Non-zero topological charge is prohibited in the chiral limit of gauge-fermion systems because any instanton would create a zero mode of the Dirac operator. On the lattice, however, the geometric $Q_\text{geom}=\langle F{\tilde F}\rangle /32π^2$ definition of the topological charge does not necessarily vanish even when the gauge fields are smoothed for example with gradient flow. Small vacuum fluctuations (dislocations) not seen by the fermions may be promoted to instanton-like objects by the gradient flow. We demonstrate that these artifacts of the flow cause the gradient flow renormalized gauge coupling to increase and run faster. In step-scaling studies such artifacts contribute a term which increases with volume. The usual $a/L\to 0$ continuum limit extrapolations can hence lead to incorrect results. In this paper we investigate these topological lattice artifacts in the SU(3) 10-flavor system with domain wall fermions and the 8-flavor system with staggered fermions. Both systems exhibit nonzero topological charge at the strong coupling, especially when using Symanzik gradient flow. We demonstrate how this artifact impacts the determination of the renormalized gauge coupling and the step-scaling $β$ function.
Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh（爱丁堡大学物理与天文学院希格斯理论物理中心）；Theoretische Physik 1, Center for Particle Physics Siegen, Naturwissenschaftlich-Technische Fakultät, Universität Siegen（锡根大学自然科学技术学院粒子物理中心理论物理一系）；Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen University（亚琛工业大学理论粒子物理与宇宙学研究所）；Department of Physics, University of Colorado, Boulder（科罗拉多大学博尔德分校物理系）；IMADA and Quantum Theory Center, University of Southern Denmark（南丹麦大学IMADA及量子理论中心）
Perturbative analysis of the gradient flow in non-abelian gauge theories
R^4上的非阿贝尔规范理论中的梯度流由一个局域扩散方程定义，该方程以规范协变的方式将规范场演化为流动时间的函数。与朗之万方程的情况类似，含时场的关联函数也可以在微扰论中展开，费曼规则是R^4 x [0，oo）上可重整化的场论的规则。对于任何物质的多重态和所有的循环顺序，我们表明，关联函数是有限的，即不需要额外的重整化，一旦在四维的理论是重整化在通常的方式。因此，该流将规范场映射为一个单参数光滑重整化场族。
