FeynCalc manual (development version)

FCLoopFindSubtopologies[topo] finds all scalefull subtopologies of the FCTopology topo.

Each subtopology receives a marker that specifies the topology from which it was derived. The symbol denoting the marker is specified via the option SubtopologyMarker. Setting it to False will disable the inclusion of the markers

See also

Overview, FCTopology, FCLoopFindTopologies, FCLoopFindTopologyMappings, SubtopologyMarker.

Examples

res = FCLoopFindSubtopologies[FCTopology[TRI, {SFAD[{{p1, 0}, {0, 1}, 1}], 
      SFAD[{{p2, 0}, {0, 1}, 1}], SFAD[{{p1 + Q1, 0}, {0, 1}, 1}], SFAD[{{p1 + p2 + Q1, 0}, 
        {0, 1}, 1}], SFAD[{{-p1 + Q2, 0}, {0, 1}, 1}], SFAD[{{-p1 - p2 + Q2, 0}, {0, 1}, 1}]}, 
     {p1, p2}, {Q1, Q2}, {}, {}]];

res // Length

19

Show the first three subtopologies of this 2-loop self-energy topology

res[[1 ;; 3]]

\left\{\text{FCTopology}\left(\text{TRI},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{((\text{p1}+\text{Q1})^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{Q1})^2+i \eta )},\frac{1}{((\text{Q2}-\text{p1})^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+\text{Q2})^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{\text{Q1},\text{Q2}\},\{\},\{\}\right),\text{FCTopology}\left(\text{TRIR1},\left\{\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{((\text{p1}+\text{Q1})^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{Q1})^2+i \eta )},\frac{1}{((\text{Q2}-\text{p1})^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+\text{Q2})^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{\text{Q1},\text{Q2}\},\{\},\{\text{FCGV}(\text{SubtopologyOf})\to \;\text{TRI}\}\right),\text{FCTopology}\left(\text{TRIR2},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p1}+\text{Q1})^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{Q1})^2+i \eta )},\frac{1}{((\text{Q2}-\text{p1})^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+\text{Q2})^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{\text{Q1},\text{Q2}\},\{\},\{\text{FCGV}(\text{SubtopologyOf})\to \;\text{TRI}\}\right)\right\}

res = FCLoopFindSubtopologies[FCTopology[topo1, {SFAD[{{p3, 0}, {0, 1}, 1}], 
      SFAD[{{p2, 0}, {0, 1}, 1}], SFAD[{{p1, 0}, {0, 1}, 1}], 
      SFAD[{{p2 + p3, 0}, {0, 1}, 1}], SFAD[{{p2 - Q, 0}, {0, 1}, 1}], 
      SFAD[{{p1 - Q, 0}, {0, 1}, 1}], SFAD[{{p2 + p3 - Q, 0}, {0, 1}, 1}], 
      SFAD[{{p1 + p3 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 + p2 + p3 - Q, 0}, 
        {0, 1}, 1}]}, {p1, p2, p3}, {Q}, {}, {}], FCE -> True];

res // Length

36

Show the first three subtopologies of this 3-loop self-energy topology

res[[1 ;; 3]]

\left\{\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right),\text{FCTopology}\left(\text{topo1R1},\left\{\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\text{FCGV}(\text{SubtopologyOf})\to \;\text{topo1}\}\right),\text{FCTopology}\left(\text{topo1R2},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\text{FCGV}(\text{SubtopologyOf})\to \;\text{topo1}\}\right)\right\}