Example : Fully massless nonplanar double pentagon

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To provide an example on how to use our package, we calculate the Baikov matrix of the fully massless nonplanar double pentagon.

We define the graph G from the list of vertices and list of edges. The direction of momenta are taken from the direction of edges. All external momenta are taken to be outgoing.

julia> G=simple_graph([1,2,3,4,5,6,7],[(6,1),(6,4),(1,2),(3,7),(4,3),(2,7),(5,6),(7,5),1,2,3,4,5]);

We then assign polynomial variables $q[i]$, at bounded edges and function field variables $p[i]$ at the unbounded edges over a prime filed of characteristic 0.


julia> G=labelGraph(G,0);

Then we use balancing condition of the graph (realtions of momenta which are obtained by applying momentum conservation law at each vertex of the graph and to the whole graph) to rewrite each dependent momenta in terms of the eliments in the ordered set $V$ of external momenta and loop momenta. Here we use invlex ordering on $p[1],...,p[E],q[1],...,q[L]$ to choose independent external momenta and independent loop momenta. G.elimVars will store the eliminated variables.

julia> G=eliminateVariables(G);
julia> printLabeledGraph(G);

julia> G.elimvar 7-element Vector{QQMPolyRingElem}: p[5] q[3] q[4] q[5] q[6] q[7] q[8]

Then the irreducible scalar products associated to $G$ can be printed as follows:

julia>ISP(G)

Ideal generated by p[3]q[1] p[4]q[1] p[1]*q[2]


We should remove the eliminated variables from $G$, in order to compute the Baikov matrix of $G$.

julia julia> G=removeElimVars(G); QQMPolyRingElem[p[1], p[2], p[3], p[4], p[5], q[1], q[2]]



We then calculate the Baikov matrix associated to Feynman integral of $G$. It will also print the assignment of Baikov variables $z[i]$ to each inverse propagators and irreducible scalar products of $G$.

julia julia> G=computeBaikovMatrix(G);

labels used for Gram matrix of external loop momenta:
["p[1]*p[2] => 1//2*t[1]"]
["p[1]*p[3] => 1//2*t[2]"]
["p[2]*p[3] => 1//2*t[3]"]
["p[1]*p[4] => 1//2*t[4]"]
["p[2]*p[4] => 1//2*t[5]"]
["p[3]*p[4] => -1//2*t[1] - 1//2*t[2] - 1//2*t[3] - 1//2*t[4] - 1//2*t[5]"]
Assignment of Baikov variables (Z_i) are:
["z[1] => p[3]*q[1]"]
["z[2] => p[4]*q[1]"]
["z[3] => q[1]^2"]
["z[4] => -2*p[1]*q[1] + q[1]^2"]
["z[5] => 2*p[1]*p[2] - 2*p[1]*q[1] - 2*p[2]*q[1] + q[1]^2"]
["z[6] => p[1]*q[2]"]
["z[7] => q[2]^2"]
["z[8] => -2*p[1]*p[2] - 2*p[1]*p[3] - 2*p[1]*p[4] - 2*p[2]*p[3] - 2*p[2]*p[4] - 2*p[3]*q[2] - 2*p[4]*q[2] + q[2]^2"]
["z[9] => -2*p[4]*q[2] + q[2]^2"]
["z[10] => q[1]^2 + 2*q[1]*q[2] + q[2]^2"]
["z[11] => -2*p[1]*q[1] - 2*p[1]*q[2] - 2*p[2]*q[1] - 2*p[2]*q[2] - 2*p[3]*q[1] - 2*p[3]*q[2] - 2*p[4]*q[1] - 2*p[4]*q[2] + q[1]^2 + 2*q[1]*q[2] + q[2]^2"]

julia> G.baikovmatrix
6×6 Matrix{RingElem}:
 0                      …  z[6]
 1//2*t[1]                 1//2*t[2] + 1//2*t[3] + 1//2*t[4] + 1//2*t[5] - z[1] - z[2] - 1//2*z[3] + 1//2*z[5] - z[6] - 1//2*z[7] + 1//2*z[8] + 1//2*z[10] - 1//2*z[11]
 1//2*t[2]                 -1//2*t[1] - 1//2*t[2] - 1//2*t[3] - 1//2*t[4] - 1//2*t[5] - 1//2*z[8] + 1//2*z[9]
 1//2*t[3]                 1//2*z[7] - 1//2*z[9]
 1//2*z[3] - 1//2*z[4]     -1//2*z[3] - 1//2*z[7] + 1//2*z[10]
 z[6]                   …  z[7]