# Perturbative Quantum Field Theory

and Applications

The main tool used in predictions for particle collisions at high-energy colliders is the perturbative evaluation of quantum field theory, more precisely the order-by-order evaluation of quantum-mechanical transition matrix elements. In a given perturbative order all contributions to transition amplitudes can be graphically represented by so-called Feynman diagrams, which are characterized by external lines representing the initial and final states of the considered transition and by a fixed number of loops that reflects the perturbative order. The complexity of explicit calculations rapidly increases both with the number of external legs and/or internal loops.

The theoretical description of most particle reactions starts at the tree level, i.e. no loops occur. In this leading order (LO) of perturbation theory, the progress since the mid 1980s rendered predictions based on full transition matrix elements possible for particle reactions with up to roughly 10 particles in the final states. The situation can be considered fully under control, since even several automated Monte Carlo generators for such predictions exist with corresponding interfaces to parton shower and hadronization. At next-to-leading order (NLO) level, which involves one-loop diagrams, predictions with up to 5 external particles became standard. Reactions of the type $2\to4,5$ particles define the current technical frontier. At the next-to-next-to-leading order (NNLO) level, first complete calculations both for electron-positron and proton-proton scattering have been presented within QCD in recent years. Complete NNLO calculations with many mass scales, as e.g. required by electroweak theory, are still beyond present technical possibilities.

Higher-order perturbative calculations do not only involve loop diagrams, but also emission processes involving more and more additional massless or at least light particles (photons, gluons, leptons, light quarks). Only a thorough combination of virtual (loop) corrections and real emission contributions delivers phenomenologically sensible predictions, because both the virtual exchange and the real emission of low-energetic (``soft'') particles involve infrared-divergent contributions which, however, cancel in the sum of virtual and real corrections. At NLO this step of cancelling infrared singularities is well understood, and powerful techniques exist for practical calcuations, at NNLO the constriction of such techniques is still subject of current research.

As explained above, fixed-order perturbative predictions for particle reactions involve a fixed number of particles in the final state. For instance, real emission contributions at NLO involve one additional light particle in the final state, NNLO emission one or two additional light particles, etc. Although higher perturbative orders in this way start to describe jets (i.e. directed hadronic particle showers), the first few theoretically fully acessible orders are not sufficient to describe these jets well. On top of fixed-order predictions realistic simulations of particles reactions have to include a parton shower, which exploits the universal properties of collinear splittings of $1\to2$ light particles. The accuracy of parton shower algorithms is typically logarithmic, and the merging of fixed-order predictions with such showers is a non-trivial task, because double-counting of particle branchings has to be carefully avoided. At LO this problem is solved, but practical general solutions for NLO are still in progress.

A completely different interesting issue in perturbative quantum field theory concerns the description of unstable particles. Standard perturbation theory is not able to describe particle resonances in any fixed order, because the propagator of the unstable particle tends to infinity on resonance in any finite order of perturbation theory. Performing the so-called Dyson summation of all propagator corrections leads to a reasonable finite propagator near the resonance, but the unavoidable mixing of perturbative orders via this partial summation jeopardizes important consistency relations of the amplitude that reflect the gauge invariance of the theory. Several solutions have been suggested in the literature, but not all are consistent. Beyond LO (to our understanding) only very few methods are convincing: a systematic expansion about the resonance pole in the propagator (``pole scheme''), the consistent use of complex masses and couplings for unstable particles (``complex-mass scheme''), and dedicated effective field theories for the resonance expansion.

In our group we work on the following field-theoretical issues:

- Techniques for the evaluation of one-loop diagrams with many external legs
- Infrared structure of Feynman diagrams and integrals
- Field-theoretical description of unstable particles, e.g. in the complex-mass scheme