We calculate the chiral condensate of QCD with 2, 2+1 and 3 flavors of sea quarks. Lattice QCD simulations are performed employing dynamical overlap fermions with up and down quark masses covering a range between 3 and 100 MeV. On L ~ 1.8-1.9 fm lattices at a lattice spacing ~ 0.11 fm, we calculate the eigenvalue spectrum of the overlap-Dirac operator. By matching the lattice data with the analytical prediction from chiral perturbation theory at the next-to-leading order, the chiral condensate in the massless limit of up and down quarks is determined.
Glue Condensate, Quark Condensate and Dirac Spectral Density
Institute of Nuclear Physics, Czech Academy of Sciences（捷克科学院核物理研究所）；University of Kentucky（肯塔基大学）；The George Washington University（乔治华盛顿大学）；Institute of Physics, Slovak Academy of Sciences（斯洛伐克科学院物理研究所）
The gluon condensate, $\langle \frac{α_s}π G^2 \rangle$, i.e. the leading order power correction in the operator product expansion of current correlators in QCD at short distances, is determined from $e^+ e^-$ annihilation data in the charm-quark region. This determination is based on finite energy QCD sum rules, weighted by a suitable integration kernel to (i) account for potential quark-hadron duality violations, (ii) enhance the contribution of the well known first two narrow resonances, the $J/ψ$ and the $ψ(2S)$, while quenching substantially the data region beyond, and (iii) reinforce the…
We determine the strange and light quark condensates in full lattice QCD for the first time. This is done by direct calculation of the expectation value of the trace of the quark propagator followed by subtraction of the appropriate perturbative contribution to convert to a value for the condensate in the MS-bar scheme at 2 GeV. We use lattice QCD configurations including u, d, s and c quarks in the sea with u/d quark masses going down to the physical value. We find the ratio of the strange to the light quark condensate to be 1.08(16).
Fermionic theories with a vanishing chiral condensate (in the chiral limit) have recently attracted considerable interest; in particular variants of multi-flavour QCD are candidates for this behaviour. Here we consider the 2-flavour Schwinger model as a simple theory with this property. Based on simulations with light dynamical overlap fermions, we test the hypothesis that in such models the low lying Dirac eigenvalues could be decorrelated. That has been observed in 4d Yang-Mills theories at high temperature, but it cannot be confirmed for the 2-flavour Schwinger model. We also discuss subtle…
We determine the non-perturbative gluon condensate of four-dimensional SU(3) gauge theory in a model independent way. This is achieved by carefully subtracting high order perturbation theory results from non-perturbative lattice QCD determinations of the average plaquette. No indications of dimension two condensates are found. The value of the gluon condensate turns out to be of a similar size as the intrinsic ambiguity inherent to its definition.
We give the hyperasymptotic expansion of the plaquette with a precision that includes the terminant associated to the leading renormalon. Subleading effects are also considered. The perturbative series is regulated using the principal value prescription for its Borel integral. We use this analysis to give a determination of the gluon condensate in SU(3) pure gluodynamics that is independent of the scale and renormalization scheme used for the coupling constant: $\langle G^2 \rangle_{\rm PV} (n_f=0)=3.15(18)\; r_0^{-4}$.
We perform a precise calculation of the chiral condensate in QCD using lattice QCD with 2+1 flavors of dynamical overlap quarks. Up and down quark masses cover a range between 3 and 100 MeV on a 16^3x48 lattice at a lattice spacing around 0.11 fm. At the lightest sea quark mass, the finite volume system on the lattice is in the epsilon-regime. By matching the low-lying eigenvalue spectrum of the Dirac operator with the prediction of chiral perturbation theory at the next-to-leading order, we determine the chiral condensate in 2+1-flavor QCD with strange quark mass fixed at its physical value a…
We derive the Operator Product Expansion whose vacuum expectation value gives the time-moments of the pseudoscalar heavy-light current-current correlator up to and including terms in $α_s^2$ multiplying $\langle\overlineψψ\rangle/M^3$ and terms in $α_s$ multiplying $\langle α_s G^2 \rangle/M^4$, where $M$ is the heavy-quark mass. Using lattice QCD results for heavy-strange correlators obtained for a variety of heavy quark masses on gluon field configurations including $u$, $d$ and $s$ quarks in the sea at three values of the lattice spacing, we are able to show that the contribution of the str…
We apply the probability distribution function method to the study of chiral properties of QCD with quarks in the exact massless limit. A relation among the chiral condensate, zeros of the Bessel function and eigenvalue of Dirac operator is also given. The chiral condensate in this limit can be measured with small number of eigenvalues of the massless Dirac operator and without any ambiguous mass extrapolation. Results for SU(3) gauge theory with quenched Kogut-Susskind quarks on the $10^4$ lattice are shown.
We study the effective monopole action obtained in the Maximal Abelian projection of quenched SU(2) lattice QCD. We determine the quadratic part of the lattice action using analytical blocking from continuum dual superconductor model to the lattice model. The leading contribution to the quadratic action depends explicitly on value of the monopole condensate. We show that the analytical monopole action matches the numerically obtained action in quenched SU(2) QCD with a good accuracy. The comparison of numerical and analytical results gives us the value of the monopole condensate in quenched SU…
I use the technique of Hernandez, et al (hep-lat/0106011) to convert a recent calculation of the lattice-regulated quark condensate from an overlap action to a continuum-regulated number. I find Sigma(MSbar)(mu = 2 GeV) = (282(6) MeV)-cubed times (a-inverse/1766 MeV)-cubed from a calculation with the Wilson gauge action at beta=5.9.
We present the results of our computation of the chiral condensate with $N_f=2$ and $N_f=2+1+1$ flavours of maximally twisted mass fermions. The condensate is determined from the Dirac operator spectrum, applying the spectral projector method proposed by Giusti and Luscher. We use 3 lattice spacings and several quark masses at each lattice spacing to perform the chiral and continuum extrapolations. We study the effect of the dynamical strange and charm quarks by comparing our results for $N_f=2$ and $N_f=2+1+1$ dynamical flavours.
We determine the strange quark condensate from lattice QCD for the first time and compare its value to that of the light quark and chiral condensates. The results come from a direct calculation of the expectation value of the trace of the quark propagator followed by subtraction of the appropriate perturbative contribution, derived here, to convert the non-normal-ordered $m\barψψ$ to the $\bar{MS}$ scheme at a fixed scale. This is then a well-defined physical `nonperturbative' condensate that can be used in the Operator Product Expansion of current-current correlators. The perturbative subtrac…
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 present a Lattice QCD determination of the chiral quark condensate based on a new method. We extract the quark condensate from the operator product expansion of the quark propagator at short euclidean distances, where it represents the leading contribution in the chiral limit. From this study we obtain <\bar q q>^ms(2 GeV)=-(265+-5+-22 MeV)^3$, in good agreement with determinations of this quantity based on different approaches. The simulation is performed by using the O(a)-improved Wilson action at beta=6.45 on a volume 32^3\times70 in the quenched approximation.
A recent Monte Carlo study of {\em quenched} QCD showed that the chiral condensate is non-vanishing above $T_c$ in the phase where the average of the Polyakov loop $P$ is complex. We show how this is related to the dependence of the spectrum of the Dirac operator on the boundary conditions in Euclidean time. We use a random matrix model to calculate the density of small eigenvalues and the chiral condensate as a function of $\arg P$. The chiral symmetry is restored in the $\arg P=2π/3$ phase at a higher $T$ than in the $\arg P=0$ phase. In the phase $\arg P = π$ of the $SU(2)$ gauge theory the…
The Dirac operator describing the coupling of continuum quark fields to SU(2) center vortex world-surfaces composed of elementary squares on a hypercubic lattice is constructed. It is used to evaluate the quenched Dirac spectral density in the random vortex world-surface model, which previously has been shown to quantitatively reproduce both the confinement properties and the topological susceptibility of SU(2) Yang-Mills theory. Under certain conditions on the modeling of the vortex gauge field, a behavior of the quenched chiral condensate as a function of temperature is obtained which is con…
The eigenvalues of the Dirac operator at finite volume encode whether or not chiral symmetry is spontaneously broken in a massless theory. We apply this framework in a particular BSM context, namely SU(3) gauge theory with N_f=2 massless flavors in the 2-index symmetric (sextet) representation. Our first results are at a single lattice spacing. We find that both the density of near-zero eigenvalues and the renormalization group invariant mode number indicate spontaneous symmetry breaking. Quantitatively, there is a discrepancy between the determination of the fermion condensate in the chiral l…
The spectrum of low-lying eigenvalues of overlap Dirac operator in quenched SU(2) lattice gauge theory with tadpole-improved Symanzik action is studied at finite temperatures in the vicinity of the confinement-deconfinement phase transition defined by the expectation value of the Polyakov line. The value of the chiral condensate obtained from the Banks-Casher relation is found to drop down rapidly at T = Tc, though not going to zero. At Tc' = 1.5 Tc = 480 MeV the chiral condensate decreases rapidly one again and becomes either very small or zero. At T < Tc the distributions of small eigenvalue…
We analyze the mass dependence of the chiral condensate for QCD at nonzero $θ$-angle and find that in general the discontinuity of the chiral condensate is not on the support of the Dirac spectrum. To understand this behavior we decompose the spectral density and the chiral condensate into contributions from the zero modes, the quenched part, and a remainder which is sensitive to the fermion determinant and is referred to as the dynamical part. We obtain general formulas for the contributions of the zero modes. Expressions for the quenched part, valid for an arbitrary number of flavors, and fo…
We study the quark-gluon mixed condensate g<\bar{q} sigma G q>, using the SU(3)c lattice QCD with the Kogut-Susskind fermion at the quenched level. We generate 100 gauge configurations on the 16^4 lattice with β= 6.0, and perform the measurement of the mixed condensate at 16 points in each gauge configuration for each current quark mass of m_q=21, 36, 52 MeV. Using the 1600 data for each m_q, we find the ratio between the mixed condensate and the quark condensate, m_0^2 = g<\bar{q} sigma G q> / <\bar{q}q> \simeq 2.5 GeV^2 at the lattice scale of 1/a \simeq 2 GeV in the chiral limit. The large …
In the $ε$-domain of QCD we have obtained exact analytical expressions for the eigenvalue density of the Dirac operator at fixed $θ\ne 0$ for both one and two flavors. These results made it possible to explain how the different contributions to the spectral density conspire to give a chiral condensate at fixed $θ$ that does not change sign when the quark mass (or one of the quark masses for two flavors) crosses the imaginary axis, while the chiral condensate at fixed topological charge does change sign. From QCD at nonzero density we have learnt that the discontinuity of the chiral condensate …
We derive exact analytical expressions for the spectral density of the Dirac operator at fixed θ-angle in the microscopic domain of one-flavor QCD. These results are obtained by performing the sum over topological sectors using novel identities involving sums of products of Bessel functions. Because the fermion determinant is not positive definite for negative quark mass, the usual Banks-Casher relation is not valid and has to be replaced by a different mechanism first observed for QCD at nonzero chemical potential. Using the exact results for the spectral density we explain how this mechanism…
Exploiting Virasoro constraints on the effective finite-volume partition function, we derive generalized Leutwyler-Smilga spectral sum rules of the Dirac operator to high order. By introducing $N_v$ fermion species of equal masses, we next use the Virasoro constraints to compute two (low-mass and large-mass) expansions of the partially quenched chiral condensate through the replica method of letting $N_v \to 0$. The low-mass expansion can only be pushed to a certain finite order due to de Wit-'t Hooft poles, but the large-mass expansion can be carried through to arbitrarily high order. Results…
Using the eigenmodes of the overlap-Dirac operator, we study the topological structure of the QCD vacuum. We investigate the space-time profile of the low-lying eigenmodes and their contribution to the vacuum action density and chiral condensate under the existence of static color sources. We demonstrate that the low-lying Dirac eigenmode shows the flux-tube structure, which suggests the relevance to confinement. We also analyze the chiral condensate in the flux-tube. Chiral symmetry is partially restored inside the flux, and the reduction of the condensate is about 20% at the center of the tu…
In the epsilon-regime of lattice QCD one can get an accurate measurement of the pion decay constant F_pi by monitoring how just one single Dirac operator eigenvalue splits into two when subjected to two different external vector sources. Because we choose imaginary chemical potentials our Dirac eigenvalues remain real. Based on the relevant chiral Random Two-Matrix Theory we derive individual eigenvalue distributions in terms of density correlations functions to leading order in the finite-volume epsilon-expansion. As a simple byproduct we also show how the associated individual Dirac eigenval…
We present a detailed study of the interplay between chiral symmetry and spectral properties of the Dirac operator in lattice gauge theories. We consider, in the framework of the Schwinger model, the fixed point action and a fermion action recently proposed by Neuberger. Both actions show the remnant of chiral symmetry on the lattice as formulated in the Ginsparg-Wilson relation. We check this issue for practical implementations, also evaluating the fermion condensate in a finite volume by a subtraction procedure. Moreover, we investigate the distribution of the eigenvalues of a properly defin…
We study the singular values of the Dirac operator in dense QCD-like theories at zero temperature. The Dirac singular values are real and nonnegative at any nonzero quark density. The scale of their spectrum is set by the diquark condensate, in contrast to the complex Dirac eigenvalues whose scale is set by the chiral condensate at low density and by the BCS gap at high density. We identify three different low-energy effective theories with diquark sources applicable at low, intermediate, and high density, together with their overlapping domains of validity. We derive a number of exact formula…
We compute the chiral condensate in 2+1-flavor QCD through the spectrum of low-lying eigenmodes of Dirac operator. The number of eigenvalues of the Dirac operator is evaluated using a stochastic method with an eigenvalue filtering technique on the background gauge configurations generated by lattice QCD simulations including the effects of dynamical up, down and strange quarks described by the Mobius domain-wall fermion formulation. The low-lying spectrum is related to the chiral condensate, which is one of the leading order low-energy constants in chiral effective theory, as dictated by the B…
We present the comparison of the analytical microscopic spectral density for lattice QCD with $N_{\rm f}=2$ twisted mass fermions with the one obtained on the lattice utilizing configurations produced by the ETM collaboration. We extract estimates for the chiral condensate as well as the low-energy constant $W_8$ of Wilson $χ$-PT by employing spectral information of the Wilson Dirac operator with fixed index at finite volume.
We present the first nonperturbative large-$N$ calculation of the $\mathcal{N}=1$ supersymmetric (SUSY) $\mathrm{SU}(N)$ Yang$-$Mills gluino condensate obtained by means of numerical simulations of the lattice-discretized theory, exploiting large-$N$ twisted volume reduction. We present two different determinations based, respectively, on the Banks$-$Casher formula and on the Gell-Mann$-$Oakes$-$Renner relation, both giving perfectly consistent results. By expressing the lattice results in the Novikov$-$Shifman$-$Vainshtein$-$Zakharov (NSVZ) scheme, we are able for the first time to compare nu…
In a sector of fixed topological charge, the chiral condensate has a discontinuity given by the Banks-Casher formula also in the case of one-flavor QCD. However, at fixed θ-angle, the chiral condensate remains constant when the quark mass crosses zero. To reconcile these contradictory observations, we have evaluated the spectral density of one-flavor QCD at θ=0. For negative quark mass, it becomes a strongly oscillating function with a period that scales as the inverse space-time volume and an amplitude that increases exponentially with the space-time volume. As we have learned from QCD at non…
We construct a new order parameter for finite temperature QCD by considering the quark condensate for U(1)-valued temporal boundary conditions for the fermions. Fourier transformation with respect to the boundary condition defines the dual condensate. This quantity corresponds to an equivalence class of Polyakov loops, thereby being an order parameter for the center symmetry. We explore the duality relation between the quark condensate and these dressed Polyakov loops numerically, using quenched lattice QCD configurations below and above the QCD phase transition. It is demonstrated that the Di…
With two degenerate quarks, the chiral condensate exhibits a jump as the quark masses pass through zero. I discuss how this single transition splits into two Ising like transitions when the quarks are made non-degenerate. The order parameter is the expectation of the neutral pion field. The transitions represent long distance coherent phenomena occuring without the Dirac operator having vanishingly small eigenvalues.
We consider the effect of a non-zero lattice spacing on the low-energy effective theory of Wilson fermions with one flavor. Analytical results are given for both the chiral condensate and the microscopic spectral density of the Wilson Dirac operator. A subtle mechanism ensures that a constant chiral condensate is recovered, once the sum over sectors of fixed index is performed.
SU(2) gauge theory with one Dirac flavour in the adjoint representation is investigated on a lattice. Initial results for the gluonic and mesonic spectrum, static potential from Wilson and Polyakov loops, and the anomalous dimension of the fermionic condensate from the Dirac mode number are presented. The results found are not consistent with conventional confining behaviour, instead tentatively pointing towards a theory lying within or very near the onset of the conformal window, with the anomalous dimension of the fermionic condensate in the range $0.9 \lesssim γ_* \lesssim 0.95$. The implic…
We study quenched SU(2) lattice gauge theory with adjoint fermions in a wide range of temperatures. We focus on spectral quantities of the Dirac operator and use the temporal fermionic boundary conditions as a tool to probe the system. We determine the deconfinement temperature through the Polyakov loop, and the chiral symmetry restoration temperature for adjoint fermions through the gap in the Dirac spectrum. This chiral transition temperature is about four times larger than the deconfinement temperature. In between the two transitions we find that the system is characterized by a non-vanishi…
We calculate numerically the eigenvalue distribution of the overlap Dirac operator in the quenched Schwinger model on a lattice. The distribution does not fit any of the three universality classes of spontaneous chiral symmetry breaking, and its strong volume dependence indicates that the chiral condensate in the quenched theory is an ill-defined and divergent quantity. When we reweight configurations with the Dirac determinant to study the theory with N_f=1, we obtain a distribution of eigenvalues that is well-behaved and consistent with the theory of explicit symmetry breaking due to the ano…
The Dirac spectrum of QCD with dynamical fermions at nonzero chemical potential is characterized by three regions, a region with a constant eigenvalue density, a region where the eigenvalue density shows oscillations that grow exponentially with the volume and the remainder of the complex plane where the eigenvalue density is zero. In this paper we derive the phase diagram of the Dirac spectrum from a chiral Lagrangian. We show that the constant eigenvalue density corresponds to a pion condensed phase while the strongly oscillating region is given by a kaon condensed phase. The normal phase wi…
A continuum expression for the trace of the massive dressed-quark propagator is used to explicate a connection between the infrared limit of the QCD Dirac operator's spectrum and the quark condensate appearing in the operator product expansion, and the connection is verified via comparison with a lattice-QCD simulation. The pseudoscalar vacuum polarisation provides a good approximation to the condensate over a larger range of current-quark masses.
We compute the low lying spectrum of the overlap Dirac operator in the deconfined phase of finite-temperature quenched gauge theory. It suggests the existence of a chiral condensate which we confirm with a direct stochastic estimate. We show that the part of the spectrum responsible for the chiral condensate can be understood as arising from a dilute gas of instantons and anti-instantons.
We present some results for SU(2) with one adjoint Dirac flavour from lattice studies. Data for the spectroscopy, the static potential, topological charge, and the anomalous dimension of the fermionic condensate are included. Our findings are found to be in- consistent with conventional confining behaviour, instead pointing tentatively towards a theory lying within or very near the onset of the conformal window, with an anomalous dimension of the fermionic condensate of almost 1. Implications of these findings on the building of models of strongly-interacting dynamics beyond the standard model…
In this paper we study how dynamical chiral symmetry breaking is affected by nonzero chiral chemical potential in Dirac semimetals. To perform this study we applied lattice quantum Monte Carlo simulations of Dirac semimetals. Within lattice simulation we calculated the chiral condensate for various fermion masses, the chiral chemical potentials and effective coupling constants. For all parameters under consideration we have found that the chiral condensate is enhanced by chiral chemical potential. Thus our results confirms that in Dirac semimetals the chiral chemical potential plays a role of …
We study spontaneous chiral symmetry breaking and the spectral properties of the staggered lattice Dirac operator using quenched gauge configurations for the exceptional group G2, which has a trivial center. In particular we study the system below and above the finite temperature transition and use the temporal boundary conditions of the fermions to probe the system. We evaluate several observables: The spectral density at the origin, the spectral gap, the chiral condensate and the recently proposed dual chiral condensate. We show that chiral symmetry is broken at low temperatures and is resto…
We compute the condensate in QCD with two flavors of dynamical fermions using numerical simulation. The simulations use overlap fermions, and the condensate is extracted by fitting the distribution of low lying eigenvalues of the Dirac operator in sectors of fixed topological charge to the predictions of Random Matrix Theory.
At a critical temperature QCD in the chiral limit undergoes a chiral restoration phase transition. Above the phase transition the quark condensate vanishes. The Banks-Casher relation connects the quark condensate to a density of the near-zero modes of the Dirac operator. In the Nambu-Goldstone mode the quasi-zero modes condense around zero, λ\rightarrow 0, and provide a nonvanishing quark condensate. The chiral restoration phase transition implies that above the critical temperature there is no any longer a condensation of the Dirac modes around zero. If a U(1)_A symmetry is also restored and …
We estimate the quark condensate in one-flavor massless QCD from the known value of the gluino condensate in SUSY Yang-Mills theory using our newly proposed "orientifold" large-N expansion. The numerical result for the quark condensate renormalized at the scale 2 GeV is then given as a function of alpha_s(2 GeV) and of possible corrections from sub-leading terms. Our value can be compared with the quark condensate in (quenched) lattice QCD or with the one extracted from the Gell-Mann--Oakes--Renner relation by virtue of non-lattice determinations of the quark masses. In both cases we find quit…
We study, by numerical simulations on a lattice, the behaviour of the gauge-invariant quark-antiquark nonlocal condensates in the QCD vacuum with dynamical fermions. A determination is also done in the quenched approximation and the results are compared with the full-QCD case. The fermionic correlation length is extracted and compared with the analogous gluonic quantity.
The relation between the spectral density of the QCD Dirac operator at nonzero baryon chemical potential and the chiral condensate is investigated. We use the analytical result for the eigenvalue density in the microscopic regime which shows oscillations with a period that scales as 1/V and an amplitude that diverges exponentially with the volume $V=L^4$. We find that the discontinuity of the chiral condensate is due to the whole oscillating region rather than to an accumulation of eigenvalues at the origin. These results also extend beyond the microscopic regime to chemical potentials $μ\sim …
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.
