Matter alters its properties remarkably when confronted with extreme conditions such as temperatures as high as in the early universe. The emergence of the Quark-Gluon Plasma and restoration of electroweak symmetry through phase transitions are but the most prominent phenomena to invigorate studies of gauge theories at finite temperatures. If the temperature is sufficiently high, static observables are effectively described in a reduced dimension by a framework known as Dimensional Reduction.

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... e computer algebraic multi-loop treatment of perturbation theory for finite-temperature theories is at the core of this thesis. It adopts sophisticated tools from zero temperature to decimate typically vast numbers of Feynman integrals with the objective to automate the dimensional reduction. To accomplish this, integration-by-parts identities pertinent to both massless and massive loops at finite temperature are illuminated. Additionally, an inclusion of higher-dimensional operators in these theories is first motivated and then generalised. The developed tools are applied to review the advancements of [1] in chapter 4 and [2] in chapter 5. There, we analyse the dimensionally reduced theories of high-temperature QCD, namely electrostatic and magnetostatic QCD. We inspect three-loop contributions stemming from non-static modes to the magnetostatic coupling in dimensionally reduced hot Yang-Mills theory [1] . By including dimension-six operators the result is found to be infrared finite and influenced by all scales in the QCD hierarchy. Incorporating also electrostatic effects indicates a non-perturbative ultrasoft gauge coupling at O(α 3/2 s ). Based on its relevance in cosmology, we determine another low-energy coefficient in electrostatic QCD, the Debye mass. By including effects from massive fermions up to two loops [2], energy ranges of (1 GeV-10 TeV) are scanned to show the smooth crossing of quark mass thresholds. course patience throughout the various stages of collaboration during the evolution of this work. I thank Gert Aarts for agreeing to co-referee this thesis and being the external expert on the defence committee. I specially thank York Schröder for hosting me during my visit in Chile, our fruitful collaboration, and the many FORM language lessons. Additionally, I want to emphasise former and current members of the Thermal Field Theory and Particle Cosmology group at the University of Bern namely Greg, Tuomas, Simone, and Germano for their dedicated conversations, programming advice, and inspirations during the preparation of my dissertation. However, none of this could have worked out without my friends Andrew, Conny, Irina, Markus, Tanja, Vivi, and Rose that constantly succeeded to put everything back into perspective. Finally, I wish to thank my parents for their unconditional support that opened the door for many opportunities during my studies -I could not appreciate it more.