Regularisation of Feynman integrals on complexified configuration spaces
Typ
Examensarbete för masterexamen
Program
Publicerad
2021
Författare
Svensson, Ludvig
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
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.
Beskrivning
Ämne/nyckelord
regularisation , divergent integrals , configuration space , blowups , currents , meromorphic continuation