FeynCalc manual (development version)

FCLoopAugmentTopology

FCLoopAugmentTopology[topo, {extraProps}] augments the topology topo by adding new propagators extraProps to the basis. This is usually needed when a tensor reduction requires us to introduce an auxiliary vector that will appear in scalar products involving loop momenta.

The input topologies do not have to be complete.

The output of this routine contains augmented topologies and a list of replacement rules for converting GLIs depending on the old topologies into new ones.

See also

Overview, FCLoopTensorReduce.

Examples

topo = FCTopology["topo1", {SFAD[{q1, m^2}], SFAD[{q1 + p1}], 
    SFAD[{q1 + p2}]}, {q1}, {p1, p2}, {Hold[SPD][p1] -> 0, Hold[SPD][p2] -> 0, 
    Hold[SPD][p1, p2] -> 0}, {}]

FCTopology(topo1,{1(q12−m2+iη),1((p1+q1)2+iη),1((p2+q1)2+iη)},{q1},{p1,p2},{Hold[SPD][p1]→0,Hold[SPD][p2]→0,Hold[SPD][p1,p2]→0},{})\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{(\text{q1}^2-m^2+i \eta )},\frac{1}{((\text{p1}+\text{q1})^2+i \eta )},\frac{1}{((\text{p2}+\text{q1})^2+i \eta )}\right\},\{\text{q1}\},\{\text{p1},\text{p2}\},\{\text{Hold}[\text{SPD}][\text{p1}]\to 0,\text{Hold}[\text{SPD}][\text{p2}]\to 0,\text{Hold}[\text{SPD}][\text{p1},\text{p2}]\to 0\},\{\}\right)

The option AugmentedTopologyMarker denotes a symbol that is usually introduced by FCLoopTensorReduce when the reduction requires an auxiliary vector. Therefore, it will appear on the right hand side of the GLI-replacement rules. This can be disabled by setting this option to False

FCLoopAugmentTopology[topo, {SFAD[{{0, q1 . n}}]}, 
  FinalSubstitutions -> {Hold[SPD][n] -> 0, Hold[SPD][n, p1] -> np1, 
    Hold[SPD][n, p2] -> np2}]

{FCTopology(topo1A,{1(q12−m2+iη),1((p1+q1)2+iη),1((p2+q1)2+iη),1(n⋅  q1+iη)},{q1},{p1,p2,n},{Hold[SPD][p1]→0,Hold[SPD][p2]→0,Hold[SPD][p1,p2]→0,Hold[SPD][n]→0,Hold[SPD][n,p1]→  np1,Hold[SPD][n,p2]→  np2},{}),FCGV(AddPropagators)({n})Gtopo1(n1_,n2_,n3_):→Gtopo1A(n1,n2,n3,0)}\left\{\text{FCTopology}\left(\text{topo1A},\left\{\frac{1}{(\text{q1}^2-m^2+i \eta )},\frac{1}{((\text{p1}+\text{q1})^2+i \eta )},\frac{1}{((\text{p2}+\text{q1})^2+i \eta )},\frac{1}{(n\cdot \;\text{q1}+i \eta )}\right\},\{\text{q1}\},\{\text{p1},\text{p2},n\},\{\text{Hold}[\text{SPD}][\text{p1}]\to 0,\text{Hold}[\text{SPD}][\text{p2}]\to 0,\text{Hold}[\text{SPD}][\text{p1},\text{p2}]\to 0,\text{Hold}[\text{SPD}][n]\to 0,\text{Hold}[\text{SPD}][n,\text{p1}]\to \;\text{np1},\text{Hold}[\text{SPD}][n,\text{p2}]\to \;\text{np2}\},\{\}\right),\text{FCGV}(\text{AddPropagators})(\{n\}) G^{\text{topo1}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$}):\to G^{\text{topo1A}}(\text{n1},\text{n2},\text{n3},0)\right\}

FCLoopAugmentTopology[topo, {SFAD[{{0, q1 . n}}]}, 
  FinalSubstitutions -> {Hold[SPD][n] -> 0, Hold[SPD][n, p1] -> np1, 
    Hold[SPD][n, p2] -> np2}, AugmentedTopologyMarker -> False]

{FCTopology(topo1A,{1(q12−m2+iη),1((p1+q1)2+iη),1((p2+q1)2+iη),1(n⋅  q1+iη)},{q1},{p1,p2,n},{Hold[SPD][p1]→0,Hold[SPD][p2]→0,Hold[SPD][p1,p2]→0,Hold[SPD][n]→0,Hold[SPD][n,p1]→  np1,Hold[SPD][n,p2]→  np2},{}),Gtopo1(n1_,n2_,n3_):→Gtopo1A(n1,n2,n3,0)}\left\{\text{FCTopology}\left(\text{topo1A},\left\{\frac{1}{(\text{q1}^2-m^2+i \eta )},\frac{1}{((\text{p1}+\text{q1})^2+i \eta )},\frac{1}{((\text{p2}+\text{q1})^2+i \eta )},\frac{1}{(n\cdot \;\text{q1}+i \eta )}\right\},\{\text{q1}\},\{\text{p1},\text{p2},n\},\{\text{Hold}[\text{SPD}][\text{p1}]\to 0,\text{Hold}[\text{SPD}][\text{p2}]\to 0,\text{Hold}[\text{SPD}][\text{p1},\text{p2}]\to 0,\text{Hold}[\text{SPD}][n]\to 0,\text{Hold}[\text{SPD}][n,\text{p1}]\to \;\text{np1},\text{Hold}[\text{SPD}][n,\text{p2}]\to \;\text{np2}\},\{\}\right),G^{\text{topo1}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$}):\to G^{\text{topo1A}}(\text{n1},\text{n2},\text{n3},0)\right\}