Abstract
We outline the evaluation of $n$-dimensional fermion traces ($n\in\mathbb{N}$) built by products of Dirac-$\gamma$ matrices suitable for a uniform dimensional continuation. Such a continuation is needed for calculations employing a dimensional regulator whenever intrinsically integer dimensional tensors yield non-vanishing contributions. A prime example for such a tensor is given by $\gamma_5$ for $n=4$. The main difference between Dimensional Regularization (DREG) and a Dimensionally Continued Regularization (DCREG) is that DCREG does not attempt to lift the algebra to continuous $d$ dimensions ($d\in\mathbb{R}$). As a consequence one has to properly deal with evanescent structures in order to ensure the uniform application of the regulator. In basic steps we identify evanescent structures in fermion traces and show that their proper treatment is crucial for example when calculating the $VVA$ anomaly in four dimensions. We checked that the performed considerations enable the evaluation of Standard Model (SM) $Z$-factors within DCREG up to including three loops.
Highlights
Dimensional regularization (DREG) [1] is a very powerful regularization scheme that keeps internal and external symmetries like Lorentz symmetry intact, without increasing the number of scales present in Feynman integrals, as is the case for example in a Pauli-Villars regularization
The lack of a consistent treatment of ε tensors within DREG is the main reason why up to now the Standard Model (SM) β functions are fully known at three-loop order [4,5,6,7,8], but only partially known up to four loops [9,10,11], the computational abilities nowadays allow for a five-loop evaluation for example of the QCD β function [12,13,14,15,16,17,18,19,20,21,22,23,24]
In this article we have been investigating how the results of fermion trace evaluations in integer dimensions N V can be continued to noninteger dimensions NV in a uniform way. Such a continuation is required for a dimensional regulator in the presence of γNVþ1
Summary
Dimensional regularization (DREG) [1] is a very powerful regularization scheme that keeps internal and external symmetries like Lorentz symmetry intact, without increasing the number of scales present in Feynman integrals, as is the case for example in a Pauli-Villars regularization. The lack of a consistent treatment of ε tensors within DREG is the main reason why up to now the Standard Model (SM) β functions are fully known at three-loop order [4,5,6,7,8], but only partially known up to four loops [9,10,11], the computational abilities nowadays allow for a five-loop evaluation for example of the QCD β function [12,13,14,15,16,17,18,19,20,21,22,23,24] In this context one has to mention that it has already been proven that a dimensional regulator can be used for a dimensional renormalization in the presence of γ5 including the aforementioned ε-tensor contributions for arbitrary loop orders in Ref. IX to discuss the implications following from our analysis and conclude with a summary in the last section
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