Numerical evaluation of tensor Feynman integrals in Euclidean kinematics

We present pySecDec, a new version of the program SecDec, which performs the factorisation of dimensionally regulated poles in parametric integrals, and the subsequent numerical evaluation of the finite coefficients. The algebraic part of the program is now written in the form of python modules, which allow a very flexible usage. The optimization of the C++ code, generated using FORM, is improved, leading to a faster numerical convergence. The new version also creates a library of the integrand functions, such that it can be linked to user-specific codes for the evaluation of matrix elements in a way similar to analytic integral libraries. (2015) 470-491. Nature of the problem: Extraction of ultraviolet and infrared singularities from parametric integrals appearing in higher order perturbative calculations in quantum field theory. Numerical integration in the presence of integrable singularities (e.g. kinematic thresholds). Solution method: Algebraic extraction of singularities within dimensional regularization using iterated sector decomposition. This leads to a Laurent series in the dimensional regularization parameter (and optionally other regulators), where the coefficients are finite integrals over the unit-hypercube. Those integrals are evaluated numerically by Monte Carlo integration. The integrable singularities are handled by choosing a suitable integration contour in the complex plane, in an automated way. The parameter integrals forming the coefficients of the Laurent series in the regulator(s) are provided in the form of libraries which can be linked to the calculation of (multi-) loop amplitudes. Restrictions: Depending on the complexity of the problem, limited by memory and CPU time. Running time: Between a few seconds and several days, depending on the complexity of the problem.

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“…Other implementations of the sector decomposition algorithm can be found in Refs. [13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

pySecDec: A toolbox for the numerical evaluation of multi-scale integrals

Borowka

Heinrich

Jahn

et al. 2018

Computer Physics Communications

297

265

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“…Besides using automated tools such as MB [58,59] and AMBRE [60][61][62] we have to overcome the problem that it is not straightforward to derive valid MB representations for crossed four-loop topologies with the loop-by-loop approach. For this purpose we constructed an in-house MATHEMATICA routine based on a hybrid of the loop-by-loop approach and using the F and U graph polynomials.…”

Section: Numerical Integration and Error Analysismentioning

confidence: 99%

The nonplanar cusp and collinear anomalous dimension at four loops in ${\mathcal N} = 4$ SYM theory

Boels¹,

Huber²,

Yang³

2018

Proceedings of 13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology) —

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We present numerical results for the nonplanar lightlike cusp and collinear anomalous dimension at four loops in N = 4 SYM theory, which we infer from a calculation of the Sudakov form factor. The latter is expressed as a rational linear combination of uniformly transcendental integrals for arbitrary colour factor. Numerical integration in the nonplanar sector reveals explicitly the breakdown of quadratic Casimir scaling at the four-loop order. A thorough analysis of the reported numerical uncertainties is carried out.13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology)

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“…A thorough study of the scope and limitations of MB.m remains to be done, but a first order examination shows that the accuracy of its results varies with the mass configuration and parameter values of the integral being evaluated. (See [41], however, for investigations into the efficiency of some aspects of these packages. )…”

Section: Numerical Analysismentioning

confidence: 99%

An analytic approach to sunset diagrams in chiral perturbation theory: Theory and practice

Ananthanarayan¹,

Bijnens²,

Ghosh³

et al. 2016

Eur. Phys. J. A

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We demonstrate the use of several code implementations of the Mellin-Barnes method available in the public domain to derive analytic expressions for the sunset diagrams that arise in the two-loop contribution to the pion mass and decay constant in three-flavoured chiral perturbation theory. We also provide results for all possible two-mass configurations of the sunset integral, and derive a new one-dimensional integral representation for the one mass sunset integral with arbitrary external momentum. Thoroughly annotated Mathematica notebooks are provided as ancillary files, which may serve as pedagogical supplements to the methods described in this paper.

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Numerical evaluation of tensor Feynman integrals in Euclidean kinematics

Cited by 76 publications

References 39 publications

pySecDec: A toolbox for the numerical evaluation of multi-scale integrals

pySecDec: A toolbox for the numerical evaluation of multi-scale integrals

The nonplanar cusp and collinear anomalous dimension at four loops in ${\mathcal N} = 4$ SYM theory

An analytic approach to sunset diagrams in chiral perturbation theory: Theory and practice

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