# CONFLUENTES MATHEMATICI

Doubling bialgebras of graphs and Feynman rules
Confluentes Mathematici, Tome 8 (2016) no. 1, pp. 3-30.

In this article, we define a doubling procedure for the bialgebra of specified Feynman graphs introduced in a previous paper [1]. This is the vector space generated by the pairs $\left(\overline{\Gamma },\overline{\gamma }\right)$ where $\overline{\Gamma }$ is a locally $1PI$ specified graph of a perturbation theory $𝒯$ with $\overline{\gamma }\subset \overline{\Gamma }$ locally $1PI$ and where $\overline{\Gamma }/\overline{\gamma }$ is a specified graph of $𝒯$. We also define a convolution product on the characters of this new bialgebra with values in an endomorphism algebra, equipped with a commutative product compatible with the composition. We then express in this framework the renormalization as formulated by A. Smirnov [13, §8.5, 8.6], adapting the approach of A. Connes and D. Kreimer for two renormalization schemes: the minimal renormalization scheme and the Taylor expansion scheme. Finally, we determine the finite parts of Feynman integrals using the BPHZ algorithm after dimensional regularization procedure, by following the approach by P. Etingof [9] (see also [11]).

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/cml.26
Classification : 05C90,  81T15,  16T05,  16T10
Mots clés : Bialgebra, Hopf algebra, Feynman Graphs, Birkhoff decomposition, renormalization, dimensional regularization
@article{CML_2016__8_1_3_0,
author = {Mohamed Belhaj Mohamed},
title = {Doubling bialgebras of graphs and {Feynman} rules},
journal = {Confluentes Mathematici},
pages = {3--30},
publisher = {Institut Camille Jordan},
volume = {8},
number = {1},
year = {2016},
doi = {10.5802/cml.26},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.26/}
}
Belhaj Mohamed, Mohamed. Doubling bialgebras of graphs and Feynman rules. Confluentes Mathematici, Tome 8 (2016) no. 1, pp. 3-30. doi : 10.5802/cml.26. https://cml.centre-mersenne.org/articles/10.5802/cml.26/

[1] M. Belhaj Mohamed, D. Manchon, Bialgebra of specified graphs and external structures, Ann. Inst. Henri Poincaré, D, Volume 1, Issue 3, pp. 307-335 (2014).

[2] J. Bernstein, Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients, Functional Analysis and Its Applications 5 (2): 89-101 (1971).

[3] N. N. Bogoliubov and O. S. Parasiuk, On the multiplication of causal functions in the quantum theory of fields. Acta Math., 97, 227-266, (1957).

[4] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210, ${n}^{\circ }1$, 249-273 (2000).

[5] S. C. Coutinho, A primer of algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press (1995).

[6] K. Ebrahimi-Fard, J. Gracia-Bondia, F. Patras, A Lie theoretic approach to renormalization, Comm. Math. Phys. 276, 519-549 (2007).

[7] K. Ebrahimi-Fard, F. Patras, Exponential renormalization, Ann. Henri Poincaré 11, 943-971 (2010).

[8] K. Ebrahimi-Fard, F. Patras, Exponential renormalization II. Bogoliubov’s R-operation and momentum subtraction schemes, J. Math. Phys. 53, 083505 (2012).

[9] P. Etingof, A note on dimensional regularization. In Quantum Fields and Strings : A Course for Mathematicians, American Mathematical Society, (2000).

[10] K. Hepp, Proof of the bogoliubov-Parasiuk theorem on renormalization, Comm. Math. Phys. 2, 301-326 (1966).

[11] R. Meyer, Dimensional Regularization, Lecture at the workshop Theory of Renormalization and Regularization, Hesselberg 2002.

[12] E. Panzer, Hopf algebraic Renormalization of Kreimer’s toy model. Arxiv: math.QA: 1202.3552, v1 (2012).

[13] V. A. Smirnov, Renormalization and asymptotic expansions, Birkhauser, Basel (1991).

[14] W.D. van Suijlekom, The Hopf algebra of Feynman graphs in QED, letters in Math. Phys. 77, 265-281 (2006).

[15] M. Veltman, Diagrammatica - The path to Feynman diagrams, Cambridge University Press, (1994).

[16] W. Zimmermann, Convergence of Bogoliubov’s method of renormalization in momentum space, Comm. in Math. Phys. 15, 208-234, (1969).