A new article in the Encyclopedia of Nuclear Physics offers a pedagogical introduction to functional approaches to Quantum Chromodynamics (QCD) at finite temperature and chemical potential. The study focuses on the phase diagram of strongly interacting matter, a map that describes the different states of matter under extreme conditions of temperature and density, such as those found in the interior of neutron stars or in the early stages of the universe. Understanding this diagram is crucial for unraveling the fundamental nature of the strong force, which binds quarks and gluons to form protons and neutrons.

The work highlights the complementarity of functional methods, such as Dyson-Schwinger equations (DSE) and the functional renormalization group (fRG), with other first-principles approximations for non-perturbative QCD. These approaches are powerful theoretical tools that allow investigation of the behavior of quark-gluon matter in regimes where perturbative approximations are not valid. By combining these methodologies, physicists can obtain a more complete and robust picture of the phase transitions experienced by strongly interacting matter.

The article discusses selected results obtained with DSE and fRG, providing a general overview of the QCD phase diagram. These methods have allowed exploration of the existence of phases such as the quark-gluon plasma, a primordial soup of elementary particles believed to have existed shortly after the Big Bang, and other exotic phases of nuclear matter. The publication is designed to be accessible to both students and researchers not specialized in functional methods, serving as a concise guide to the more advanced literature in this field of fundamental research.