Numerical evaluation of master integrals from differential equations

Caffo, M.; Czyż, Henryk; Remiddi, E.

doi:10.1016/s0920-5632(03)80212-8

Cited by 17 publications

(18 citation statements)

References 17 publications

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“…So one can integrate the differential equations from the initial conditions at s = 0 to the desired value of s using well-known numerical techniques such as Runge-Kutta. For the integrals of the type S, T, U , this has already been done and explained in detail in [45]- [48]. Here, I will extend these results to include the master integral M , and present results for S, T, U integrals in a different basis which may be more convenient for some purposes.…”

Section: Introductionmentioning

confidence: 85%

“…Standard computer numerical methods (for example, Runge-Kutta, or improvements thereof) are used to evolve the differential equations from s = 0 to the desired s. Since the physical-sheet s is always taken to have an infinitesimal real imaginary part, and branch cuts lie along the real s axis, one should take the contour of integration to lie in the upper-half complex plane. Reference [48] suggests using a rectangular contour going from 0 to ih to s+ih to s+iε, where h is chosen large enough to stay away from singularities on the real s axis. Independence of the choice of h, and more generally on the choice of contour in the upper half-plane, provides a useful check on the numerical convergence.…”

Section: Numerical Evaluation By Differential Equationsmentioning

confidence: 99%

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Evaluation of two-loop self-energy basis integrals using differential equations

2003

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I study the Feynman integrals needed to compute two-loop self-energy functions for general masses and external momenta. A convenient basis for these functions consists of the four integrals obtained at the end of Tarasov's recurrence relation algorithm. The basis functions are modified here to include one-loop and two-loop counterterms to render them finite; this simplifies the presentation of results in practical applications. I find the derivatives of these basis functions with respect to all squared-mass arguments, the renormalization scale, and the external momentum invariant, and express the results algebraically in terms of the basis. This allows all necessary two-loop selfenergy integrals to be efficiently computed numerically using the differential equation in the external momentum invariant. I also use the differential equations method to derive analytic forms for various special cases, including a four-propagator integral with three distinct non-zero masses.

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Section: Introductionmentioning

confidence: 85%

Section: Numerical Evaluation By Differential Equationsmentioning

confidence: 99%

Evaluation of two-loop self-energy basis integrals using differential equations

2003

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“…It considerably reduces the complexity of the calculation and allows for new qualitative and quantitative results for this important particular class of Feynman integrals [25,37,38,39,40,41]. Configuration space techniques can be used to verify known results obtained within other techniques both analytically and numerically [42,43,44,45] and to investigate some general features of Feynman diagram calculation [46,30,39,47,48,49,50,51]. We list features of the x-space technique in turn with the aim to show how efficient the method is.…”

Section: Introductionmentioning

confidence: 99%

On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order

2007

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We review recently developed new powerful techniques to compute a class of Feynman diagrams at any loop order, known as sunrise-type diagrams. These sunrise-type topologies have many important applications in many different fields of physics and we believe it to be timely to discuss their evaluation from a unified point of view. The method is based on the analysis of the diagrams directly in configuration space which, in the case of the sunrise-type diagrams and diagrams related to them, leads to enormous simplifications as compared to the traditional evaluation of loops in momentum space. We present explicit formulae for their analytical evaluation for arbitrary mass configurations and arbitrary dimensions at any loop order. We discuss several limiting cases of their kinematical regimes which are e.g. relevant for applications in HQET and NRQCD. We completely solve the problem of renormalization using simple formulae for the counterterms within dimensional regularization. An important application is the computation of the multi-particle phase space in D-dimensional space-time which we discuss. We present some examples of their functions. We discuss the use of recurrence relations naturally emerging in configuration space for the calculation of special series of integrals of the sunrise topology. We finally report on results for the computation of an extension of the basic sunrise topology, namely the spectacle topology and the topology with an irreducible loop addition.

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“…For TSIL, we instead use the differential equations method [51,52] to evaluate the integrals numerically. For the S, T and U integrals, this strategy was proposed and implemented in [53][54][55][56][57]. The method was rewritten in terms of the S, T, U integrals and extended to M in Ref.…”

Section: Introductionmentioning

confidence: 99%

TSIL: a program for the calculation of two-loop self-energy integrals

Martin¹,

Robertson²

2006

Computer Physics Communications

126

143

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TSIL is a library of utilities for the numerical calculation of dimensionally regularized two-loop self-energy integrals. A convenient basis for these functions is given by the integrals obtained at the end of O.V. Tarasov's recurrence relation algorithm. The program computes the values of all of these basis functions, for arbitrary input masses and external momentum. When analytical expressions in terms of polylogarithms are available, they are used. Otherwise, the evaluation proceeds by a Runge-Kutta integration of the coupled first-order differential equations for the basis integrals, using the external momentum invariant as the independent variable. The starting point of the integration is provided by known analytic expressions at (or near) zero external momentum. The code is written in C, and may be linked from C, C++, or Fortran. A Fortran interface is provided. We describe the structure and usage of the program, and provide a simple example application. We also compute two new cases analytically, and compare all of our notations and conventions for the two-loop self-energy integrals to those used by several other groups.Comment: 31 pages. Updated to reflect new functionality through v1.4 May 2016 and new information about use with C++. Source code and documentation are available at http://www.niu.edu/spmartin/TSIL or http://faculty.otterbein.edu/DRobertson/tsil

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Numerical evaluation of master integrals from differential equations

Cited by 17 publications

References 17 publications

Evaluation of two-loop self-energy basis integrals using differential equations

Evaluation of two-loop self-energy basis integrals using differential equations

On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order

TSIL: a program for the calculation of two-loop self-energy integrals

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