Graphical functions in parametric space

Golz, Marcel; Panzer, Erik; Schnetz, Oliver

doi:10.1007/s11005-016-0935-6

Cited by 25 publications

(35 citation statements)

References 26 publications

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“…1. They belong to a broader family of conformal integrals which has attracted much attention over the years [17][18][19][20][21][22][23][24][25]. The black lines in the figure provide the position-space interpretation of the "fishnet" diagram, as a contribution to the correlation function G m,n (x i ) = φ We are only interested in the first planar graph contributing to this correlator.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Gluing Ladder Feynman Diagrams into Fishnets

2017

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We use integrability at weak coupling to compute fishnet diagrams for four-point correlation functions in planar φ 4 theory. The results are always multi-linear combinations of ladder integrals, which are in turn built out of classical polylogarithms. The Steinmann relations provide a powerful constraint on such linear combinations, leading to a natural conjecture for any fishnet diagram as the determinant of a matrix of ladder integrals.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Gluing Ladder Feynman Diagrams into Fishnets

2017

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“…There exist many identities for graphical functions. A Fourier-identity relating planar duals was proved in [31]. Like in the case of periods there also exists a twist identity.…”

Section: 5mentioning

confidence: 99%

“…It also follows from the parametric representation of graphical functions [31]. Property (G2) is proved in [31] while (G3) will be handled in [50]. Conjecture 4.12 gives additional information on the leading terms of the above expansions.…”

Section: Functionsmentioning

confidence: 99%

Numbers and functions in quantum field theory

2018

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We review recent results in the theory of numbers and single-valued functions on the complex plane which arise in quantum field theory. These results are the basis for a new approach to high-loop-order calculations. As concrete examples, we provide scheme-independent counterterms of primitive logdivergent graphs in ϕ 4 theory up to eight loops and the renormalization functions β, γ, γ m of dimensionally regularized ϕ 4 theory in the minimal subtraction scheme up to seven loops.

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“…Of the many recent advances made in the evaluation of Feynman integrals, parametric integration is one of the most powerful methods for p-integrals; surpassed only by the position-space approach of graphical functions [58] and the combination [34] of both techniques used in [52,59].…”

Section: Parametric Integrationmentioning

confidence: 99%

Renormalization group functions of $\phi^4$ theory in the MS-scheme to six loops

Kompaniets¹,

Panzer²

2016

Proceedings of Loops and Legs in Quantum Field Theory — PoS(LL2016)

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Subdivergences constitute a major obstacle to the evaluation of Feynman integrals and an expression in terms of finite quantities can be a considerable advantage for both analytic and numeric calculations. We report on our implementation of the suggestion by F. Brown and D. Kreimer, who proposed to use a modified BPHZ scheme where all counterterms are single-scale integrals. Paired with parametric integration via hyperlogarithms, this method is particularly well suited for the computation of renormalization group functions and easily automated. As an application of this approach we compute the 6-loop beta function and anomalous dimensions of the φ 4 model. Loops and Legs in Quantum Field Theory

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Graphical functions in parametric space

Cited by 25 publications

References 26 publications

Gluing Ladder Feynman Diagrams into Fishnets

Gluing Ladder Feynman Diagrams into Fishnets

Numbers and functions in quantum field theory

Renormalization group functions of $\phi^4$ theory in the MS-scheme to six loops

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