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Algebraic renormalization and Feynman integrals in configuration spaces

Ceyhan, Özgür and Marcolli, Matilde (2014) Algebraic renormalization and Feynman integrals in configuration spaces. Advances in Theoretical and Mathematical Physics, 18 (2). pp. 469-511. ISSN 1095-0761. doi:10.4310/ATMP.2014.v18.n2.a5.

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This paper continues our previous study of Feynman integrals in configuration spaces and their algebro-geometric and motivic aspects. We consider here both massless and massive Feynman amplitudes, from the point of view of potential theory. We consider a variant of the wonderful compactification of configuration spaces that works simultaneously for all graphs with a given number of vertices and that also accounts for the external structure of Feynman graph. As in our previous work, we consider two version of the Feynman amplitude in configuration space, which we refer to as the real and complex versions. In the real version, we show that we can extend to the massive case a method of evaluating Feynman integrals, based on expansion in Gegenbauer polynomials, that we investigated previously in the massless case. In the complex setting, we show that we can use algebro-geometric methods to renormalize the Feynman amplitudes, so that the renormalized values of the Feynman integrals are given by periods of a mixed Tate motive. The regularization and renormalization procedure is based on pulling back the form to the wonderful compactification and replace it with a cohomologous one with logarithmic poles. A complex of forms with logarithmic poles, endowed with an operator of pole subtraction, determine a Rota-Baxter algebra on the wonderful compactifications. We can then apply the renormalization procedure via Birkhoff factorization, after interpreting the regularization as an algebra homomorphism from the Connes-Kreimer Hopf algebra of Feynman graphs to the Rota-Baxter algebra. We obtain in this setting a description of the renormalization group.We also extend the period interpretation to the case of Dirac fermions and gauge bosons.

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Additional Information:© 2014 International Press. The first author is partially supported by PCIG11-GA-2012-322154. The second author acknowledges support from NSF grants DMS-0901221, DMS-1007207, DMS-1201512, and PHY-1205440 and the hospitality and support of the Mathematical Sciences Research Institute in Berkeley and of the Kavli Institute for Theoretical Physics China and the Morningside Center for Mathematics in Beijing. The second author also thanks Li Guo for useful conversations.
Funding AgencyGrant Number
European Research Council (ERC)PCIG11-GA-2012-322154
Mathematical Sciences Research Institute (MSRI)UNSPECIFIED
Kavli Institute for Theoretical PhysicsUNSPECIFIED
Morningside Center for MathematicsUNSPECIFIED
Issue or Number:2
Record Number:CaltechAUTHORS:20141212-091347163
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:52619
Deposited By: Tony Diaz
Deposited On:12 Dec 2014 17:28
Last Modified:10 Nov 2021 19:43

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