High-energy complete quantum field theories in four dimensions beyond gauge theories

• Applicant: Dr. Hannes Keppler
• Subject Area: Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 580518761

Quantum field theory (QFT) forms the foundation of modern fundamental physics and provides a highly successful framework for the Standard Model of particle physics. Yet, examples of mathematically consistent interacting QFTs in four dimensions (4D) are lacking. While a subsector of the Standard Model, quantum chromodynamics (QCD), is widely believed to be mathematically consistent, this has not yet been proven rigorously due to the intricacies of dealing rigorously with gauge symmetry. For mathematical consistency, a theory must be well-defined up to arbitrarily high energy scales (high-energy complete). This can be achieved if the theory becomes effectively non-interacting (asymptotically free) at high energies, as in QCD. Despite this simplicity at high energies, QCD becomes strongly interacting at low energies, and quarks and gluons confine to form hadrons. Rigorously constructing mathematically consistent QFTs is the aim of constructive field theory based on functional integrals and the Wilson-Polchinski renormalization group. To date, all the QFTs that have been constructed are either super-renormalizable or asymptotically free, like the Gross-Neveu model in 2D which is asymptotically free and undergoes dimensional transmutation at low energies. Finding and understanding 4D asymptotically free QFTs is highly desirable. This project aims to study two recently uncovered very promising families of such models: relativistic Luttinger Fermions (which are self-interacting non-Dirac Fermions) on the one hand, and large-N tensor field theory (which is a field theory with tensorial degrees of freedom) on the other. Preliminary studies of relativistic Luttinger Fermions showed that, despite being massless at high energies, a mass is generated non-perturbatively, and a Fermionic condensate forms at low energies. Current studies, however, neglect interaction channels and use rough approximation schemes. My goal is to rigorously demonstrate the non-perturbative mass generation in these models, including all the interaction channels that are generated by the renormalization group, proving estimates, and controlling rest terms. Such a result would establish this theory as a prime example of a mathematically consistent interacting QFT in 4D. For the tensor field theory, I aim to uncover low-energy phases with spontaneous symmetry breaking and bound state formation. In these theories, the so called large N limit of very large tensors allows a non-perturbative analysis, even at strong coupling. I aim to use functional methods (Schwinger-Dyson equations or fRG) to explore the model across all energy scales. This project will establish two controlled 4D QFTs that can serve as theoretical laboratories for exploring the interplay between high-energy completeness and low-energy dynamics.

• DFG Programme: Position