FeynCalc manual (development version)

FCLoopApplyTopologyMappings

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.

See also

Overview, FCTopology, GLI, FCLoopFindTopologies, FCLoopFindTopologyMappings.

Examples

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}]]

gliProduct(cc1(p1⋅Q),Gfctopology1(1,1,1,1,1,1,1,1,1))+gliProduct(cc2(p1⋅  p2),Gfctopology2(1,1,1,1,1,1,1,1,1))+gliProduct(cc3  p22,Gfctopology3(1,1,1,1,1,1,1,1,1))+gliProduct(cc4(p1⋅  p2),Gfctopology4(1,1,1,1,1,1,1,1,1))+gliProduct(cc5(p2⋅Q),Gfctopology5(1,1,1,1,1,1,1,1,1))+gliProduct(cc6  p12,Gfctopology1(1,1,2,1,1,1,1,1,1))\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}]}}

(  FCTopology(fctopology3,{1(p32+iη),1(p22+iη),1(p12+iη),1((p2+p3)2+iη),1((p1+p3)2+iη),1((p2−Q)2+iη),1((p2+p3−Q)2+iη),1((p1+p3−Q)2+iη),1((p1+p2+p3−Q)2+iη)},{p1,p2,p3},{Q},{},{}){p1→−p1−p3+Q,p2→−p2−p3+Q,p3→  p3}Gfctopology3(n1_,n7_,n8_,n5_,n6_,n4_,n2_,n3_,n9_):→Gfctopology1(n1,n2,n3,n4,n5,n6,n7,n8,n9)  FCTopology(fctopology4,{1(p32+iη),1(p22+iη),1(p12+iη),1((p2+p3)2+iη),1((p1+p3)2+iη),1((p2−Q)2+iη),1((p1−Q)2+iη),1((p1+p3−Q)2+iη),1((p1+p2+p3−Q)2+iη)},{p1,p2,p3},{Q},{},{}){p1→Q−p2,p2→Q−p1,p3→−p3}Gfctopology4(n1_,n6_,n5_,n8_,n7_,n3_,n2_,n4_,n9_):→Gfctopology1(n1,n2,n3,n4,n5,n6,n7,n8,n9)  FCTopology(fctopology5,{1(p32+iη),1(p22+iη),1(p12+iη),1((p1+p3)2+iη),1((p2−Q)2+iη),1((p1−Q)2+iη),1((p1+p3−Q)2+iη),1((p1+p2−Q)2+iη),1((p1+p2+p3−Q)2+iη)},{p1,p2,p3},{Q},{},{}){p1→  p2,p2→  p1,p3→  p3}Gfctopology5(n1_,n3_,n2_,n4_,n6_,n5_,n7_,n8_,n9_):→Gfctopology2(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}, {}, {}]}

{FCTopology(fctopology1,{1(p32+iη),1(p22+iη),1(p12+iη),1((p2+p3)2+iη),1((p2−Q)2+iη),1((p1−Q)2+iη),1((p2+p3−Q)2+iη),1((p1+p3−Q)2+iη),1((p1+p2+p3−Q)2+iη)},{p1,p2,p3},{Q},{},{}),FCTopology(fctopology2,{1(p32+iη),1(p22+iη),1(p12+iη),1((p2+p3)2+iη),1((p2−Q)2+iη),1((p1−Q)2+iη),1((p2+p3−Q)2+iη),1((p1+p2−Q)2+iη),1((p1+p2+p3−Q)2+iη)},{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]

12Gfctopology1(1,1,1,1,1,1,1,1,1)(cc1Q2+cc4Q2+2  cc6)+12(cc1−cc4)Gfctopology1(1,1,0,1,1,1,1,1,1)−12(cc1−cc4)Gfctopology1(1,1,1,1,1,0,1,1,1)+12Q2(cc2+cc5)Gfctopology2(1,1,1,1,1,1,1,1,1)−12(cc2+cc5)Gfctopology2(1,1,1,1,1,0,1,1,1)−12  cc2Gfctopology2(1,1,1,1,0,1,1,1,1)+12  cc2Gfctopology2(1,1,1,1,1,1,1,0,1)+cc3Gfctopology1(1,1,1,1,1,1,0,1,1)+12  cc4Gfctopology1(0,1,1,1,1,1,1,1,1)−12  cc4Gfctopology1(1,1,1,0,1,1,1,1,1)−12  cc4Gfctopology1(1,1,1,1,1,1,1,0,1)+12  cc4Gfctopology1(1,1,1,1,1,1,1,1,0)+12  cc5Gfctopology2(1,1,0,1,1,1,1,1,1)\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]

gliProduct(cc3(−p2−p3+Q)2,Gfctopology1(1,1,1,1,1,1,1,1,1))+gliProduct(cc4((Q−p1)⋅(Q−p2)),Gfctopology1(1,1,1,1,1,1,1,1,1))+gliProduct(cc1(p1⋅Q),Gfctopology1(1,1,1,1,1,1,1,1,1))+gliProduct(cc2(p1⋅  p2),Gfctopology2(1,1,1,1,1,1,1,1,1))+gliProduct(cc5(p1⋅Q),Gfctopology2(1,1,1,1,1,1,1,1,1))+gliProduct(cc6  p12,Gfctopology1(1,1,2,1,1,1,1,1,1))\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]

gliProduct(12  cc1(Gfctopology1(0,0,−1,0,0,0,0,0,0)−Gfctopology1(0,0,0,0,0,−1,0,0,0)+Q2),Gfctopology1(1,1,1,1,1,1,1,1,1))+gliProduct(12  cc2(−Gfctopology2(0,0,0,0,−1,0,0,0,0)−Gfctopology2(0,0,0,0,0,−1,0,0,0)+Gfctopology2(0,0,0,0,0,0,0,−1,0)+Q2),Gfctopology2(1,1,1,1,1,1,1,1,1))+gliProduct(cc3Gfctopology1(0,0,0,0,0,0,−1,0,0),Gfctopology1(1,1,1,1,1,1,1,1,1))+gliProduct(cc4(12(−Gfctopology1(0,−1,0,0,0,0,0,0,0)+Gfctopology1(0,0,0,0,−1,0,0,0,0)−Q2)+12(−Gfctopology1(0,0,−1,0,0,0,0,0,0)+Gfctopology1(0,0,0,0,0,−1,0,0,0)−Q2)+12(Gfctopology1(−1,0,0,0,0,0,0,0,0)+Gfctopology1(0,−1,0,0,0,0,0,0,0)−Gfctopology1(0,0,0,−1,0,0,0,0,0)−Gfctopology1(0,0,0,0,−1,0,0,0,0)−Gfctopology1(0,0,0,0,0,0,0,−1,0)+Gfctopology1(0,0,0,0,0,0,0,0,−1)+Q2)+Q2),Gfctopology1(1,1,1,1,1,1,1,1,1))+gliProduct(12  cc5(Gfctopology2(0,0,−1,0,0,0,0,0,0)−Gfctopology2(0,0,0,0,0,−1,0,0,0)+Q2),Gfctopology2(1,1,1,1,1,1,1,1,1))+gliProduct(cc6Gfctopology1(0,0,−1,0,0,0,0,0,0),Gfctopology1(1,1,2,1,1,1,1,1,1))\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)