Lattice Field Theory

Applications of the gradient flow

Last Modified: 7 November 2023

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:

• Lattice determination of the QCD running coupling by the ALPHA collaboration [3]. The theoretical foundation of this work was laid down in [4].

• Scale setting in CLS Nf=2+1 configurations [5]. The use of the auxiliary quantity $t_0$ as an intermediate quantity to set the scale in QCD simulations was proposed in [1]. Given our ability to calculate $t_0/a^2$ very precisely from lattice simulations, this strategy (or variants of it) has imediately been adopted by many collaborations.

• Renormalization of the energy-momentum tensor in QCD via the small-flow time expansion [6]. The theoretical foundation of this work was laid down in [7]. See also [8] for an alternative, but related, approach.

• Determination of the topological susceptibility in SU(3) Yang-Mills at finite temperature [9]. The use of the gradient flow as a tool to define the topological charge has a long history, and was put on solid theoretical basis in [1] and [2]. The proof that the topological density defined with the gradient flow coincides in the continuum limit with the expression that appears in the anomalous axial Ward identity is given in [10].

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 $t_0^2 E(t_0) = 0.3$.

Projects

• Brandon Parfimczyk is studying a modification of the gradient flow, given by the equation $$ \frac{\partial}{\partial t} B_k(t,x) = \sum_{j=1}^3 D_j G_{jk}(t,x) \ . $$ This project has two main goals: to understand the renormalization properties of this 3d gradient flow, and to quantify cutoff effects to simple observables defined with this 3d gradient flow.

• Jonathan Zert has studied the theoretical properties and the continuum limit of a modification of QCD in which the gauge field in the Dirac operator is replaced with some smeared version of it. This setup is often used in lattice simulations because it is computationally cheaper and numerically more stable. In this project, the gradient flow is used as a particular type of smearing.

Bibliography

[1]	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]
[2]	Martin Luscher, Peter Weisz Perturbative analysis of the gradient flow in non-abelian gauge theories JHEP 02 (2011) 051 arXiv: 1101.0963 [hep-th]
[3]	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]
[4]	Patrick Fritzsch, Alberto Ramos The gradient flow coupling in the Schrödinger functional JHEP 10 (2013) 008 arXiv: 1301.4388 [hep-lat]
[5]	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]
[6]	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]
[7]	Hiroki Makino, Hiroshi Suzuki Lattice energy-momentum tensor from the Yang-Mills gradient flow -- a simpler prescription arXiv: 1404.2758 [hep-lat]
[8]	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]
[9]	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]
[10]	Martin Lüscher On the chiral anomaly and the Yang–Mills gradient flow Phys.Lett.B 823 (2021) 136725 arXiv: 2109.07965 [hep-lat]