Abstract
The gradient flow in non-abelian gauge theories on \( {\mathbb{R}^4} \) is defined by a local diffusion equation that evolves the gauge field as a function of the flow time in a gauge-covariant manner. Similarly to the case of the Langevin equation, the correlation functions of the time-dependent field can be expanded in perturbation theory, the Feynman rules being those of a renormalizable field theory on \( {\mathbb{R}^4} \times \left[ {0,\infty } \right) \). For any matter multiplet and to all loop orders, we show that the correlation functions are finite, i.e. do not require additional renormalization, once the theory in four dimensions is renormalized in the usual way. The flow thus maps the gauge field to a one-parameter family of smooth renormalized fields.
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References
M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [arXiv:1006.4518] [SPIRES].
M. Lüscher, Topology, the Wilson flow and the HMC algorithm, PoS(Lattice2010) 015 [arXiv:1009.5877] [SPIRES].
J. Zinn-Justin, Renormalization and stochastic quantization, Nucl. Phys. B 275 (1986) 135 [SPIRES].
J. Zinn-Justin and D. Zwanziger, Ward identities for the stochastic quantization of gauge fields, Nucl. Phys. B 295 (1988) 297 [SPIRES].
C. Becchi, A. Rouet and R. Stora, Renormalization of the abelian Higgs-Kibble model, Commun. Math. Phys. 42 (1975) 127 [SPIRES].
C. Becchi, A. Rouet and R. Stora, Renormalization of gauge theories, Annals Phys. 98 (1976) 287 [SPIRES].
K. Symanzik, Schrödinger representation and Casimir effect in renormalizable quantum field theory, Nucl. Phys. B 190 (1981) 1 [SPIRES].
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ArXiv ePrint:1101.0963
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Lüscher, M., Weisz, P. Perturbative analysis of the gradient flow in non-abelian gauge theories. J. High Energ. Phys. 2011, 51 (2011). https://doi.org/10.1007/JHEP02(2011)051
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DOI: https://doi.org/10.1007/JHEP02(2011)051