Singular Manual: FeynmanIntegralo

| #### D.15.25.10 FeynmanIntegralo Procedure from library `tropicalEllipticCovers.lib` (see tropicalEllipticCovers_lib). Usage: FeynmanIntegralo(G,aa,O,l,k,t[,gg]); G graph, aa list, O 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, O is an ordering of the vertices 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: Feynman integral 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: Feynman integral 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: Feynman integral for a fixed ordering of a Feynman graph G with vertex contributions as in [BGM1] possibly with self-looping edges and leaks. Q_3: Feynman integral 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 of the Feynman integral for a given ordering of the vertices of the graph G. Otherwise, returns a list showing the ordering and the coefficient of the Feynman integral for the corresponding ordering. 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)));
    FeynmanIntegralo(G,list (0,2,1,0,0,1), list (x1,x3,x4,x2),list(0,0,0,0,0,0),1,0);
    //This is another example:
    ring r2=0,(x1,x2,x3,z1,z2,z3),dp;
    graph GG = makeGraph(list(1,2,3),list(list(1,1),list(1,2),list(2,3),list(3,1)));
    FeynmanIntegralo(GG, list (2,0,0,1), list (x1,x2,x3),list (0,0,0), 0,1);
    //Yet another example
    ring r3=0,(x1,x2,x3,x4,x5),dp;
    graph P=makeGraph(list(1,2,3,4),list(list(1,2),list(2,3),list(3,4),list(4,1)));
    //graph P is a pearl chain
    FeynmanIntegralo(P,list (1,0,0,1),list (x1,x2,x3,x4), list (0,0,0,0), 0,3);

User manual for Singular version 4.4.0, 2024, generated by texi2html.