Standard Model parameters and observables from gradient flow

12 May 2026, 08:45 → 15 May 2026, 17:35 Europe/London

Lecture theatre B, 3rd Floor (James Clerk Maxwell Building)

Matthew Black (University of Edinburgh), Jorge Luis Golan Dasilva, Curtis Peterson, Luigi Del Debbio (The University of Edinburgh)

Gradient flow is a diffusion-like smearing transformation that smoothly suppresses short-distance ultraviolet fluctuations, thus introducing a simple non-perturbative method to renormalise strongly-coupled theories such as Quantum Chromodynamics (QCD). 

It has become an invaluable tool in lattice simulations of both QCD and theories beyond the Standard Model, providing a method for e.g. scale setting and the topological structure of the vacuum.

Many recent developments have identified the deep potential of the gradient flow as a renormalisation procedure, opening new methodologies to calculate running couplings, the renormalisation group beta function, and renormalised composite operators.

These applications often depend on perturbative calculations in the short-flow-time expansion to match the gradient flow to other renormalisation schemes and provide phenomenologically-relevant results, for observables such as the energy-momentum tensor, quark masses, and parton distribution functions.

This workshop will convene the leading experts on both the perturbative and non-perturbative aspects of gradient flow to tackle the current theoretical and computational challenges, and to chart the future directions of this growing field. 

Confirmed speakers

Martin Lüscher (CERN)

Robert Harlander (RWTH Aachen)

Anna Hasenfratz (University of Colorado) Martin Beneke (Technical University of Munich)

Alberto Ramos (University of Valencia)

Oliver Witzel (Siegen University)

Fabian Lange (University of Zurich & PSI)

Robert Mason (RWTH Aachen)

Jonas Kohnen (RTWH Aachen)

Guilherme Catumba (Milan Bicocca University)

Antonio Rago (University of Southern Denmark)

Sofie Martins (University of Graz)

Chik Him Wong (University of Wuppertal) Julian Mayer-Steudte (Technical University of Munich) Oscar Lara Crosas (University of Zurich) Andrea Shindler (RWTH Aachen) Nathan Mackey (University of Colorado)

 

All participants are expected to arrange and pay for their own travel and accommodation.

 

Nordic Lattice: Some participants may also be interested in the Nordic Lattice meeting taking place in Edinburgh after this workshop, 18th-20th May (see https://indico.ph.ed.ac.uk/e/NOLA26).

 

Please note that participation is moderated as we can only host a limited number of participants.

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Tuesday 12 May

08:45 → 09:00 Greetings

Speakers: Matthew Black (University of Edinburgh), Curtis Peterson, Jorge Dasilva

09:00 → 10:00 Gradient flow in perturbation theory

I will give an overview of methods, tools, and applications of the gradient-flow formalism at higher orders in perturbation theory.

Speaker: Robert Harlander

10:00 → 10:30 Gradient-flowed operator product expansion of the Adler function without IR renormalons

A long-standing problem concerns the question how to consistently combine perturbative expansions in QCD with power corrections in the context of the operator product expansion (OPE), since the former exhibit ambiguities due to infrared renormalons, which are of the same order as the power corrections. We propose to use the gradient flow time 1/\sqrt{t} as a hard factorization scale and to express the OPE in terms of IR renormalon-free subtracted perturbative expansions and unambiguous matrix elements of gradient-flow regularized local operators. We show on the example of the Adler function and its leading power correction from the gluon condensate that this method dramatically improves the convergence of the perturbative expansion. We employ lattice data on the action density to estimate the gradient-flowed gluon condensate, and obtain the Adler function with non-perturbative accuracy and significantly reduced theoretical uncertainty, enlarging the predicitivity at low Q^2.

Speaker: Martin Beneke

10:30 → 11:00 Coffee Break

11:00 → 11:30 Quark mass effects in flowed action density

The gradient flow provides a consistent scheme for matching perturbation theory and lattice field theory. In the continuum this has typically been done with the simplifying assumption that quarks are massless. However, this neither reflects lattice computations nor physical reality. In this talk we discuss the computation of three quantities fundamental to the gradient flow, the vacuum expectation values of the flowed fermion and gluon condensates and the fermion kinetic operator, and consider their quark mass effects to the three loop level. We then briefly discuss the idea of Takaura et. al. to use these mass effects with lattice data for precision estimates of the quark masses.

Speaker: Robert Mason (RWTH Aachen University)

11:30 → 12:00 The SFTX of LEFT

Speaker: Oscar Lara Crosas

12:00 → 12:30 The perturbative matching of four-quark operators between gradient flow and MSbar

Speaker: Jonas Kohnen

12:30 → 13:00 Lattice determination of bag parameters using gradient flow

Speaker: Antonio Rago

13:00 → 14:30 Lunch

14:30 → 15:30 Discussion Session

15:30 → 16:00 Coffee Break

16:00 → 16:30 The perturbative Ricci flow

One candidate for a quantum theory of gravity is the Asymptotic Safety conjecture, which postulates the existence of a non-trivial fixed point for the gravitational couplings. We investigate the search for such a fixed point within the framework of perturbation theory.

For this, we employ the Ricci flow as the central tool, pursuing a perturbative scheme analogous to the gradient flow formalism in QCD. In QCD, the gradient flow was introduced to bridge the gap between lattice simulations and perturbative calculations, allowing for a definition of a renormalisation scheme that is accessible in both settings. Applied to gravity the gradient flow is known as Ricci flow. The core idea is to utilize this framework to compute the beta function of the gravitational coupling by explicitly defining it within the Ricci flow scheme.

In this talk, I will present the first steps towards constructing a perturbative implementation of the Ricci flow and the definition of Newton's coupling in this scheme.

Speaker: Henry Werthenbach (TTK, RWTH Aachen University, 52056 Aachen, Germany)

Wednesday 13 May

09:00 → 10:00 Lattice Gradient Flow

Speaker: Anna Hasenfratz

10:00 → 10:30 Quark masses using SFTX

Speaker: Oliver Witzel

10:30 → 11:00 Coffee Break

11:00 → 11:30 A new approach to quark masses using gradient flow

Speaker: Fabian Lange

11:30 → 12:00 Quark masses

Speaker: Akhil Chauhan (University of Illinois Urbana Champaign)

12:00 → 13:00 Discussion Session

13:00 → 14:30 Lunch

14:30 → 15:30 Higgs Centre Colloquium -- Demystifying lattice QCD computations of alpha_s

Lattice QCD permits the strong coupling constant $\alpha_s$ to be determined from the masses and decay constants of the light hadrons. The precision achieved in these computations is competitive with the one of the world average of the experimental measurements of the coupling and is likely to improve in the coming years. In this colloquium, the aim is to explain, in terms that do not assume any expert knowledge of lattice QCD, how exactly the hadronic regime of QCD is connected to high-energy collider physics and how the lattice calculations of $\alpha_s$ proceed.

Higgs Centre Link

Speaker: Martin Luscher

15:30 → 16:00 Coffee Break

16:00 → 16:30 The running coupling

Speaker: Alberto Ramos

19:30 → 22:00 Conference Dinner

Thursday 14 May

09:30 → 10:00 Anomalous dimensions

Speaker: Nathan Mackey (University of Colorado Boulder)

10:00 → 10:30 Lattice Determination of Lambda_QCD

In recent years, gradient flow has been utilized in lattice simulations to determine Lambda QCD . In this talk, we discuss our results at Nf=3 massless fermions, and briefly outline our plan to proceed to the physical point.

Speaker: Chik Him Wong

10:30 → 11:00 Coffee Break

11:00 → 11:30 Classically perfect gradient flow

Classically perfect fixed-point lattice actions preserve continuum classical properties while reducing lattice artifacts at the quantum level. They allow the extraction of continuum physics from coarser lattices and hence provide an effective way to overcome the challenges of critical slowing down and topological freezing as the continuum limit is approached. In this talk we show that fixed-point actions can be used to define classically perfect gradient-flow observables which are free of tree-level lattice artifacts to all orders. We demonstrate the effectiveness of this approach using a fixed-point action for four-dimensional SU(3) gauge theory obtained from a machine-learned gauge-covariant convolutional neural network.

Speaker: Urs Wenger (University of Bern)

11:30 → 12:00 Adjoint chromoelectric correlators with lattice QCD

In various effective field theory descriptions, adjoint chromoelectric correlators play a crucial role. At finite temperatures, these correlators are linked to the diffusion of heavy quarks and quarkonium, which are important for modeling non-equilibrium dynamics. At zero temperature, the correlator is used in the pNRQCD framework to describe inclusive decays of quarkonium. However, to achieve a complete understanding, non-perturbative lattice computations are necessary. In these lattice computations, gradient flow enhances the signal and helps in renormalizing the operators. In this presentation, we will show and discuss the recent results from our calculations of chromoelectric correlators.

Speaker: Julian Mayer-Steudte (Technical University of Munich)

12:00 → 12:30 Gradient flow scale setting with the pion decay constant in QCD+QED

We present the latest determination of the gradient flow scale $w_{0}$ from the BMW Collaboration. The value of $w_{0}$ is determined from the pion decay constant $f_{\pi}$ using QCD ensembles with tree-level improved Lüscher-Weisz gauge action and N$_{f}$=2+1+1 4stout staggered quarks at the physical point. We use multiple lattice spacings, ranging from $0.11$ to $0.048$ fm, to extrapolate to the continuum limit. In addition, we employ two lattice definitions of the gradient flow, the Wilson flow and the improved version, the Zeuthen flow, to have more control over cutoff effects. To check for finite volume effects, we simulate different finite boxes. Isospin-breaking corrections are computed from first principles and included in the final result.

Speaker: Alessandro Cotellucci

12:30 → 13:00 Scale setting in finite-temperature QCD via gradient-flow rescaling

We discuss how to set the scale of finite-temperature lattice QCD via step-scaling. In practice, starting from a reference lattice with known scale, we want to tune the bare coupling and quark masses of a second lattice so that its lattice spacing is a chosen fraction of the reference one, while quark masses and temperature are fixed in physical units, up to lattice artifacts. This can be achieved using the gradient-flow renormalized coupling to tune the bare coupling, and mesonic screening masses to tune the bare quark masses. In particular, the ratio of lattice spacings can be set by rescaling the flow time as a function of the renormalized coupling for the two lattices. We test this method in the pure-gauge theory and present some preliminary results for QCD with dynamical fermions. The end goal is to use the resulting lines of constant physics for high-temperature determinations of the topological susceptibility.

Speaker: Andrea Giorgieri (University of Pisa)

13:00 → 14:30 Lunch

14:30 → 15:30 Discussion Session

15:30 → 16:00 Coffee Break

16:00 → 16:30 Lambda

Speaker: Yash Mandlecha

Friday 15 May

09:00 → 09:30 Extracting the gluon momentum fraction of the nucleon via the gradient flow

The gluon momentum fraction of the nucleon is a key measure of hadronic structure. It characterises the average hadron momentum carried by gluons and is directly related to the energy-momentum tensor, which is central to the origin of mass in quantum chromodynamics (QCD). I report on a recent calculation [arXiv:2602.14260] by the HadStruc Collaboration that uses the gradient flow as a novel nonperturbative renormalisation procedure to extract the gluon momentum fraction from lattice QCD. We determined the gluon momentum fraction on a single Wilson-clover ensemble using Nf = 2+1 flavors with pion mass 358 MeV and lattice spacing 0.094 fm. Our final result, evaluated at a scale of 2 GeV in the MS-bar scheme, is <x>_g = 0.482(35), where we quote only statistical uncertainties.

Speaker: Chris Monahan (Colorado College)

09:30 → 10:00 Pion PDFs / EDM

Speaker: Andrea Shindler

10:00 → 10:30 Lattice Study of the Pion Vector Form Factor in the Instanton Liquid Model Using Gradient Flow

Instanton liquid model is believed to capture the main features of vacuum QCD dynamics. Recently, multiple predictions for hadron structure functions have been derived and compared with experimental measurements and lattice QCD calculations, finding a general agreement. In order to explore the precision of the instanton liquid model, one has to compare its predictions with non-perturbative simulations in a regime dominated by instanton dynamics.
This has been performed for two gluon-sensitive observables: the gluon Green's function and the strong running coupling constant [1]. In this contribution, we propose to study a fermionic observable, the pion vector form factor, for which instanton liquid model predictions have been discussed in [1]. We use the gradient flow to single out the dominant contribution from the instantons out of a lattice QCD configuration ensemble. We describe the details of our numerical setup, and some first, preliminary results.

References:
1. Wei-Yang Liu, Edward Shuryak, and Ismail Zahed, Phys. Rev. D 109,
074029 (2024). doi:10.1103/PhysRevD.109.074029

Speaker: Vaibhav Chahar (Jagiellonian University)

10:30 → 11:00 Coffee Break

11:00 → 11:30 Renormalization Constants of the Effective Electroweak Hamiltonian via Flowed Fermion Fields

Non-perturbative determinations of renormalization coefficients are an integral part of the continuum limit of lattice calculations. We explore an alternative strategy of determining the renormalization constants of the effective electroweak Hamiltonian. The method uses composite operators at positive flow time as probes together with finite axial Ward identities to resolve the mixing due to the breaking of chiral symmetry generated by Wilson-like fermions. The resulting renormalization conditions lead to relations between gauge-invariant off-shell two-point functions in coordinate space. As a simplified example, I will present the aforementioned strategy in the setting of twisted mass fermions to determine the renormalization constant of the axial current, including numerical results. The extension to the effective electroweak Hamiltonian will be discussed, with hopes to be used in an ongoing project to perform a measurement of the isospin-breaking corrections of the tau decay rate, which is discussed by Erik Bäske in this workshop.

Speaker: Lukas Holan (Humboldt University of Berlin)

11:30 → 12:00 Isospin-breaking corrections to the tau decay-rate

The inclusion of isospin-breaking corrections is necessary for high precision measurements of hadronic observables. One such observable with phenomenological relevance is the tau decay-rate, which is connected to the CKM-matrix elements. The isospin-symmetry is broken by both the interaction of the quarks with the photon and their mass differences. In the following we would like to present one possible approach that we plan to follow in order to improve upon the currently available iso-symmetric calculations. The breaking of the symmetry and the inclusion of the additional photon field poses new challenges. The first of these is the renormalization of the effective electro-weak Hamiltonian that mediates the decay. The initially large $5\times 5$ mixing can be reduced by employing Wilson averages for the valence OS twisted-mass fermions. This enables us to reasonably compute the relevant 4-point function, for which the effective mixing pattern is reduced to just $2\times 2$. We plan to calculate the renormalization constants using flowed probe operators at finite flow time, see L. Holan's talk for more information. Furthermore, to include electromagnetic effects we use $C^*$ boundary conditions, which allows us to treat QED fully non-perturbatively in a finite volume. The final challenge is to obtain the real time-decay rate or spectral density from the euclidean data. We plan to address this ill-posed inverse Laplace transform with the HLT-method.

Speaker: Erik Bäske (Humboldt Universität zu Berlin)

12:00 → 12:30 A non-perturbative definition of Magnetostatic QCD

High-temperature QCD can be described, through dimensional reduction,
by a 3D effective field theory. Electrostatic QCD consists of a
gauged-scalar theory, while magnetostatic QCD of a pure 3D gauge
theory. The use of dimensionally reduced effective theories to
predict 4D observables relies on the matching between the 3D and 4D
theories, which is known only perturbatively, and is thought to be
highly non-trivial due to the behavior of perturbation theory at
finite temperature. We have devised a strategy to perform the first
non-perturbative computation of the matching, for which we consider
gradient flow quantities in a finite volume setup, allowing us to keep
all lattice systematics under control, and perform the matching over a
large range of temperatures with high precision. As a first step in
this study, we investigate the non-perturbative matching between
high-temperature pure gauge theory and MQCD for which we report the
corresponding results.

Speaker: Dr Guilherme Catumba (University Milano-Bicocca)

12:30 → 13:00 Smoothing properties of the gradient flow at large $N_c$

Tunnelling between distinct topological sectors, described by instantons, plays a central role in understanding decorrelation in lattice gauge theory simulations. The gradient flow allows the smoothing of gauge configurations and, hence, the precise examination of these topological features. In this presentation, we systematically analyse the discretisation effects on topological quantities induced by the gradient flow using a theory with a single fermion in the two-index antisymmetric representation as an example. This particular theory serves as a proxy for supersymmetric models at large $N_c$ and could naively exhibit fractional topological charge. By improving the reliability and consistency of topological charge measurement and scale setting, we probe properties at the centre of the confinement mechanism for theories beyond QCD, which is particularly relevant for controlling simulations beyond the Standard Model on the lattice.

See published results in 2501.16043 (PoS), 2504.10197 (PRD) and 2603.05155.

Speaker: Sofie Martins (University of Graz)

13:00 → 14:30 Lunch

14:30 → 15:30 Discussion Session

15:30 → 16:00 Coffee Break

16:00 → 16:30 The gradient flow coupling of three- and four-dimensional QED

Connecting low-energy lattice results to high-energy continuum calculations is highly nontrivial, as they rely on fundamentally different regularizations. The gradient flow offers an elegant way to bridge this gap, since it can be implemented both on the lattice and in the continuum and naturally defines renormalized couplings and observables.

While the GF up to now has found most of its applications in QCD, I consider its implementations in other theories such as scalar QCD, or theories with U$(1)$ gauge factor. I present explicit results for the perturbative gradient-flow coupling for QED in (3+1) and (2+1) dimensions. QED$_4$ serves as a clean Abelian testing ground and limiting case of QCD, while QED$_3$ shares important qualitative features with QCD such as chiral symmetry breaking and the presence of an infrared fixed point in the large-$N_f$ expansion.

Speaker: Lars Georg (TTK RWTH Aachen University)