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)