The massless two-loop two-point function and zeta functions in counterterms of Feynman diagrams [Elektronische Ressource] / Isabella Bierenbaum

Sujets

• Physik

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2005
• Nombre de lectures: 166
• Langue: English

Extrait

The Massless Two-loop Two-point Function
and Zeta Functions
in Counterterms of Feynman Diagrams
Dissertation
zur Erlangung des Grades
\Doktor der Naturwissenschaften"
am Fachbereich Physik
der Johannes-Gutenberg-Universit at
in Mainz
Isabella Bierenbaum
geboren in Landau/Pfalz
Mainz, Februar 2005Datum der mundlic hen Prufung: 6. Juni 2005
D77 (Diss. Universit at Mainz)Contents
1 Introduction 1
2 Renormalization 3
2.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Feynman diagrams and power counting . . . . . . . . . . . . . . . . . . . . . 5
2.3 Dimensional regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Counterterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.1 One-loop diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.2 Multi-loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Hopf and Lie algebra 15
3.1 The Hopf algebras H and H . . . . . . . . . . . . . . . . . . . . . . . . . 15R FG
3.1.1 The Hopf algebra of rooted trees H . . . . . . . . . . . . . . . . . . . 16R
3.1.2 The Hopf of Feynman graphs H . . . . . . . . . . . . . . . 20FG
3.1.3 Correspondence between H , H , and ZFF . . . . . . . . . . . . . . 20R FG
3.2 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 The massless two-loop two-point function | an overview 27
4.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Integer exponents | the triangle relation . . . . . . . . . . . . . . . . . . . 29i
4.3 Non-integer exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32i
5 Nested sums 37
5.1 Nested sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.1.1 From polylogarithms to multiple polylogarithms . . . . . . . . . . . . 38
5.1.2 Algebraic relations: shu e and quasi-shu e . . . . . . . . . . . . . . . 41
5.2 nestedsums | the theory and some of its functions . . . . . . . . . . . . . . . 45
5.2.1 nestedsums | the general idea . . . . . . . . . . . . . . . . . . . . . . 46
5.2.2 neste | the four classes of functions . . . . . . . . . . . . . . . . 48
6 The massless two-loop two-point function 53
(2;5)^6.1 The expansion of I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
(2;5)^6.1.1 Decomposition of I . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
(1;3)6.1.2 Calculation of the Mellin-Barnes integrals for I . . . . . . . . . . . 55
6.1.3 Collecting residues | general remarks . . . . . . . . . . . . . . . . . . 60
6.1.4 | the di eren t cases . . . . . . . . . . . . . . . . . 64
(2;5)^6.2 The expansion of I | the program . . . . . . . . . . . . . . . . . . . . . . 71
iiiiv CONTENTS
(2;5)^6.3 The expansion of I | the results . . . . . . . . . . . . . . . . . . . . . . . 73
7 Building massless Feynman diagrams 77
7.1 One-loop diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.1.1 Yukawa theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.1.2 QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 The non-planar vertex correction . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.2.1 The theory behind the programs . . . . . . . . . . . . . . . . . . . . . 86
7.2.2 The programs for calculating the non-planar vertex correction . . . . . 91
8 Counterterms of massless Feynman diagrams 95
8.1 The programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.2 How to use the programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.3 Counterterms and zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . 98
9 Summary and outlook 107
A Algebras 109
A.1 Monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.2 Free algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.3 Hopf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.4 Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.5 Universal enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
B The gamma function and related functions 117
B.1 The gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
B.2 Mellin-Barnes integrals and gamma functions . . . . . . . . . . . . . . . . . . 120
C Integrals 123
C.1 Scalar integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.2 Vector integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
C.3 Tensor integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
(2;5)^D Relations for I 127
E The classes 131Chapter 1
Introduction
The necessity to improve theoretical predictions for scattering processes includes the demand
for extending calculations in perturbative quantum eld theories (pQFTs) to higher loop
orders in the perturbation series. Using dimensional regularization, this implies the need
to expand the analytic expressions of various Feynman diagrams to higher orders in the
regularization parameter ", with D = 4 2", as we will explain in the following chapters.
Much e ort is invested into these higher-order calculations, including attempts to nd new
ways for addressing this problem. The main part of this thesis provides the description
of a new method for expanding the massless two-loop two-point function. Unlike earlier
calculations, this method will enable us to expand the integral in principle up to an arbitrary
order in ".
The massless two-loop two-point function is interesting in di eren t respects. On the one hand
+it is needed for instance in calculations for the process e e ! hadrons in orders of [GKLs
1991,SuSa 1991]. On the other hand there is a number theoretical question associated with
this integral: It was shown by di eren t authors that the low-order " expansion of this integral
involves rational numbers and multiple zeta values. A discussion of this issue can be found
in Chapter 4. However, it was not clear whether multiple zeta values are su cien t for the
expansion of this two-loop function to all orders in " [Broa 2003]. We will solve this problem
and give an answer to this question in Chapter 6.
These number theoretical considerations are related to investigations into mathematical struc-
tures underlying perturbative quantum eld theories. Although the results of calculations
using Feynman diagrams in pQFTs are in good correspondence with experimental results
there are a lot of mathematical problems concerning the perturbation series itself that are
not yet solved. The high predictive power nevertheless suggests that the mathematical prob-
lems might be due to a lack of understanding of the series and that we did not nd the correct
mathematical formulations so far. A step towards a better understanding of the mathematical
structures was done by Dirk Kreimer when he found the Hopf algebra of renormalization in
renormalizable QFTs [Krei 1998b,CoKr 1998]. Since then, the search for further mathema-
tical structures in Feynman diagrams brought up connections to several mathematical elds.
The hope is that improving the knowledge about the structures underlying the perturbation
series, one will also be able to understand the results of calculations better. We will make use
of such mathematical structures in the following chapters and emphasize their appearance in
our calculations.
12 1. Introduction
This thesis is organized as follows: Chapters 2 and 3 provide the basic de nitions necessary for
subsequent chapters. We start in Chapter 2 with a short introduction to perturbative quantum
eld theories, only mentioning the very basics. Chapter 3 then de nes the Hopf algebra
and Lie algebra of rooted trees and Feynman diagrams, emphasizing on the one hand the
relation between the antipode of the Hopf algebra and the counterterms of a given Feynman
diagram and on the other hand the insertion operation of Feynman graphs that allows us to
build Feynman diagrams out of certain building blocks [Krei 1998b,CoKr 1998,CoKr 2000].
However, both chapters 2 and 3 are not intended to give a full review of renormalization of
perturbative quantum eld theories and its Hopf algebra structure, nor will they fully explain
the Hopf algebra and Lie algebra occurring in this context in all of their aspects. We will
neglect everything that would distract us from the main path, which necessarily implies that
we will omit many aspects that would be interesting in their own right.
Chapters 4 to 6 are dedicated to the calculation of the massless two-loop two-point function,
starting in Chapter 4 with a short introduction to some earlier works on the expansion of this
function. In 5 we will then take a closer look at the functions that typically occur
in the calculation of Feynman diagrams, and provide an overview of polylogarithms, multiple
zeta values, multiple polylogarithms, and related functions. Investigations into these functions
and their occurrence in analytic expressions of Feynman diagrams were done in [MUW 2002],
where the authors describe a way to expand sums and double sums of fractions of gamma
functions in an expansion para