FCTopology
FCTopology[id, {prop1, prop2, ...}, {l1, l2, ...}, {p1, p2, ...}, {kRule1, kRule2, ...}, {}]
denotes a topology with the identifier id
that is characterized by the propagators {prop1, prop2, ...}
. The propagators in the list do not necessarily have to form a valid basis, i.e. the basis may also be incomplete or overdetermined. The lists {l1, l2, ...}
and {p1, p2, ...}
stand for the loop and external momenta respectively. Furthermore, {kRule1, kRule2, …} denotes replacement rules for kinematic invariants.
The last argument (an empty list) is reserved for future improvements.
See also
Overview, FCLoopFromGLI, FCLoopValidTopologyQ, GLI.
Examples
A 2-loop topology with one external momentum Q
FCTopology[topo1, {SFAD[p1], SFAD[p2], SFAD[Q - p1 - p2], SFAD[Q - p2], SFAD[Q - p1]}, {p1, p2}, {Q}, {
Hold[SPD[Q]] -> qq}, {}]
FCTopology(topo1,{(p12+iη)1,(p22+iη)1,((−p1−p2+Q)2+iη)1,((Q−p2)2+iη)1,((Q−p1)2+iη)1},{p1,p2},{Q},{Hold[SPD(Q)]→qq},{})
A 3-loop topology with one external momentum Q
topo = FCTopology[topo2, {SFAD[p1], SFAD[p2], SFAD[p3], SFAD[Q - p1 - p2 - p3], SFAD[Q - p1 - p2],
SFAD[Q - p1], SFAD[Q - p2], SFAD[p1 + p3], SFAD[p2 + p3]}, {p1, p2, p3}, {Q}, {}, {}]
FCTopology(topo2,{(p12+iη)1,(p22+iη)1,(p32+iη)1,((−p1−p2−p3+Q)2+iη)1,((−p1−p2+Q)2+iη)1,((Q−p1)2+iη)1,((Q−p2)2+iη)1,((p1+p3)2+iη)1,((p2+p3)2+iη)1},{p1,p2,p3},{Q},{},{})
Use FCLoopValidTopologyQ
to check if the syntax of the given topology is correct.
FCLoopValidTopologyQ[topo]
True
The list of propagators in the topology essentially defines the propagator representation of a GLI
. Notice that propagators are allowed to have symbolical or numerical prefactors, as long as those do not depend on loop or external momenta
topo2 = FCTopology[topoTest, {a SFAD[p1], b SFAD[p2], c SFAD[Q - p1 - p2], d SFAD[Q - p2],
e SFAD[Q - p1]}, {p1, p2}, {Q}, {}, {}]
FCTopology(topoTest,{(p12+iη)a,(p22+iη)b,((−p1−p2+Q)2+iη)c,((Q−p2)2+iη)d,((Q−p1)2+iη)e},{p1,p2},{Q},{},{})
FCLoopFromGLI[GLI[topoTest, {1, 1, 1, 1, 1}], topo2]
(p12+iη)(p22+iη)((Q−p1)2+iη)((Q−p2)2+iη)((−p1−p2+Q)2+iη)abcde