Graph Theory and Feynman Integrals

Question

In Vladimir A. Smirnov's book Analytic Tools for Feynman Integrals, Section 2.3, the alpha representation of general Feynman integral takes the form

$$ F_{\Gamma}(q_1,\ldots,q_n;d) = \frac{i^{-a-h}\pi^{2h}}{\prod_l\Gamma(a_l)} \int_0^{\infty}\mathrm{d}\alpha_1 \ldots \int_0^{\infty}\mathrm{d}\alpha_L \prod_l\alpha_l^{a_l-1} \mathcal{U}^{-2} Z e^{i\mathcal{V}/\mathcal{U} - i\sum m_l^2\alpha_l} $$ where $\mathcal{U}$ and $\mathcal{V}$ are defined as sums running over trees and 2-trees of the given Feynman graph. I know that $\mathcal{U}$ is equivalent to $\det{A}$ in the $4h$-dimensional Gauss integrals, but I can't figure out how it can be expressed in the language of graph theory. Could anyone provide some help? References on the topic of graph theory and Feynman integrals are also desired. Thanks a lot!

This post imported from StackExchange Physics at 2014-09-03 18:16 (UCT), posted by SE-user soliton

Comments

It seems that there is a weighted graph notion which is the (edge-)weight product \(w(graph)\). So it seems that \(\mathcal U\) (see formula 14 in the Scholarpedia article: http://www.scholarpedia.org/article/Multiloop_Feynman_integrals )could be written as

 \(\mathcal U =\)\(\sum\limits_{tree\, subgraphs}\frac{w(graph)}{w(tree\, subgraphs)}\)