We study an effective theory for QCD at finite temperature and density which contains the leading center symmetric and center symmetry breaking terms. The effective theory is studied in a flux representation where the complex phase problem is absent and the model becomes accessible to Monte Carlo techniques also at finite chemical potential. We simulate the system using a generalized Prokof'ev-Svistunov worm algorithm and compare the results to a low temperature expansion. The phase diagram is determined as a function of temperature, chemical potential and quark mass. Shape and quark mass dependence of the phase boundaries are as expected for QCD. The transition into the deconfined phase is smooth throughout, without any discontinuities or critical points.
Properties of local Polyakov loops are studied in finite temperature lattice QCD and SU(3) lattice gauge theory. We evaluate local Polyakov loops, identify the closest center element for each loop and investigate cluster properties of these center phases. For a suitable definition of the clusters we find that the deconfinement transition of pure SU(3) gauge theory may be characterized by percolation of the center clusters. For the case of full QCD cluster observables show a behavior which seems to be compatible with a smooth crossover type of transition.
In a numerical experiment, we remove center vortices from an ensemble of lattice SU(2) gauge configurations. This removal adds short-range disorder. Nevertheless, we observe long-range order in the modified ensemble: confinement is lost and chiral symmetry is restored (together with trivial topology), proving that center vortices are responsible for both phenomena. As for the Abelian monopoles, they survive but their percolation properties are lost.
Common lore suggests that $N$-color QCD with massive quarks has no useful order parameters which can be non-trivial at zero baryon density. However, such order parameters do exist when there are $n_f$ quark flavors with a common mass and $d\equiv\gcd(n_f,N) > 1$. These theories have a $\mathbb Z_d$ color-flavor center symmetry arising from intertwined color center transformations and cyclic flavor permutations. The symmetry realization depends on the temperature, baryon chemical potential and value of $n_f/N$, with implications for conformal window studies and dense quark matter.
We review lattice evidence showing that center-vortex condensation is a serious candidate for the mechanism of colour confinement in quantum chromodynamics.
For the $Z_{3}$-symmetric lattice QCD-like theory ($Z_3$-QCD), in which $SU(3)$ gauge theory is coupled with three fundamental Wilson quarks with flavor-dependent twisted boundary conditions, we calculate the expectation values of Polyakov loop and chiral condensate as functions of temperature on $16^3 \times4$ and $20^3 \times 4$ lattices with $m_{PS}/m_{V}=0.70$ fixed. We find the first-order phase transition with respect to the $Z_{3}$ center symmetry, where the Polyakov loop exhibits a hysteresis depending on the initial condition of thermalization process. We also show that the crossover …
We present evidence that the center vortex model of confinement is more consistent with lattice results than other currently available models. In particular we show that Abelian field distributions predicted by monopole plasma, caloron gas or dual superconductor models cannot reproduce the area-law falloff of double winding Wilson loops in full $SU(2)$ and center vortex only gauge fields.
We study the center structure of full dynamical QCD at finite temperatures and nonzero values of the background magnetic field using continuum extrapolated lattice data. We concentrate on two particular observables characterizing center clusters: their fractality and the probability for percolation. For temperatures below and around the transition region, the fractal dimension is found to be significantly smaller than three, leading to a vanishing mean free path inside the cluster structure. This finding might be relevant for center symmetry-based models of heavy-ion collisions. In addition, t…
Center vortices are unambiguously identified after Laplacian Center Gauge fixing and their influence on confinement and chiral symmetry breaking is investigated on a sample of SU(2) configurations at zero and finite temperature.
We investigate QCD-like theory with exact center symmetry, with emphasis on the finite-temperature phase transition concerning center and chiral symmetries. On the lattice, we formulate center symmetric $SU(3)$ gauge theory with three fundamental Wilson quarks by twisting quark boundary conditions in a compact direction ($Z_3$-QCD model). We calculate the expectation value of Polyakov loop and the chiral condensate as a function of temperature on 16^3 x 4 and 20^3 x 4 lattices along the line of constant physics realizing $m_{PS}/m_{V}=0.70$. We find out the first-order center phase transition,…
In this lecture I will explain the established pictures of the QCD vacuum and, in particular, the underlying confinement mechanism. These are: the magnetic monopole condensation (dual Meißner effect), the center vortex picture and the Gribov--Zwanziger picture. I will start by giving a survey of the common order and disorder parameters of confinement: the temporal and spatial Wilson loop, the Polyakov loop and the 't Hooft loop. Next the dual Meißner effect, which assumes a condensate of magnetic monopoles, will be explained as a picture of confinement. I will also show how magnetic monopoles …
Using the effective center model of QCD we test series expansions for finite chemical potential $μ$. In particular we study two variants of Taylor expansion as well as the fugacity series. The effective center model has a dual representation where the sign problem is absent and reliable Monte Carlo simulations are possible at arbitrary $μ$. We use the results from the dual simulation as reference data to assess the Taylor- and fugacity series approaches. We find that for most of parameter space fugacity expansion is the best (but also numerically most expensive) choice for reproducing the dual…
Exact center symmetry and first-order phase transition in QCD with three degenerate dynamical quarks
ELTE Eötvös Loránd University（厄特沃什·罗兰大学）；Technische Universität Darmstadt（达姆施塔特工业大学）；Universität Bielefeld（比勒费尔德大学）；Universität Wuppertal（伍珀塔尔大学）；Goethe Universität Frankfurt（法兰克福歌德大学）
We construct a non Abelian model for SU(2) QCD in Euclidean three-dimensional spacetime and study its different phases. The model contains a center vortex sector coupled to a dual effective field encoding information about how the vortices are paired in the ensemble. The possible phases in parameter space are interpreted in terms of the proliferation of either closed center vortices or closed chains, where the endpoints of open vortices are attached in pairs to monopole-like defects.
We investigate the properties of quarks and gluons above the chiral phase transition temperature $T_c,$ using the RG improved gauge action and the Wilson quark action with two degenerate quarks mainly on a $32^3\times 16$ lattice. In the one-loop perturbation theory, the thermal ensemble is dominated by the gauge configurations with effectively $Z(3)$ center twisted boundary conditions, making the thermal expectation value of the spatial Polyakov loop take a non-trivial $Z(3)$ center. This is in agreement with our lattice simulation of high temperature QCD. We further observe that the temporal…
We study the thermal confinement/deconfinement and non-thermal quantum phase transitions or rapid cross-overs in QCD and QCD-like theories in external magnetic fields. At large magnetic fields, while the contribution of gauge fluctuations to Wilson-line potential remains unaltered at one-loop order, the contribution of fermions effectively becomes two lower dimensional and is enhanced by the density of states of the lowest Landau level (LLL). In a spatial compactification and for heavy adjoint fermions, this enhancement leads to a calculable zero temperature quantum phase transition on R^3*S^1…
The center vortex model of the QCD vacuum is very successful in explaining the non-perturbative properties of QCD, especially confinement, chiral symmetry breaking and the topological charge of vacuum configurations. On the other hand, the center vortex model still suffers from a Gribov problem: Direct maximal center gauge and center projection can lead to an underestimation of the string tension in smooth configurations or after persistent simulated annealing. We discuss methods to identify center regions, whose boundaries evaluate to center elements, and want to improve the vortex detection:…
It is usual to study confinement via quantum chromodynamics (QCD) alone. The deconfinement transition of the pure gauge theory (i.e. with static quarks) is then characterized by the breaking of center symmetry. Center vortices offer an intuitive and quantitative description of the transition. Dynamical quarks explicitly break center symmetry, and the phase transition becomes a crossover. However, it may be misleading to study QCD in isolation. Quarks also carry fractional electric charge. This bestows the Standard Model with a global center symmetry that combines color center phases with an ap…
In this study, we extend the HAL QCD method to a case where a total momentum of a two-particle system is non-zero and apply it to the $I=2$ S-wave $ππ$ scattering in order to confirm its validity. We derive a fundamental relation of an energy-independent non-local potential defined in the center of mass frame with NBS wave functions in a laboratory frame. Based on the relation, we propose the time-dependent method to extract potentials, often used in practice for the HALQCD method in the center of mass frame. For numerical simulations in the $I=2$ $ππ$ system, we employ (2+1)-flavor gauge conf…
Recently, it was argued that the thermal deconfinement transition in pure Yang-Mills theory is continuously connected to a quantum phase transition in softly-broken N=1 SYM theory on R^3 x S^1. The transition is semiclassically calculable at small S^1 size L, occurs as the soft mass m_soft and L vary, and is driven by a competition between perturbative effects and nonperturbative topological molecules. These are correlated instanton--anti-instanton tunneling events, whose constituents are monopole-instantons "bound" by attractive long-range forces. The mechanism driving the transition is unive…
We further pursue an approach to the sign problem of quantum chromodynamics at nonzero chemical potential, in which configurations of the lattice path integral are gathered into subsets. In the subset construction we multiply each temporal link by center elements independently and in a first step neglect the gauge action. The positivity of the subset weights -- shown for 0+1 dimensions in an earlier study -- extends to larger lattices: for two sites in the temporal direction and arbitrary spatial extent we give a proof of the positivity by decomposing the subset weight in positive summands. Fr…
We present a subset method which solves the sign problem for QCD at nonzero quark chemical potential in 0+1 dimensions. The subsets gather gauge configurations based on the center symmetry of the SU(3) group. We show that the sign problem is solved for one to five quark flavors and that it slowly reappears for a larger number of flavors. We formulate an extension of the center subsets that solves the sign problem for a larger number of flavors as well. We also derive some new analytical results for this toy model.
According to the present understanding, the observed diversity of the strong interaction phenomena is described by Quantum Chromodynamics, a gauge field theory with only very few parameters. One of the fundamental questions in this context is how precisely the world of mesons and nucleons is related to the properties of the theory at high energies, where quarks and gluons are the important degrees of freedom. The lattice formulation of QCD combined with numerical simulations and standard perturbation theory are the tools that allow one to address this issue at a quantitative level.
We perform a lattice investigation of QCD with three colors and 2 flavors of Dirac (staggered) fermions in the adjoint representation, defined on a 4d space with one spatial dimension compactified, and study the phase structure of the theory as a function of the size Lc of the compactified dimension. We show that four different phases take place, corresponding to different realizations of center symmetry: two center symmetric phases, for large or small values of Lc, separated by two phases in which center symmetry is broken in two different ways; the dependence of these results on the quark ma…
A brief overview of the QCD phase diagram at nonzero temperature and density is provided. It is explained why standard lattice QCD techniques are not immediately applicable for its determination, due to the sign problem. We then discuss a selection of recent lattice approaches that attempt to evade the sign problem and classify them according to the underlying principle: constrained simulations (density of states, histograms), holomorphicity (complex Langevin, Lefschetz thimbles), partial summations (clusters, subsets, bags) and change in integration order (strong coupling, dual formulations).
We study the thermodynamics of two-color QCD at high temperature and/or density using a dimensionally reduced superrenormalizable effective theory, formulated in terms of a coarse grained Wilson line. In the absence of quarks, the theory is required to respect the Z(2) center symmetry, while the effects of quarks of arbitrary masses and chemical potentials are introduced via soft Z(2) breaking operators. Perturbative matching of the effective theory parameters to the full theory is carried out explicitly, and it is argued how the new theory can be used to explore the phase diagram of two-color…
In quantum chromodynamics with static quarks the confinement-deconfinement phase transition is connected to the spontaneous breaking of the global Z3 center symmetry. This symmetry is lost when one considers dynamical quarks. Owing to the fractional electric charge of quarks, we recover a global Z6 center symmetry when QCD is regarded as a part of the Standard Model. We present results from QCD-like theories extended by electromagnetic interactions and show that the weak coupling limit of the QED part of the model results in a center-like symmetry with disorder in the vacuum. This can be seen …
The spectrum of the Upsilon system is investigated using the Nonrelativistic Lattice QCD approach to heavy quarks and ignoring light quark vacuum polarization. We find good agreement with experiment for the Upsilon(1S), Upsilon(2S), Upsilon(3S) and for the center of mass and fine structure of the chi_b states. The lattice calculations predict b-bbar D-states with center of mass at (10.20 +/- 0.07 +/- 0.03)GeV. Fitting procedures aimed at extracting both ground and excited state energies are developed. We calculate a nonperturbative dispersion mass for the Upsilon(1S) and compare with tadpole-i…
Consequences of gauging exact $Z_k^C$ center symmetries in several simple $SU(N)$ gauge theories, where $k$ is a divisor of $N$, are investigated. Models discussed include: the $SU(N)$ gauge theory with $N_f$ copies of Weyl fermions in self-adjoint single-column antisymmetric representation, the well-discussed adjoint QCD, QCD-like theories in which the quarks are in a two-index representation of $SU(N)$, and a chiral $SU(N)$ theory with fermions in the symmetric as well as in anti-antisymmetric representations but without fundamentals. Mixed 't Hooft anomalies between the 1-form $Z_k^C$ symme…
We suggest a tetracritical fixed point to naturally occur in strongly interacting theories. As a fundamental example we analyze the temperature--quark chemical potential phase diagram of QCD with fermions in the adjoint representation of the gauge group (i.e. adjoint QCD). Here we show that such a non trivial multicritical point exists and is due to the interplay between the spontaneous breaking of a global U(1) symmetry and the center group symmetry associated to confinement. Our results demonstrate that taking confinement into account is essential for understanding the critical behavior as w…
We study chemical-potential dependence of confinement and mass gap in QCD with adjoint fermions in spacetime with one spatial compact direction. By calculating the one-loop effective potential for the Wilson line in the presence of chemical potential, we show that a center-symmetric phase and a center-broken phase alternate when the chemical potential in unit of the compactification scale is increased. In the center-symmetric phase we use semiclassical methods to show that photons in the magnetic bion plasma acquire a mass gap that grows with the chemical potential as a result of anisotropic i…
We investigate QCD with adjoint Dirac fermions on $\mathbb{R}^3\times S^1$ with generic boundary conditions for fermions along $S^1$. By means of perturbation theory, semiclassical methods and a chiral effective model, we elucidate a rich phase structure in the space spanned by the $S^1$ compactification scale $L$, twisted fermionic boundary condition $ϕ$ and the fermion mass $m$. We found various phases with or without chiral and center symmetry breaking, separated by first- and second-order phase transitions, which in specific limits ($ϕ=0$, $ϕ=π$, $L\to 0$ and $m\to \infty$) reproduce known…
We show analytically that the QCD potential can be expressed, up to an O(Lambda_QCD^3 r^2) uncertainty, as the sum of a ``Coulomb'' potential (with log corrections at short distances) and a linear potential, within an approximation based on perturbative expansion in alpha_S and the renormalon dominance picture. The expansion of V_QCD(r) is truncated at O(alpha_S^N) [N = 6 pi/(beta_0 alpha_S)], where the term becomes minimal according to the estimate by NLO renormalon, and is studied for N >> 1. Analytic expressions for the linear potential are obtained in some cases.
We construct a novel observable for finite temperature QCD that relates confinement and chiral symmetry. It uses phases as boundary conditions for the fermions. We discuss numerical and analytical aspects of this observable, like its spectral behavior below and above the critical temperature, as well as the connection to chiral condensate and center symmetry.
According to the Nielsen-Ninomiya No-Go theorem, the doubling of fermions on the lattice cannot be suppressed in a chiral theory. Whereas Wilson and staggered fermions suppress doublers with explicit breaking of chiral symmetry, the random lattice does so by spontaneous chiral symmetry breaking even in the free theory. I present results for meson masses, the chiral condensate and fermionic eigenvalues from simulations of quenched QCD on random lattices in four dimensions, focusing on chiral symmetry breaking.
We study the scaling behaviour of the pseudo-critical couplings for the chiral phase transition in two-flavour QCD. We show that all existing results from lattice simulations on lattices with temporal extent $N_τ= 4$, 6 and 8 can be mapped onto a universal scaling curve. The relevant combination of critical exponents, $βδ$, is consistent with the scaling behaviour expected for a second order phase transition with $O(4)$ exponents. At present, scaling according to the $O(2)$ symmetry group can, however, not be ruled out.
We present the results of extensive simulations regarding the critical behavior at the endpoint of the Roberge-Weiss transition for N_f = 2 QCD. We confirm early evidence, presented in arXiv:0909.0254, according to which the Roberge-Weiss endpoint is first order in the limit of large or small quark masses, and second order for intermediate masses. A systematic study of the transition strength as a function of the quark mass in the first order regions, permits us to estimate the tricritical values of the quark mass separating the second order region from the first order ones.
At imaginary values of the quark chemical potential $μ$, Quantum Chromodynamics shows an interesting phase structure due to an exact center, or Roberge-Weiss (RW), symmetry. This can be used to constrain QCD at real $μ$, where the sign problem prevents Monte Carlo simulations of the lattice theory. In previous studies of this region with staggered fermions it was found that the RW endpoint, where the center transition changes from first-order to a crossover, depends non-trivially on the quark mass: for high and low masses, it is a triple point connecting to the deconfinement and chiral transit…
We study the one-loop potential of large-N QCD with adjoint Dirac fermions. Space-time is a discretization of R^3 x S^1 where the compact direction consists of a single lattice site. We use Wilson fermions with different values of the quark mass m and set the lattice spacings in the compact and non-compact directions to be a_t and a_s respectively. Extending the results of JHEP 0906:091,2009, we prove that if the ratio xi = a_s/a_t obeys 0<xi<2, then the minimum of the one-loop lattice potential for one or more Dirac flavors is Z_N symmetric at the chiral point. For xi=0 our formulas reduce to…
In order to understand the puzzle of the free energy of an individual quark in QCD, we explicitly construct ensembles with quark numbers $N_V\neq 0\!\mod 3$, corresponding to non-zero triality in a finite subvolume $V$ on the lattice. We first illustrate the basic idea in an effective Polyakov-loop theory for the heavy-dense limit of QCD, and then extend the construction to full Lattice QCD, where the electric center flux through the surface of $V$ has to be fixed at all times to account for Gauss's law. This requires introducing discrete Fourier transforms over closed center-vortex sheets aro…
It is commonly believed that in confining vector-like gauge theories the center and chiral symmetry realizations are parametrically entangled, and if phase transitions occur, they must take place around the strong scale $Λ^{-1}$ of the gauge theory. We demonstrate that (non-thermal) vector-like theories formulated on ${\mathbb R}^{3} \times S^1$ where $S^1$ is a spatial circle exhibit new dynamical scales and new phenomena. There are chiral phase transitions taking place at $Λ^{-1}/N_c$ in the absence of any change in center symmetry. $Λ^{-1}/N_c$, invisible in (planar) perturbation theory, is…
We present a calculation of the scattering phase shift for the I=2 S-wave pion-pion system in the continuum limit with two-flavor full QCD. Calculations are made at three lattice spacings, using the finite volume method of Lüscher in the center of mass frame, and its extension to the laboratory frame.
QCD can be formulated using any gauge group. One particular interesting choice is to replace SU(3) by the exceptional group G2. Conceptually, this group is the simplest group with a trivial center. It thus permits to study the conjectured relevance of center degrees of freedom for QCD. Practically, since all its representation are real, it is possible to perform lattice simulations for this theory also at finite baryon densities. It is thus an excellent environment to test methods and to investigate general properties of gauge theories at finite densities. We review the status of our understan…
Probing QCD instantons using jet correlation observables in proton-proton collisions at the LHC
The Institute of Mathematical Sciences（数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Indian Institute of Science Education and Research (IISER)（印度科学与教育研究所）
We study the phase diagram of two-flavor massless two-color QCD (QC$_2$D) under the presence of quark chemical potentials and imaginary isospin chemical potentials. At the special point of the imaginary isospin chemical potential, called the isospin Roberge--Weiss (RW) point, two-flavor QC$_2$D enjoys the $\mathbb{Z}_2$ center symmetry that acts on both quark flavors and the Polyakov loop. We find a $\mathbb{Z}_2$ 't Hooft anomaly of this system, which involves the $\mathbb{Z}_2$ center symmetry, the baryon-number symmetry, and the isospin chiral symmetry. Anomaly matching, therefore, constrai…
We study the large-N volume reduction of QCD with adjoint quarks regularized on the lattice. Specifically, we use Wilson fermions, and while our d-dimensional lattice has (d-1) infinite dimensions, the remaining direction is reduced to a point. We perform a weak-coupling one-loop calculation of the free energy as a function of the holonomy in the reduced direction, and map the regimes in the bare lattice parameter space where the holonomy averages to zero and a Z_N-center symmetric configuration is the ground state. For d=4 and N_f=1/2,1 and 2 Dirac flavors we see that the center symmetry is i…
We discuss an effective theory for QCD at finite chemical potential and non-zero temperature, where QCD is reduced to its center degrees of freedom. The effective action can be mapped to a flux representation, where the complex phase problem is solved and the theory accessible to Monte Carlo techniques. In this work, we use a generalized Prokof'ev-Svistunov worm algorithm to perform the simulations and determine the phase diagram as a function of temperature, quark mass and chemical potential. It turns out that the transition is qualitatively as expected for QCD.
Quenched QCD at zero baryonic chemical potential undergoes a first-order deconfinement phase transition at a critical temperature $T_c$, which is related to the spontaneous breaking of the global center symmetry. Including heavy, dynamical quarks breaks the center symmetry explicitly and weakens the first-order phase transition. For decreasing quark masses the first-order phase transition turns into a smooth crossover at a $Z_2$-critical point. The critical quark mass corresponding to this point has been examined with $N_\text{f} = 2$ Wilson fermions for several $N_τ$ in a recent study within …
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.
We construct a novel observable for finite temperature QCD that relates confinement and chiral symmetry. It uses phases as boundary conditions for the fermions. We discuss numerical and analytical aspects of this observable, like its spectral behavior below and above the critical temperature, as well as the connection to chiral condensate, center symmetry and the canonical ensemble.
We report on simulation results with overlap hypercube fermions (overlap HF) - a type of exactly chiral lattice fermions - and their link to chiral perturbation theory. We first sketch the construction of the overlap HF and discuss its high level of locality. Next we show applications in the p-regime of QCD, where we evaluate m_π, m_ρ, the quark mass according to the PCAC relation, the renormalisation constant Z_A and the pion decay constant F_π as functions of the bare quark mass. F_π is then reconsidered at even smaller quark masses in the epsilon-regime, along with the scalar condensate Σ. …
We present static meson-meson and baryon--anti-baryon potentials in Z(2) and Z(3) random center vortex models for the infrared sector of Yang-Mills theory, i.e., hypercubic lattice models of random vortex world-surfaces. In particular, we calculate Polyakov loop correlators of two static mesons resp. (anti-)baryons in a center vortex background and observe that their expectation values follow the minimal area law and show bond rearrangement behavior. The static meson-meson and baryon--anti-baryon potentials are compared with theoretical predictions and lattice QCD simulations.
According to the Leutwyler-Smilga relation, in Quantum Chromodynamics (QCD) the topological susceptibility vanishes linearly with the quark masses. Calculations of the topological susceptibility in the context of lattice QCD, extrapolated to zero quark masses, show a remnant non-zero value as a lattice artefact. Employing the Atiyah-Singer theorem in the framework of Symanzik's effective action and chiral perturbation theory, we show the validity of the Leutwyler-Smilga relation in lattice QCD with lattice artefacts of order $a^2$ in the lattice spacing $a$.
The most computationally demanding part of Lattice QCD simulations is solving quark propagators. Quark propagators are typically obtained with a linear equation solver utilizing HPC machines. The CCS QCD Benchmark is a benchmark program solving the Wilson-Clover quark propagator, and is developed at the Center for Computational Sciences (CCS), University of Tsukuba. We optimized the benchmark program for a \Intel \XeonPhi (Knights Corner, KNC) system named "COMA (PACS-IX)" at CCS Tsukuba under the Intel Parallel Computing Center program. A single precision BiCGStab solver with the overlapped R…
We review recent lattice QCD results for the surface tension at the finite temperature quark-hadron phase transition and discuss their implications on the possible scale of inhomogeneities. In the quenched approximation the average distance between nucleating centers is smaller than the diffusion length of a protron, so that inhomogeneities are washed out by the time nucleosynthesis sets in. Consequently the baryon density fluctuations formed by a QCD phase transition in the early universe cannot significantly affect standard big-bang nucleosynthesis calculations and certainly cannot allow bar…
