FeynCalc manual (development version)

FCLoopIntegralToGraph

FCLoopIntegralToGraph[int, {q1, q2, ...}] constructs a graph representation of the loop integral int that depends on the loop momenta q1, q2, .... The function returns a list of the form {edges,labels,props,pref}, where edges is a list of edge rules representing the loop integral int, labels is a list of lists containing the line momentum, multiplicity and the mass term of each propagator, props is a list with the original propagators and pref is the piece of the integral that was ignored when constructing the graph representation (e.g. scalar products or vectors in the numerator) .

Use FCLoopGraphPlot to visualize the output of FCLoopIntegralToGraph.

A quick and simple way to plot the graph is to evaluate GraphPlot[List @@@ Transpose[output[[1 ;; 2]]]] or GraphPlot[Labeled @@@ Transpose[output[[1 ;; 2]]]]. The visual quality will not be that great, though. To obtain a nicer plot one might use GraphPlot with a custom EdgeTaggedGraph or export the output to a file and visualize it with an external tool such as dot/neato from graphviz.

It is also possible to invoke the function as FCLoopIntegralToGraph[GLI[...], FCTopology[...]] or FCLoopIntegralToGraph[FCTopology[...]].

See also

Overview, FCLoopGraphPlot.

Examples

out = FCLoopIntegralToGraph[FAD[{q - k1}, k1, q - k2, k2, {k2 - k3, mb}, {k1 - k3, mb}], {k1, k2, k3}]

{{−3→2,−1→1,1→3,1→4,2→3,2→4,3→4,3→4},{−q,q,{k2,1,0},{q−k2,1,0},{k1,1,0},{q−k1,1,0},{k2−k3,1,−mb2},{k1−k3,1,−mb2}},{0,0,1(k22+iη),1(k12+iη),1((q−k2)2+iη),1((q−k1)2+iη),1((k2−k3)2−mb2+iη),1((k1−k3)2−mb2+iη)},1}\left\{\{-3\to 2,-1\to 1,1\to 3,1\to 4,2\to 3,2\to 4,3\to 4,3\to 4\},\left\{-q,q,\{\text{k2},1,0\},\{q-\text{k2},1,0\},\{\text{k1},1,0\},\{q-\text{k1},1,0\},\left\{\text{k2}-\text{k3},1,-\text{mb}^2\right\},\left\{\text{k1}-\text{k3},1,-\text{mb}^2\right\}\right\},\left\{0,0,\frac{1}{(\text{k2}^2+i \eta )},\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{((q-\text{k2})^2+i \eta )},\frac{1}{((q-\text{k1})^2+i \eta )},\frac{1}{((\text{k2}-\text{k3})^2-\text{mb}^2+i \eta )},\frac{1}{((\text{k1}-\text{k3})^2-\text{mb}^2+i \eta )}\right\},1\right\}

FCLoopGraphPlot[out]

0bfm86ewusrdi

Labeled @@@ Transpose[out[[1 ;; 2]]]

01p715ugi4jrv

GraphPlot[List @@@ Transpose[out[[1 ;; 2]]]]

0blv15cgm9d0u

FCLoopIntegralToGraph[FAD[{q - k1}, k1, q - k2, k2, {k2 - k3, mb}, {k1 - k3, mb}], {k1, k2,k3}]

{{−3→2,−1→1,1→3,1→4,2→3,2→4,3→4,3→4},{−q,q,{k2,1,0},{q−k2,1,0},{k1,1,0},{q−k1,1,0},{k2−k3,1,−mb2},{k1−k3,1,−mb2}},{0,0,1(k22+iη),1(k12+iη),1((q−k2)2+iη),1((q−k1)2+iη),1((k2−k3)2−mb2+iη),1((k1−k3)2−mb2+iη)},1}\left\{\{-3\to 2,-1\to 1,1\to 3,1\to 4,2\to 3,2\to 4,3\to 4,3\to 4\},\left\{-q,q,\{\text{k2},1,0\},\{q-\text{k2},1,0\},\{\text{k1},1,0\},\{q-\text{k1},1,0\},\left\{\text{k2}-\text{k3},1,-\text{mb}^2\right\},\left\{\text{k1}-\text{k3},1,-\text{mb}^2\right\}\right\},\left\{0,0,\frac{1}{(\text{k2}^2+i \eta )},\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{((q-\text{k2})^2+i \eta )},\frac{1}{((q-\text{k1})^2+i \eta )},\frac{1}{((\text{k2}-\text{k3})^2-\text{mb}^2+i \eta )},\frac{1}{((\text{k1}-\text{k3})^2-\text{mb}^2+i \eta )}\right\},1\right\}

FAD[q - k1, k1, q - k2, k2, {k2 - k3, mb}, {k1 - k3, mb}]

1(q−k1)2.k12.(q−k2)2.k22.((k2−k3)2−mb2).((k1−k3)2−mb2)\frac{1}{(q-\text{k1})^2.\text{k1}^2.(q-\text{k2})^2.\text{k2}^2.\left((\text{k2}-\text{k3})^2-\text{mb}^2\right).\left((\text{k1}-\text{k3})^2-\text{mb}^2\right)}

If the input is given as a list of propagators, their ordering will be preserved when constructing the graph

FCLoopIntegralToGraph[FCTopology[topo1, {FAD[q - k1], FAD[k1], FAD[q - k2], FAD[k2], 
    FAD[{k2 - k3, mb}], FAD[{k1 - k3, mb}]}, {k1, k2, k3}, {q}, {}, {}]]

{{−3→2,−1→1,1→3,1→4,2→3,2→4,3→4,3→4},{−q,q,{q−k1,1,0},{k1,1,0},{q−k2,1,0},{k2,1,0},{k2−k3,1,−mb2},{k1−k3,1,−mb2}},{0,0,1((q−k1)2+iη),1(k12+iη),1((q−k2)2+iη),1(k22+iη),1((k2−k3)2−mb2+iη),1((k1−k3)2−mb2+iη)},1}\left\{\{-3\to 2,-1\to 1,1\to 3,1\to 4,2\to 3,2\to 4,3\to 4,3\to 4\},\left\{-q,q,\{q-\text{k1},1,0\},\{\text{k1},1,0\},\{q-\text{k2},1,0\},\{\text{k2},1,0\},\left\{\text{k2}-\text{k3},1,-\text{mb}^2\right\},\left\{\text{k1}-\text{k3},1,-\text{mb}^2\right\}\right\},\left\{0,0,\frac{1}{((q-\text{k1})^2+i \eta )},\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{((q-\text{k2})^2+i \eta )},\frac{1}{(\text{k2}^2+i \eta )},\frac{1}{((\text{k2}-\text{k3})^2-\text{mb}^2+i \eta )},\frac{1}{((\text{k1}-\text{k3})^2-\text{mb}^2+i \eta )}\right\},1\right\}

FCLoopIntegralToGraph[GLI[topo1, {1, 1, 1, 1, 1, 1}], 
  FCTopology[topo1, {FAD[q - k1], FAD[k1], FAD[q - k2], FAD[k2], 
    FAD[{k2 - k3, mb}], FAD[{k1 - k3, mb}]}, {k1, k2, k3}, {q}, {}, {}]]

{{−3→2,−1→1,1→3,1→4,2→3,2→4,3→4,3→4},{−q,q,{k2,1,0},{q−k2,1,0},{k1,1,0},{q−k1,1,0},{k2−k3,1,−mb2},{k1−k3,1,−mb2}},{0,0,1(k22+iη),1(k12+iη),1((q−k2)2+iη),1((q−k1)2+iη),1((k2−k3)2−mb2+iη),1((k1−k3)2−mb2+iη)},1}\left\{\{-3\to 2,-1\to 1,1\to 3,1\to 4,2\to 3,2\to 4,3\to 4,3\to 4\},\left\{-q,q,\{\text{k2},1,0\},\{q-\text{k2},1,0\},\{\text{k1},1,0\},\{q-\text{k1},1,0\},\left\{\text{k2}-\text{k3},1,-\text{mb}^2\right\},\left\{\text{k1}-\text{k3},1,-\text{mb}^2\right\}\right\},\left\{0,0,\frac{1}{(\text{k2}^2+i \eta )},\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{((q-\text{k2})^2+i \eta )},\frac{1}{((q-\text{k1})^2+i \eta )},\frac{1}{((\text{k2}-\text{k3})^2-\text{mb}^2+i \eta )},\frac{1}{((\text{k1}-\text{k3})^2-\text{mb}^2+i \eta )}\right\},1\right\}

If the second argument contains multiple topologies, the function will automatically select the relevant ones.

FCLoopIntegralToGraph[GLI[topo1, {1, 1, 1, 0, 0, 0}], 
  {FCTopology[topo1, {FAD[q - k1], FAD[k1], FAD[q - k2], FAD[k2], 
     FAD[{k2 - k3, mb}], FAD[{k1 - k3, mb}]}, {k1, k2, k3}, {q}, {}, {}], 
   FCTopology[topo2, {FAD[q - k1], FAD[k1], FAD[q - k2], FAD[k2], 
     FAD[{k2 - k3, mg}], FAD[{k1 - k3, mg}]}, {k1, k2, k3}, {q}, {}, {}] 
  }]

{{−3→2,−1→1,1→2,1→2,2→2},{−q,q,{k1,1,0},{q−k1,1,0},{q−k2,1,0}},{0,0,1(k12+iη),1((q−k2)2+iη),1((q−k1)2+iη)},1}\left\{\{-3\to 2,-1\to 1,1\to 2,1\to 2,2\to 2\},\{-q,q,\{\text{k1},1,0\},\{q-\text{k1},1,0\},\{q-\text{k2},1,0\}\},\left\{0,0,\frac{1}{(\text{k1}^2+i \eta )},\frac{1}{((q-\text{k2})^2+i \eta )},\frac{1}{((q-\text{k1})^2+i \eta )}\right\},1\right\}