FCLoopFindTopologyMappings
FCLoopFindTopologyMappings[{topo1, topo2, ...}]
finds mappings between topologies (written as FCTopology
objects) topo1, topo2, ...
. For each source topology the function returns a list of loop momentum shifts and a GLI
replacement rule needed to map it to the given target topology. If you need to map everything to a particular set of target topologies, you can specify them via the PreferredTopologies
option.
The output is a list of two lists, the former containing the mappings and the latter enumerating the final contributing topologies
To enable shifts in the external momenta you need to set the option Momentum
to All
.
See also
Overview , FCTopology , GLI , FCLoopFindTopologies .
Examples
Here we have a set of 5 topologies
topos1 = {
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[ 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 }, {}, {}],
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 }, {}, {}],
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 }, {}, {}]} ;
3 of them can be mapped to the other two
mappings1 = FCLoopFindTopologyMappings[ topos1] ;
FCLoopFindTopologyMappings: Found 3 mapping relations \text{FCLoopFindTopologyMappings: }\;\text{Found }3\text{ mapping relations } FCLoopFindTopologyMappings: Found 3 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 2 \text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }2 FCLoopFindTopologyMappings: Final number of independent topologies: 2
( FCTopology ( fctopology3 , { 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p1 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p2 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) { p1 → − p1 − p3 + Q , p2 → − p2 − p3 + Q , p3 → p3 } G fctopology3 ( n1 _ , n7 _ , n8 _ , n5 _ , n6 _ , n4 _ , n2 _ , n3 _ , n9 _ ) : → G fctopology1 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 ) FCTopology ( fctopology4 , { 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p1 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p1 − Q ) 2 + i η ) , 1 ( ( p1 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) { p1 → Q − p2 , p2 → Q − p1 , p3 → − p3 } G fctopology4 ( n1 _ , n6 _ , n5 _ , n8 _ , n7 _ , n3 _ , n2 _ , n4 _ , n9 _ ) : → G fctopology1 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 ) FCTopology ( fctopology5 , { 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p1 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p1 − Q ) 2 + i η ) , 1 ( ( p1 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) { p1 → p2 , p2 → p1 , p3 → p3 } G fctopology5 ( n1 _ , n3 _ , n2 _ , n4 _ , n6 _ , n5 _ , n7 _ , n8 _ , n9 _ ) : → G 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) FCTopology ( fctopology3 , { ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p1 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p2 + p3 − Q ) 2 + i η ) 1 , (( p1 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } ) FCTopology ( fctopology4 , { ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p1 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p1 − Q ) 2 + i η ) 1 , (( p1 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } ) FCTopology ( fctopology5 , { ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p1 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p1 − Q ) 2 + i η ) 1 , (( p1 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } ) { p1 → − p1 − p3 + Q , p2 → − p2 − p3 + Q , p3 → p3 } { p1 → Q − p2 , p2 → Q − p1 , p3 → − p3 } { p1 → p2 , p2 → p1 , p3 → p3 } G fctopology3 ( n1_ , n7_ , n8_ , n5_ , n6_ , n4_ , n2_ , n3_ , n9_ ) :→ G fctopology1 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 ) G fctopology4 ( n1_ , n6_ , n5_ , n8_ , n7_ , n3_ , n2_ , n4_ , n9_ ) :→ G fctopology1 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 ) G fctopology5 ( n1_ , n3_ , n2_ , n4_ , n6_ , n5_ , n7_ , n8_ , n9_ ) :→ G fctopology2 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 )
And these are the final topologies
{ FCTopology ( fctopology1 , { 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p1 − Q ) 2 + i η ) , 1 ( ( p2 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) , FCTopology ( fctopology2 , { 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p1 − Q ) 2 + i η ) , 1 ( ( p2 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) } , { 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\} { FCTopology ( fctopology1 , { ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p1 − Q ) 2 + i η ) 1 , (( p2 + p3 − Q ) 2 + i η ) 1 , (( p1 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } ) , FCTopology ( fctopology2 , { ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p1 − Q ) 2 + i η ) 1 , (( p2 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } ) }
Here is another example
topos2 = { FCTopology[ fctopology1, { SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1 + q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p + q1, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ p - q2, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}],
FCTopology[ fctopology2, { SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ p + q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p - q1, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}],
FCTopology[ fctopology3, { SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ p - q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p - q1 + q2, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}]}
{ FCTopology ( fctopology1 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( q1 + q2 ) 2 + i η ) , 1 ( ( p + q1 ) 2 + i η ) , 1 ( ( p − q2 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( fctopology2 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( p + q2 ) 2 + i η ) , 1 ( ( p − q1 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( fctopology3 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( p − q1 ) 2 + i η ) , 1 ( ( p − q1 + q2 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) } \left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((\text{q1}+\text{q2})^2+i \eta )},\frac{1}{((p+\text{q1})^2+i \eta )},\frac{1}{((p-\text{q2})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology2},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p+\text{q2})^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta )},\frac{1}{((p-\text{q1}+\text{q2})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right)\right\} { FCTopology ( fctopology1 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( q1 + q2 ) 2 + i η ) 1 , (( p + q1 ) 2 + i η ) 1 , (( p − q2 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( fctopology2 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( p + q2 ) 2 + i η ) 1 , (( p − q1 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( fctopology3 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( p − q1 ) 2 + i η ) 1 , (( p − q1 + q2 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) }
Yet this time we have some preferred set of topologies and want to match to them (if possible)
preferredTopos2 = { FCTopology[ prop2L, { SFAD[{{ q1, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1 - q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ - p + q1, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ - p + q2, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}],
FCTopology[ prop2LX1, { SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1 - q2, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ - p + q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ - p + q2, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}],
FCTopology[ prop2LX3, { SFAD[{{ q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ - p + q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ - p + q2, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}],
FCTopology[ prop2LX15, { SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1 - q2, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ - p + q1, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}]}
{ FCTopology ( prop2L , { 1 ( q1 2 + i η ) , 1 ( q2 2 + i η ) , 1 ( ( q1 − q2 ) 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX1 , { 1 ( q2 2 + i η ) , 1 ( ( q1 − q2 ) 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX3 , { 1 ( q1 2 + i η ) , 1 ( q2 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX15 , { 1 ( q2 2 + i η ) , 1 ( ( q1 − q2 ) 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) } \left\{\text{FCTopology}\left(\text{prop2L},\left\{\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX1},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX3},\left\{\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX15},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right)\right\} { FCTopology ( prop2L , { ( q1 2 + i η ) 1 , ( q2 2 + i η ) 1 , (( q1 − q2 ) 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX1 , { ( q2 2 + i η ) 1 , (( q1 − q2 ) 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX3 , { ( q1 2 + i η ) 1 , ( q2 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX15 , { ( q2 2 + i η ) 1 , (( q1 − q2 ) 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) }
mappings2 = FCLoopFindTopologyMappings[ topos2, PreferredTopologies -> preferredTopos2] ;
FCLoopFindTopologyMappings: Found 3 mapping relations \text{FCLoopFindTopologyMappings: }\;\text{Found }3\text{ mapping relations } FCLoopFindTopologyMappings: Found 3 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 3 \text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }3 FCLoopFindTopologyMappings: Final number of independent topologies: 3
( FCTopology ( fctopology1 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( q1 + q2 ) 2 + i η ) , 1 ( ( p + q1 ) 2 + i η ) , 1 ( ( p − q2 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) { q1 → − q2 , q2 → q1 } G fctopology1 ( n1 _ , n2 _ , n3 _ , n5 _ , n4 _ ) : → G prop2L ( n1 , n2 , n3 , n4 , n5 ) FCTopology ( fctopology2 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( p + q2 ) 2 + i η ) , 1 ( ( p − q1 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) { q1 → q2 , q2 → − q1 } G fctopology2 ( n1 _ , n2 _ , n3 _ , n4 _ ) : → G prop2LX3 ( n1 , n2 , n3 , n4 ) FCTopology ( fctopology3 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( p − q1 ) 2 + i η ) , 1 ( ( p − q1 + q2 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) { q1 → q2 , q2 → q2 − q1 } G fctopology3 ( n2 _ , n1 _ , n4 _ , n3 _ ) : → G prop2LX1 ( n1 , n2 , n3 , n4 ) ) \left(
\begin{array}{ccc}
\;\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((\text{q1}+\text{q2})^2+i \eta )},\frac{1}{((p+\text{q1})^2+i \eta )},\frac{1}{((p-\text{q2})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right) & \{\text{q1}\to -\text{q2},\text{q2}\to \;\text{q1}\} & G^{\text{fctopology1}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n5$\_$},\text{n4$\_$}):\to G^{\text{prop2L}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5}) \\
\;\text{FCTopology}\left(\text{fctopology2},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p+\text{q2})^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right) & \{\text{q1}\to \;\text{q2},\text{q2}\to -\text{q1}\} & G^{\text{fctopology2}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n4$\_$}):\to G^{\text{prop2LX3}}(\text{n1},\text{n2},\text{n3},\text{n4}) \\
\;\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta )},\frac{1}{((p-\text{q1}+\text{q2})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right) & \{\text{q1}\to \;\text{q2},\text{q2}\to \;\text{q2}-\text{q1}\} & G^{\text{fctopology3}}(\text{n2$\_$},\text{n1$\_$},\text{n4$\_$},\text{n3$\_$}):\to G^{\text{prop2LX1}}(\text{n1},\text{n2},\text{n3},\text{n4}) \\
\end{array}
\right) FCTopology ( fctopology1 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( q1 + q2 ) 2 + i η ) 1 , (( p + q1 ) 2 + i η ) 1 , (( p − q2 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) FCTopology ( fctopology2 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( p + q2 ) 2 + i η ) 1 , (( p − q1 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) FCTopology ( fctopology3 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( p − q1 ) 2 + i η ) 1 , (( p − q1 + q2 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) { q1 → − q2 , q2 → q1 } { q1 → q2 , q2 → − q1 } { q1 → q2 , q2 → q2 − q1 } G fctopology1 ( n1_ , n2_ , n3_ , n5_ , n4_ ) :→ G prop2L ( n1 , n2 , n3 , n4 , n5 ) G fctopology2 ( n1_ , n2_ , n3_ , n4_ ) :→ G prop2LX3 ( n1 , n2 , n3 , n4 ) G fctopology3 ( n2_ , n1_ , n4_ , n3_ ) :→ G prop2LX1 ( n1 , n2 , n3 , n4 )
And these are the final occurring topologies
{ FCTopology ( prop2L , { 1 ( q1 2 + i η ) , 1 ( q2 2 + i η ) , 1 ( ( q1 − q2 ) 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX1 , { 1 ( q2 2 + i η ) , 1 ( ( q1 − q2 ) 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX3 , { 1 ( q1 2 + i η ) , 1 ( q2 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) } \left\{\text{FCTopology}\left(\text{prop2L},\left\{\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX1},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX3},\left\{\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right)\right\} { FCTopology ( prop2L , { ( q1 2 + i η ) 1 , ( q2 2 + i η ) 1 , (( q1 − q2 ) 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX1 , { ( q2 2 + i η ) 1 , (( q1 − q2 ) 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX3 , { ( q1 2 + i η ) 1 , ( q2 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) }
If we need to match subtopologies into larger topologies, we first need to generate all possible subtopologies for each relevant topology.
topos3 = {
FCTopology[ fctopology1, {
SFAD[{{ l1 + l2 - q1, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ l2, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }],
SFAD[{{ l1, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }],
SFAD[{{ l2 + q2, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }],
SFAD[{{ l1 - q1, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }],
SFAD[{{ l1 - q1 - q2, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }]}, { l1, l2}, { q1, q2}, {}, {}],
FCTopology[ fctopology9, {
SFAD[{{ l1 + l2 + q2, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ l2, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }],
SFAD[{{ l1, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }],
SFAD[{{ l1 + q2, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }],
SFAD[{{ l1 - q1, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }]}, { l1, l2}, { q1, q2}, {}, {}]
}
{ FCTopology ( fctopology1 , { 1 ( ( l1 + l2 − q1 ) 2 + i η ) , 1 ( l2 2 − m t 2 + i η ) , 1 ( l1 2 − m t 2 + i η ) , 1 ( ( l2 + q2 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 − q2 ) 2 − m t 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { } , { } ) , FCTopology ( fctopology9 , { 1 ( ( l1 + l2 + q2 ) 2 + i η ) , 1 ( l2 2 − m t 2 + i η ) , 1 ( l1 2 − m t 2 + i η ) , 1 ( ( l1 + q2 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 ) 2 − m t 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { } , { } ) } \left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{((\text{l1}+\text{l2}-\text{q1})^2+i \eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta )},\frac{1}{(\text{l1}^2-m_t^2+i \eta )},\frac{1}{((\text{l2}+\text{q2})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1}-\text{q2})^2-m_t^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology9},\left\{\frac{1}{((\text{l1}+\text{l2}+\text{q2})^2+i \eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta )},\frac{1}{(\text{l1}^2-m_t^2+i \eta )},\frac{1}{((\text{l1}+\text{q2})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right)\right\} { FCTopology ( fctopology1 , { (( l1 + l2 − q1 ) 2 + i η ) 1 , ( l2 2 − m t 2 + i η ) 1 , ( l1 2 − m t 2 + i η ) 1 , (( l2 + q2 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 − q2 ) 2 − m t 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { } , { } ) , FCTopology ( fctopology9 , { (( l1 + l2 + q2 ) 2 + i η ) 1 , ( l2 2 − m t 2 + i η ) 1 , ( l1 2 − m t 2 + i η ) 1 , (( l1 + q2 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 ) 2 − m t 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { } , { } ) }
subTopos3 = Flatten [ FCLoopFindSubtopologies[ topos3]] ;
37 37 37
Now we can match a smaller topology into a larger topology
mappings3 = FCLoopFindTopologyMappings[ topos3, PreferredTopologies -> subTopos3] ;
FCLoopFindTopologyMappings: Found 1 mapping relations \text{FCLoopFindTopologyMappings: }\;\text{Found }1\text{ mapping relations } FCLoopFindTopologyMappings: Found 1 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 1 \text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1 FCLoopFindTopologyMappings: Final number of independent topologies: 1
( FCTopology ( fctopology9 , { 1 ( ( l1 + l2 + q2 ) 2 + i η ) , 1 ( l2 2 − m t 2 + i η ) , 1 ( l1 2 − m t 2 + i η ) , 1 ( ( l1 + q2 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 ) 2 − m t 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { } , { } ) { l1 → q1 − l1 , l2 → − l2 − q2 } G fctopology9 ( n1 _ , n3 _ , n4 _ , n5 _ , n2 _ ) : → G fctopology1 ( n1 , 0 , n2 , n3 , n4 , n5 ) ) \left(
\begin{array}{ccc}
\;\text{FCTopology}\left(\text{fctopology9},\left\{\frac{1}{((\text{l1}+\text{l2}+\text{q2})^2+i \eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta )},\frac{1}{(\text{l1}^2-m_t^2+i \eta )},\frac{1}{((\text{l1}+\text{q2})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right) & \{\text{l1}\to \;\text{q1}-\text{l1},\text{l2}\to -\text{l2}-\text{q2}\} & G^{\text{fctopology9}}(\text{n1$\_$},\text{n3$\_$},\text{n4$\_$},\text{n5$\_$},\text{n2$\_$}):\to G^{\text{fctopology1}}(\text{n1},0,\text{n2},\text{n3},\text{n4},\text{n5}) \\
\end{array}
\right) ( FCTopology ( fctopology9 , { (( l1 + l2 + q2 ) 2 + i η ) 1 , ( l2 2 − m t 2 + i η ) 1 , ( l1 2 − m t 2 + i η ) 1 , (( l1 + q2 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 ) 2 − m t 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { } , { } ) { l1 → q1 − l1 , l2 → − l2 − q2 } G fctopology9 ( n1_ , n3_ , n4_ , n5_ , n2_ ) :→ G fctopology1 ( n1 , 0 , n2 , n3 , n4 , n5 ) )
{ FCTopology ( fctopology1 , { 1 ( ( l1 + l2 − q1 ) 2 + i η ) , 1 ( l2 2 − m t 2 + i η ) , 1 ( l1 2 − m t 2 + i η ) , 1 ( ( l2 + q2 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 − q2 ) 2 − m t 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { } , { } ) } \left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{((\text{l1}+\text{l2}-\text{q1})^2+i \eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta )},\frac{1}{(\text{l1}^2-m_t^2+i \eta )},\frac{1}{((\text{l2}+\text{q2})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1}-\text{q2})^2-m_t^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right)\right\} { FCTopology ( fctopology1 , { (( l1 + l2 − q1 ) 2 + i η ) 1 , ( l2 2 − m t 2 + i η ) 1 , ( l1 2 − m t 2 + i η ) 1 , (( l2 + q2 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 − q2 ) 2 − m t 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { } , { } ) }
Mapping the following two topologies onto each other requires shifts in the external momenta due to the chosen kinematic constraints.
topos4 = {
FCTopology[ topo1, {
SFAD[{{ l1 + q1, 0 }, { m ^ 2 , 1 }, 1 }],
SFAD[{{ l1 - l2, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ l2 + q1, 0 }, { m ^ 2 , 1 }, 1 }],
SFAD[{{ l2 - q2, 0 }, { m ^ 2 , 1 }, 1 }],
SFAD[{{ l2, 0 }, { 0 , 1 }, 1 }]}, { l1, l2}, { q1, q2}, { SPD[ q1, q1] -> 0 , SPD[ q2, q2] -> 0 , SPD[ q1, q2] -> s / 2 }, {}],
FCTopology[ topo2, {
SFAD[{{ l1 - l2, 0 }, { m ^ 2 , 1 }, 1 }],
SFAD[{{ l1 - q2, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ l2 - q2, 0 }, { m ^ 2 , 1 }, 1 }],
SFAD[{{ l2 + q1, 0 }, { m ^ 2 , 1 }, 1 }],
SFAD[{{ l2, 0 }, { 0 , 1 }, 1 }]}, { l1, l2}, { q1, q2}, { SPD[ q1, q1] -> 0 , SPD[ q2, q2] -> 0 , SPD[ q1, q2] -> s / 2 }, {}]}
{ FCTopology ( topo1 , { 1 ( ( l1 + q1 ) 2 − m 2 + i η ) , 1 ( ( l1 − l2 ) 2 + i η ) , 1 ( ( l2 + q1 ) 2 − m 2 + i η ) , 1 ( ( l2 − q2 ) 2 − m 2 + i η ) , 1 ( l2 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → s 2 } , { } ) , FCTopology ( topo2 , { 1 ( ( l1 − l2 ) 2 − m 2 + i η ) , 1 ( ( l1 − q2 ) 2 + i η ) , 1 ( ( l2 − q2 ) 2 − m 2 + i η ) , 1 ( ( l2 + q1 ) 2 − m 2 + i η ) , 1 ( l2 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → s 2 } , { } ) } \left\{\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{((\text{l1}+\text{q1})^2-m^2+i \eta )},\frac{1}{((\text{l1}-\text{l2})^2+i \eta )},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta )},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta )},\frac{1}{(\text{l2}^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to 0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to \frac{s}{2}\right\},\{\}\right),\text{FCTopology}\left(\text{topo2},\left\{\frac{1}{((\text{l1}-\text{l2})^2-m^2+i \eta )},\frac{1}{((\text{l1}-\text{q2})^2+i \eta )},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta )},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta )},\frac{1}{(\text{l2}^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to 0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to \frac{s}{2}\right\},\{\}\right)\right\} { FCTopology ( topo1 , { (( l1 + q1 ) 2 − m 2 + i η ) 1 , (( l1 − l2 ) 2 + i η ) 1 , (( l2 + q1 ) 2 − m 2 + i η ) 1 , (( l2 − q2 ) 2 − m 2 + i η ) 1 , ( l2 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → 2 s } , { } ) , FCTopology ( topo2 , { (( l1 − l2 ) 2 − m 2 + i η ) 1 , (( l1 − q2 ) 2 + i η ) 1 , (( l2 − q2 ) 2 − m 2 + i η ) 1 , (( l2 + q1 ) 2 − m 2 + i η ) 1 , ( l2 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → 2 s } , { } ) }
mappings4 = FCLoopFindTopologyMappings[ topos4, Momentum -> All ] ;
FCLoopFindTopologyMappings: Found 1 mapping relations \text{FCLoopFindTopologyMappings: }\;\text{Found }1\text{ mapping relations } FCLoopFindTopologyMappings: Found 1 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 1 \text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1 FCLoopFindTopologyMappings: Final number of independent topologies: 1
( FCTopology ( topo2 , { 1 ( ( l1 − l2 ) 2 − m 2 + i η ) , 1 ( ( l1 − q2 ) 2 + i η ) , 1 ( ( l2 − q2 ) 2 − m 2 + i η ) , 1 ( ( l2 + q1 ) 2 − m 2 + i η ) , 1 ( l2 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → s 2 } , { } ) { l1 → − l1 + l2 − q1 , l2 → l2 , q1 → − q2 , q2 → − q1 } G topo2 ( n1 _ , n2 _ , n3 _ , n4 _ , n5 _ ) : → G topo1 ( n1 , n2 , n3 , n4 , n5 ) ) \left(
\begin{array}{ccc}
\;\text{FCTopology}\left(\text{topo2},\left\{\frac{1}{((\text{l1}-\text{l2})^2-m^2+i \eta )},\frac{1}{((\text{l1}-\text{q2})^2+i \eta )},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta )},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta )},\frac{1}{(\text{l2}^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to 0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to \frac{s}{2}\right\},\{\}\right) & \{\text{l1}\to -\text{l1}+\text{l2}-\text{q1},\text{l2}\to \;\text{l2},\text{q1}\to -\text{q2},\text{q2}\to -\text{q1}\} & G^{\text{topo2}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n4$\_$},\text{n5$\_$}):\to G^{\text{topo1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5}) \\
\end{array}
\right) ( FCTopology ( topo2 , { (( l1 − l2 ) 2 − m 2 + i η ) 1 , (( l1 − q2 ) 2 + i η ) 1 , (( l2 − q2 ) 2 − m 2 + i η ) 1 , (( l2 + q1 ) 2 − m 2 + i η ) 1 , ( l2 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → 2 s } , { } ) { l1 → − l1 + l2 − q1 , l2 → l2 , q1 → − q2 , q2 → − q1 } G topo2 ( n1_ , n2_ , n3_ , n4_ , n5_ ) :→ G topo1 ( n1 , n2 , n3 , n4 , n5 ) )
Otherwise no mappings exist
FCLoopFindTopologyMappings[ topos4][[ 1 ]]
FCLoopFindTopologyMappings: Found 0 mapping relations \text{FCLoopFindTopologyMappings: }\;\text{Found }0\text{ mapping relations } FCLoopFindTopologyMappings: Found 0 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 2 \text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }2 FCLoopFindTopologyMappings: Final number of independent topologies: 2
{ } \{\} { }
Topologies containing eikonal or other nonstandard propagators may introduce additional challenges. Even though two such topologies can be recognized to be identical, the code still would not be able to work out the correct momentum shifts without some additional input.
topoEik1 = FCTopology[ mytopo67, { SFAD[{{ k2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ k1, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ k1 + k2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ 0 , - k1 . nb}, { 0 , 1 }, 1 }],
SFAD[{{ k2, - meta u0b k2 . nb}, { 0 , 1 }, 1 }], SFAD[{{ k1 + k2, - 2 gkin meta u0b (k1 + k2) . n },
{ 0 , 1 }, 1 }], SFAD[{{ k1, - 2 gkin meta k1 . n + meta u0b k1 . nb}, { 2 gkin meta^ 2 u0b, 1 }, 1 }]},
{ k1, k2}, { n , nb}, { Hold [ SPD][ n ] -> 0 , Hold [ SPD][ nb] -> 0 , Hold [ SPD][ n , nb] -> 2 }, {}] ;
topoEik2 = FCTopology[ mytopo79, { SFAD[{{ k2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ k1, 0 }, { 0 , 1 }, 1 }],
SFAD[{{ 0 , k1 . nb}, { 0 , 1 }, 1 }], SFAD[{{ k2, - meta u0b k2 . nb}, { 0 , 1 }, 1 }],
SFAD[{{ k1 + k2, - meta u0b (k1 + k2) . nb}, { 0 , 1 }, 1 }], SFAD[{{ k1,
2 gkin meta k1 . n - meta u0b k1 . nb}, { 2 gkin meta^ 2 u0b, 1 }, 1 }],
SFAD[{{ k1 + k2, 2 gkin meta u0b (k1 + k2) . n - meta u0b (k1 + k2) . nb},
{ 2 gkin meta^ 2 u0b^ 2 , 1 }, 1 }]}, { k1, k2}, { n , nb}, { Hold [ SPD][ n ] -> 0 ,
Hold [ SPD][ nb] -> 0 , Hold [ SPD][ n , nb] -> 2 }, {}] ;
DataType[ meta, FCVariable] = True ;
DataType[ u0b, FCVariable] = True ;
At first sight these two topologies are independent from each other
FCLoopFindTopologyMappings[{ topoEik1, topoEik2}] ;
FCLoopFindTopologyMappings: Found 0 mapping relations \text{FCLoopFindTopologyMappings: }\;\text{Found }0\text{ mapping relations } FCLoopFindTopologyMappings: Found 0 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 2 \text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }2 FCLoopFindTopologyMappings: Final number of independent topologies: 2
However, if we tell the code how some eikonal propagators can be brought into a quadratic form, then an explicit mapping can be found
eikRule = { SFAD[{{ k2, - meta u0b k2 . nb}, { 0 , 1 }, 1 }] -> SFAD[ k2 - meta u0b/ 2 nb]}
{ 1 ( k2 2 − meta u0b ( k2 ⋅ nb ) + i η ) → 1 ( ( k2 − meta u0b nb 2 ) 2 + i η ) } \left\{\frac{1}{(\text{k2}^2-\text{meta} \;\text{u0b} (\text{k2}\cdot \;\text{nb})+i \eta )}\to \frac{1}{((\text{k2}-\frac{\text{meta} \;\text{u0b} \;\text{nb}}{2})^2+i \eta )}\right\} { ( k2 2 − meta u0b ( k2 ⋅ nb ) + i η ) 1 → (( k2 − 2 meta u0b nb ) 2 + i η ) 1 }
eikMappings = FCLoopFindTopologyMappings[{ topoEik1, topoEik2},
InitialSubstitutions -> eikRule] ;
FCLoopFindTopologyMappings: Found 1 mapping relations \text{FCLoopFindTopologyMappings: }\;\text{Found }1\text{ mapping relations } FCLoopFindTopologyMappings: Found 1 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 1 \text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1 FCLoopFindTopologyMappings: Final number of independent topologies: 1
eikMappings[[ 1 ]][[ 1 ]][[ 2 ;;]]
{ { k1 → − k1 , k2 → 1 2 ( meta nb u0b − 2 k2 ) } , G mytopo79 ( n5 _ , n2 _ , n4 _ , n1 _ , n3 _ , n7 _ , n6 _ ) : → G mytopo67 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 ) } \left\{\left\{\text{k1}\to -\text{k1},\text{k2}\to \frac{1}{2} (\text{meta} \;\text{nb} \;\text{u0b}-2 \;\text{k2})\right\},G^{\text{mytopo79}}(\text{n5$\_$},\text{n2$\_$},\text{n4$\_$},\text{n1$\_$},\text{n3$\_$},\text{n7$\_$},\text{n6$\_$}):\to G^{\text{mytopo67}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7})\right\} { { k1 → − k1 , k2 → 2 1 ( meta nb u0b − 2 k2 ) } , G mytopo79 ( n5_ , n2_ , n4_ , n1_ , n3_ , n7_ , n6_ ) :→ G mytopo67 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 ) }