scholarly journals SYSTEMATIC IMPLEMENTATION OF IMPLICIT REGULARIZATION FOR MULTILOOP FEYNMAN DIAGRAMS

2011 ◽  
Vol 26 (15) ◽  
pp. 2591-2635 ◽  
Author(s):  
A. L. CHERCHIGLIA ◽  
MARCOS SAMPAIO ◽  
M. C. NEMES
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Feynman Diagram ◽  
Lorentz Invariance ◽  
Recursion Formula ◽  
Feynman Diagrams ◽  
Quantum Field ◽  
Loop Momentum ◽  
Loop Order ◽  
Fundamental Symmetry

Implicit Regularization (IReg) is a candidate to become an invariant framework in momentum space to perform Feynman diagram calculations to arbitrary loop order. In this work we present a systematic implementation of our method that automatically displays the terms to be subtracted by Bogoliubov's recursion formula. Therefore, we achieve a twofold objective: we show that the IReg program respects unitarity, locality and Lorentz invariance and we show that our method is consistent since we are able to display the divergent content of a multiloop amplitude in a well-defined set of basic divergent integrals in one-loop momentum only which is the essence of IReg. Moreover, we conjecture that momentum routing invariance in the loops, which has been shown to be connected with gauge symmetry, is a fundamental symmetry of any Feynman diagram in a renormalizable quantum field theory.

scholarly journals A Brief Review of Implicit Regularization and Its Connection with the BPHZ Theorem

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 956
Author(s):  
Dafne Carolina Arias-Perdomo ◽  
Adriano Cherchiglia ◽  
Brigitte Hiller ◽  
Marcos Sampaio
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Gauge Symmetry ◽  
Particle Physics ◽  
Analytic Extension ◽  
Quantum Field ◽  
Loop Order ◽  
Time Dimension ◽  
The Core ◽  
Striking Success

Quantum Field Theory, as the keystone of particle physics, has offered great insights into deciphering the core of Nature. Despite its striking success, by adhering to local interactions, Quantum Field Theory suffers from the appearance of divergent quantities in intermediary steps of the calculation, which encompasses the need for some regularization/renormalization prescription. As an alternative to traditional methods, based on the analytic extension of space–time dimension, frameworks that stay in the physical dimension have emerged; Implicit Regularization is one among them. We briefly review the method, aiming to illustrate how Implicit Regularization complies with the BPHZ theorem, which implies that it respects unitarity and locality to arbitrary loop order. We also pedagogically discuss how the method complies with gauge symmetry using one- and two-loop examples in QED and QCD.

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.

Evaluation of loop diagrams using quantum mechanics

1992 ◽  
Vol 70 (8) ◽  
pp. 652-655 ◽  
Author(s):  
D. G. C. McKeon
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Matrix Element ◽  
Proper Time ◽  
Quantum Field ◽  
Loop Momentum ◽  
Loop Order ◽  
Point Function ◽  
Integral Formalism ◽  
Time Formalism

In using the proper time formalism, Schwinger demonstrated that one-loop processes in quantum field theory can be expressed in terms of a matrix element whose form is encountered in quantum mechanics, and which can be evaluated using the Heisenberg formalism. We demonstrate how instead this matrix element can be computed using standard results in the path-integral formalism. The technique of operator regularization allows one to extend this approach to arbitrary loop order. No loop-momentum integrals are ever encountered. This technique is illustrated by computing the two-point function in [Formula: see text] theory to one-loop order.

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.

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.

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