues if these topologies are meant to be processed using tools such as FIRE, KIRA or Fermat.}