We study the Polyakov loop and the correlator of two Polyakov loops at finite temperature in the weak-coupling regime. We calculate the Polyakov loop at order g^4. The calculation of the correlator of two Polyakov loops is performed at distances shorter than the inverse of the temperature and for electric screening masses larger than the Coulomb potential. In this regime, it is accurate up to order g^6. We also evaluate the Polyakov-loop correlator in an effective field theory framework that takes advantage of the hierarchy of energy scales in the problem and makes explicit the bound-state dynamics. In the effective field theory framework, we show that the Polyakov-loop correlator is at leading order in the multipole expansion the sum of a colour-singlet and a colour-octet quark-antiquark correlator, which are gauge invariant, and compute the corresponding colour-singlet and colour-octet free energies.
We present the complete next-to-next-to-leading-order calculation of the correlation function of two Polyakov loops for temperatures smaller than the inverse distance between the loops and larger than the Coulomb potential. We discuss the relationship of this correlator with the singlet and octet potentials which we obtain in an Effective Field Theory framework based on finite-temperature potential Non-Relativistic QCD, showing that the Polyakov loop correlator can be re-expressed, at the leading order in a multipole expansion, as a sum of singlet and octet contributions. We also revisit the c…
We study the Polyakov loop correlator in the weak coupling expansion and show how the perturbative series re-exponentiates into singlet and adjoint contributions. We calculate the order $g^7$ correction to the Polyakov loop correlator in the short distance limit. We show how the singlet and adjoint free energies arising from the re-exponentiation formula of the Polyakov loop correlator are related to the gauge invariant singlet and octet free energies that can be defined in pNRQCD, namely we find that the two definitions agree at leading order in the multipole expansion, but differ at first or…
The asymptotic behavior of the correlator for Polyakov loop operators separated by a large distance $R$ is determined for high temperature QCD. It is dominated by nonperturbative effects related to the exchange of magnetostatic gluons. To analyze the asymptotic behavior, the problem is formulated in terms of the effective field theory of QCD in 3 space dimensions. The Polyakov loop operator is expanded in terms of local gauge-invariant operators constructed out of the magnetostatic gauge field, with coefficients that can be calculated using resummed perturbation theory. The asymptotic behavior…
Polyakov loop eigenvalues and their N-dependence are studied in 2 and 4 dimensional SU(N) YM theory. The connected correlation function of the single eigenvalue distributions of two separated Polyakov loops in 2D YM is calculated and is found to have a structure differing from the one of corresponding hermitian random matrix ensembles. No large $N$ non-analyticities are found for two point functions in the confining regime. Suggestions are made for situations in which large-N phase transitions involving Polyakov loops might occur.
We investigate chiral and deconfinement transitions in the strong coupling lattice QCD for color SU(3). We combine the leading order Polyakov loop effective action of the strong coupling expansion and the next-to-next-to-leading order (1/g^4) fermionic effective action with one species of unrooted staggered fermion. Two approximation schemes are adopted to evaluate the Polyakov loop effects; a Haar measure method (no fluctuation from the mean field) and a Weiss mean-field method (with fluctuations). The Polyakov loop is found to suppress the chiral condensate and to reduce the chiral transitio…
We review our present knowledge of the Polyakov loop, the correlator of Polyakov loops and the singlet correlator in thermal QCD from the point of view of perturbation theory and lattice QCD.
We discuss a non-perturbative renormalization of n-point Polyakov loop correlation functions by explicitly introducing a renormalization constant for the Polyakov loop operator on a lattice deduced from the short distance properties of 2-point correlators. We calculate this constant for the SU(3)gauge theory.
A popular observable in finite-temperature lattice QCD is the so-called singlet quark-antiquark free energy, conventionally defined in Coulomb gauge. In an effort to interpret the existing numerical data on this observable, we compute it at order O(alpha_s^2) in continuum, and analyze the result at various distance scales. At short distances (r << 1/pi T) the behaviour matches that of the gauge-independent zero-temperature potential; on the other hand at large distances (r >> 1/pi T) the singlet free energy appears to have a gauge-fixing related power-law tail. At infinite distance the result …
It appears well accepted in the literature that the correlator of Polyakov loops in a finite temperature system decays with the "average" free energy of the static quark-antiquark system, and can be decomposed into singlet and adjoint (or octet for QCD) contributions. By fixing a gauge respecting the transfer matrix, attempts have been made to extract those contributions separately. In this paper we point out that the "average" and "adjoint" channels of Polyakov loop correlators are misconceptions. We show analytically that all channels receive contributions from singlet states only, and give …
We calculate the next-to-next-to-leading correction to the expectation value of the Polyakov loop or equivalently to the free energy of a static charge. This correction is of order $g^5$. We show that up to this order the free energy of the static charge is proportional to the quadratic Casimir operator of the corresponding representation. We also compare our perturbative result with the most recent lattice results in SU(3) gauge theory.
A non-perturbative method of Field Correlators is applied to calculate the Polyakov loop dependence on temperature, $L(T)$, in the 2+1 flavor QCD with small quark masses, so the only relevant scale is the color-electric string tension $σ(T)$. Polyakov loop in the temperature range 100 MeV $\lesssim T\lesssim$ 400 MeV is expressed via the heavy-light meson mass that decreases with $T$ as $\sqrt{σ(T)}$. The latter is deduced from a gradually vanishing chiral condensate. The resulting $L(T)$ is in good agreement with recent lattice data.
We study Polyakov loop as well as correlators of real and imaginary parts of the Polyakov loop in 2+1 flavor QCD at finite temperature. We use hypercubic (HYP) smearing to improve the signal in the lattice calculations and to obtain reliable results for the correlators at large distances. From the large distance behavior of the correlators we estimate the chromo-electric screening length to be (0.38-44)/T. Furthermore, we show that the short distance distortions due to HYP smearing do not affect the physics of interest
Polyakov loops $L_a(T), a=3,8,...$ are shown to give the most important nonperturbative contribution to the thermodynamic potentials. Derived from the gluonic field correlators they enter as factors into free energy. It is shown in the $SU(3)$ case that $L_a (T)$ define to a large extent the behavior of the free energy and the trace anomaly $I(T)$, most sensitive to nonperturbative effects.
We give a short overview of the renormalization properties of rectangular Wilson loops, the Polyakov loop correlator and the cyclic Wilson loop. We then discuss how to renormalize loops with more than one intersection, using the simplest non-trivial case as an illustrative example. Our findings expand on previous treatments. The generalized exponentiation theorem is applied to the Polyakov loop correlator and used to renormalize linear divergences in the cyclic Wilson loop.
Brambilla, Escobedo, Soto, and Vairo have derived an effective description of quarkonium with two parameters; a momentum diffusion term and a real self-energy term. We point out that there is a similar real self-energy term for a single open heavy flavor and that it can be expressed directly in terms of Euclidean electric field correlators along a Polyakov line. This quantity can be directly studied on the lattice without the need for analytical continuation. We show that Minkowski-space calculations of this correlator correspond with the known NLO Euclidean value of the relevant electric fiel…
We investigate chiral and deconfinement transitions in the framework of the strong coupling lattice QCD for color SU(3) with one species of unrooted staggered fermion at finite temperature and quark chemical potential. We take account of the leading order Polyakov loop terms as well as the next-to-next-to-leading order (1/g^4) fermionic terms of the strong coupling expansion in the effective action. We investigate the Polyakov loop effects by comparing two approximation schemes, a Haar measure method (no fluctuation from the mean field) and a Weiss mean-field method (with fluctuations). The ef…
Using vector and axial-vector correlators within finite energy sum rules with inputs from a chiral quark model, coupled to the Polyakov loop, with nonlocal vector interactions, we extend our previous work to confirm the equivalence between the continuum threshold $s_0$ and the trace of the Polyakov loop $Φ$ as order parameters for the deconfinement transition at finite temperature $T$ and quark chemical potential $μ$. The obtained results are in agreement with our initial conclusion, where we showed that $s_0(T,μ)$ and $Φ(T,μ)$ provide the same information for the QCD deconfinement transition.
We discuss divergences of loop functions in thermal QCD and compute perturbatively the Polyakov loop, the Polyakov loop correlator and the cyclic Wilson loop. We show how these functions get mixed under renormalization.
The Schroedinger equation for the charmonium and bottomonium states at finite temperature is solved by employing an effective temperature dependent potential given by a linear combination of the color singlet free and internal energies obtained on the lattice from the Polyakov loop correlation functions. The melting temperatures and other properties of the quarkonium states are evaluated. The consistency of the potential model approach with the available lattice data on the quarkonium temporal correlators and spectral functions is explored.
We report the nonperturbative behavior of the twisted Polyakov loop (TPL) coupling constant for the SU(3) gauge theories defined by the ratio of Polyakov loop correlators in finite volume with twisted boundary condition. Carrying out the numerical simulations, we determine the nonperturbative running coupling constant in this renormalization scheme for Nf=12 SU(3) gauge theory. According to the perturbative two loop analysis, the Nf=12 SU(3) gauge theory might have a conformal fixed point in the infrared region. However, the recent lattice studies show controversial results for the existence o…
The correlator of two field strengths is computed from an effective action for Yang Mills theories, which contains both gluons and an auxiliary antisymmetric tensor field for the field strength as local variables. This action allows to relate explicitly many different approaches to confinement, and is computed using Wilsonian renormalization group equations with the bare Yang Mills action as starting point. Due to the inclusion of a higher dimensional operator in the ghost sector the running gauge coupling becomes vanishingly small at a critical scale k_c, and the resulting low energy action r…
We investigate the properties of scalar and pseudo-scalar mesons at finite temperature and quark chemical potential in the framework of the Nambu-Jona-Lasinio (NJL) model coupled to the Polyakov loop (PNJL model) with the aim of taking into account features of both chiral symmetry breaking and deconfinement. The mesonic correlators are obtained by solving the Schwinger-Dyson equation in the RPA approximation with the Hartree (mean field) quark propagator at finite temperature and density. In the phase of broken chiral symmetry a narrower width for the sigma meson is obtained with respect to th…
We calculate three-loop corrections to correlation functions of heavy-quark currents in the low- and high-energy regions. We present 30 coefficients both in the low-energy and the high-energy expansion of the scalar and the vector correlator with non-diagonal flavour structure. In addition we compute 30 coefficients in the high-energy expansion of the diagonal vector, axial-vector, scalar and pseudo-scalar correlators. Possible applications of our new results are improvements of lattice-based quark-mass determinations and the approximate reconstruction of the full momentum dependence of the co…
An attempt to adapt the study of color flux tubes to the case of finite temperature has been made. The field is measured both through the correlator of two Polyakov loops, one of which connected to a plaquette, and through a connected correlator of Wilson loop and plaquette in the spatial sublattice. Still the profile of the flux tube resembles the transverse field distribution around an isolated vortex in an ordinary superconductor. The temperature dependence of all the parameters characterizing the flux tube is investigated.
The gauge invariant two-point correlation function of the gauge field strength tensor is calculated in perturbation theory at the next-to-leading order. Besides a direct calculation in perturbative QCD we also present a derivation of the correlation function in the heavy quark effective theory. Our results are briefly compared with recent determinations of the field strength correlator on the lattice.
Hopping parameter expansion is a useful tool to investigate heavy dynamical quarks in lattice QCD, while the range of its applicability has been sometimes questioned. We study the convergence and the valid range of the hopping parameter expansion in the determination of the critical point (critical quark mass) of QCD with heavy quarks at finite temperature and density. On lattices with sufficiently large spatial extent, the terms in the hopping parameter expansion are classified into Wilson loop terms and Polyakov-type loop terms. We first study the case of the worst convergence in which all t…
Spectral sums of the Dirac-Wilson operator and their relation to the Polyakov loop are thoroughly investigated. The approach by Gattringer is generalized to mode sums which reconstruct the Polyakov loop locally. This opens the possibility to study the mode sum approximation to the Polyakov loop correlator. The approach is re-derived for the ab initio continuum formulation of Yang-Mills theories, and the convergence of the mode sum is studied in detail. The mode sums are then explicitly calculated for the Schwinger model and SU(2) gauge theory in a homogeneous background field. Using SU(2) latt…
Corrections of order $T^4$ to vector and axial current correlators in QCD at a finite temperature $T<T_c$ are obtained using dispersion relations for the amplitudes of deep inelastic scattering on pions. Their relation with the operator product expansion is presented. An interpretation of the results in terms of $T$-dependent meson masses is given: masses of $ρ$ and $a_1$ start to move with temperature in order $T^4$.
We compute the next-to-leading order perturbative QCD corrections to the correlators of nucleon interpolating currents in relativistic nuclear matter. The main new result is the calculation of the O(alpha_s) perturbative corrections to the coefficient functions of the vector quark condensate in matter. This condensate appears in matter due to the violation of Lorentz invariance. The NLO perturbative QCD corrections turn out to be large which implies that the NLO corrections must be included in a sum rule analysis of the properties of both bound nucleons and relativistic nuclear matter.
We present an operator based factorization formula for the transverse energy-energy correlator (TEEC) hadron collider event shape in the back-to-back (dijet) limit. This factorization formula exhibits a remarkably symmetric form, being a projection onto a scattering plane of a more standard transverse momentum dependent factorization. Soft radiation is incorporated through a dijet soft function, which can be elegantly obtained to next-to-next-to-leading order (NNLO) due to the symmetries of the problem. We present numerical results for the TEEC resummed to next-to-next-to-leading logarithm (NN…
We study the Sivers asymmetry in semi-inclusive hadron production in deep inelastic scattering. We concentrate on the contribution from the photon-gluon fusion channel at ${\cal O}(α_{em}^2 α_s)$, where three-gluon correlation functions play a major role within the twist-3 collinear factorization formalism. We establish the correspondence between such a formalism with three gluon correlation functions and the usual transverse momentum dependent (TMD) factorization formalism at moderate hadron transverse momenta. We derive the coefficient functions used in the usual TMD evolution formalism rela…
The investigation of the scalar gluonium correlator is interesting because it carries the quantum numbers of the vacuum and the relevant hadronic current is related to the anomalous trace of the QCD energy-momentum tensor in the chiral limit. After reviewing the purely perturbative corrections known up to next-next-to-leading order, the behaviour of the correlator is studied to all orders by means of the large-beta_0 approximation. Similar to the QCD Adler function, the large-order behaviour is governed by the leading ultraviolet renormalon pole. The structure of infrared renormalon poles, bei…
We report progress towards computing the heavy quark momentum diffusion coefficient from the correlator of two chromo-electric fields attached to a Polyakov loop in pure SU(3) gauge theory. Using a multilevel algorithm and tree-level improvement, we study the behavior of the diffusion coefficient as a function of temperature in the wide range $1.1 < T / T_c < 10^4$ in order to compare it to perturbative expansions at high temperature. We find that within errors the lattice results are remarkably compatible with the next-to-leading order perturbative result.
We study the dynamics of light quantum scalar fields in de Sitter space on superhorizon scales. We compute the self-energy of an O(N) symmetric theory at next-to-leading order in a 1/N expansion in the regime of superhorizon momenta, and we obtain an exact analytical solution of the corresponding Dyson-Schwinger equations for the two-point correlator. This amounts to resumming the infinite series of nonlocal self-energy insertions, which typically generate spurious infrared and/or secular divergences. The potentially large de Sitter logarithms resum into well-behaved power laws from which we e…
We report the nonperturbative behavior of the twisted Polyakov loop (TPL) coupling constant for the SU(3) gauge theories defined by the ratio of Polyakov loop correlators in finite volume with twisted boundary condition. We reveal the vacuum structures and the phase structure for the lattice gauge theory with the twisted boundary condition. Carrying out the numerical simulations, we determine the nonperturbative running coupling constant in this renormalization scheme for the quenched QCD and Nf=12 SU(3) gauge theories. At first, we study the quenched QCD theory using the plaquette gauge actio…
We compute the finite temperature ghost and gluon propagators of Yang-Mills theory in the Landau-DeWitt gauge. The background field that enters the definition of the latter is intimately related with the (gauge-invariant) Polyakov loop and serves as an equivalent order parameter for the deconfinement transition. We use an effective gauge-fixed description where the nonperturbative infrared dynamics of the theory is parametrized by a gluon mass which, as argued elsewhere, may originate from the Gribov ambiguity. In this scheme, one can perform consistent perturbative calculations down to infrar…
The dual superconductivity is a promising mechanism for quark confinement. We have proposed the non-Abelian dual superconductivity picture for SU(3) Yang-Mills theory, and showed the restricted field dominance (called conventionally Abelian dominance), and non-Abelian magnetic monopole dominance in the string tension. We have further demonstrated by measuring the chromoelectric flux that the non-Abelian dual Meissner effect exists and determined that the dual superconductivity for SU(3) case is of type I, which is in sharp contrast to the SU(2) case: the border of type I and type II. In this t…
We represent Polyakov loops and their correlators as spectral sums of eigenvalues and eigenmodes of the lattice Dirac operator. The deconfinement transition of pure gauge theory is characterized as a change in the response of moments of eigenvalues to varying the boundary conditions of the Dirac operator. We argue that the potential between static quarks is linked to spatial correlations of Dirac eigenvectors.
The liquid droplet formula is applied to an analysis of the properties of geometrical (anti)clusters formed in SU(2) gluodynamics by the Polyakov loops of the same sign. Using this approach, we explain the phase transition in SU(2) gluodynamics as a transition between two liquids during which one of the liquid droplets (the largest cluster of a certain Polyakov loop sign) experiences a condensation, while the droplet of another liquid (the next to the largest cluster of the opposite sign of Polyakov loop) evaporates. The clusters of smaller sizes form two accompanying gases, which behave oppos…
Energy Correlators in $V + X$ as a Benchmark Observable for Precision QCD
Cavendish Laboratory, University of Cambridge（剑桥大学卡文迪许实验室）；High Energy Physics Division, Argonne National Laboratory（阿贡国家实验室高能物理部）；Department of Physics, Yale University（耶鲁大学物理系）；Institute of Nuclear Physics（核物理研究所）；Center for Theoretical Physics - a Leinweber Institute, Massachusetts Institute of Technology（麻省理工学院莱因韦伯理论物理中心）
We compute next-to-leading order (NLO) perturbative QCD corrections to the correlators of interpolating pentaquark currents. We employ modular techniques in configuration space which saves us from the onus of having to do loop calculations. The modular technique is explained in some detail. We present explicit NLO results for several interpolating pentaquark currents that have been written down in the literature. Our modular approach is easily adapted to the case of NLO corrections to multiquark correlators with an arbitrary number of quarks/antiquarks.
The spectral function related to the correlator of two colour-electric fields along a Polyakov loop determines the momentum diffusion coefficient of a heavy quark near rest with respect to a heat bath. We compute this spectral function at next-to-leading order, O(alpha_s^2), in the weak-coupling expansion. The high-frequency part of our result (omega >> T), which is shown to be temperature-independent, is accurately determined thanks to asymptotic freedom; the low-frequency part of our result (omega << T), in which Hard Thermal Loop resummation is needed in order to cure infrared divergences, …
Benchmarking the Nearside Energy-Energy Correlators with Mellin Transform
Nuclear Science Division, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室核科学部）；Physics Department, Brookhaven National Laboratory（布鲁克海文国家实验室物理系）；RIKEN BNL Research Center, Brookhaven National Laboratory（布鲁克海文国家实验室理化学研究所BNL研究中心）
We perform a complete next-to-leading order calculation of the non-Abelian electric field correlator in a SU($N_c$) plasma, which encodes properties of the plasma relevant for heavy particle bound state formation and dissociation, and is different from the correlator for the heavy quark diffusion coefficient. The calculation is carried out in the real-time formalism of thermal field theory and includes both vacuum and finite temperature contributions. By working in the $R_ξ$ gauge, we explicitly show the results are gauge independent, infrared and collinear safe. The renormalization group equa…
The B-decay constant $f_B$ is an important component for studying $B$-meson decays, which can be studied through QCD sum rules. We make a detailed discussion on $f_B$ from two sum rules, i.e. sum rules I and II, which are derived from the conventional correlator and the correlator with chiral currents respectively. It is found that these two sum rules are consistent with each other. However, the sum rules II has less uncertainty sources than that of sum rules I, and then it can be more accurate if we know the dimension-four gluon condensate well. It is found that $f_B$ decreases with the incre…
We study quark confinement by computing the Polyakov loop potential in Yang--Mills theory within different non-perturbative functional continuum approaches [1]. We extend previous studies in the formalism of the functional renormalisation group and complement those with findings from Dyson--Schwinger equations and two-particle-irreducible functionals. These methods are formulated in terms of low order Green functions. This allows to identify a criterion for confinement solely in terms of the low-momentum behaviour of correlators.
The Debye screening mass for a quark-gluon plasma at high temperature is calculated to next-to-leading order in the QCD coupling constant from the correlator of two Polyakov loops. The result agrees with the screening mass defined by the location of the pole in the gluon propagator as calculated by Rebhan. It is logarithmically sensitive to nonperturbative effects associated with the screening of static chromomagnetic fields.
We present analytical next-to-leading order results for the correlator of baryonic currents at the three loop level with one finite mass quark. We obtain the massless and the HQET limit of the correlator from the general formula as particular cases. We also give explicit expressions for the moments of the spectral density.
Using configuration space techniques, we calculate next-to-leading order (NLO) five-loop QCD corrections to the correlators of interpolating pentaquark currents in the limit of massless quarks. We obtain very large NLO corrections to the spectral density which makes a standard sum rule analysis problematic. However, the NLO corrections to the correlator in configuration space are reasonable. We discuss the implications of our results for the phenomenological sum rule analysis of pentaquark states.
Nonperturbative treatment of quark-hadron transition at nonzero temperature T and chemical potential mu in the framework of Field Correlator Method is generalized to the case of nonzero magnetic field B. A compact form of the quark pressure for arbitrary B, mu, T is derived. As a result the transition temperature is found as a function of B and mu, which depends on only parameters: vacuum gluonic condensate G_2 and the field correlator D_1^E(x), which defines the Polyakov loops and it is known both analytically and on the lattice. A moderate (25%) decrease of T_c(mu=0) for eB changing from zer…
We highlight the principal results of a computation in the Color Glass Condensate effective field theory (CGC EFT) of the next-to-leading order (NLO) impact factor for inclusive photon+dijet production at Bjorken $x_{\rm Bj} \ll 1$ in deeply inelastic electron-nucleus (e+A DIS) collisions. When combined with extant results for next-to-leading log $x_{\rm Bj}$ JIMWLK renormalization group (RG) evolution of gauge invariant two-point ("dipole") and four-point ("quadrupole") correlators of light-like Wilson lines, the inclusive photon+dijet e+A DIS cross-section can be determined to $\sim 10$\% ac…
The isospin-breaking correlator of the product of flavor octet vector currents, $V^3_μ$ and $V^8_ν$, $Π^{38}_{μν}(q^2)$ is computed to next-to-next- to-leading (two-loop) order in Chiral Perturbation Theory. Large corrections to both the magnitude and $q^2$-dependence of the one-loop result are found, and the reasons for the slow convergence of the chiral series for the correlator given. The two-loop expression involves a single ${\cal O}(q^6)$ counterterm, present also in the two-loop expressions for $Π^{33}_{μν}(q^2)$ and $Π^{88}_{μν}(q^2)$, which counterterm contributes a constant to the sc…
