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Renormalization of position space amplitudes in a massless QFT

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Abstract

Ultraviolet renormalization of position space massless Feynman amplitudes has been shown to yield associate homogeneous distributions. Their degree is determined by the degree of divergence while their order—the highest power of logarithm in the dilation anomaly—is given by the number of (sub)divergences. In the present paper we review these results and observe that (convergent) integration over internal vertices does not alter the total degree of (superficial) ultraviolet divergence. For a conformally invariant theory internal integration is also proven to preserve the order of associate homogeneity. The renormalized 4-point amplitudes in the φ4 theory (in four space-time dimensions) are written as (non-analytic) translation invariant functions of four complex variables with calculable conformal anomaly.

Our conclusion concerning the (off-shell) infrared finiteness of the ultraviolet renormalized massless φ4 theory agrees with the old result of Lowenstein and Zimmermann [23].

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References

  1. S. Bloch, “Applications of the dilogarithm functions in algebraic K-theory and algebraic geometry”, in Proceedings of the International Symposium on Algebraic Geometry (Kinokuniya, Tokyo, 1978).

    Google Scholar 

  2. S. Bloch, H. Esnault, and D. Kreimer, “On motives and graph polynomials”, Commun. Math. Phys. 267, 181–225 (2006), arXiv:math/0510011.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. S. Bloch and D. Kreimer, “Mixed Hodge structures and renormalization in physics”, Commun. Number Theory Phys. 2, 637–718 (2008); arXiv:0804.4399[hepth]. “Feynman amplitudes and Landau singularities for 1-loop graphs”, arXiv:1007.0338[hep-th].

    Article  MathSciNet  MATH  Google Scholar 

  4. N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, 3d Ed. (Wiley, 1980) (Russian edition, 1957).

    Google Scholar 

  5. D. J. Broadhurst, “Summation of an infinite series of ladder diagrams”, Phys. Lett. B 307, 132–139 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  6. D. J. Broadhurst, “Multiple Deligne values: A data mine with empirically tamed denominators”, arXiv:1409.7204[hep-th]. ?4

  7. D. J. Broadhurst and D. Kreimer, “Knots and numbers in to 7 loops and beyond”, Int. J. Mod. Phys. 6C, 519–524 (1995). “Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops”, Phys. Lett. B 393, 403–412 (1997), arXiv:hep-th/9609128.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. D. Broadhurst and O. Schnetz, “Algebraic geometry informs perturbative quantum field theory”, Proc. Sci. 211, 078 (2014), arXiv:1409.5570.

    Google Scholar 

  9. F. Brown, “Single valued multiple polylogarithms in one variable”, C. R. Acad. Sci. Paris Ser. I 338, 522–532 (2004).

    Article  MATH  Google Scholar 

  10. F. Brown, “Single-valued periods and multiple zeta values”, arXiv:1309.5309[math.NT].

  11. F. Brown and D. Kreimer, “Angles, scales and parametric renormalization”, arXiv:1112.1180[hep-th].

  12. F. Brown and O. Schnetz, “Proof of the Zig-zag conjecture”, arXiv:1208.1890v2[math.NT].

  13. K. G. Chetyrkin, A. I. Kataev, and F. V. Tkachov, “New approach to evaluation of multiloop Feynman integrals: The Gegenbauer polynomial x-space technique”, Nucl. Phys. B 174, 345–377 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  14. C. Duhr, “Hopf algebras, coproducts and symbols: Application to Higgs boson amplitudes”, arXiv:1203.0454[hep-ph].

  15. F. J. Dyson, “Missed opportunities”, Bull. Am. Math. Soc. 78 (5), 635–652 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Epstein and V. Glaser, “The role of locality in perturbation theory”, Ann. Inst. H. Poincaré A 19 (3), 211–295 (1973).

    MathSciNet  MATH  Google Scholar 

  17. K. Frednhagen and K. Rejzner, “QFT on curved spacetimes: Axiomatic framework and examples”, J. Math. Phys., 57 (2016) 031101, arXiv:1412.5125v2[math-ph]; “Perturbative construction of models of algebraic Quantum Field Theory”, arXiv:1503.07814[math-ph].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. J. M. Gracia-Bondia, “Systematic renormalization at all orders in the DiffRen and improved Epstein-Glaser schemes”, in Proceedings of the Conference “Quantum Mathematical Physics” (Regensburg, 2014), arXiv:1507.06493[hep-th].

    Google Scholar 

  19. J. M. Gracia-Bondia, H. Gutierrez-Garro, and J. C. Varilly, “Imroved Epstein-Glaser renormalization in x-space, III Versus differential renormalization”, Nucl. Phys. B 886, 824–869 (2014), arXiv:1403.1785v3.

    Article  ADS  MATH  Google Scholar 

  20. S. Hollands and R. M. Wald, “Axiomatic quantum field theory on curved spacetime”, Commun. Math. Phys. 293, 85–125 (2010); arXiv:0803.2003[gr-qc]; “Quantum field theory on curved spacetime, operator product expansion, and Dark energy”, Gen. Rel. Grav. 40, 2051–2059 (2008), arXiv:0805.3419[gr-qc].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. L. Hörmander, The Analysis of Linear Partial Differential Operators, I: Distribution Theory and Fourier Analysis, 2d Ed. (Springer, 1990).

    MATH  Google Scholar 

  22. M. Kontsevich and D. Zagier, “Periods”, in Mathematics- 20101 and Beyond, Ed. by B. Engquist and W. Schmid (Springer, Berlin, 2001), pp. 771–808.

    Google Scholar 

  23. J. H. Lowenstein and W. Zimmermann, “On the formulation of theories with zero-mass propagators”, Nucl. Phys. B 86, 77–103 (1975).

    Article  ADS  Google Scholar 

  24. J. W. Milnor, “Hyperbolic geometry: The first 150 years”, Bull. Am. Math. Soc. 6 (1), 9–24 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  25. S. Müller-Stach, “What is a period?”, Notices AMS (2014), arXiv:1407.2388[math.NT].

    Google Scholar 

  26. N. M. Nikolov, R. Stora, and I. Todorov, Euclidean Configuration Space Renormalization, Residues and Dilation Anomaly, Geneva/Annecy Preprint CERN-THPH/ 2012-076, LAPTH-Conf-016/12; in Proceedings of the Varna Workshop “Lie Theory and Its Applications in Physics” (LT9), Ed. by V. K. Dobrev (Springer, Japan, Tokyo, 2013), pp. 127–147.

    Chapter  Google Scholar 

  27. N. M. Nikolov, R. Stora, and I. Todorov, “Renormalization of massless Feynman amplitudes as an extension problem for associate homogeneous distributions”, Rev. Math. Phys. 26 (4), 1430002 (2014); CERN-TH-PH/2013-107, arXiv:1307.6854[hep-th].

    Article  MathSciNet  MATH  Google Scholar 

  28. E. Panzer, “Feynman integrals via hyperlogarithms”, Proc. Sci. 211, 049 (2014), arXiv:1407.0074[hep-ph].

    Google Scholar 

  29. O. Schnetz, “Quantum periods: A census of transcendentals”, Commun. Number Theory Phys. 4 (1), 1–48, arXiv:0801.2856v2.

  30. O. Schnetz, “Graphical functions and single-valued multiple polylogarithms”, Commun. Number Theory Phys. 8 (4), 589–685 (2014), arXiv:1302.6445[math.NT].

    Article  MathSciNet  MATH  Google Scholar 

  31. I. Todorov, “Polylogarithms and multizeta values in massless Feynman amplitudes”, in Lie Theory and Its Applications in Physics (LT10), Ed. by V. Dobrev; Springer Proceedings in Mathematics and Statistics, Vol. 111 (Springer, Tokyo, 2014), pp. 155–176; Buressur-Yvette Preprint IHES/P/14/10.

    Google Scholar 

  32. F. Wilczek, “Origins of mass”, arXiv:1206.7114v2[hep-ph].

  33. D. Zagier, “The dilogarithm function”, in Frontiers in Number Theory, Physics and Geometry II, Ed. by P. Cartier (Springer, Berlin, 2006), pp. 3–65.

    Google Scholar 

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  1. Institute for Nuclear Research and Nuclear Energy Tsarigradsko Chaussee 72, BG-1784, Sofia, Bulgaria

    Ivan Todorov

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  1. Ivan Todorov
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Correspondence to Ivan Todorov.

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Todorov, I. Renormalization of position space amplitudes in a massless QFT. Phys. Part. Nuclei 48, 227–236 (2017). https://doi.org/10.1134/S1063779617020083

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