FCLoopApplyTopologyMappings[expr, {mappings, topos}] applies mappings between topologies obtained using FCLoopFindTopologyMappings to the output of FCLoopFindTopologies denoted as expr. The argument topos denotes the final set of topologies present in the expression.
Instead of {mappings, topos} one can directly use the output FCLoopFindTopologyMappings.
By default the function will attempt to rewrite all the occurring loop integrals as GLIs. If you just want to apply the mappings without touching the remaining scalar products, set the option FCLoopCreateRulesToGLI to False. Even when all scalar products depending on loop momenta are rewritten as GLIs, you can still suppress the step of multiplying out products of GLIs by setting the option GLIMultiply to False.
Overview, FCTopology, GLI, FCLoopFindTopologies, FCLoopFindTopologyMappings.
This is a trial expression representing some loop amplitude that has already been processed using FCFindTopologies
ex = gliProduct[cc6*SPD[p1, p1], GLI[fctopology1, {1, 1, 2, 1, 1, 1, 1, 1, 1}]] +
gliProduct[cc2*SPD[p1, p2], GLI[fctopology2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] +
gliProduct[cc4*SPD[p1, p2], GLI[fctopology4, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] +
gliProduct[cc1*SPD[p1, Q], GLI[fctopology1, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] +
gliProduct[cc3*SPD[p2, p2], GLI[fctopology3, {1, 1, 1, 1, 1, 1, 1, 1, 1}]] +
gliProduct[cc5*SPD[p2, Q], GLI[fctopology5, {1, 1, 1, 1, 1, 1, 1, 1, 1}]]\text{gliProduct}\left(\text{cc1} (\text{p1}\cdot Q),G^{\text{fctopology1}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc2} (\text{p1}\cdot \;\text{p2}),G^{\text{fctopology2}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc3} \;\text{p2}^2,G^{\text{fctopology3}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc4} (\text{p1}\cdot \;\text{p2}),G^{\text{fctopology4}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc5} (\text{p2}\cdot Q),G^{\text{fctopology5}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc6} \;\text{p1}^2,G^{\text{fctopology1}}(1,1,2,1,1,1,1,1,1)\right)
These mapping rules describe how the 3 topologies “fctopology3”, “fctopology4” and “fctopology5” are mapped to the topologies “fctopology1” and “fctopology2”
mappings = {
{FCTopology[fctopology3, {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[{{p1 + p3, 0}, {0, 1}, 1}],
SFAD[{{p2 - 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}, {}, {}], {p1 -> -p1 - p3 + Q, p2 -> -p2 - p3 + Q, p3 -> p3},
GLI[fctopology3, {n1_, n7_, n8_, n5_, n6_, n4_, n2_, n3_, n9_}] :>
GLI[fctopology1, {n1, n2, n3, n4, n5, n6, n7, n8, n9}]},
{FCTopology[fctopology4, {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[{{p1 + p3, 0}, {0, 1}, 1}], SFAD[{{p2 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 - 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}, {}, {}], {p1 -> -p2 + Q, p2 -> -p1 + Q, p3 -> -p3},
GLI[fctopology4, {n1_, n6_, n5_, n8_, n7_, n3_, n2_, n4_, n9_}] :>
GLI[fctopology1, {n1, n2, n3, n4, n5, n6, n7, n8, n9}]},
{FCTopology[fctopology5, {SFAD[{{p3, 0}, {0, 1}, 1}], SFAD[{{p2, 0}, {0, 1}, 1}], SFAD[{{p1, 0}, {0, 1}, 1}],
SFAD[{{p1 + p3, 0}, {0, 1}, 1}], SFAD[{{p2 - Q, 0}, {0, 1}, 1}],SFAD[{{p1 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 + p3 - Q, 0}, {0, 1}, 1}],
SFAD[{{p1 + p2 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 + p2 + p3 - Q, 0}, {0, 1}, 1}]}, {p1, p2, p3}, {Q}, {}, {}], {p1 -> p2, p2 -> p1, p3 -> p3},
GLI[fctopology5, {n1_, n3_, n2_, n4_, n6_, n5_, n7_, n8_, n9_}] :>
GLI[fctopology2, {n1, n2, n3, n4, n5, n6, n7, n8, n9}]}}\left( \begin{array}{ccc} \;\text{FCTopology}\left(\text{fctopology3},\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{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-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{p1}\to -\text{p1}-\text{p3}+Q,\text{p2}\to -\text{p2}-\text{p3}+Q,\text{p3}\to \;\text{p3}\} & G^{\text{fctopology3}}(\text{n1$\_$},\text{n7$\_$},\text{n8$\_$},\text{n5$\_$},\text{n6$\_$},\text{n4$\_$},\text{n2$\_$},\text{n3$\_$},\text{n9$\_$}):\to G^{\text{fctopology1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\ \;\text{FCTopology}\left(\text{fctopology4},\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{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-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{p1}\to Q-\text{p2},\text{p2}\to Q-\text{p1},\text{p3}\to -\text{p3}\} & G^{\text{fctopology4}}(\text{n1$\_$},\text{n6$\_$},\text{n5$\_$},\text{n8$\_$},\text{n7$\_$},\text{n3$\_$},\text{n2$\_$},\text{n4$\_$},\text{n9$\_$}):\to G^{\text{fctopology1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\ \;\text{FCTopology}\left(\text{fctopology5},\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{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}-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{p1}\to \;\text{p2},\text{p2}\to \;\text{p1},\text{p3}\to \;\text{p3}\} & G^{\text{fctopology5}}(\text{n1$\_$},\text{n3$\_$},\text{n2$\_$},\text{n4$\_$},\text{n6$\_$},\text{n5$\_$},\text{n7$\_$},\text{n8$\_$},\text{n9$\_$}):\to G^{\text{fctopology2}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\ \end{array} \right)
These are the two topologies onto which everything is mapped
finalTopos = {
FCTopology[fctopology1, {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}, {}, {}],
FCTopology[fctopology2, {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 + p2 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 + p2 + p3 - Q, 0}, {0, 1}, 1}]}, {p1, p2, p3}, {Q}, {}, {}]}\left\{\text{FCTopology}\left(\text{fctopology1},\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{fctopology2},\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{p2}-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)\right\}
FCLoopApplyTopologyMappings applies the given mappings to the expression creating an output that is ready to be processed further
FCLoopApplyTopologyMappings[ex, {mappings, finalTopos}, Head -> gliProduct, FCVerbose -> 0]\frac{1}{2} G^{\text{fctopology1}}(1,1,1,1,1,1,1,1,1) \left(\text{cc1} Q^2+\text{cc4} Q^2+2 \;\text{cc6}\right)+\frac{1}{2} (\text{cc1}-\text{cc4}) G^{\text{fctopology1}}(1,1,0,1,1,1,1,1,1)-\frac{1}{2} (\text{cc1}-\text{cc4}) G^{\text{fctopology1}}(1,1,1,1,1,0,1,1,1)+\frac{1}{2} Q^2 (\text{cc2}+\text{cc5}) G^{\text{fctopology2}}(1,1,1,1,1,1,1,1,1)-\frac{1}{2} (\text{cc2}+\text{cc5}) G^{\text{fctopology2}}(1,1,1,1,1,0,1,1,1)-\frac{1}{2} \;\text{cc2} G^{\text{fctopology2}}(1,1,1,1,0,1,1,1,1)+\frac{1}{2} \;\text{cc2} G^{\text{fctopology2}}(1,1,1,1,1,1,1,0,1)+\text{cc3} G^{\text{fctopology1}}(1,1,1,1,1,1,0,1,1)+\frac{1}{2} \;\text{cc4} G^{\text{fctopology1}}(0,1,1,1,1,1,1,1,1)-\frac{1}{2} \;\text{cc4} G^{\text{fctopology1}}(1,1,1,0,1,1,1,1,1)-\frac{1}{2} \;\text{cc4} G^{\text{fctopology1}}(1,1,1,1,1,1,1,0,1)+\frac{1}{2} \;\text{cc4} G^{\text{fctopology1}}(1,1,1,1,1,1,1,1,0)+\frac{1}{2} \;\text{cc5} G^{\text{fctopology2}}(1,1,0,1,1,1,1,1,1)
This just applies the mappings without any further simplifications
FCLoopApplyTopologyMappings[ex, {mappings, finalTopos}, Head -> gliProduct, FCLoopCreateRulesToGLI -> False]\text{gliProduct}\left(\text{cc3} (-\text{p2}-\text{p3}+Q)^2,G^{\text{fctopology1}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc4} ((Q-\text{p1})\cdot (Q-\text{p2})),G^{\text{fctopology1}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc1} (\text{p1}\cdot Q),G^{\text{fctopology1}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc2} (\text{p1}\cdot \;\text{p2}),G^{\text{fctopology2}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc5} (\text{p1}\cdot Q),G^{\text{fctopology2}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc6} \;\text{p1}^2,G^{\text{fctopology1}}(1,1,2,1,1,1,1,1,1)\right)
This applies the mappings and eliminates the numerators but still keeps products of GLIs in the expression
FCLoopApplyTopologyMappings[ex, {mappings, finalTopos}, Head -> gliProduct, FCLoopCreateRulesToGLI -> True, GLIMultiply -> False]\text{gliProduct}\left(\frac{1}{2} \;\text{cc1} \left(G^{\text{fctopology1}}(0,0,-1,0,0,0,0,0,0)-G^{\text{fctopology1}}(0,0,0,0,0,-1,0,0,0)+Q^2\right),G^{\text{fctopology1}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\frac{1}{2} \;\text{cc2} \left(-G^{\text{fctopology2}}(0,0,0,0,-1,0,0,0,0)-G^{\text{fctopology2}}(0,0,0,0,0,-1,0,0,0)+G^{\text{fctopology2}}(0,0,0,0,0,0,0,-1,0)+Q^2\right),G^{\text{fctopology2}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc3} G^{\text{fctopology1}}(0,0,0,0,0,0,-1,0,0),G^{\text{fctopology1}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc4} \left(\frac{1}{2} \left(-G^{\text{fctopology1}}(0,-1,0,0,0,0,0,0,0)+G^{\text{fctopology1}}(0,0,0,0,-1,0,0,0,0)-Q^2\right)+\frac{1}{2} \left(-G^{\text{fctopology1}}(0,0,-1,0,0,0,0,0,0)+G^{\text{fctopology1}}(0,0,0,0,0,-1,0,0,0)-Q^2\right)+\frac{1}{2} \left(G^{\text{fctopology1}}(-1,0,0,0,0,0,0,0,0)+G^{\text{fctopology1}}(0,-1,0,0,0,0,0,0,0)-G^{\text{fctopology1}}(0,0,0,-1,0,0,0,0,0)-G^{\text{fctopology1}}(0,0,0,0,-1,0,0,0,0)-G^{\text{fctopology1}}(0,0,0,0,0,0,0,-1,0)+G^{\text{fctopology1}}(0,0,0,0,0,0,0,0,-1)+Q^2\right)+Q^2\right),G^{\text{fctopology1}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\frac{1}{2} \;\text{cc5} \left(G^{\text{fctopology2}}(0,0,-1,0,0,0,0,0,0)-G^{\text{fctopology2}}(0,0,0,0,0,-1,0,0,0)+Q^2\right),G^{\text{fctopology2}}(1,1,1,1,1,1,1,1,1)\right)+\text{gliProduct}\left(\text{cc6} G^{\text{fctopology1}}(0,0,-1,0,0,0,0,0,0),G^{\text{fctopology1}}(1,1,2,1,1,1,1,1,1)\right)