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27
RotaBaxter algebras in renormalization of perturbative quantum field theory
 Fields Institute Communications v. 50, AMS
"... Abstract. Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularize ..."
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Abstract. Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularized Hopf algebra characters. In this context the notion of Rota–Baxter algebras enters the scene. We review several aspects of Rota–Baxter algebras as they appear in other sectors also relevant to perturbative renormalization, for instance multiplezetavalues and matrix differential equations.
An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
, 2005
"... ..."
The Hopf algebra structure of multiple harmonic sums
, 2004
"... Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of their properties that can be exploited in calculations. ..."
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Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of their properties that can be exploited in calculations.
A Symbolic Summation Approach to Feynman Integral Calculus
"... Given a Feynman parameter integral, depending on a single discrete variable N and a real parameter ε, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in ε. In a first step, the integrals are expressed by hypergeometric multisums by means of sy ..."
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Given a Feynman parameter integral, depending on a single discrete variable N and a real parameter ε, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in ε. In a first step, the integrals are expressed by hypergeometric multisums by means of symbolic transformations. Given this sum format, we develop new summation tools to extract the first coefficients of its series expansion whenever they are expressible in terms of indefinite nested productsum expressions. In particular, we enhance the known multisum algorithms to derive recurrences for sums with complicated boundary conditions, and we present new algorithms to find formal Laurent series solutions of a given recurrence relation.
STRUCTURAL THEOREMS FOR SYMBOLIC SUMMATION
"... Starting with Karr’s structural theorem for summation —the discrete version of Liouville’s structural theorem for integration — we work out crucial properties of the underlying difference fields. This leads to new and constructive structural theorems for symbolic summation. E.g., these results can b ..."
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Starting with Karr’s structural theorem for summation —the discrete version of Liouville’s structural theorem for integration — we work out crucial properties of the underlying difference fields. This leads to new and constructive structural theorems for symbolic summation. E.g., these results can be applied for harmonic sums which arise frequently in particle physics.
Multiple (inverse) binomial sums of arbitrary weight and depth and the allorder epsilonexpansion of generalized hypergeometric functions with one halfinteger value of parameter
, 2007
"... Multiple (inverse) binomial sums of arbitrary weight and depth and the allorder εexpansion of generalized hypergeometric functions with one halfinteger value of parameter ..."
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Cited by 2 (2 self)
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Multiple (inverse) binomial sums of arbitrary weight and depth and the allorder εexpansion of generalized hypergeometric functions with one halfinteger value of parameter
Automated calculations for multileg processes
"... The search for signals of new physics at the forthcoming LHC experiments involves the analysis of final states characterised by a high number of hadronic jets or identified particles. Precise theoretical predictions for these processes require the computation of scattering amplitudes with a large n ..."
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The search for signals of new physics at the forthcoming LHC experiments involves the analysis of final states characterised by a high number of hadronic jets or identified particles. Precise theoretical predictions for these processes require the computation of scattering amplitudes with a large number of external particles and beyond leading order in perturbation theory. The complexity of a calculation grows with the number of internal loops as well as with the number of external legs. Automatisation of at least nexttoleading order calculations for LHC processes is therefore a timely task. I will discuss various approaches.
arXiv:0902.4091 [hepph]
, 902
"... Determining the closed forms of the O(a3 s) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra ..."
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Determining the closed forms of the O(a3 s) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra
NUMBER THEORY IN PHYSICS
"... always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation ..."
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always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation to physics. In the past decade, mostly through the influence of string theory, algebraic geometry also began to play a major role in this interaction. Recent years have seen an increasing number of results suggesting that number theory also is beginning to play an essential part on the scene of contemporary theoretical and mathematical physics. Conversely, ideas from physics, mostly from quantum field theory and string theory, have started to influence work in number theory. In describing significant occurrences of number theory in physics, we will, on the one hand, restrict our attention to quantum physics, while, on the other hand, we will assume a somewhat extensive definition of number theory, that will allow us to include arithmetic algebraic geometry. The territory is vast and an extensive treatment would go beyond the size limits imposed by the encyclopaedia. The