Lattice Field Theory

Institut für Physik
Mathematisch-Naturwissen­schaft­liche Fakultät
Humboldt-Universität zu Berlin

Applications of the gradient flow

The gradient flow [1] is a technique which allows to construct composite operators with simplified renormalization properties [2]. The gradient flow has already found several applications in Quantum Chromodynamics (QCD), and in particular in lattice QCD calculations, but its definition can be easily generalized to other Quantum Field Theories as well.

A selection of interesting applications, with a grossly incomplete bibliography, follows:

Given the gauge field $A_\mu(x)$ in four-dimensional Euclidean space time, one introduces a fifth fictitious coordinate $t$ and defines the gauge field $B_\mu(t,x)$ at positive flowtime $t$ by means of the following partial differential equation: $$ \frac{\partial}{\partial t} B_\mu(t,x) = D_\rho G_{\rho\mu}(t,x) \ , $$ with initial condition $B_\mu(0,x) = A_\mu(x)$. In the above equation, $G_{\rho\mu}(t,x)$ is the field tensor at positive flowtime, i.e. $$ G_{\rho\mu}(t,x) = \partial_\rho B_\mu(t,x) - \partial_\mu B_\rho(t,x) + i [B_\rho(t,x), B_\mu(t,x)] \ , $$ and its covariant derivative is given by $$ D_\rho G_{\rho\mu}(t,x) = \partial_\rho G_{\rho\mu}(t,x) + i [ B_\rho(t,x) , G_{\rho\mu}(t,x) ] \ . $$ One of the simplest observables that is defined at positive flowtime is $$ E(t) = \frac{1}{2} \langle \mathrm{tr}\ G_{\mu\nu}^2(t,0) \rangle \ . $$ This quantity is UV finite once the parameters of the action (i.e. the coupling constant and the masses) are renormalized as usual. The non-trivial part of this statement is that the operator $G_{\mu\nu}^2(t,x)$ itself does not need to be renormalized as long as $t>0$, while this is not true at $t=0$. As a consequence $E(t)$ calculated in lattice QCD has a finite continuum limit. The quantity $E(t)$ is useful to define the auxiliary scale $t_0$ via the equation $t_0^2 E(t_0) = 0.3$.



Martin Lüscher
Properties and uses of the Wilson flow in lattice QCD
JHEP 08 (2010) 071 , JHEP 03 (2014) 092
arXiv: 1006.4518 [hep-lat]
Martin Luscher, Peter Weisz
Perturbative analysis of the gradient flow in non-abelian gauge theories
JHEP 02 (2011) 051
arXiv: 1101.0963 [hep-th]
Mattia Bruno, Mattia Dalla Brida, Patrick Fritzsch, Tomasz Korzec, Alberto Ramos, Stefan Schaefer, Hubert Simma, Stefan Sint, Rainer Sommer
QCD Coupling from a Nonperturbative Determination of the Three-Flavor $\Lambda$ Parameter
Phys.Rev.Lett. 119 (2017) 102001
arXiv: 1706.03821 [hep-lat]
Patrick Fritzsch, Alberto Ramos
The gradient flow coupling in the Schrödinger Functional
JHEP 10 (2013) 008
arXiv: 1301.4388 [hep-lat]
Mattia Bruno, Tomasz Korzec, Stefan Schaefer
Setting the scale for the CLS $2 + 1$ flavor ensembles
Phys.Rev.D 95 (2017) 074504
arXiv: 1608.08900 [hep-lat]
Hiroshi Suzuki, Hiromasa Takaura
$t \to 0$ extrapolation function in the small flow time expansion method for the energy–momentum tensor
PTEP 2021 (2021) 073B02
arXiv: 2102.02174 [hep-lat]
Hiroki Makino, Hiroshi Suzuki
Lattice energy-momentum tensor from the Yang-Mills gradient flow -- a simpler prescription
arXiv: 1404.2758 [hep-lat]
Luigi Del Debbio, Agostino Patella, Antonio Rago
Space-time symmetries and the Yang-Mills gradient flow
JHEP 11 (2013) 212
arXiv: 1306.1173 [hep-th]
Leonardo Giusti, Martin Lüscher
Topological susceptibility at $T>T_{\rm c}$ from master-field simulations of the SU(3) gauge theory
Eur.Phys.J.C 79 (2019) 207
arXiv: 1812.02062 [hep-lat]
Martin Lüscher
On the chiral anomaly and the Yang–Mills gradient flow
Phys.Lett.B 823 (2021) 136725
arXiv: 2109.07965 [hep-lat]