Numerical evaluation of the general massive 2-loop sunrise self-mass master integrals from differential equations

The system of 4 differential equations in the external invariant satisfied by the 4 master integrals of the general massive 2-loop sunrise self-mass diagram is solved by the Runge-Kutta method in the complex plane. The method offers a reliable and robust approach to the direct and precise numerical evaluation of Feynman graph integrals.The relevance of the higher order calculations for the comparison with nowadays precision measurements in high energy physics is well known and comprehensively presented by G. Passarino in this conference.Therefore can be of some interest the exploitation of an alternative method (but still in the context of the integration by part identities and master integrals (MI) [1]) to the more common direct integration method for the numerical evaluation of the MI.The method uses directly the differential equations. Starting from the integral representation of the MI, related to a certain Feynman graph, by derivation with respect to one of the internal masses [2] or one of the external invariants [3] and with the repeated use of the integration by part identities, a system of independent first order partial differential equations is obtained in a number equal to the number of the MI (master differential equations).Enlarging the number of loops and legs grows the number of parameters, MI and equations, but does not change or spoil the method.

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“…Comparisons are done in [5] with some values present in the literature [10,11] with excellent agreement.…”

mentioning

confidence: 65%

“…From these equations the analytic expressions for their first order expansion were completed at the special points [4,8,9,5]:…”

mentioning

confidence: 99%

Numerical evaluation of general massive 2-loop self-mass master integrals from differential equations

Caffo

Czyż

Remiddi

2003

Nuclear Instruments and Methods in Physics Research Section A:

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“…To test the method we have chosen to start from the simple, but not trivial, 2-loop sunrise graph with arbitrary masses [5,6], shown in Fig.1. This graph is one of the topologies of the 2-loop self-mass and has 4 MI.…”

Section: Results: Sunrise mentioning

confidence: 99%

“…From these equations the analytic expressions for their first order expansion were completed around the special points [5,10,11,6]:…”

Section: Results: Sunrise mentioning

confidence: 99%

Numerical evaluation of master integrals from differential equations

Caffo

Czyż

Remiddi

2003

Nuclear Physics B - Proceedings Supplements

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The 4-th order Runge-Kutta method in the complex plane is proposed for numerically advancing the solutions of a system of first order differential equations in one external invariant satisfied by the master integrals related to a Feynman graph. The particular case of the general massive 2-loop sunrise self-mass diagram is analyzed. The method offers a reliable and robust approach to the direct and precise numerical evaluation of master integrals.

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“…In the 2 F 1 solutions having appeared so far in single scale Feynman diagram calculations to higher order [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], one could always find elliptic integral representations with either the appearance of only the complete elliptic integral of the first kind K or of both the complete elliptic integrals of the first and second kind K and E in the single scale case. The solutions of the second order differential equations read…”

Section: The Functional Structure Of Feynman Integrals In the Single mentioning

confidence: 99%

Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations

Bluemlein¹

2018

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

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Various of the single scale quantities in massless and massive QCD up to 3-loop order can be expressed by iterative integrals over certain classes of alphabets, from the harmonic polylogarithms to root-valued alphabets. Examples are the anomalous dimensions to 3-loop order, the massless Wilson coefficients and also different massive operator matrix elements. Starting at 3-loop order, however, also other letters appear in the case of massive operator matrix elements, the so called iterative non-iterative integrals, which are related to solutions based on complete elliptic integrals or any other special function with an integral representation that is definite but not a Volterratype integral. After outlining the formalism leading to iterative non-iterative integrals,we present examples for both of these cases with the 3-loop anomalous dimension γ (2) qg and the structure of the principle solution in the iterative non-interative case of the 3-loop QCD corrections to the ρ-parameter.

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Numerical evaluation of the general massive 2-loop sunrise self-mass master integrals from differential equations

Cited by 52 publications

References 20 publications

Numerical evaluation of general massive 2-loop self-mass master integrals from differential equations

Numerical evaluation of general massive 2-loop self-mass master integrals from differential equations

Numerical evaluation of master integrals from differential equations

Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations

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