Pion dynamics in a soft-wall AdS-QCD model

Pseudo-Goldstone modes appear in many physical systems and display robust universal features. First, their mass m obeys the so-called Gell-Mann-Oakes-Renner (GMOR) relation f2m2 = Hσ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H\overline{\sigma} $$\end{document}, with f the Goldstone stiffness, H the explicit breaking scale and σ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\sigma} $$\end{document} the spontaneous condensate. More recently, it has been shown that their damping Ω is constrained to follow the relation Ω = m2Dφ, where Dφ is the Goldstone diffusivity in the purely spontaneous phase. Pions are the most paradigmatic example of pseudo-Goldstone modes and they are related to chiral symmetry breaking in QCD. In this work, we consider a bottom-up soft-wall AdS-QCD model with broken SU(2)L × SU(2)R symmetry and we study the nature of the associated pseudo-Goldstone modes — the pions. In particular, we perform a detailed investigation of their dispersion relation in presence of dissipation, of the role of the explicit breaking induced by the quark masses and of the dynamics near the critical point. Taking advantage of the microscopic information provided by the holographic model, we give quantitative predictions for all the coefficients appearing in the effective description. In particular, we estimate the finite temperature behavior of the kinetic parameter r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{r} $$\end{document}2 defined as the ratio between the Goldstone diffusivity Dφ and the pion attenuation constant DA. Interestingly, we observe important deviations from the value r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{r} $$\end{document}2 = 3/4 computed in chiral perturbation theory in the limit of zero temperature.