Effective Field Theories (EFTs) are used to describe physics beyond the Standard Model in a generic way. However, already in the simplest case, the EFT derived from the Standard Model contains around 100 field operators. A proper analysis requires the renormalization of these operators, by which they can mix with each other. However, not all operators mix with one another if one is only interested in the first few terms of perturbation theory.

The goal of this project is to systematically determine the sectors among the operators which mix with one another under renormalization.

In this project, you will learn the basic concepts of Effective Field Theories and renormalization.

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. At lowest order, this can be studied analytically, which shall then be cross-checked with a numerical and an approximate analytical approach.

You will learn:

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

Cross-section calculations play a central role in particle physics. Numerical integration is vital for making these prediction. The most commonly used method for integration is the Monte-Carlo integration. As the integrals get more and more complicated, the basic Monte-Carlo integration comes to its limits and has to be improved by variance reduction. Commonly used are importance sampling techniques like the VEGAS algorithm but recently also Machine Learning methods in the form of density estimators began to gain attention in the community.

The goal of this project is to study and implement variance reduction methods into our Monte-Carlo integrator. You will learn the basics of Monte-Carlo integration and variance reduction and gain insights into how cross sections are calculated. Here, an affinity to computer programming and familiarity with C or C++ is helpful.

FeynGame is a tool to learn about the concept of Feynman diagrams in a playful way. It allows to easily produce high-quality images of Feynman diagrams. In this project, you will design and implement a new type of game into FeynGame, where the player must construct valid Feynman diagrams from a given set of vertices.

You will learn:

• The algorithmic structure of Feynman diagrams.

Requirements:

• Interest in Feynman diagrams.
• Affinity to computer programming.