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D.15.25.12 FeynmanIntegralA
Procedure from library tropicalEllipticCovers.lib (see tropicalEllipticCovers_lib).
- Usage:
- FeynmanIntegralA(G,O,d,l,k,t[,gg]); G graph, O list, d int, l list, k int, t int, gg list
- Assume:
- G is a graph (a Feynman graph or a pearl chain) of the degree d, O is an ordering
of vertices of G, d is the degree of G, l is a list representing the leaky edges
of G, k is any integer, gg is a list representing the genus function and t is
one of 0,1,2 or 3.
- Return:
- number or list Q_t (depending on k) of Feynman integral for a fixed ordering of
vertices of G, the results are as follows:
Q_0: sum of Feynman integrals (over all partitions of d) for a fixed ordering of a
Feynman graph G as in [BBM], i.e. a graph without any self-looping edges, leaks
or vertex contributions.
Q_1: sum of Feynman integrals (over all partitions of d) for a fixed ordering of a
Feynman graph G without vertex contributions as in [BGM1], i.e. A graph that may
have self-looping edges and leaks.
Q_2: sum of Feynman integrals (over all partitions of d) for a fixed ordering of a
Feynman graph G with vertex contributions as in [BGM1] possibly with self-looping
edges and leaks.
Q_3: sum of Feynman integrals (over all partitions of d) for a fixed ordering of a
pearl chain G as in [BGM2], i.e. graph G may have leaks.
- Theory:
- If k is zero it returns the coefficient which is a sum of Feynman integrals over
all partitions of d. Otherwise, returns a list showing the partition from which
the Feynman integral is computed, the respective coefficient of the Feynman
integral for the given ordering and the sum over all partitions.
Example:
| LIB "tropicalEllipticCovers.lib";
ring r1=0, (x1,x2,x3,x4),dp;
graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4)));
FeynmanIntegralA(G,list(x1,x3,x4,x2),4,list(0,0,0,0,0,0),0,0);
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