Field Correlator Method Research Articles

Colour-Electric and Colour-Magnetic Confinement

The basic properties of the confinement mechanism in QCD—the temperature dependence of the spatial and temporal string tensions (\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\sigma }_{{\ ext{s}}}}(T)$$\\end{document} and \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\sigma }_{{\ ext{E}}}}(T)$$\\end{document})—are studied in the framework of the Field Correlator Method (FCM). It is shown that both functions are connected respectively to the spatial and temporal parts of the vacuum gluon energy \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\epsilon }_{{\ ext{s}}}}$$\\end{document} and \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\epsilon }_{{\ ext{E}}}}$$\\end{document} which define their equal values at \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T = 0$$\\end{document}. However at \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T > 0$$\\end{document} the spatial part is growing with T while the temporal part is destroyed by the hadronic pressure at \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T = {{T}_{{\ ext{c}}}}$$\\end{document} (the deconfinement). Both properties are derived within the same method and are in a good agreement with the corresponding lattice data.

Non-perturbative suppression of chiral vortical effect in hot (s)QGP for hyperons spin polarization in heavy ion collisions

With the field correlator method (FCM) for QCD, we show that the chiral vortical effect (CVE) in hot (strongly-interacting) quark-gluon plasma ((s)QGP) is modified by non-perturbative interactions – by color-magnetic confinement, and by remnant color-electric interaction, which is encoded in the Polyakov line. The obtained result demonstrates numerical suppression of CVE comparable to the phenomenological suppression used for numerical simulations of RHIC-STAR data on hyperons spin polarization in non-central heavy ion collision (HIC). The parameters range in the temperature – quark chemical potential plane is expected to cover ALICE and RHIC data. The chiral current is calculated for the rigidly rotating model of (s)QGP in the linear order in angular velocity at the rotation axis with account of non-perturbative interactions.

The spatial string tension from the field correlator method

The behaviour of spatial string tension sigma _s(T) as a function of the temperature T is found in the framework of the Field Correlator Method (FCM). Here the string tension is calculated using the gluelump Green’s function, where gluons in the gluelump are interacting via the same spatial string tension. The resulting T-dependence was obtained without extra parameters in the region T_c< T < 5 T_c using the formalism of elliptic functions theta _3, demonstrating good agreement with available lattice data.

Effect of interactions on the topological expression for the chiral separation effect

In the absence of interactions the conductivity of chiral separation effect (CSE) in the system of massless fermions is given by topological expression. Interactions might change the pattern drastically. However, we prove that the CSE conductivity is still given by the topological invariant composed of the Green functions at zero temperature as long as the chiral symmetry is present, and if the renormalized axial current is considered. This allows to predict its appearance with the standard value of conductivity per Dirac fermion $\sigma_{CSE} = \frac{1}{2 \pi^2}$ in quark - gluon matter at $T = 0$ and sufficiently large baryon chemical potential, in the hypothetical phase with restored chiral symmetry and without color superconductivity. This phase may be realized inside the neutron stars. We also argue that the same topological expression for the CSE may be observed in Weyl semimetals, which realize the system of interacting relativistic fermions in solid state systems. In order to estimate the non - perturbative corrections to $\sigma_{CSE}$ within QCD at finite temperatures we apply method of field correlators developed by Yu.A.Simonov. As expected, above the deconfinement crossover the topological expression is approached within the quark - gluon plasma phase, when the quark chemical potential is sufficiently large. However, we observe that this occurs only when quark chemical potential is much larger than the thermal (Debye) mass. This range of parameters appears to be far out of the region accessible at the modern colliders.

Chiral Separation Effect in a Quark–Gluon Plasma

An expression for the chiral separation effect in a quark–gluon plasma has been derived using the method of field correlators by expanding in Landau levels.

Chiral Separation Effect in a Quark–Gluon Plasma

An expression for the chiral separation effect in a quark–gluon plasma has been derived using the method of field correlators by expanding in Landau levels.

The Spatial String Tension and the Nonperturbative Debye Mass from the Field Correlator Method

The phenomenon of the almost linear growth of the square root of spatial string tension $$\sqrt{\sigma_{s}(T)}=c_{\sigma}g^{2}T$$ and of the Debye mass $$m_{D}(T)$$ was found both in lattice and in theory, based on the Field Correlator Method (FCM). In the latter the string tension (both spatial and colorelectric) is expressed as an integral of the two gluon Green’s function calculated with the same string tension: $$\sigma=\int G^{(2g)}_{\sigma}$$ . This relation allows to check the self-consistency of the theory and at high $$T$$ it allows also to calculate $$c_{\sigma}$$ and hence $$\sigma_{s}$$ . We calculate below in the paper the corresponding coefficients $$c_{\sigma}$$ and $$c_{D}$$ numerically in the FCM method and compare the results with lattice data finding a good agreement. This justifies the use of the FCM in the space-like region and in high- $$T$$ thermodynamics without extra parameters.

The Colormagnetic Confinement in QCD

Colormagnetic confinement as a natural component of the QCD confinement is explained and treated in the framework of the Field Correlator Method (FCM). For quarks and gluons in hadrons the effects of the colormagnetic confinement are discussed at zero temperature,where it contributes to the spectrum properties and can create its ownbound states, while at nonzero temperature in the EoS of the quark gluon plasma the colormagnetic confinement plays a dominating role.Its properties in the QCD thermodynamics are discussed in detail.In particular the CM string tension and the Debye screening mass calculated in FCM are compared with lattice data.

The Fundamental Scale of QCD

There are several scales in the QCD as the theory of strong interaction: the vacuum gluonic condensate (as the divergence of the dilatation current), the nucleon mass as the basic mass scale in our universe, which is connected to the string tension $\sigma$, and the perturbative QCD scale $\Lambda_{\rm QCD}$, which defines the scale of the renormalized perturbative expansions. In this paper we connect all these scales to the vacuum condensate, using the field correlator method, where the string tension is expressed in terms of nonlocal field correlators, which are connected to the local condensates, while the perturbative $\Lambda_V$ is connected to $\sigma$ in the framework of the Background Perturbation Theory (BPT). We also demonstrate how the IR singularities and IR renormalons disappear in BPT making the resulting theory internally consistent.

Hybrid Stars with Color Superconducting Cores in an Extended FCM Model

We investigate the influence of repulsive vector interactions and color superconductivity on the structure of neutron stars using an extended version of the field correlator method (FCM) for the description of quark matter. The hybrid equation of state is constructed using the Maxwell description, which assumes a sharp hadron-quark phase transition. The equation of state of hadronic matter is computed for a density-dependent relativistic lagrangian treated in the mean-field approximation, with parameters given by the SW4L nuclear model. This model described the interactions among baryons in terms of σ, ω, ρ, σ*, and ϕ mesons. Quark matter is assumed to be in either the CFL or the 2SC+s color superconducting phase. The possibility of sequential (hadron-quark, quark-quark) transitions in ultra-dense matter is investigated. Observed data related to massive pulsars, gravitational-wave events, and NICER are used to constrain the parameters of the extended FCM model. The successful equations of state are used to explore the mass-radius relationship, radii, and tidal deformabilities of hybrid stars. A special focus lies on investigating consequences that slow or fast conversions of quark-hadron matter have on the stability and the mass-radius relationship of hybrid stars. We find that if slow conversion should occur, a new branch of stable massive stars would exist whose members have radii that are up to 1.5 km smaller than those of conventional neutron stars of the same mass. Such objects could be possible candidates for the stellar high-mass object of the GW190425 binary system.

Low mass strange stars and the compact star 1E1207.4-5209 in the field correlator method

We investigate the possible existence of anomalous mass defects in the low mass region of stellar sequences of strange stars. We employ the nonperturbative equation of state derived in the framework of the field correlator method to describe the hydrostatic equilibrium of the strange matter. The large distance static $Q\overline{Q}$ potential ${V}_{1}$ and the gluon condensate ${G}_{2}$ are the main parameters of the model. We use the surface gravitational redshift measurements as a probe to determine the ratio $(\mathcal{P}/\mathcal{E}{)}_{C}$ at the center of strange stars. For ${V}_{1}=0$ and ${G}_{2}\ensuremath{\gtrsim}0.035\text{ }\text{ }{\mathrm{GeV}}^{4}$, we show that $(\mathcal{P}/\mathcal{E}{)}_{C}\ensuremath{\simeq}0.262$ and the corresponding redshift ${z}_{S}\ensuremath{\simeq}0.47$ are limiting values at the maximum mass of the highest mass stellar sequence. As a direct application of our study, we try to determine the values of ${V}_{1}$ and ${G}_{2}$ from astrophysical observations of the compact star 1E 1207.4-5209. Due to the uncertainties in the surface redshift determination, we made two attempts to obtain the model parameters. Our findings show that $(\mathcal{P}/\mathcal{E}{)}_{C}=0.07{3}_{+0.024}^{+0.029}$ at 68% confidence, ${V}_{1}=0.44\ifmmode\pm\else\textpm\fi{}0.10\text{ }\text{ }\mathrm{GeV}$ at 90% confidence and ${G}_{2}=0.008\ifmmode\pm\else\textpm\fi{}0.001\text{ }\text{ }{\mathrm{GeV}}^{4}$ at 95% confidence in the first attempt; and $(\mathcal{P}/\mathcal{E}{)}_{C}=0.087\ifmmode\pm\else\textpm\fi{}0.028$ at 71% confidence, ${V}_{1}=0.43\ifmmode\pm\else\textpm\fi{}0.085\text{ }\text{ }\mathrm{GeV}$ at 94% confidence and ${G}_{2}=0.0093\ifmmode\pm\else\textpm\fi{}0.00092\text{ }\text{ }{\mathrm{GeV}}^{4}$ at 94% confidence in the second attempt. These values of ${V}_{1}$ and ${G}_{2}$ are in reasonable agreement with the lattice and QCD sum rules calculations. As a consequence of the high values of ${V}_{1}$ and ${G}_{2}$, the anomalous mass defects of 1E 1207.4-5209 are $|{\mathrm{\ensuremath{\Delta}}}_{2}M|\ensuremath{\simeq}2.56\ifmmode\times\else\texttimes\fi{}{10}^{53}\text{ }\text{ }\mathrm{erg}$ in the first attempt and $|{\mathrm{\ensuremath{\Delta}}}_{2}M|\ensuremath{\simeq}2.94\ifmmode\times\else\texttimes\fi{}{10}^{53}\text{ }\text{ }\mathrm{erg}$ in the second attempt.

Structure and tidal deformability of a hybrid star within the framework of the field correlator method

The structure of hybrid stars within the nonperturbative framework of the field correlator method, extended to zero-temperature limit as a quark model, has been studied. For the hadronic sector, we have used the lowest-order constraint variational method by employing AV18 two-body nucleon-nucleon interaction supplemented by the phenomenological Urbana-type three-body force. For an adapted value of the gluon condensate, G2 = 0:006 GeV4, which gives the critical temperature of about Tc ? 170 MeV, stable hybrid stars with a maximum mass of 2:04M? are predicted. The stability of hybrid star has been investigated for a wide range of gluon condensate value, G2, and quark-antiquark potential, V1. A hybrid equation of state fulfills the constraints on tidal deformability and hence on the radii of the stars, extracted from the binary GW170817. Moreover, tidal deformability for different chirp masses and different binary mass ratios of hybrid stars have been studied. The mass-radius relation satisfies the new constraint obtained from the neutron star interior composition explorer (NICER). A comprehensive analysis on the structure of a hybrid star and also its compactness, tidal Love number, and tidal deformability has been conducted for several parameter sets of the quark equation of state. The influence of different crustal equations of state on the mentioned quantities has been studied. Our calculations suggest the value of quark-antiquark potential, V1, to be around 0.08 GeV. The results achieved in this study are in strong concurrence with the other calculations reported on this subject.

Dense quark\u2013gluon plasma in strong magnetic fields

A non-perturbative (np) method of Field correlators (FCM) was applied to study QCD at temperatures above the deconfinement transition (1<T/T_c<3,~T_csim 0.16~mathrm{GeV}) and nonzero baryon densities (baryon chemical potential mu _B<0.5~mathrm{GeV}) in an external uniform magnetic field (eB<0.5~mathrm{GeV}^2). Within FCM, the np high-temperature dynamics is embodied in the Polyakov loop and in the Debye mass due to the Color-Magnetic confinement. Analytic expressions for quark pressure and magnetic susceptibility were obtained. The expressions were represented as series and in integral form. Magnetic susceptibility was found to increase rapidly with temperature and slowly with density. The results at the zero density limit are in agreement with lattice data.

Thermodynamics of a quark-gluon plasma at finite baryon density

Properties of the quark-gluon plasma(QGP) in the presence of the baryon chemical potential $\mu_B$ are studied using the Field Correlator Method(FCM). The nonperturbative FCM dynamics includes the Polyakov line, computed via colorelectric string tension $\sigma^E(T)$ and the quark and gluon Debye masses, defined by the colormagnetic string tension $\sigma^H(T)$. The resulting QGP thermodynamics at $\mu_B \le 400$ MeV is in a good agreement with the available lattice data,both pressure and the sound velocity do not show any sign of a critical behavior in this region.

Field correlator method for the confinement in QCD

The theory of confinement based on the stochastic field mechanism, known as the Field Corrleator Method (FCM) is discussed in detail. Experimental and lattice data have accumulated a vast amount of material on the properties of confinement in QCD. We enumerateall these properties as 1)-7), and discuss beyond FCM two existing approaches: monopole based Dual Ginzburg-Landau (DGL) theory,and Gribov-Zwanziger model, from this point of view. It is shown that the FCM satisfies all required criteria. We also prove its selfconsistency; in particular, it is shown that the string tension {\sigma} is the only scaleful parameter in the theory beyond fermion masses, and {\Lambda}_QCD is calculated explicitly to the lowest order in terms of {\sigma}. We also formulate physical consequences of confinement, such as string breaking,Regge trajectories, role of confinement in the perturbation theory, chiral symmetry breaking, confinement in the boosted systems etc. It is demonstrated that the FCM is a suitable tool for the solution of these problems.

Speed of sound in the QGP and an SU(3) Yang-Mills theory

The speed of sound $C_{s}$ in the SU(3) and (2+1)QCD is calculated within the Field Correlator Method using the nonperturbative colour magnetic confinement and Polyakov loop interaction in the deconfined region.The resulting $C_{s}$ displays a discontinuity at $T=T_{c}$ in the SU(3) case. It is shown numerically and analytically that $ C^{2}_{s}$ never exceeds $\frac{1}{3}$ both for SU(3) and (2+1) QCD for vanishing chemical potential.A good agreement is found of our numerical results with the corresponding lattice data.

Nonperturbative quark–gluon thermodynamics at finite density

Thermodynamics of the quark–gluon plasma at finite density is studied in the framework of the Field Correlator Method, where thermodynamical effects of Polyakov loops and color magnetic confinement are taken into account. Having found good agreement with numerical lattice data for zero density, we calculate pressure [Formula: see text], for [Formula: see text] MeV and [Formula: see text] MeV. For the first time, the explicit integral form is found in this region, demonstrating analytic structure in the complex [Formula: see text] plane. The resulting multiple complex branch points are found at the Roberge–Weiss values of [Formula: see text], with [Formula: see text] defined by the values of Polyakov lines and color magnetic confinement.

Single-camera displacement field correlation method for centrosymmetric 3D dynamic deformation measurement

Three-dimensional (3D) deformation measurements are a key issue in experimental mechanics. In this paper, a displacement field correlation (DFC) method to measure centrosymmetric 3D dynamic deformation using a single camera is proposed for the first time. When 3D deformation information is collected by a camera at a tilted angle, the measured displacement fields are coupling fields of both the in-plane and out-of-plane displacements. The features of the coupling field are analysed in detail, and a decoupling algorithm based on DFC is proposed. The 3D deformation to be measured can be inverted and reconstructed using only one coupling field. The accuracy of this method was validated by a high-speed impact experiment that simulated an underwater explosion. The experimental results show that the approach proposed in this paper can be used in 3D deformation measurements with higher sensitivity and accuracy, and is especially suitable for high-speed centrosymmetric deformation. In addition, this method avoids the non-synchronisation problem associated with using a pair of high-speed cameras, as is common in 3D dynamic measurements.

Color screening in flux tubes and in the color Coulomb potential from QCD field correlators

Colorelectric and Colormagnetic structure of the flux tubes, connecting heavy quark and antiquark, is investigated in the framework of the Field Correlator method which describes all resulting fields in terms of correlators $D^E$ and $D^E_1$. The latter have been computed via gluelumps, which allows to predict the resulting distribution of color fields $\mathbf{E} (\mathbf{r}),$ and colormagnetic currents $\mathbf{k} (\mathbf{r})$ in the flux tubes. It is shown, that at large distances $r\gg \lambda \approx 0.2$ fm the whole structure of fields and relations between them is similar to that of the dual superconductor theory, but the basic dynamics, including small distances, is given by field correlators of the real stochastic vacuum. The important contradiction between the strong screening of color fields in the width of flux tubes and almost no screening in the perturbative $Q\bar Q$ potential is resolved.

The CSS parametrization for Hybrid Stars with the Field Correlator Method

We explore the structure of hybrid stars based on a nuclear matter equation of state (EoS) built with the microscopic Brueckner-Hartree-Fock many-body theory, and a quark matter EoS derived with the Field Correlator Method (FCM), which can be accurately represented by the CSS (constant speed of sound) parametrization. We find that the main features of the hadron-quark phase transition are directly related to the FCM parameters, i.e. the quark-antiquark potential V1, the gluon condensate G2 and the color-flavour superconducting gap ∆, whose values range can be determined by the observational data on neutron star (NS) masses.