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Renormalization and Mellin Transforms

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Computer Algebra in Quantum Field Theory

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

We study renormalization in a kinetic scheme (realized by subtraction at fixed external parameters as implemented in the BPHZ and MOM schemes) using the Hopf algebraic framework, first summarizing and recovering known results in this setting. Then we give a direct combinatorial description of renormalized amplitudes in terms of Mellin transform coefficients, featuring the universal property of rooted trees H R . In particular, a special class of automorphisms of H R emerges from the action of changing Mellin transforms on the Hochschild cohomology of perturbation series. Furthermore, we show how the Hopf algebra of polynomials carries a refined renormalization group property, implying its coarser form on the level of correlation functions. Application to scalar quantum field theory reveals the scaling behaviour of individual Feynman graphs.

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Notes

  1. 1.

    We consider unordered trees

    and forests

    sometimes called non-planar.

  2. 2.

    By wv we denote the subtree of w rooted at the node v ∈ V (w).

  3. 3.

    Even if the divergence of a Feynman graph does depend on external momenta as happens for higher degrees of divergence, the Hopf algebra is defined such that the counterterms are evaluations on certain external structures, given by distributions in [9]. So in any case, ϕ − maps to scalars.

  4. 4.

    This already implies Ï• to be a morphism of Hopf algebras.

  5. 5.

    This combinatorial relation among tree factorials, noted in [17], thus drops out of \({\varDelta }^{}\varphi = {(}^{}\varphi {\otimes }^{}\varphi )\circ \varDelta\).

  6. 6.

    As x0 =1, for arbitrary p the series \({\left [X(g)\right ]}^{p} \mathop{:}=\sum _{n\in \mathbb{N}_{0}}\binom{p}{n}{\left [X(g) -1\!\!1\right ]}^{n} \in H_{R}[[g]]\) is well defined.

  7. 7.

    A proof of (30) may be found in[11] and [12, 13] study systems of Dyson-Schwinger equations.

  8. 8.

    Counting the number of corresponding ordered trees.

  9. 9.

    For this generality we need decorated rooted trees as commented on in Sect. 6.1

  10. 10.

    We prefer to work in the parametric representation as introduced in [14, Sect. 6-2-3].

  11. 11.

    This simple form circumvents the decomposition into one-scale graphs utilized in [6] and therefore holds in the original renormalization Hopf algebra H.

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Acknowledgements

Alexander von Humboldt Chair in Mathematical Physics, supported by the Alexander von Humboldt Foundation and the BMBF.

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  1. Departments of Mathematics and Physics, Humboldt University, Unter den Linden 6, 10099, Berlin, Germany

    Dirk Kreimer & Erik Panzer

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  1. Dirk Kreimer
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Correspondence to Dirk Kreimer .

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Editors and Affiliations

  1. RISC, Johannes Kepler University, Linz, Austria

    Carsten Schneider

  2. DESY, Zeuthen, Germany

    Johannes Blümlein

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Kreimer, D., Panzer, E. (2013). Renormalization and Mellin Transforms. In: Schneider, C., Blümlein, J. (eds) Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1616-6_8

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  • DOI: https://doi.org/10.1007/978-3-7091-1616-6_8

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