scholarly journals Calculation of master integrals in terms of elliptic multiple polylogarithms

2020 ◽  
Vol 35 (13) ◽  
pp. 2050063
Author(s):  
Maxim A. Bezuglov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Twentieth Century ◽  
Feynman Diagrams ◽  
Heavy Quarkonium ◽  
Quantum Field ◽  
Loop Level ◽  
Multiple Polylogarithms ◽  
Class Of Functions

In modern quantum field theory, one of the most important tasks is the calculation of loop integrals. Loop integrals appear when evaluating the Feynman diagrams with one or more loops by integrating over the internal momenta. Even though this problem has already been in place since the mid-twentieth century, we not only do not understand how to calculate all classes of these integrals beyond one loop, we do not even know in what class of functions the answer is expressed. To partially solve this problem, different variations of new functions usually called elliptic multiple polylogarithms have been introduced in the last decade. In this paper, we explore the possibilities and limitations of this class of functions. As a practical example, we chose the processes associated with the physics of heavy quarkonium at the two-loop level.

scholarly journals ON BORWEIN’S CONJECTURES FOR PLANAR UNIFORM RANDOM WALKS

2019 ◽  
Vol 107 (3) ◽  
pp. 392-411 ◽  
Author(s):  
YAJUN ZHOU
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Random Walks ◽  
Euclidean Plane ◽  
Feynman Diagrams ◽  
Quantum Field ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Uniformly Convergent

Let $p_{n}(x)=\int _{0}^{\infty }J_{0}(xt)[J_{0}(t)]^{n}xt\,dt$ be Kluyver’s probability density for $n$-step uniform random walks in the Euclidean plane. Through connection to a similar problem in two-dimensional quantum field theory, we evaluate the third-order derivative $p_{5}^{\prime \prime \prime }(0^{+})$ in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that $p_{n}(x),0\leq x\leq 1$ admits a uniformly convergent Maclaurin expansion for all odd integers $n\geq 5$, thus settling another conjecture of Borwein.

Quantum Field Theory

2020 ◽  
pp. 237-288
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Theory ◽  
Canonical Quantization ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Coordinate Space ◽  
Matrix Formalism ◽  
Quantum Field ◽  
Transfer Matrix Formalism

Chapter 7 covers the main reasons for adopting the methods of quantum field theory (QFT) to study the critical phenomena. It presents both the canonical quantization and the path integral formulation of the field theories as well as the analysis of the perturbation theory. The chapter also covers transfer matrix formalism and the Euclidean aspects of QFT, the field theory of the Ising model, Feynman diagrams, correlation functions in coordinate space, the Minkowski space and the Legendre transformation and vertex functions. Everything in this chapter will be needed sooner or later, since it highlights most of the relevant aspects of quantum field theory.

FIRST NON-VANISHING SELF-LINKING OF KNOTS (I) COMBINATORIC AND DIAGRAMMATIC STUDY

2011 ◽  
Vol 20 (12) ◽  
pp. 1637-1648 ◽  
Author(s):  
CHUN-CHUNG HSIEH
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Knot Theory ◽  
Feynman Diagrams ◽  
Quantum Field ◽  
Borromean Rings ◽  
Diagrammatic Representation ◽  
Perturbative Quantum ◽  
Perturbative Quantum Field Theory ◽  
Chern Simons

In this paper, following the scheme of [Borromean rings and linkings, J. Geom. Phys.60 (2010) 823–831; Combinatoric and diagrammatic study in knot theory, J. Knot Theory Ramifications16 (2007) 1235–1253; Massey–Milnor linking = Chern–Simons–Witten graphs, J. Knot Theory Ramifications17 (2008) 877–903], we study the first non-vanishing self-linkings of knots, aiming at the study of combinatorial formulae and diagrammatic representation. The upshot of perturbative quantum field theory is to compute the Feynman diagrams explicitly, though it is impossible in general. Along this line in this paper we could not only compute some Feynman diagrams, but also give the explicit and combinatorial formulae.

scholarly journals BOGOLUBOV'S RECURSION AND INTEGRABILITY OF EFFECTIVE ACTIONS

2001 ◽  
Vol 16 (09) ◽  
pp. 1531-1558 ◽  
Author(s):  
A. GERASIMOV ◽  
A. MOROZOV ◽  
K. SELIVANOV
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hopf Algebra ◽  
Moduli Space ◽  
Feynman Diagrams ◽  
Coupling Constants ◽  
Quantum Field ◽  
Integrable Structure ◽  
Standard Formalism

The Hopf algebra of Feynman diagrams, analyzed by A. Connes and D. Kreimer, is considered from the perspective of the theory of effective actions and generalized τ-functions, which describes the action of diffeomorphism and shift groups in the moduli space of coupling constants. These considerations provide additional evidence of the hidden group (integrable) structure behind the standard formalism of quantum field theory.

RENORMALIZATION IN p-ADIC QUANTUM FIELD THEORY

1991 ◽  
Vol 06 (15) ◽  
pp. 1421-1427 ◽  
Author(s):  
V.A. SMIRNOV
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Space Time ◽  
Feynman Diagrams ◽  
Quantum Field ◽  
Feynman Integrals ◽  
New Type ◽  
Euclidean Quantum Field Theory ◽  
Low Order

A version of p-adic perturbative Euclidean quantum field theory is presented. It is based on a new type of the propagator which happens to be rather natural for p-adic space-time. Low-order Feynman diagrams are explicitly calculated and typical renormalization schemes are introduced: analytic, dimensional, and BPHZ renormalizations. The calculations show that in p-adic Feynman integrals only logarithmical divergences appear.

scholarly journals Gauge properties of hadronic structure of nucleon in neutron radiative beta decay to order O(α/π) in standard V − A effective theory with QED and linear sigma model of strong low-energy interactions

2018 ◽  
Vol 33 (33) ◽  
pp. 1850199 ◽  
Author(s):  
A. N. Ivanov ◽  
R. Höllwieser ◽  
N. I. Troitskaya ◽  
M. Wellenzohn ◽  
Ya. A. Berdnikov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Weak Interactions ◽  
Feynman Diagrams ◽  
Low Energy ◽  
Effective Theory ◽  
Quantum Field ◽  
Gauge Invariant ◽  
Hadronic Structure ◽  
Loop Approximation

Within the standard [Formula: see text] theory of weak interactions, Quantum electrodynamics (QED) and the linear [Formula: see text]-model (L[Formula: see text]M) of strong low-energy hadronic interactions, we analyze gauge properties of hadronic structure of the neutron and proton in the neutron radiative [Formula: see text]-decay. We show that the Feynman diagrams, describing contributions of hadronic structure to the amplitude of the neutron radiative [Formula: see text]-decay in the tree-approximation for strong low-energy interactions in the L[Formula: see text]M, are gauge invariant. In turn, the complete set of Feynman diagrams, describing the contributions of hadron–photon interactions in the one-hadron-loop approximation, is not gauge invariant. In the infinite limit of the scalar [Formula: see text]-meson, reproducing the current algebra results (S. Weinberg, Phys. Rev. Lett. 18, 188 (1967)), and to leading order in the large nucleon mass expansion the Feynman diagrams, violating gauge invariance, do not contribute to the amplitude of the neutron radiative [Formula: see text]-decay in agreement with Sirlin’s analysis of strong low-energy interactions in neutron [Formula: see text] decays. We assert that the problem of appearance of gauge noninvariant Feynman diagrams of hadronic structure of the neutron and proton is related to the following. The vertex of the effective [Formula: see text] weak interactions does not belong to the combined quantum field theory including the L[Formula: see text]M and QED. We argue that gauge invariant set of Feynman diagrams of hadrons, coupled to real and virtual photons in neutron [Formula: see text] decays, can be obtained within the combined quantum field theory including the Standard Electroweak Model (SEM) and the L[Formula: see text]M, where the effective [Formula: see text] vertex of weak interactions is a result of the [Formula: see text]-electroweak boson exchange.

scholarly journals One-loop calculations in quantum field theory: From Feynman diagrams to unitarity cuts

2012 ◽  
Vol 518 (4-5) ◽  
pp. 141-250 ◽  
Author(s):  
R. Keith Ellis ◽  
Zoltan Kunszt ◽  
Kirill Melnikov ◽  
Giulia Zanderighi
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Feynman Diagrams ◽  
Quantum Field

Renormalization

2021 ◽  
pp. 171-222
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Dimensional Regularization ◽  
General Concept ◽  
Feynman Diagrams ◽  
Quantum Field ◽  
Subtraction Procedure ◽  
Renormalization Group Equations

This chapter describes in detail the general concept of renormalization. It starts with a discussion of the regularization of Feynman diagrams. After that, the subtraction procedure is explained in detail, followed by an introduction to the notion of a superficial degree of divergence of the diagram. On this basis, the models of quantum field theory are classified as renormalizable or non-renormalizable theories. The main arbitrariness of the subtraction procedure is fixed by imposing renormalization conditions. Special sections of this chapter are devoted to renormalization in dimensional regularization and renormalization group equations.

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