Regularisation of Feynman integrals on complexified configuration spaces

Abstract

We present a regularisation procedure for divergent configuration space Feynman integrals coming from a complexified euclidean scalar quantum field theory on a complex manifold X. The inspiration for the thesis as well as the setting is provided by recent works of Ceyhan and Marcolli that proposes a construction of the configuration space and a complex generalisation of a Feynman amplitude dictated by a Feynman graph 􀀀, reminiscent of a set of Feynman rules in physics. Furthermore, Ceyhan and Marcolli describe a compactification of the configuration space of a given graph as an iterated sequence of blowups along certain diagonals in a product space where the amplitude associated to the graph in general has non-integrable singularities. We identify the possibility of the amplitude also having singularities at infinity and propose a construction, complementary to that of Ceyhan and Marcolli, with the desired result that the singular locus of the pullback of the amplitude constitutes a normal crossings divisor. This property allows for the application of techniques from the theory of currents in complex analysis. We consider a regularisation of the divergent integral, which has a Laurent series expansion in the regularisation parameter with current coefficients. We define the degree of divergence as the leading order of the expansion. We go through the regularisation procedure for three explicit Feynman graphs, with X = CPD. We give upper bounds for their respective degrees of divergence and for one of the graphs, in the special case D = 2, we show that the leading order term vanishes.