Finite temperature field theory for the masses

The Standard Model of particle physics is the most successful theory ever invented by humankind.At the same time, it is fundamentally incomplete.Among other things, it does not account for dark matter, which constitutes about 85% of the matter content of the universe.Many constraints on physics beyond the Standard Model come from astrophysics and cosmology.While collider physics computations can assume that processes happen in vacuum, our universe is far from being a vacuum, and this approximation will not hold in many astrophysical environments such as stars or the early universe.In these physical settings, an accurate description requires that we take into account thermal effects generated by the hot and dense medium.The Standard Model photon is particularly affected by in-medium effects.Indeed, densities of charged particles can be high in nature, and the photon's coupling to these charged particles is big enough to maintain a high interaction rate.In particular, the photon acquires an effective mass, or self-energy, in a plasma.Furthermore, it acquires a new, longitudinal degree of freedom, in addition to the two transverse ones present in vacuum.Approximately 30 years ago, Braaten and Segel derived beautiful analytic approximations for the photon self-energy, more specifically for the longitudinal and transverse modes in a homogeneous and isotropic plasma.These expressions have since been used by many researchers in the high-energy physics phenomenology community.However, these dispersion relations are only valid in the on-shell regime, meaning for actual photons that can propagate Hugo Schrer iv Abstract through the medium for an arbitrarily high distance, or in the soft photon momentum regime.There exists no analytic approximation for the general photon self-energy, which would also be generally valid off shell, meaning for virtual photons which are intermediate states in particle physics processes and which can have any energy and momentum.In this thesis, I first review some context about the search for physics beyond the standard model using cosmology and astrophysics, and show how the photon self-energy is a central element in a particular model, the dark photon.I then review some finite temperature field theory formalism and use it to compute the general photon self-energy in a homogeneous and isotropic medium to first order in the fine-structure constant.This is a general expression in the sense that it is valid on shell and off shell for any photon energy and momentum.I derive analytic approximations for the general self-energy, and I evaluate how good they are compared to numerical computations.

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