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On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions

J. Math. Phys. 49, 043510 (2008); doi:10.1063/1.2902996

Published 9 April 2008

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Mark W. Coffey
Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA
Perturbative quantum field theory for the Ising model at the three-loop level yields a tetrahedral Feynman diagram C(a,b) with masses a and b and four other lines with unit mass. The completely symmetric tetrahedron CTet[equivalent]C(1,1) has been of interest from many points of view, with several representations and conjectures having been given in the literature. We prove a conjectured exponentially fast convergent sum for C(1,1), as well as a previously empirical relation for C(1,1) as a remarkable difference of Clausen function values. Our presentation includes propositions extending the theory of the dilogarithm Li2 and Clausen Cl2 functions, as well as their relation to other special functions of mathematical physics. The results strengthen connections between Feynman diagram integrals, volumes in hyperbolic space, number theory, and special functions and numbers, specifically including dilogarithms, Clausen function values, and harmonic numbers. ©2008 American Institute of Physics
History: Received 13 February 2008; accepted 27 February 2008; published 9 April 2008
Permalink: http://link.aip.org/link/?JMAPAQ/49/043510/1
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KEYWORDS and PACS

Keywords
PACS
  • 11.10.Cd
    Axiomatic approach in field theory
  • 05.50.+q
    Lattice theory and statistics
  • 02.30.Gp
    Special functions
  • YEAR: 2008

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PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
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