The lifetime and mixing parameters of bound quark states is a prime example for the application of the gradient flow. In this project, you will provide important input to the determination of these parameters.

You will learn the concepts of the gradient flow, effective field theories, flavor physics, and perturbative calculations. We are working in close cooperation with lattice experts on this topic, so you will also get insight into this field.

The gradient flow formalism provides a way relate perturbative calculations to lattice calculations in QCD. Up to now, the perturbative calculations have been mostly restricted to massless quarks. However, quark mass effects can become important for large flow times.

The goal of this project is to study the gradient flow of QCD with massive quarks. While such problems can be dealt with in a purely numerical way, you are going to approach the problem analytically at the level of asymptotic expansions.

You will learn:

• The general method of the gradient flow.
• Approaches to calculating non-standard Feynman integrals.

Applying the concept of the gradient flow to gravity leads to the so-called Ricci flow. It is a well-known concept in topology and was used by Grigori Perelman to prove the Poincaré conjecture.

In this project, you are going to consider the Ricci flow from a perturbative perspective. This corresponds to a novel regularization of the UV divergences in gravity and may lead to new insights concerning its renormalization group structure. Specifically, you will couple quantum-electrodynamics to gravity in this framework. In this way, you may contribute to the long-standing questions whether QED becomes asymptotically free when coupled to gravity.

• Lars Georg (September 2025)
  The Short-Flow-Time Expansion of the Dimension-Four Operators in Scalar QCD
• Lucas Cesinger (September 2025)
  The Short-Flow-Time Expansion of the CP-Odd Gluonic Dimension-Six Operator
• Nils Felten (October 2024)
  The gradient flow for scalar QCD at next-to-next-to leading order
• Jonas Kohnen (October 2023, RWTH Aachen)
  Small-flow-time expansion of quark bilinears at next-to-next-to-leading order QCD
• Janosch Borgulat (April 2022, RWTH Aachen)
  Towards the Full Energy-Momentum Tensor in the Gradient Flow Formalism

• R.V. Harlander, Y. Kluth, J.T. Kohnen, H. Werthenbach,
  The perturbative Ricci flow in gravity
  [arXiv:2604.18678]
  
• L. Georg, R.V. Harlander, R.H. Mason,
  The gradient-flow coupling of three- and four-dimensional QED
  Phys. Rev. D 113 (2026) 076016 [arXiv:2601.13914]
  
• M. Black, R.V. Harlander, A. Hasenfratz, A. Rago, O. Witzel,
  Renormalized quark masses using gradient flow
  Phys. Rev. D 113 (2026) 094504 [arXiv:2506.16327]
  
• M. Black, R. Harlander, F. Lange, A. Rago, A. Shindler, O. Witzel,
  Gradient Flow Renormalisation for Meson Mixing and Lifetimes
  PoS LATTICE2024 (2025) 243 [arXiv:2409.18891]
  
• J. Borgulat, R.V. Harlander, J.T. Kohnen, F. Lange,
  Short-flow-time expansion of quark bilinears through next-to-next-to-leading order QCD
  JHEP 05 (2024) 179 [arXiv:2311.16799]
  
• R.V. Harlander, Y. Kluth, and F. Lange,
  The two-loop energy-momentum tensor within the gradient-flow formalism
  Eur. Phys. J. C 78 (2018) 944 [err. ibid. C79 (2019) 858]
  [arXiv:1808.09837]
  

AutoEFT consistently generates the physical operators of an effective field theory. However, at higher orders, it is often necessary to include unphysical operators at intermediate steps of the calculation. Among them are the so-called BRST-exact operators which are usually very cumbersome to find. The goal of this project is to construct BRST-operators systematically and to develop an algorithm which could be implemented in AutoEFT.

In this project, you will learn the concepts of Effective Field Theories, BRST-invariance, operator renormalization, and related topics.

The representation of an Effective Field Theory is not unique. In order to be able to compare experimental results to theoretical predictions, it is necessary to be able to translate one operator basis into the other.

In this project, you will work out a conversion method for operator bases and supply AutoEFT with the capability to express arbitrary operators in terms of a particular basis.

You will learn the concepts of effective field theories and group theory.

• Lars Bündgen (September 2025)
  Automatic construction of Green's bases for effective field theories
• Nick Michaelis (March 2026)
  Factorizable Operators in Effective Field Theories
• Maximilian Rzehak (November 2023, RWTH Aachen)
  Representation of operators in effective field theories
• Tim Kempkens (October 2021, RWTH Aachen)
  Automated Generation of EFT Operators

• R.V. Harlander and M.C. Schaaf,
  AutoEFT: Automated Operator Construction for Effective Field Theories
  Comput. Phys. Commun. 300 (2024) 109198 [arXiv:2309.15783]
  
• R.V. Harlander, T. Kempkens, M.C. Schaaf,
  Standard Model Effective Field Theory up to Mass Dimension 12
  Phys. Rev. D 108 (2023) 055020 [arXiv:2305.06832]