Feynman integrals and hypergeometric functions

2014 ◽  
Vol 05 (supp01) ◽  
pp. 1441001
Author(s):  
Héctor Luna García ◽  
Luz María García
Keyword(s):  
Hypergeometric Functions ◽  
Dimensional Regularization ◽  
Feynman Integrals ◽  
Regularization Scheme

We review Davydychev method for calculating Feynman integrals for massive and no massive propagators, by employing Mellin–Barnes transformation and the dimensional regularization scheme, same that lead to hypergeometric functions. In particular, an example is calculated explicitly from such a method.

scholarly journals A prescription for projectors to compute helicity amplitudes in D dimensions

2021 ◽  
Vol 81 (5) ◽  
Author(s):  
Long Chen
Keyword(s):  
Dimensional Regularization ◽  
Tensor Decomposition ◽  
Regularization Scheme ◽  
Lorentz Index ◽  
Dimensional Splitting ◽  
Lorentz Tensor ◽  
Helicity Amplitudes ◽  
Amplitude Level ◽  
Infrared Singularities

AbstractThis article discusses a prescription to compute polarized dimensionally regularized amplitudes, providing a recipe for constructing simple and general polarized amplitude projectors in D dimensions that avoids conventional Lorentz tensor decomposition and avoids also dimensional splitting. Because of the latter, commutation between Lorentz index contraction and loop integration is preserved within this prescription, which entails certain technical advantages. The usage of these D-dimensional polarized amplitude projectors results in helicity amplitudes that can be expressed solely in terms of external momenta, but different from those defined in the existing dimensional regularization schemes. Furthermore, we argue that despite being different from the conventional dimensional regularization scheme (CDR), owing to the amplitude-level factorization of ultraviolet and infrared singularities, our prescription can be used, within an infrared subtraction framework, in a hybrid way without re-calculating the (process-independent) integrated subtraction coefficients, many of which are available in CDR. This hybrid CDR-compatible prescription is shown to be unitary. We include two examples to demonstrate this explicitly and also to illustrate its usage in practice.

scholarly journals Modern Technique for Real Photon Radiative Correction Calculations

2021 ◽  
Vol 24 (2) ◽  
pp. 184-191
Author(s):  
I. A. Shershan ◽  
T. V. Shishkina
Keyword(s):  
Gauge Boson ◽  
Cross Sections ◽  
Dimensional Regularization ◽  
Real Photon ◽  
Boson Production ◽  
Radiative Corrections ◽  
Hard Photon ◽  
Regularization Scheme ◽  
Modern Technique ◽  
Bremsstrahlung Contribution

The problem of the bremsstrahlung contribution calculation as a part of the radiative corrections in the case of single gauge boson production was discussed. It was shown that the hard photon bremsstrahlung contribution can be divided into the finite and divergent terms. The exact calculation of soft photon bremsstrahlung and infrared part of hard photon bremsstrahlung was presented in frame of the dimensional regularization scheme. Numerical analysis of radiative corrections to the cross sections of single gauge boson production was performed.

scholarly journals The diagrammatic coaction beyond one loop

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Samuel Abreu ◽  
Ruth Britto ◽  
Claude Duhr ◽  
Einan Gardi ◽  
James Matthew
Keyword(s):  
Differential Equations ◽  
Hypergeometric Functions ◽  
Differential Forms ◽  
Feynman Graph ◽  
Loop Order ◽  
Feynman Integrals ◽  
Mathematical Properties ◽  
Multiple Polylogarithms ◽  
Complete Set ◽  
General Tool

Abstract The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the ϵ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.

scholarly journals REGULARIZATION PARAMETER INDEPENDENT ANALYSIS IN NAMBU–JONA-LASINIO MODEL

2013 ◽  
Vol 28 (31) ◽  
pp. 1350164 ◽  
Author(s):  
T. INAGAKI ◽  
D. KIMURA ◽  
H. KOHYAMA ◽  
A. KVINIKHIDZE
Keyword(s):  
Regularization Parameter ◽  
Dimensional Regularization ◽  
Low Energy ◽  
Leading Order ◽  
Regularization Scheme ◽  
Independent Analysis

Nambu–Jona-Lasinio model used to investigate low energy phenomena is nonrenormalizable, therefore the results depend on the regularization parameter in general. A possibility of the finite in four-dimensional limit and even the in regularization parameter (this is dimension in the dimensional regularization scheme) independent analysis is shown in the leading order of the 1/Nc expansion.

Hypergeometric expression for a three-loop vacuum integral

2020 ◽  
Vol 35 (19) ◽  
pp. 2050089
Author(s):  
Zhi-Hua Gu ◽  
Hai-Bin Zhang ◽  
Tai-Fu Feng
Keyword(s):  
Finite Element ◽  
Hypergeometric Functions ◽  
Dimensional Regularization ◽  
Feynman Integral ◽  
Numerical Continuation ◽  
Residue Theorem ◽  
Linear Partial Differential Equations ◽  
Multidimensional Residue ◽  
Stationary Conditions ◽  
Scalar Integral

Using the corresponding Mellin–Barnes representation, we derive holonomic hypergeometric system of linear partial differential equations (PDEs) satisfied by Feynman integral of a three-loop vacuum with five propagators. Through the multidimensional residue theorem in dimensional regularization, the scalar integral can be written as the summation of multiple hypergeometric functions, whose convergent regions can be obtained by the Horn’s convergent theory. The numerical continuation of the scalar integral from convergent regions to whole kinematic regions can be accomplished with the finite element methods, when the system of PDEs can be treated as the stationary conditions of a functional under the restrictions.

scholarly journals GEOMETRICALLY RELATING MOMENTUM CUT-OFF AND DIMENSIONAL REGULARIZATION

2012 ◽  
Vol 10 (02) ◽  
pp. 1250081 ◽  
Author(s):  
SUSAMA AGARWALA
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Hopf Algebras ◽  
Dimensional Regularization ◽  
Mass Scale ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Coupling Constant ◽  
Regularization Scheme ◽  
Β Function

The β function for a scalar field theory describes the dependence of the coupling constant on the renormalization mass scale. This dependence is affected by the choice of regularization scheme. I explicitly relate the β functions of momentum cut-off regularization and dimensional regularization on scalar field theories by a gauge transformation using the Hopf algebras of the Feynman diagrams of the theories.

SUPERSYMMETRY AND THE FOUR-DIMENSIONAL COUNTERPART OF DIMENSIONAL REGULARIZATION

1992 ◽  
Vol 07 (01) ◽  
pp. 41-59
Author(s):  
J. NOVOTNÝ
Keyword(s):  
Gauge Invariance ◽  
Dimensional Reduction ◽  
Dimensional Regularization ◽  
Explicit Calculation ◽  
Regularization Scheme

A supersymmetric generalization of the natural four-dimensional counterpart of canonical dimensional regularization developed recently is discussed for SUSY QED. The gauge invariance of the regularization scheme is proved and the method illustrated by simple examples of explicit calculation of one-loop supergraphs. The relation to the regularization by dimensional reduction is briefly discussed.

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