An extension of the method of brackets. Part 2

González, Iván; Jiu, Lin; Moll, Victor H.

doi:10.1515/math-2020-0062

Cited by 7 publications

(14 citation statements)

References 9 publications

(11 reference statements)

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“…Method 4. The final form uses the representations 1 These four bracket series lead to the terms (see [9] for the concept of divergent expansion in the context of the method of brackets). Using this series and the Mellin-Barnes representation of K 0…”

Section: Flexibility Of the Methods Of Bracketsmentioning

confidence: 99%

“…These examples have been evaluated in [9] by a different procedure. The Mellin-Barnes representations of the factors in the integrand…”

Section: Direct Computations Of Mellin Transformsmentioning

confidence: 99%

“…The examples below illustrates this process. Integrals involving the Bessel function K ν (x) have been presented in [9]. The computation there is based on the concept of totally null/divergent representations.…”

Section: An Integral Involving Exponentials and Bessel Modified Funct...mentioning

confidence: 99%

“…In spite of the lack of rigor, these series have shown to be useful in the evaluation of definite integrals. See [9] for details. A second technique used before is the integral representation of K ν (x) such as (8.4)…”

Section: An Integral Involving Exponentials and Bessel Modified Funct...mentioning

confidence: 99%

“…x ν ∞ 0 cos t dt (x 2 + t 2 ) ν+1/2 , and then apply the methods of brackets. The method present next is an improvement over those employed in [9]. Naturally the parameter b can be scaled out, but it is instructive to leave it as is.…”

Section: An Integral Involving Exponentials and Bessel Modified Funct...mentioning

confidence: 99%

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Mellin-Barnes and the method of brackets

Gonzalez,

Kondrashuk,

Moll

et al. 2021

Preprint

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The method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin-Barnes integral. The fundamental idea is to transform these integral representations into a bracket series to obtain their values. The expansion of the gamma function in such a series constitute the main part of this new application. The power and flexibility of this procedure is illustrated with a variety of examples.

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Section: Flexibility Of the Methods Of Bracketsmentioning

confidence: 99%

“…These examples have been evaluated in [9] by a different procedure. The Mellin-Barnes representations of the factors in the integrand…”

Section: Direct Computations Of Mellin Transformsmentioning

confidence: 99%

Section: An Integral Involving Exponentials and Bessel Modified Funct...mentioning

confidence: 99%

Section: An Integral Involving Exponentials and Bessel Modified Funct...mentioning

confidence: 99%

Section: An Integral Involving Exponentials and Bessel Modified Funct...mentioning

confidence: 99%

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Mellin-Barnes and the method of brackets

Gonzalez,

Kondrashuk,

Moll

et al. 2021

Preprint

Self Cite

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Celestial insights into the S-matrix bootstrap

Ghosh

Raman

Sinha

2022

J. High Energ. Phys.

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We consider 2-2 scattering in four spacetime dimensions in Celestial variables. Using the crossing symmetric dispersion relation (CSDR), we recast the Celestial amplitudes in terms of crossing symmetric partial waves. These partial waves have spurious singularities in the complex Celestial variable, which need to be removed in local theories. The locality constraints (null constraints) admit closed form expressions, which lead to novel bounds on partial wave moments. These bounds allow us to quantify the degree of low spin dominance(LSD) for scalar theories. We study a new kind of positivity that seems to be present in a wide class of theories. We prove that this positivity arises only in theories with a spin-0 dominance. The crossing symmetric partial waves with spurious singularities removed, dubbed as Feynman blocks, have remarkable properties in the Celestial variable, namely typically realness, in the sense of Geometric Function Theory (GFT). Using GFT techniques we derive non-projective bounds on Wilson coefficients in terms of partial wave moments.

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Hypergeometric structures in Feynman integrals

Blümlein

Saragnese

Schneider

2023

Ann Math Artif Intell

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For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

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An extension of the method of brackets. Part 2

Cited by 7 publications

References 9 publications

Mellin-Barnes and the method of brackets

Mellin-Barnes and the method of brackets

Celestial insights into the S-matrix bootstrap

Hypergeometric structures in Feynman integrals

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