TVID 2: evaluation of planar-type three-loop self-energy integrals with arbitrary masses
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Springer
Received: September 17, Revised: October 24, Accepted: December 5, Published: January 7,
2019 2019 2019 2020
Stefan Bauberger,a Ayres Freitasb and Daniel Wiegandc,d a
Hochschule f¨ ur Philosophie, Philosophische Fakult¨ at S.J., Kaulbachstr. 31, 80539 M¨ unchen, Germany b Pittsburgh Particle-physics Astro-physics & Cosmology Center (PITT-PACC), Department of Physics & Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. c HEP Division, Argonne National Laboratory, Argonne, Illinois 60439, U.S.A. d Department of Physics & Astronomy, Northwestern University, Evanston, Illinois 60208, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: We present TVID 2, a program to numerically evaluate an important class of planar three-loop self-energy master integrals with arbitrary masses. As with the predecessor version (TVID 1) the integrals are separated into a known piece, containing the UV divergencies, and a finite piece that is integrated numerically, implemented in C. The set of master integrals under consideration was found with self-energy diagrams containing two closed fermion loops in mind. Two techniques are employed in deriving the expressions for the finite pieces that are then numerically integrated: (a) Sub-loop dispersion relations in the case of topologies containing sub-bubbles, and (b) a modification of the procedure suggested by Ghinculov for integrals with only sub-loop triangles. Keywords: Higgs Physics, Quark Masses and SM Parameters ArXiv ePrint: 1908.09887
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)024
JHEP01(2020)024
TVID 2: evaluation of planar-type three-loop self-energy integrals with arbitrary masses
Contents 1
2 Planar three-loop self-energy topologies and master integrals
2
3 Examples 3.1 Double-bubble integrals: U5b 3.2 Planar master topology: U8a 3.3 Planar 7-propagator topology: U7a
4 4 6 7
4 Implementation in TVID 2
7
5 Conclusions
13
A Subtraction of divergent terms
14
B TVID 2 manual B.1 Numerical part B.2 Algebraic part B.3 Installation
19 19 20 23
1
Introduction
The calculation of higher-order radiative corrections is important for the interpretation of precision measurements at the LHC and various e+ e− machines, such as SuperKEKB and planned future Higgs and Z factories. Multi-loop contributions in the full Standard Model or models beyond the Standard Model (BSM) are particularly challenging due to the presence of many independent mass and momentum scales [1]. General loop integrals beyond the one-loop level cannot be solved analytically in terms of elementary functions. This observation prompted the investigation of new classes of special functions, such as harmonic polylogarithms [2], generalized harmonic polylogarithms [3–6], and elliptic polylogarithms [7–15], see e.g. ref. [16] for a recent review. However, it is not clear if any multi-loop integral can be represented by these classes of functions, in particular beyond the two-loop level. Th
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