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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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REFERENCES (28)
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  1. Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC, 1972).
  2. Arnold, P. and Yaffe, L. G., “epsilon expansion analysis of very weak first-order transitions in the cubic anisotropy model. I,” Phys. Rev. D 55, 7760 (1997).
  3. Berndt, B. C., Ramanujan's Notebooks (Springer, New York, 1985), Pt. I.
  4. Broadhurst, D. J., “A dilogarithmic 3-dimensional Ising tetrahedron,” Eur. Phys. J. C 8, 363 (1999),
    5. e-print arXiv:/hep-th/9805025.
  5. Broadhurst, D. J., e-print arXiv:/hep-th/9806174.
  6. Choy, T. C., Sherrington, D., Thomsen, M., and Thorpe, M. F., “Local magnetic field distributions. II. Further results,” Phys. Rev. B 31, 7355 (1985).
  7. Coffey, M. W., e-print arXiv:/math-ph/0801.0272.
  8. Coxeter, H. S. M., “The functions of Schläfli and Labatschefsky,” Q. J. Math. 6, 13 (1935).
  9. de Doelder, P. J., “On the Clausen integral Cl2(theta) and a related integral,” J. Comput. Appl. Math. 11, 325 (1984).
  10. Devoto, A. and Duke, D. W., “Table of integrals and formulae for Feynman diagram calculations,” Riv. Nuovo Cimento 7, 1 (1984).
  11. Espinosa, J. R. and Zhang, R.-J., “Complete two-loop dominant corrections to the mass of the lightest CP-even Higgs boson in the minimal supersymmetric standard model,” Nucl. Phys. B 586, 3 (2000).
  12. Fromm, D. M. and Hill, R. N., “Analytic evaluation of three-electron integrals,” Phys. Rev. A 36, 1013 (1987).
  13. Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products (Academic, New York, 1980).
  14. Grosjean, C. C., “Formulae concerning the computation of the Clausen integral Cl2(theta),” J. Comput. Appl. Math. 11, 331 (1984).
  15. Hansen, E. R., A Table of Series and Products (Prentice-Hall, Englewood Cliffs, NJ, 1975).
  16. Harris, F. E., “Analytic evaluation of three-electron atomic integrals with Slater wave functions,” Phys. Rev. A 55, 1820 (1997).
  17. Holdeman, Jr., J. T., “A method for the approximation of functions defined by formal series expansions in orthogonal polynomials,” Math. Comput. 23, 275 (1969). Note that in Eq. (52) of this reference, (2n−1) should read (2n+1), as given in Ref. 15 [formula (46.2.21)].
  18. Lewin, L., Polylogarithms and Associated Functions (North Holland, Amsterdam, 1981).
  19. Li, E. H. and Tin, S. K., “The Lobachevskiy's function and related integrals,” Bull. Inst. Math. Appl. 27, 175 (1991).
  20. Lunev, F. A., “Evaluation of two-loop self-energy diagram with three propagators,” Phys. Rev. D 50, 7735 (1994). It appears that in Eq. (18), the integrand factor (1−lambday) should read (1−lambdax).
  21. Milnor, J., “Hyperbolic geometry: The first 150  years,” Bull., New Ser., Am. Math. Soc. 6, 9 (1981).
  22. Nielsen, N., “Der Eulersche Dilogarithmus und seine Verallgemeinerungen,” Nova Acta Leopold. 90, 154 (1909).
  23. Pachucki, K. and Puchalski, M., “Extended Hylleraas three-electron integral,” Phys. Rev. A 71, 032514 (2005);
    24. Pachucki, K., Puchalski, M., and Remiddi, E., “Recursion relations for the generic Hylleraas three-electron integral,” ibid. 70, 032502 (2004);
    Pachucki, K. and Komasa, J., “Three-electron integral in a Gaussian basis set with linear terms,” ibid. 70, 022513 (2004).
  24. Rainville, E. D., Special Functions (Macmillan, London, 1960).
  25. Rajantie, A. K., “Feynman diagrams to three loops in three-dimensional field theory,” Nucl. Phys. B 480, 729 (1996).
  26. Remiddi, E., “Analytic value of the atomic three-electron correlation integal with Slater wave functions,” Phys. Rev. A 44, 5492 (1991);
    27. Harris, F. E. Frolov, A. M., and Smith, Jr., V. H., ibid. 69, 056501 (2004);
    Sims, J. S. and Hagstrom, S. A., ibid. 68, 016501 (2003).
  27. Srivastava, H. M. and Choi, J., Series Associated With the Zeta and Related Functions (Kluwer Academic, Norwell, MA, 2001).
  28. Wannier, G. H., “Antiferromagnetism. The triangular Ising net,” Phys. Rev. 79, 357 (1950);
    29. Phys. Rev. B 7, 5017(E) (1973).

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