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© 2002 Amer Physical Soc This work is partially supported by Research Contract No. PID2019-106080 GB-C21 (Spanish "Ministerio de Ciencia e Innovacion"), and the European Union Horizon 2020 research and innovation program under Grant Agreement No. 824093. A. V.-R. acknowledges support from a fellowship of the UCM predoctoral program., We revisit the most general effective Lagrangian within chiral perturbation theory at nonzero isospin chemical potential mu(I) up to O(p(4)). In addition to the contributions already considered in the literature, we discuss the effects of new terms allowed by the symmetries derived within the external source method including spurion fields, as well as linear-field corrections relevant to O(p(4)). We study the influence of those new contributions to the free energy density at zero temperature and observables derived from it, such as the pion and quark condensates and the isospin density. Corrections are shown to be compatible with lattice results, which favor nonzero values for the low-energy constants (LECs) multiplying the new O(p(2)) and O(p(4)) field operators in the Lagrangian. In particular, the O(p(4)) LECs are renormalized to render the free energy density finite. Constraints on the LECs arise from preserving the physical condition n(I)(mu(I) < mu(c)) = 0, while mu(c) = M-pi, still holds to leading order and can be maintained to next-to-leading order through an additional constraint requiring the new LECs., Unión Europea. H2020, Ministerio de Ciencia e Innovación (MICINN), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2020 MDPI Work partially supported by research contract FPA2016-75654-C2-2-P (spanish "Ministerio de Economia y Competitividad"). This work has also received funding from the European Union Horizon 2020 research and innovation programme under grant agreement No 824093., We review recent advances in the understanding of the Quantum Chromodynamics (QCD) transition and its nature, paying special attention to the analysis of chiral symmetry restoration within different approaches based on effective theories. After presenting some of the main aspects of the current knowledge of the phase diagram from the theoretical, experimental and lattice sides, we discuss some recent problems where approaches relying on effective theories have been particularly useful. In particular, the combination of ideas such as Chiral Perturbation Theory, unitarity and Ward Identities allows us to describe successfully several observables of interest. This is particularly relevant for quantities expected to be dominated by the light meson components of the hadron gas such as the scalar and topological susceptibilities. In addition, ward identities and effective Lagrangians provide systematic results regarding chiral andU(1)Apartner degeneration properties which are of great importance for the interplay between those two transitions and the nature of chiral symmetry restoration. Special attention is paid to the connection of this theoretical framework with lattice simulations., Unión Europea. Horizonte 2020, Ministerio de Economía y Competitividad (MINECO), Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub