Mathematical aspects of Feynman integrals [Elektronische Ressource] / Christian Bogner

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• Physik

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• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2009
• Nombre de lectures: 17
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Mathematical Aspects of Feynman Integrals
Dissertation
zur Erlangung des Grades
“Doktor der Naturwissenschaften”
am Fachbereich Physik
¨der Johannes Gutenberg-Universitat
in Mainz
Christian Bogner
geboren in Mainz
Mainz, August 200923
“What can be said at all can be said clearly; and whereof one cannot speak
thereof one must be silent.”
(Ludwig Wittgenstein)Contents
Chapter 1. Introduction 7
1.1. General Introduction 7
1.2. Outline 8
Chapter 2. Preliminaries on Feynman Integrals 11
2.1. Theories, Graphs and Rules 11
2.2. Dimensional Regularization 20
2.3. Feynman Parameters 24
2.4. From Tensors to Scalars 28
Chapter 3. Integration-by-Parts for Feynman Integrals 31
3.1. The IBP-Approach 32
3.2. Translation to Feynman Parameters 41
Chapter 4. Feynman Graph Polynomials 49
4.1. Graphs, Feynman Graphs and Associated Matrices 50
4.2. Symanzik Polynomials and Theorems of the Matrix-Tree-
Type 64
4.3. The Multivariate Tutte Polynomial 73
Chapter 5. Sector Decomposition and Resolution of Singularities 79
5.1. Sector Decomposition 80
5.2. Hironaka’s Polyhedra Game 96
5.3. Resolution of Singularities 107
Chapter 6. Periods 117
6.1. Periods, Nested Sums and Iterated Integrals 118
6.2. Periods and Feynman Integrals 130
6.3. Outlook on Periods and Feynman Integrals 135
Chapter 7. Conclusions 139
Appendix A: Auxiliary Definitions 141
Appendix B: The Two-Loop Equal Mass Sunrise Graph 143
Appendix C: Periods and Prefactors of Feynman Integrals 147
Bibliography 149
5CHAPTER 1
Introduction
1.1. General Introduction
The idea that matter consists of smallest elements belongs to human culture since
the ancient Greeks. The physics of the last few decades has given deep insight into the
propertiesofthesesupposedlyelementary particles. This illumination ofthemicrocosmos
is a story of success for two scientific concepts: the scattering experiment and quantum
field theory.
Scattering experiments, from Rutherford’s scattering of α-particles off gold atoms in
1909 to the collision of highly accelerated protons at the Large Hadron Collider, expected
to begin in 2009, have provided crucial information on the building blocks of matter. The
success of a physical theory of elementary particles has to be judged by its capability to
predict the values of quantities which can be measured in scattering experiments. The
typical quantity to be measured in this kind of experiments is the so-called cross-section,
from which in turn characteristic properties of the particles involved and their interaction
can be deduced.
The evolution of physical theories, probed on these experimental data, led to a model
for the fundamental strong, electromagnetic and weak interactions of the known elemen-
tary particles. This so-called Standard Model is in good agreement with a wealth of
observed phenomena. Nevertheless, the model is expected to be extended, depending on
1eagerly expected results from the Large Hadron Collider in the near future .
The Standard Model is a highly successful quantum field theory whose application
to the precise evaluation of an observable is usually far from trivial. In order to apply
the model to the quantitative prediction of a cross-section one requires a perturbative
formulation. Thebasicidea ofperturbation theoryisthe assumption, thattheinteraction
energies between the particles are relatively small compared to the energy of their free
motion. The strength of an interaction is scaled by a so-called coupling parameter, which
is assumed to be a relatively small quantity. An observable is then evaluated as a power
series in the coupling parameter. This infinite series is truncated at a certain order and
the precision of the result depends on the number of orders to be taken into account.
The evaluation of the coefficients of this power series is in general highly elaborate.
With increasing order of the perturbative expansion it becomes more and more difficult
and it is therefore for many cases possible only to consider the first one or two orders. At
higher orders the computational techniques of present days, involving the use of advanced
computer programs, always have reached the limits of computer capabilities very soon.
However, because of the increasing precision of the experiments and the nature of the
physical effects to be observed, the consideration of higher orders becomes relevant.
The topics of this dissertation are related to calculational problems which hinder
higher-order calculations. The complications to be considered here do not depend on
the underlying physical model. They hamper the evaluations in the Standard Model but
1By the time of writing, the first collisions at the Large Hadron Collider are expected for the fall of
2009. A possible discovery of the Higgs boson is expected to take place not before 2010.
78 1. INTRODUCTION
they are also present in Quantum Electrodynamics and even in simple toy models like
4the so-called φ -theory.
The difficulties to be treated lie in the nature of Feynman integrals and therefore in
the nature of the common perturbative formulation of quantum field theories in general.
A good understanding of this particular kind of integrals is inevitable for the efficient
calculation of observables. An additional motivation to study the properties of these
integrals arose over the last ten years from pure mathematics, as it turned out that
Feynman integrals tend to evaluate to certain numbers which are significant in algebraic
geometry. Theachievements ofthisdissertationarerelatedtobothaspectsandmotivated
from both points of view.
For this reason we want this dissertation to be readable for the physicist and the
mathematician. Feynman integrals will be considered as objects of mathematical interest
in their own right, without referring to a particular physical model or even a physical
process. Thanks to Richard Feynman one does not need to know quantum field theory in
ordertoobtaintheseintegrals. Theycanbeconstructedfromcertainintuitivegraphs, the
Feynman graphs, using a set of so-called Feynman rules which incorporate the properties
of the chosen physical model. One does not need to be a physicist to study Feynman
integrals. A physicist just has the strongest reason to do so.
1.2. Outline
In the following we apply (algebraic) graph theory, algebraic geometry and number
theory to Feynman integrals. Not all aspects to be discussed in the following are closely
related to each other. However, Ariadne’s thread through the dissertation may be seen
in the course of the calculation of a Feynman integral:
The above general introduction and the present outline form chapter 1.
Chapter 2: We begin with a brief introduction of Feynman integrals and Feynman
rules. The framework of dimensional regularization is explained and some further stan-
dard techniques are described, such that the problem ofcalculating anarbitrary Feynman
integral is reduced to the evaluation of regularized, scalar Feynman integrals in the con-
venient Feynman parametric representation.
Chapter 3: Not all Feynman integrals really need to be calculated separately. Most
of them can be expressed in terms of simpler integrals, so-called master integrals, by the
so-called IBP-identities (integration-by-parts identities). We briefly review this technique
and give a handy formulation of some of these identities in the Feynman parametric
representation.
Chapter 4: For many cases the Feynman parametric representation is a convenient
starting point for the actual evaluation of a master integral. In this representation the
integrandisexpressedintermsoftwocertainpolynomials,theSymanzikpolynomials. We
dedicatechapter4toagraphtheoreticstudyofthesegraphpolynomialsandderiveanovel
relation for them. This is achieved by the use of a generalized theorem of the so-called
matrix-tree-type. We furthermore explain the correspondence between the Symanzik
polynomials and the multivariate Tutte polynomial.
Chapter 5: A method for the numerical evaluation of an arbitrary Feynman integral,
possibly a master integral, in a certain momentum region was given by the sector de-
composition algorithm of Binoth and Heinrich, which is briefly reviewed. We expose the
problem that thisalgorithm does not terminate in the general case and solve thisproblem
by mapping the combinatorics of the algorithm to the abstract polyhedra game of Hi-
ronaka. We extend the algorithm by use of so-called winning strategies of this game and
obtain a version which always terminates. We briefly report on an implementation of this1.2. OUTLINE 9
improved algorithm. Furthermore we add an explanation in terms of algebraic geometry
on the relation between sector decomposition and the so-called resolution of singularities
by a sequence of blow-ups, formalized by Hironaka’s game. The work presented in this
chapter was previously published in joint work with Stefan Weinzierl [BW08].
Chapter 6: Having finally obtained a result for a Feynman integral, one may be
interested in the mathematical nature of the result. Chapter 6 is dedicated to the classifi-
cation ofnumbers and functions one obtains in the Laurent coefficients ofa dimensionally
regularized Feynman integral. We briefly discuss zeta values, multiple zeta values, poly-
logarithms and elliptic integrals and their presence in loop calculations. Then we prove a
sta