D.15.25.11 FeynmanIntegralO

Usage:

FeynmanIntegralO(G,aa,l,k,t[,gg]); G graph, aa list, l list, k int, t int, gg list

Assume:

G is a graph (a Feynman graph or a pearl chain) of the degree d, aa is a partition of degree d, 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 all possible orderings of vertices of G, the results are as follows: Q_0: Feynman integral for all orderings of a Feynman graph G as in [BBM], i.e., a graph without any self-looping edges, leaks or vertex contributions. Q_1: Feynman integral for all orderings 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: Feynman integral for all orderings of a Feynman graph G with vertex contributions as in [BGM1] possibly with self-looping edges and leaks. Q_3: Feynman integral for all orderings of a pearl chain G as in [BGM2], i.e., a graph G may have leaks.

Theory:

If k is zero it returns the coefficient which is a sum of Feynman integrals over all orderings of the vertices of the graph G. Otherwise, returns a list showing the ordering, the coefficient of the Feynman integral for the corresponding ordering and the sum.

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)));
FeynmanIntegralO(G,list (0,2,1,0,0,1),list (0,0,0,0,0,0),1,0);