We discuss the QCD phase structure at finite temperature and chemical potential for $2$-flavour and $2+1$-flavour QCD. The results are achieved by computing QCD correlation functions within a generalised functional approach that combines Dyson-Schwinger equations (DSE) and the functional renormalisation group (fRG). In this setup fRG precision data from arXiv:1706.06326 for the vacuum quark-gluon vertex and gluon propagator of $2$-flavour QCD are used as input, and the respective DSEs are expanded about this input. While the vacuum results for other correlation functions serve as a self-consistency check for functional approaches, the results at finite temperature and density are computed, for the first time, without the need of phenomenological infrared parameters.
We review recent work by the ALPHA and UKQCD Collaborations where masses and matrix elements were computed in lattice QCD using Schrödinger functional boundary conditions and where the strange quark mass was determined in the quenched approximation. We emphasize the general concepts and our strategy for the computation of quark masses.
Extracting freeze-out conditions in beam energy scan via functional QCD
Peking University（北京大学）；Justus-Liebig-Universität Gießen（吉森大学）；GSI Helmholtzzentrum für Schwerionenforschung（德国重离子研究中心）；Beijing Institute of Technology（北京理工大学）；Center for High Energy Physics, Peking University（北京大学高能物理中心）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；Universität Heidelberg（海德堡大学）；ExtreMe Matter Institute EMMI, GSI（GSI极端物质研究所）
Background field method is used to perform renormalization group transformations for Schrödinger equation in QCD. The dependence of the ground state wave functional on rapidly oscillating fields is found.
In a Euclidean space functional integral treatment of the free energy of QCD, a chemical potential enters only through the functional determinant of the Dirac operator which for any flavor is $\dslash + m - μ_f γ_0$ (where $μ_f$ is the chemical potential for the given flavor). Any nonzero $μ$ alters all of the eigenvalues of the Dirac operator relative to the $μ=0$ value, leading to a naive expectation that the determinant is altered and which thereby alters the free energy. Phenomenologically, this does not occur at T=0 for sufficiently small $μ$, in contradiction to this naive expectation. T…
The understanding of the phase structure and the fundamental properties of QCD matter from its microscopic description requires appropriate first-principle approaches. Here I review the progress towards a quantitative first-principle continuum approach within the framework of the Functional Renormalization group established by the fQCD collaboration. I focus on recent quantitative results for quenched QCD and Yang-Mills theory in the vacuum before addressing the calculation of dynamical quantities such as spectral functions and transport coefficients in this framework.
A real-time path integral for ultrasoft QCD is formulated. It exhibits a Feynman's influence functional. The statistical properties of the theory and the gauge symmetry are explicit. The correspondence is established with the alternative version, where a noise term enters a transport equation.
The effective amplitudes for gluon momentum p<<gT in hot QCD exhibit damping as a result of collisions. The whole set of n-point amplitudes is shown to be generated from one functional K(x,y;A), in addition to the induced current j(x;A).
The lepton flavour asymmetries of the Universe are observationally almost unconstrained before the onset of neutrino oscillations. We calculate the cosmic trajectory during the cosmic QCD epoch in the presence of large lepton flavour asymmetries. By including QCD thermodynamic quantities derived from functional QCD methods in our calculation our work reveals for the first time the possibility of a first-order cosmic QCD transition. We specify the required values of the lepton flavour asymmetries for which a first-order transition occurs for a number of different benchmark scenarios.
We study a functional method to extract the V-A condensate of dimension 6 from a comparison of tau-decay data with the asymptotic space-like QCD prediction. Our result is in agreement within errors with that from conventional analyses based on finite energy sum rules.
Walking technicolor theory attempts to realize electroweak symmetry breaking as the spontaneous chiral symmetry breakdown caused by the gauge dynamics with slowly varying gauge coupling constant and large mass anomalous dimension. Many-flavor QCD is one of the candidates owning these features. We focus on the SU(3) gauge theory with ten flavors of massless fermions in the fundamental representation, and compute the gauge coupling constant in the Schrödinger functional scheme. Numerical simulation is performed with $O(a)$-unimproved lattice action, and the continuum limit is taken in linear in …
A study of QCD at finite extension is presented and the relation to QCD at finite temperature and in the infinite momentum frame is discussed. The dynamics of Polyakov loops is investigated and shown to be described by functional integrals with finite range of integration. Consequences of this non-Gaussian form of the generating functional concerning the QCD vacuum, gluonic excitations and confinement are discussed.
Dalian University of Technology（大连理工大学）；Universität Heidelberg（海德堡大学）；ExtreMe Matter Institute EMMI, GSI（极端物质研究所EMMI，GSI）；Justus-Liebig-Universität Gießen（吉森大学）；University of Chinese Academy of Sciences（中国科学院大学）
The present knowledge of the QCD phase diagram based on simulations of lattice QCD is summarised. The main questions are whether there is a critical point in the QCD phase diagram and whether it is related to a chiral phase transition. It is shown that QCD at imaginary chemical potentials has a rich phase structure, which can be determined in a controlled way without sign problem and which severely constrains the phase structure at real chemical potentials.
We first review the method to calculate the spectral functions in the functional renormalization group (FRG) approach, which has been recently developed. We also provide the numerical stability conditions given by the present authors for a generic nonlinear evolution equation that are necessary for obtaining the accurate effective potential from the flow equation in the FRG. As an interesting example, we report the recent calculation of the spectral functions of the mesonic and particle-hole excitations using a chiral effective model of Quantum Chromodynamics (QCD); we extract the dispersion r…
We summarize recent results obtained in the Dyson-Schwinger formalism to study the chiral and deconfinement phase transitions of quenched and unquenched QCD at finite temperature and chemical potential. In the quenched case we compare SU(2) and SU(3) gauge theories by taking lattice data for the gluon as an input for the quark Dyson-Schwinger equation. As compared to previous investigations we find a clearer distinction between the second order transition of the two-color theory and the (weak) first order transition of the three-color gauge theory. We then extend this study to unquenched QCD a…
In this paper we consider the simplest functional Schroedinger equation of a quantum field theory (in particular QCD) and study its solutions. We observe that the solutions to this equation must possess a number of properties. Its Taylor coefficients are multivalued functions with rational and logarithmic branchings and essential singularities of exponential type. These singularities occur along a locus defined by polynomial equations. The conditions we find define a class of functions that generalizes to multiple dimensions meromorphic functions with finite Nevanlinna type. We note that in pe…
This paper reviews some recent progress on QCD functional integrals at nonzero chemical potentials. One issue discussed is the use of QCD inequalities for this regime. In particular, the positivity of the integrand of particular Euclidean space functional integrals for two-flavor QCD with degenerate quark masses is used to demonstrate that the free energy per unit volume for QCD with a baryon chemical potential $μ_B$ (and zero isospin chemical potential) is necessarily greater than the free energy with isospin chemical potential $μ_I = \frac{2 μ_B}{N_c} $ (and zero baryon chemical potential). …
QCD equations for generating functionals are solved at coinciding momenta of particles. As a result, the relations for $q$-particle correlation functions at equal momenta are obtained. They are directly connected to previously derived results about the factorial and cumulant moments of multiplicity distributions. It is predicted that the correlations at coinciding points decrease, first, as a function of the rank $q$, and then start oscillating when represented in a form of the ratio of cumulant to factorial correlations. In particular, the cumulant function of the fifth rank should become neg…
We show how the functional renormalization group can be applied to a study of chiral symmetry breaking in QCD. We discuss the running gauge coupling at finite temperature for QCD. For scales smaller than the corresponding temperature, the coupling is governed by a fixed point of the 3-dimensional theory. Chiral symmetry breaking is approached if the running coupling drives the quark sector to criticality. We analyze the phase boundary in the plane of temperature and number of flavors. For small flavor-numbers, we find good agreement with lattice results. For flavor numbers near the critical fl…
Consideration of the analytic properties of pion-induced baryon self-energies leads to new functional forms for the extrapolation of light baryon masses. These functional forms reproduce the leading non-analytic behavior of chiral perturbation theory, the correct non-analytic behavior at the $N π$ threshold and the appropriate heavy-quark limit. They involve only three unknown parameters, which may be obtained by fitting to lattice data. Recent dynamical fermion results from CP-PACS and UKQCD are extrapolated using these new functional forms. We also use these functions to probe the limit of a…
The generating functional is suggested for multiparticle generation processes. In mean field approximation of high density QCD two equations for new generating functional are derived: linear functional equation for an arbitrary initial condition and non-linear one for a specific initial condition. The non-linear equation has the form of Kovchegov-Levin equation for diffraction production and gives its generalization on the processes with fixed multiplicities of produced particles.
We perform a variational calculation in the SU(N) Yang Mills theory in 3+1 dimensions. Our trial variational states are explicitly gauge invariant, and reduce to simple Gaussian states in the zero coupling limit. Our main result is that the energy is minimized for the value of the variational parameter away form the perturbative value. The best variational state is therefore characterized by a dynamically generated mass scale $M$. This scale is related to the perturbative scale $Λ_{QCD}$ by the following relation: $α_{QCD}(M)={π\over 4}{1\over N}$. Taking the one loop QCD $β$- function and $Λ_…
We study the QCD phase transition based on the statistical treatment with the bag-model picture of hadrons, and derive a phenomenological relation among the low-lying hadron masses, the hadron sizes and the critical temperature of the QCD phase transition. We apply this phenomenological relation to both full QCD and quenched QCD, and compare these results with the corresponding lattice QCD results. Whereas such a statistical approach works well in full QCD, it results in an extremely large estimate of the critical temperature in quenched QCD, which indicates a serious problem in understanding …
We investigate QCD at large mu/T by using Z_3-symmetric SU(3) gauge theory, where mu is the quark-number chemical potential and T is temperature. We impose the flavor-dependent twist boundary condition on quarks in QCD. This QCD-like theory has the twist angle theta as a parameter, and agrees with QCD when theta=0 and becomes symmetric when theta=2π/3. For both QCD and the Z_3-symmetric SU(3) gauge theory, the phase diagram is drawn in mu--T plane with the Polyakov-loop extended Nambu--Jona-Lasinio model. In the Z_3-symmetric SU(3) gauge theory, the Polyakov loop varphi is zero in the confined…
Neither the chiral limit nor finite baryon density can be simulated directly in lattice QCD, which severely limits our understanding of the QCD phase diagram. In this review I collect results for the phase structure in an extended parameter space of QCD, with varying numbers of flavours, quark masses, colours, lattice spacings, imaginary and isospin chemical potentials. Such studies help in understanding the underlying symmetries and degrees of freedom, and are beginning to provide a consistent picture constraining the possibilities for the physical phase diagram.
We review results for the phase diagram of QCD, the properties of quarks and gluons and the resulting properties of strongly interacting matter at finite temperature and chemical potential. The interplay of two different but related transitions in QCD, chiral symmetry restoration and deconfinement, leads to a rich phenomenology when external parameters such as quark masses, volume, temperature and chemical potential are varied. We discuss the progress in this field from a theoretical perspective, focusing on non-perturbative QCD as encoded in the functional approach via Dyson-Schwinger and Bet…
Functional renormalization group study of the jet quenching parameter near the QCD critical end point
School of Physics, Dalian University of Technology（大连理工大学物理学院）；Institut für Theoretische Physik, Justus-Liebig-Universität Gießen（吉森大学理论物理研究所）；School of Physical Science and Technology, Lanzhou University（兰州大学物理科学与技术学院）；Energy Singularity Fusion Power Technology (SH) Ltd.（能量奇点融合电力技术（上海）有限公司）
The transition between the hadronic phase and the quark gluon plasma phase at nonzero temperature and quark chemical potentials is studied within the large-Nc expansion of QCD.
Nankai University（南开大学）；Dalian University of Technology（大连理工大学）；Universität Heidelberg（海德堡大学）；ExtreMe Matter Institute EMMI, GSI（GSI重离子研究中心极端物质研究所EMMI）；The University of Tokyo（东京大学）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心(iTHEMS)）
We present a comprehensive study of the quark sector of $2+1$ flavour QCD, based on a self-consistent treatment of the coupled system of Schwinger-Dyson equations for the quark propagator and the full quark-gluon vertex. The individual form factors of the quark-gluon vertex are expressed in a special tensor basis obtained from a set of gauge-invariant operators. The sole external ingredient used as input to our equations is the Landau gauge gluon propagator with $2+1$ dynamical quark flavours, obtained from studies with Schwinger-Dyson equations, the functional renormalisation group approach, …
We report results on the Coulomb-gauge ghost propagator and the color-Coulomb potential computed in two lattice gauge-field ensembles: (1) configurations derived from our recently proposed Yang-Mills vacuum wave functional in 2+1 dimensions, and (2) lattices generated by Monte Carlo simulations of the three-dimensional Euclidean SU(2) lattice gauge theory with the Wilson action. We observe remarkable agreement between the ghost propagators in both ensembles, but some differences in the potentials. Those originate from rare configurations with very small values of the lowest eigenvalue of the C…
We study the QCD phase diagram at nonzero baryon and isospin chemical potentials using the 1/Nc expansion. We find that there are two phase transitions between the hadronic phase and the quark gluon plasma phase. We discuss the consequences of this result for the universality class of the critical endpoint at nonzero baryon and zero isospin chemical potential.
We demonstrate that the multiplicity distribution of a system located in the vicinity of a first-order phase transition can be successfully measured in terms of its factorial cumulants with a surprisingly small number of events. This finding has direct implications for the experimental search of a QCD phase transition conjectured to be located in the high baryon density region of the QCD phase diagram.
$CP$ phase structure of QCD from functional renormalization group
Center for Theoretical Physics and College of Physics, Jilin University（吉林大学理论物理中心与物理学院）；Department of Physics and Astronomy, Kwansei Gakuin University（关西学院大学物理天文系）
We explore the phase diagram of two-flavor QCD at imaginary values of baryon and isospin chemical potentials analyzing the thermodynamic potential of QCD analytically and that of the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model numerically. QCD has no pion condensation at imaginary baryon and isospin chemical potentials, and therefore has discrete symmetries that are not present at real baryon and isospin chemical potentials. The PNJL model possesses all the discrete symmetries. The PNJL model can reproduce qualitatively lattice QCD data presented very lately.
If QCD is to undergo a second order phase transition, the light quark return probability is universal for large times at the critical point. We show that this behavior is distinct from the one expected at the mobility edge of a metal-insulator transition or a percolation transition in d$\leq 4$. Our results are accessible to current lattice QCD simulations.
Current status of theoretical researches on the QCD phase diagram at finite temperature and baryon chemical potential is reviewed with special emphasis on the origin of various phases and their symmetry breaking patterns. Topics include; quark deconfinement, chiral symmetry restoration, order of the phase transitions, QCD critical point(s), colour superconductivity, various inhomogeneous states and implications from QCD-like theories.
We argue that the chiral phase transition and the deconfinement phase transition in finite temperature QCD are just different limits of a single crossover phase transition driven by the lightest scalar field which is a mixture of the glueball and the sigma meson. The key mechanism, the level repulsion between the sigma and the glueball, gives not only a unified description of the finite temperature phase transitions, but also a consistent characterization of the entire QCD phase diagram in the temperature-quark mass plane.
The Schrodinger wave functional for the d=3+1 SU(N) vacuum is a partition function constructed in d=4; the exponent 2S in the square of the wave functional plays the role of a d=3 Euclidean action. We start from a gauge-invariant conjecture for the infrared-dominant part of S, based on dynamical generation of a gluon mass M in d=4. We argue that the exact leading term, of O(M), in an expansion of S in inverse powers of M is a d=3 gauge-invariant mass term (gauged non-linear sigma model); the next leading term, of O(1/M), is a conventional Yang-Mills action. The d=3 action that is the sum of th…
We show the universality of phase diagrams in QCD and QCD-like theories through the large-N_c equivalence. The whole phase diagrams are identical between QCD at finite isospin chemical potential and SO(2N_c) and Sp(2N_c) gauge theories at finite baryon chemical potential. In the chiral limit, they are also identical to that of QCD at finite chiral chemical potential. Outside the pion or diquark condensed phase in these theories, the universality of phase diagrams is applicable to QCD at finite baryon chemical potential. We further argue that the universality may work approximately even for N_c…
We give a general overview about the approaches to study the phase diagram of QCD. Thereafter, we examine the evolution of a fireball in a chiral fluid dynamic model including nonequilibrium effects.
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 $ρ_{σ, π}(ω,\, p)$ in the scalar ($σ$) and pseudoscalar ($π$) channels beyond the random phase approximation in the quark--meson model. At finite baryon chemical potential $μ$ with a finite quark mass, the baryon-number fluctuation is coupled to the scalar channel and the spectral function in the $σ$ channel has a support not only in the time-like ($ω\,>\…
We derive exact constraints relating QCD at nonzero baryon chemical potential and temperature to QCD at nonzero isospin chemical potential and temperature, a theory which can be simulated by conventional methods. These results challenge the consistency of dynamical models of superconductivity and superfluidity in QCD.
The phase diagram of lattice QCD in the strong coupling limit can be measured in the full $μ$-$T$ plane, also in the chiral limit. In particular, the phase diagram in the chiral limit features a tricritical point at some $(μ_c,T_c)$. This point may be related to the critical end point expected in the QCD phase diagram. We discuss the gauge corrections to the phase diagram at strong coupling and compare our findings with various possible scenarios in continuum QCD. We comment on the possibility that the tricritical point at strong coupling is connected to the tricritical point in the continuum,…
We argue that the whole or the part of the phase diagrams of QCD and QCD-like theories should be universal in the large-N_c limit through the orbifold equivalence. The whole phase diagrams, including the chiral phase transitions and the BEC-BCS crossover regions, are identical between SU(N_c) QCD at finite isospin chemical potential and SO(2N_c) and Sp(2N_c) gauge theories at finite baryon chemical potential. Outside the BEC-BCS crossover region in these theories, the phase diagrams are also identical to that of SU(N_c) QCD at finite baryon chemical potential. We give examples of the universal…
We elucidate that the phase diagram of massless $N$-flavor QCD under $\mathbb{Z}_N$ flavor-twisted boundary condition (massless $\mathbb{Z}_N$-QCD) is constrained by an 't Hooft anomaly involving two-form gauge fields. As a consequence, massless $\mathbb{Z}_N$-QCD turns out to realize persistent order at any temperatures and quark chemical potentials, namely, the symmetric and gapped phase is strictly forbidden. This is the first result on the finite-$(T,μ)$ phase diagram in QCD-type theories based on anomaly matching related to center and discrete axial symmetries.
The aim of this review is to summarize the contemporary understanding of the QCD phase diagram as a function of temperature and baryo-chemical potential. The focus is on recent theoretical developments due to lattice simulations of the phase diagram.
