Functional renormalization group analysis of the soft mode at the QCD critical point

Abstract

We make an intensive investigation of the soft mode at the quantum chromodynamics (QCD) critical point on the basis of the functional renormalization group (FRG) method in the local potential approximation. We calculate the spectral functions ${\rho }_{\sigma ,\pi }(\omega ,p)$ in the scalar ($\sigma$) and pseudoscalar ($\pi$) channels beyond the random phase approximation in the quark--meson model. At finite baryon chemical potential $\mu$ with a finite quark mass, the baryon-number fluctuation is coupled to the scalar channel and the spectral function in the $\sigma$ channel has a support not only in the time-like ($\omega >p$) but also in the space-like ($\omega <p$) regions, which correspond to the mesonic and the particle--hole phonon excitations, respectively. We find that the energy of the peak position of the latter becomes vanishingly small with the height being enhanced as the system approaches the QCD critical point, which is a manifestation of the fact that the phonon mode is the {\em soft mode} associated with the second-order transition at the QCD critical point, as has been suggested by some authors. Moreover, our extensive calculation of the spectral function in the $(\omega ,p)$ plane enables us to see that the mesonic and phonon modes have the respective definite dispersion relations ${\omega }_{\sigma .\mathrm{ph}}(p)$, and it turns out that ${\omega }_{\sigma }(p)$ crosses the light-cone line into the space-like region, and then eventually merges into the phonon mode as the system approaches the critical point more closely. This implies that the sigma-mesonic mode also becomes soft at the critical point. We also provide numerical stability conditions that are necessary for obtaining the accurate effective potential from the flow equation.

arXiv comment26 pages, 9 figures, 1 table, v2: typos corrected, references added, descriptions improved, published version

References in corpus (57)

• Exact evolution equation for the effective potential
• QCD Phenomenology based on a Chiral Effective Lagrangian
• Non-Perturbative Renormalization Flow in Quantum Field Theory and Statistical Physics
• NJL-model analysis of dense quark matter
• The phase diagram of dense QCD
• Chiral effective model with the Polyakov loop
• Aspects of the Functional Renormalisation Group
• Phases of QCD: lattice thermodynamics and a field theoretical model
• Event-by-Event Fluctuations in Heavy Ion Collisions and the QCD Critical Point
• Optimised Renormalisation Group Flows
• Non-Gaussian fluctuations near the QCD critical point
• On the Phase Diagram of QCD
• The Phase Structure of the Polyakov--Quark-Meson Model
• Maximum Entropy Analysis of the Spectral Functions in Lattice QCD
• On the sign of kurtosis near the QCD critical point
• Universality, the QCD critical/tricritical point and the quark number susceptibility
• Slowing Out of Equilibrium Near the QCD Critical Point
• Introduction to the functional RG and applications to gauge theories
• Chiral Phase Transition within Effective Models with Constituent Quarks
• Susceptibilities and the Phase Structure of a Chiral Model with Polyakov Loops
• Fluctuations as probe of the QCD phase transition and freeze-out in heavy ion collisions at LHC and RHIC
• Third moments of conserved charges as probes of QCD phase structure
• The Phase Diagram of the Quark-Meson Model
• Lattice QCD at Finite Density -- An introductory review
• Inhomogeneous phases in the Nambu-Jona-Lasino and quark-meson model
• Chiral symmetry and density wave in quark matter
• Susceptibilities near the QCD (tri)critical point
• Dynamic universality class of the QCD critical point
• Effective Action for the Quark-Meson Model
• The renormalization group and quark number fluctuations in the Polyakov loop extended quark-meson model at finite baryon density
• How many phases meet at the chiral critical point?
• Meson fluctuations and thermodynamics of the Polyakov loop extended quark-meson model
• On the phase structure and thermodynamics of QCD
• Critical endpoint in the Polyakov-loop extended NJL model
• Spectral Functions for the Quark-Meson Model Phase Diagram from the Functional Renormalization Group
• Exploring the QCD Phase Structure with Beam Energy Scan in Heavy-ion Collisions
• Sigma and hydrodynamic modes along the critical line
• Källén-Lehmann spectroscopy for (un)physical degrees of freedom
• Polyakov loop potential at finite density
• Chiral dynamics in a magnetic field from the functional renormalization group
• On the relevance of matter and glue dynamics for baryon number fluctuations
• Real-Time Correlation Functions in the O(N) Model from the Functional Renormalization Group
• Towards quantitative precision in the chiral crossover: masses and fluctuation scales
• Linking the Quark Meson Model with QCD at High Temperature
• Flow equations for spectral functions at finite external momenta
• Fluctuations and isentropes near the chiral critical endpoint
• Dyson-Schwinger study of chiral density waves in QCD
• The Functional Renormalization Group and O(4) scaling
• Net quark number probability distribution near the chiral crossover transition
• Criticality of the net-baryon number probability distribution at finite density
• Dynamical Density Fluctuations around QCD Critical Point Based on Dissipative Relativistic Fluid Dynamics-possible fate of Mach cone at the critical point-
• Phase diagram and nucleation in the Polyakov-loop-extended Quark-Meson truncation of QCD with the unquenched Polyakov-loop potential
• Weak solution of the non-perturbative renormalization group equation to describe the dynamical chiral symmetry breaking
• Functional Renormalization Group Study of Phonon Mode Effects on Chiral Critical Point
• Net baryon number fluctuations across the chiral phase transition at finite density in the strong coupling lattice QCD
• Momentum scale dependence of the net quark number fluctuations near chiral crossover
• Critical Opalescence around the QCD Critical Point and Second-order Relativistic Hydrodynamic Equations Compatible with Boltzmann Equation

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• Functional renormalization-group calculation of the equation of state of one-dimensional nuclear matter inspired by the Hohenberg--Kohn theorem
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