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Euclidean Configuration Space Renormalization, Residues and Dilation Anomaly

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Lie Theory and Its Applications in Physics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 36))

Abstract

Configuration (x-)space renormalization of Euclidean Feynman amplitudes in a massless quantum field theory is reduced to the study of local extensions of associate homogeneous distributions. Primitively divergent graphs are renormalized, in particular, by subtracting the residue of an analytically regularized expression. Examples are given of computing residues that involve zeta values. The renormalized Green functions are again associate homogeneous distributions of the same degree that transform under indecomposable representations of the dilation group.

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Notes

  1. 1.

    Whereas x-space renormalization was straightened out in all generality [5, 33, 34, 51, 58], it took some more time to settle the p-space problem [45, 46, 63, 64], resulting in what is now termed the BPHZ theory.

  2. 2.

    In view of recent interest in 3D CFT [28, 47] we explicitly include here odd D.

  3. 3.

    A notion of residue of a Feynman graph has been introduced in the momentum space approach in terms of the graph polynomial [3, 4]. It would be interesting to establish the precise relationship between that notion and ours. The notion of Poincaré residue considered in [12], on the other hand, works in a straightforward manner for simple poles in x-space, a rather unnatural restriction for ultraviolet divergences.

  4. 4.

    A similar decomposition in an overall scale and angle variables is derived and used very recently in momentum space in [9].

  5. 5.

    The fact that the maximum function R, which replaces \(\rho _{\Sigma }(\textbf{x})\) of Theorem 2.2, does not depend smoothly on the coordinates, requires, in general, a special treatment of the lower dimensional manifolds of discontinuities of its derivatives. (See Example 4.1 below.)

  6. 6.

    Usually, in perturbation theory one is dealing with Lie algebra cohomology. Group cohomology has occurred in various contexts in the early 1980s [20, 21, 52, 59].

  7. 7.

    Representations of this type have been considered back in the 1970’s [23] within a study of a spontaneous breaking of dilation symmetry.

  8. 8.

    We thank Detlev Buchholz for stressing this point to us.

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Acknowledgements

The authors thank Detlev Buchholz, Maxim Kontsevich and Dirk Kreimer for discussions. N.N. acknowledges partial support by the French-Bulgarian Project Rila under the contract Egide-Rila N112. N.N. and I.T. acknowledge partial support by grant DO 02-257 of the Bulgarian National Science Foundation. N.N. and I.T. thank the organizers of the International Workshops LT-9 and SQS’2011 for the invitation and the IHÉS and the Theory Division of CERN for hospitality during the course of this work.

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Authors and Affiliations

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

    Nikolay M. Nikolov & Ivan Todorov

  2. Laboratoire d’Annecy-le-Vieux de Physique Théorique (LAPTH), 74941, Annecy-le-Vieux Cedex, France

    Raymond Stora

  3. Theory Division, Department of Physics, CERN, 1211, Geneva 23, Switzerland

    Raymond Stora & Ivan Todorov

  4. Institut des Hautes Études Scientifiques, 91440, Bures-sur-Yvette, France

    Ivan Todorov

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  1. Nikolay M. Nikolov
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Correspondence to Ivan Todorov .

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  1. Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, Sofia, 1784, Sofiya, Bulgaria

    Vladimir Dobrev

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Nikolov, N.M., Stora, R., Todorov, I. (2013). Euclidean Configuration Space Renormalization, Residues and Dilation Anomaly. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol 36. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54270-4_9

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