text:
Charmonium at finite temperature in quenched lattice QCD
Takashi Umeda Center for Computational Physics, University of Tsukuba, Tsukuba 305-8577, Japan Kouji Nomura Department of Physics, Hiroshima University, Higashi-hiroshima 739-8526, Japan Hideo Matsufuru Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
We study charmonium correlators in pseudoscalar and vector channels at finite temperature using lattice QCD simulation in the quenched approximation. Anisotropic lattices are used in order to have sufficient numbers of degrees of freedom in the Euclidean temporal direction. We focus on the low energy structure of the spectral function, corresponding to the ground state in the hadron phase, by applying the smearing technique to enhance the contribution to the correlator from this region. We employ two analysis procedures: the maximum entropy method (MEM) for the extraction of the spectral function without assuming a specific form, to estimate the shape of the spectral function, and the standard χ2 fit analysis using typical forms in accordance with the result of MEM, for a more quantitative evaluation. To verify the applicability of the procedures, we first analyze the smeared correlators as well as the point correlators at zero temperature. We find that by shortening the t-interval used for the analysis (a situation inevitable at T>0) the reliability of MEM for point correlators is lost, while it subsists for smeared correlators. Then the smeared correlators at T≃0.9Tc and 1.1Tc are analyzed. At T≃0.9Tc, the spectral function exhibits a strong peak, well approximated by a delta function corresponding to the ground state with almost the same mass as at T=0. At T≃1.1Tc, we find that the strong peak structure still persists at almost the same place as below Tc, but with a finite width of a few hundred MeV. This result indicates that the correlators possess a nontrivial structure even in the deconfined phase.
12.38.Gc, 12.38.Mh
††preprint: UTCCP-P-132, HUPD-0210, YITP-02-66††preprint: hep-lat/0211003
The charmonium systems have been payed much attention as a signal of the QCD phase transition. Due to a change of the interquark potential by thermal effects, the J/ψ mass is expected to decrease when approaching the phase transition Has86 . Above Tc, the screening of the interquark potential may dissolve the charmonium states, and the resulting J/ψ suppression has been regarded as one of the most important signals for detecting the formation of the plasma state MS86 ; Jpsi:exp ; Jpsi:th .
In principle, lattice QCD simulations can provide information on such excitation modes based on a nonperturbative and model independent framework, since the correlation functions in the Euclidean temporal direction measured on a lattice are related to the real time retarded and advanced Green functions by analytic continuation from a single spectral function AGDF59 ; HNS93 . In practice, however, the extraction of reliable information from numerical data for correlators becomes increasingly difficult as the temperature increases. There are two restrictions on the correlators at finite temperature: the numbers of available degrees of freedom, and the physical extent in the temporal direction. The former can be improved by the use of anisotropic lattices, on which the temporal lattice spacings are finer than the spatial ones HNS93 ; TARO01 ; Ume01 . The latter restriction is, however, physically inevitable and may spoil standard procedures applicable at zero temperature.
Recently, there has been technical progress in the direct extraction of the spectral function from lattice data of correlators NAH99 ; SpF_early . The key role is played by the maximum entropy method (MEM), combined with the singular value decomposition for constructing the functional bases in the space of spectral functions. At zero temperature this technique has reproduced spectra consistent with the standard fit analyses NAH99 ; ODS00 ; WK00 ; CPPACS01a ; Fie02 . Application to systems at finite temperature is, however, not straightforward because of the two restrictions mentioned above AMR02 . In particular, shortness of temporal extent, 1/T, makes it difficult to extract the correct low energy structure of the spectral function. Therefore, one at least needs to verify at T=0 that the method produces results which are stable against variation of the t-interval. There have already been a number of applications of MEM for finite temperature Bielefeld02a ; Bielefeld02b ; Bielefeld02c ; AHN02 , but as long as such tests are not systematically performed these results may contain uncontrolled uncertainties. In fact, we will show in this paper that MEM applied to point-point correlators at T=0 with shortened t-interval fails to produce the correct spectral functions. We therefore apply the smearing technique to enhance the contribution of the low energy part to the correlators ODS00 ; TARO01 ; Ume01 . For the smeared correlators at T=0, MEM works, at least qualitatively, also with a shorter t-interval.
In this paper, charmonium correlators are investigated using lattice QCD simulations in the quenched approximation. We focus on properties of the ground state, more generally the low energy structure of the correlator, rather than the whole spectral function. This paper mainly deals with the following two subjects:
(1) A technical investigation of procedures for extraction of reliable information from the numerical data of correlators in the temporal direction. In addition to MEM, we apply the standard χ2 fit analysis assuming a peak structure for the spectral function ISM02 . The latter approach complements the former, since MEM with small numbers of data points appears not work beyond the level of qualitative estimate for the shape of spectral function.
(2) A study of temperature effects on the correlators near to the deconfining phase transition. Below Tc, the question is whether the charmonium mass shifts, as expected from the potential model approach Has86 . Above Tc, previous studies have indicated that nontrivial structures may survive TARO01 ; Ume01 . The possibility of the existence of collective modes in the plasma phase is therefore carefully examined.
We use quenched anisotropic lattices with the renormalized anisotropy ξ=aσ/aτ=4 and the spatial lattice cutoff a−1σ≃2 GeV. The phase transition occurs near Nt=28, and hence sufficiently many points are available in the vicinity of Tc. At T≃0 we try to determine the conditions for a reliable extraction of the spectral function. We find that for the point correlators, reliability of MEM with less than 24 t-points (meaning a physical t-range of less than O(0.5fm)∼0.7Tc) is not guaranteed with the present level of statistics. Therefore only the smeared correlators are analyzed at finite temperature, T≃0.9Tc and 1.1 Tc.
Here we comment on the smearing technique. Although the correlator of spatially extended operators has no direct relation to the physically observable dilepton production rate, the properties of collective excitations such as mass and width are unchanged and can be probed more efficiently. To the extent that these excitations drive the dilepton rate we can at least say something about their position and shape, if not about their strength. A disadvantage is the possibility of detecting a fake peak, which may be produced by the smearing even in the case of free quarks Bielefeld02a . To distinguish such an artifact from a genuine physical peak we use besides the smearing function based on the wave function obtained at T=0 also a narrower one of essentially half-width. We speak thereby of "smeared" and of "half-smeared" correlators (because the energy region enhanced by this narrower smearing function is wider, while the high frequency part of O(1/aτ) is still sufficiently suppressed). Comparing the results of these two types of smeared correlators, one can argue whether the observed peak structure is an artifact of free quarks or physical indication of a collective mode.
This paper is organized as follows. In Sec. II, we recall some fundamental properties of the spectral function. For later uses in the χ2 fit analysis, presumable forms of the spectral function for a collective mode are discussed. Section III reviews anisotropic lattice actions and describes the parameters used in this paper. We also show the spectral function of the correlator of a free quark-antiquark pair, for comparison with the correlators from the Monte Carlo simulation. In Sec. IV our analysis procedures are explained. Section V describes the setup of Monte Carlo simulation. The following three sections show the result of the numerical simulation. In Section VI we analyze the correlators at zero temperature, as a preparation for the more involved situations at finite temperatures. Sections VII and VIII show the results at T≃0.9Tc and 1.1Tc, respectively. In each case, we first estimate the presumable form of the spectral function with MEM, and then apply the χ2 fit analysis for a more quantitative evaluation of its structure. Section IX is devoted to our conclusions on the technical and physical implications of the results. Our preliminary results have been reported in Ref. NMUM02 .
Ii Spectral function
In this section β represents the inverse temperature and should not be confused with the coupling of gauge field which appears in later sections. For a finite lattice of temporal extent Nτ we have β=1/T=Ntaτ. The Euclidean time is represented by τ in this section to make clear the distinction between the Euclidean and Minkowski formulations. In later sections we use also t as the Euclidean time, because no confusion is expected for the lattice QCD formulated in the Euclidean space.
We consider correlators of the form:
C(τ)=∑→x⟨O(→x,τ)O†(0,0)⟩, (1)
where the operator O(→x,τ) is a quark bilinear,
O(→x,τ)=∑→yϕ(→y)¯q(→x+→y,τ)Γq(→x,τ). (2)
4×4 matrix Γ specifies the quantum number, such as, for example, Γ=γ5 for pseudoscalar and γi (i=1,2,3) for vector channels. ϕ(→y) is a smearing function defining the spatial extension of the bilinear operator. Since the smearing in general violates the gauge invariance of the correlator, one needs to fix the gauge or employ a gauge invariant form of the smearing function. We fix the configurations to the Coulomb gauge, since it is widely used and allows easy implementation of the smearing function. Source and sink operators are always identical in this work.
Point correlators correspond to a ultra local "smearing function" ϕ(→y)=δ→y,0 in Eq. (2). For the smeared correlators we use smearing function of the form
ϕ(→y)=exp(−a|→y|p), (3)
where the parameters a and p are determined by fitting the wave function measured in the numerical simulation to the above form. The smearing technique described here was already applied to the problems of the charmonium correlators at T>0 Ume01 , as well as to the light meson correlators TARO01 .
For a while let us consider the continuum Euclidean space. The Matsubara Green function is represented in terms of the spectral function as
G(νn,→p)=∫∞−∞dω2πρ(ω,→p)iνn−ω+iϵ. (4)
The spectral function ρ(ω,→p) is represented as
ρ(ω,→p) = 12Z(β)∑n,m(e−βEn−e−βEm)|⟨n|O(0)|m⟩|2 (5)
×(2π)4δ(ω−Em+En)δ(→p−→Pm+→Pn),
which is an odd real function due to the bosonic nature of the present correlator and positive for ω>0.
The retarded and advanced Green functions for real time are represented as
GR(ν,→p) = ∫∞−∞dω2πρ(ω,→p)ν−ω+iϵ (6)
GA(ν,→p) = ∫∞−∞dω2πρ(ω,→p)ν−ω−iϵ (7)
with the same spectral function, ρ(ω,→p) AGDF59 . The spectral function is then represented as
ρ(ν,→p)=−2ImGR(ν,→p)=2ImGA(ν,→p). (8)
Strictly speaking, the above integrations do not always converge, and one needs appropriate subtraction terms. On the other hand, the high frequency part of the spectral function is not practically significant in the numerical analysis, because of the strong suppression by K(τ,ω), and of the existence of the lattice cutoff. We therefore do not consider these subtractions in this paper. We note that the smearing of the operator in Eq. (2) is performed only in the spatial directions, and therefore the analytic continuation of the Matsubara Green function to the real time Green functions is independent of this operation (but, of course, the spectral function depends on the operators, hence on smearing).
In the following we are only concerned with the correlators projected on the zero momentum states and discard the argument →p. Since the spectral function ρ(ω) is an odd function in ω, the correlator (1) is represented as
C(τ)=∫∞0dω2πK(τ,ω)ρ(ω). (9)
K(τ,ω)=e−ωτ+e−βω+ωτ1−e−βω. (10)
In the numerical simulation of lattice QCD one measures the left hand side of Eq. (9). The question is how to solve the inverse problem to obtain ρ(ω) with limited data for C(τ). The procedures to extract information about ρ(ω) are described in Sec. IV.
Now we shall present several ansätze for the spectral function which will be used for later analysis of the lattice data, and discuss their physical implications. A simple representation of a collective mode is the following form of the retarded Green function:
GR(ν)=R(ν)ν−μ+iγ(ν)/2, (11)
where μ and γ(ν) express the dispersion and the width of the mode. The corresponding spectral function is the well-known Breit-Wigner form. We shall neglect the frequency dependence of the width γ for simplicity. Taking the oddness of the bosonic spectral function into account, ρ(ω) reads
ρ(ω)=γR(ω)(ω−m)2+γ2/4−γR(ω)(ω+m)2+γ2/4. (12)
If the quantum numbers of the operators used to represent the collective mode are the correct ones, the physical properties associated with the mode, characterized by μ and γ, are independent of the particular operator. Therefore the smeared operators can be used to observe these quantities. However, the smearing does change the overlap of the operator with the state, R(ω). Therefore the effects depending on R(ω) and those coming only from the peak structure (represented with μ and γ) should be distinguished. If the change of R(ω) is sufficiently gentle over the region of the observed peak, the ω dependence in R(ω) is negligible.
Instead of Eq. (12), at zero temperature one often uses the relativistic Breit-Wigner type form:
ρ(ω)=sign(ω)ω2mγ(4R/m)(ω2−m2)2+m2γ2. (13)
This form shows a similar behavior to Eq. (12) around the peak position. However, the behavior far from the peak is different. In particular (12) and (13) give different contribution to the integral (9) near vanishing ω: In the vicinity of ω=0, Eq. (12) behaves linearly in ω, while Eq. (13) behaves linearly in ω2. Since K(τ,ω) behaves as (2−βω)/βω, Eq. (12) gives a t-independent contribution to the integral (9) AMR02 . We note that there is no indication of linear behavior in ω in the small ω region of the spectral functions obtained from our numerical data with MEM, as shown in Sec. VII and VIII. Therefore, in the χ2 fit analysis, we use the form of Eq. (13) as ansatz for the spectral function to which the lattice data are fitted. While the use of Eq. (13) is valid at zero temperature, it should only be taken as a representative form of peak structure at T>0, and hence the parameters m and γ do not have a strict sense but represent only a convenient parameterization of the peak.
In the limit of γ→0, both Eqs. (12) and (13) tend towards a delta function form,
ρ(ω)=2πR[δ(ω−m)−δ(ω+m)]. (14)
This form corresponds to the standard exponential fit of correlators used in the hadron spectroscopy at zero temperature.
Iii Anisotropic lattice QCD
iii.1 Anisotropic lattice actions
For analysis of the correlators in the temporal direction at T>0 the fine temporal resolution is very important to extract meaningful information TARO01 ; Ume01 ; ISM02 . Anisotropic lattices have become a powerful tool to achieve sufficiently high temporal cutoff while keeping total computational size modest. In the following, we summarize the anisotropic lattice actions employed in this work. The parameters we use are described in the next subsection.
For the gauge field, we adopt the standard Wilson plaquette action Kar82 ,
SG = βγG∑x3∑i=1[1−13% ReTrUi4(x)] (15)
+βγG∑x3∑i<j=1[1−13ReTrUij(x)],
where Uμν is a product of link variables along a plaquette in the μ-ν plane. Here the parameters β and γG are bare coupling and bare anisotropy, respectively.
For the quark field we use the O(a) improved Wilson-type action Ume01 ; Aniso01a ; Aniso01b :
Sq=∑x,y¯q(x)K(x,y)q(y), (16)
K(x,y) = δxy (17)
−κτ[(1−γ4)U4(x)δx+^4,y+(1+γ4)U4(x−^4)δx−^4,y]
−κσ3∑i=1[(r−γi)Ui(x)δx+^i,y+(r+γi)Ui(x−^i)δx−^i,y]
−κσcE3∑i=1σ4iF4i(x)δx,y
−κσcB3∑i>j=1σijFij(x)δx,y.
where κσ and κτ are the spatial and temporal hopping parameters, respectively, r is the spatial Wilson parameter, and cE, cB are the clover coefficients for the O(a) improvement. We set r=1/ξ and cE, cB to the tadpole-improved tree-level values, and vary the two parameters κσ and κτ to change the quark mass and the fermionic anisotropy. The tadpole improvement LM93 is performed by rescaling the link variables as Ui(x)→Ui(x)/uσ and U4(x)→U4(x)/uτ, respectively, with the corresponding mean-field values of the spatial and temporal link variables, uσ and uτ. Then cE and cB with the choice r=1/ξ read as
cE=1/uσu2τ,cB=1/u3σ. (18)
Instead of (κσ,κτ), it is convenient to use the parameters
γF ≡ uτκτ/uσκσ, (19)
1κ ≡ 1uσκσ−2(γF+3r−4)=2(m0γF+4), (20)
where m0 is the bare quark mass in temporal lattice units. κ plays the same role as on isotropic lattices, and controls the bare quark mass. γF is the bare fermionic anisotropy.
The lattice quark field q(x) in the action (17) is related to the dimensionful field ψ(x) as
ψ(x)=a−3/2σZq(β,κ)K(κ)q(x), (21)
where we assume mQ≪a−1τ. K(κ) is so-called KLM normalization factor KLM ,
K(κ) = [2uτκτ(1+m0)]1/2 (22)
= [1+2κ(γF−4)1+2κ(γF+3r−4)]1/2.
We consider the tadpole-improved tree-level matching of the fields in continuum and lattice theories, and hence Zq is set to unity.
iii.2 Parameters
We here describe the parameters used in this paper. First, we discuss the calibration of gauge and quark fields. In an interacting theory, the renormalized anisotropy ξ=aσ/aτ generally differs from the bare anisotropy parameters γG and γF. Therefore one needs to tune the latter such that
ξG(γG,γF)=ξF(γG,γF)=ξ (23)
holds, where ξG and ξF are the observed anisotropies defined through the gauge and fermionic observables, respectively. In quenched simulations, one can calibrate the gauge and fermion fields separately, first the former, and then the latter.
For the calibration of the gauge field, we refer to an elaborated work by Klassen Kla98 which uses Wilson loops in the spatial-spatial and spatial-temporal planes and is performed at 1% accuracy level. Making use of his relation of γG to β and ξ, we choose β=6.10 and γG=3.2108 corresponding to the renormalized anisotropy ξ=4. These values were also adopted in Ref. Aniso01b , and correspond to the spatial lattice cutoff a−1σ=2.030(13) GeV set by the hadronic radius r0 Som94 .
For the quark field, we use the result of Ref. Aniso01b in which the calibration was performed on the same lattice as this work using the meson dispersion relation in a quark mass region including the charm quark mass. Accordingly, we adopt κ=0.112 and γF=4.00, namely mq=0.121, as the values corresponding to the charm quark mass. (In Ref. Aniso02a it was argued that the value of γF tuned for the massless quark is also applicable to the charm quark mass. For historical reason, however, we use the result of the mass dependent tuning in Ref. Aniso01b . The difference is actually small and does not cause any significant effect.)
As smearing function we use the result for the wave function in the Coulomb gauge at T=0. For the vector channel the fit to Eq. (3) yields a=0.2275(9) and p=1.258(5). These values are used for both the pseudoscalar and vector correlators.
In addition, we also use a narrower smearing function with a=0.45 and the same p. The correlators smeared with this function are called half-smeared correlators, since the smearing function has about twice the slope as the main one and hence a smaller smearing effect is expected: the low energy part of the correlator is only partially enhanced, while the high energy part of O(1/aτ) is sufficiently reduced. This intermediate smearing function plays an important role at T>Tc in examining whether the observed peak structure of the spectral function for the smeared correlator is an artifact of smearing or a genuine physical effect (Sec. VIII).
iii.3 Free quark case
Figure 1: The spectral functions for the pseudoscalar correlators composed of free quarks at T=0.
Since in the perturbative plasma regime we expect free quarks and gluons it is important to compare the result of Monte Carlo simulation with the case of free quarks on the lattice. The spectral function of the correlator (1) in the case of free quarks is obtained from the expression (5). In the case of vanishing total momentum, for positive frequency,
ρ(ω) = πZ∑→p[1−e−Nt2E(→p)]~ϕ(→p)2 (24)
×∑j,k=1,2[¯vj(−p)Γuk(p)2E(p)]2δ(ω−2E(→p))
where uj(p) and vj(p) are the quark and antiquark spinors with j-th spin, and ~ϕ(→p) is the Fourier transform of ϕ(→y). The summation in →p is taken over all modes on the finite lattice of the same size as in Monte Carlo simulation, 203. With the present lattice quark action, the energy of free quark, E(→p), satisfies the dispersion relation Ume01 ; Aniso01a
coshE(→p)=1+→¯p2+[m0+r2γF→^p2]22[1+m0+r2γF→^p2], (25)
where ¯pi=(1/γF)sinpi and ^pi=2sin(pi/2). We set the bare quark anisotropy γF=ξ=4 and the bare quark mass m0=0.121 such that the free quark has the same value as in the Monte Carlo simulation.
Figure 1 shows the spectral functions of point, smeared, and half-smeared correlators composed of free quarks in PS channel. The integral of these functions are normalized to unity. Since on the finite lattice the spectral function is a sum of delta functions, we represent each contribution at momentum →p with a Gaussian of width 0.005. As clearly shown in the figure, the smearing of the operator strongly enhances the low energy part of the spectral function, as compared to the point correlator. In the (fully) smeared correlator, several low momentum states over the range of ω about 0.1 dominate the correlator. From this correlator, therefore, an analysis with poor energy resolution in ω might produce a peak with full width of order of 0.1 instead of a sum of distinct peaks corresponding to individual momentum states. As we will see in Sec. IV.4, the χ2 fit analysis with the single Breit-Wigner type ansatz applied to the smeared correlator composed of free quarks produces a much narrower peak with the width of about 0.027 centered at ω≃0.275. This is of the same order of magnitude as the width extracted from the correlators in Monte Carlo simulation at T≃1.1Tc. Therefore this occasion must be examined carefully.
It is for this purpose that the half-smeared correlator is introduced. As shown in Fig. 1, a slightly higher and wider frequency region contributes dominantly to the half-smeared correlator than to the smeared one. If the correlator is approximated by a single peak, the peak will be centered at larger energy and have wider width as compared to those of the smeared correlator. This behavior is verified in Sec. IV by applying the χ2 fit analysis to the correlators composed of free quarks. Therefore, comparing the results for spectral functions for the smeared and half-smeared correlators can help judging whether the peak observed from the simulation data is a genuine excitation mode or an artifact brought by the smearing.
Iv Analysis procedure
iv.1 Strategy of analysis
One of the goals of this work is to investigate techniques to extract the relevant information of the spectral function from the correlators with limited numbers of the degrees of freedom. We treat an inverse problem represented as
C(t)=∫∞0dωK(t,ω)A(ω), (26)
where C(t) is the given lattice result for the correlator, and the spectral function A(ω) is what we need to obtain. Hereafter we shall denote the spectral function reconstructed from the lattice data A(ω) instead of ρ(ω) (×(2π)−1). Unless stated otherwise, the variables are in temporal lattice units in the following. The kernel K(t,ω) is given as
K(t,ω)=e−ωt+e−ω(Nt−t)1−e−Ntω. (27)
This is the continuum type kernel, and an alternative form of the kernel which explicitly incorporate the lattice structure was also applied in the literatures NAH99 ; Bielefeld02a . We have seen no significant difference in applying the two kernels, therefore we shall not further discuss this point. In practice, the above integration over the frequency ω is replaced by a summation with sufficiently fine discretization Δω and a cut off at some maximum value ωmax.
As already mentioned, one of the main analysis procedures is the maximum entropy method (MEM) NAH99 . Before applying it to finite temperature problems, one should verify its applicability under the condition which one encounters at T>0. For this we test how the result changes when varying the number of data points used in analysis for the correlators at T=0. The condition to be satisfied is that the ground state peak – which we know to exist at T=0 – is correctly reproduced, at least at a qualitative level. We stress that without such verification, the result of MEM may contain uncontrolled artifacts at finite temperature. Concerning quantitative questions, we find that with reduced number of data points, MEM hardly gives a result beyond the qualitative level, for example for the width of a peak.
Another technique in our hand is the standard χ2 fit assuming a certain ansatz for the spectral function. This method has been applied to a problem of glueballs at finite temperature ISM02 . A disadvantage of this approach is that one needs to assume a form of the spectral function, which introduces a bias. For this purpose, the result of MEM can be a good guide. Once a specific ansatz is used, the χ2 fit gives much more reliable results than MEM for the parameters, both in statistical as well as systematic sense.
Therefore MEM and the χ2 fit are complementary to each other at this stage of computational power, and in combination provide a more reliable way to analyze the structure of spectral functions than taken independently. We thus propose a two-step procedure: we first apply MEM to the correlators, for a rough estimate of the shape of spectral function. Once a presumable form is known, we then examine this form using χ2 fit, and estimate the values of parameters such as the mass and width of a peak structure.
iv.2 Maximum entropy method
Our MEM analysis basically follows Ref. NAH99 , which reviews in detail the maximum entropy method applied to data of lattice QCD simulation. Here we briefly summarize just several formulae necessary for the later description of our analysis.
To obtain A(ω) from C(t) by solving the inverse problem Eq. (26), the maximum entropy method maximizes a functional
Q(A;α)=αS[A]−L[A]. (28)
L[A] is the likelihood function,
L[A]=12∑t1,t2[C(ti)−CA(ti)]V(ti,tj)−1[C(tj)−CA(tj)], (29)
where CA(t) is the resulting correlator (26) for the trial A(ω), and V(ti,tj) is the covariance matrix of the data
V(ti,tj)=1N(N−1)N∑k=1[Ck(ti)−C(ti)][Ck(tj)−C(tj)] (30)
with Ck(t) the k-th sample of the correlator. The standard χ2 fit minimizes this functional L. The Shannon-Jaynes entropy S[A] is defined as
S[A]=∫∞0dω[A(ω)−m(ω)−A(ω)log(A(ω)m(ω))]. (31)
The function m(ω) is called the default model, and should be given as a plausible form of A(ω). The parameter α can be integrated out at the last stage of calculation. Following Ref. NAH99 , we use a form
m(ω)=mDMω2. (32)
In the case of point correlators, a natural choice of mDM is determined according to the asymptotic behavior of the meson correlators at large ω in perturbation theory. Although such an asymptotic behavior is not observed in practical simulation because of the finite lattice cutoff, this form has been successfully applied to problems at zero temperature NAH99 ; CPPACS01a . For the smeared correlators, the situation is more subtle, since the high frequency part of the correlator is suppressed by the smearing function. We therefore adopt a practical choice: we normalize the smeared correlator so as to produce the same overlap with the ground state as the point correlator. Then the same normalization is also applied to the correlators at finite temperatures, and we observe how the result changes with the change of mDM.
In the maximization step of Q(A;α) the singular value decomposition of the kernel K(t,ω) is used. Then the spectral function is represented as a linear combination of the eigenfunctions of K(t,ω). The number of degrees of freedom of A(ω) is accordingly reduced to the number of data points of the correlator. Although the coefficients of the linear combination for A(ω) can in principle be determined uniquely from the data without introducing an entropy term, the small eigenvalues of K(t,ω), θi, lead for the spectral function to a singular behavior; hence truncation at some i is practically required SpF_early . In MEM, the addition of the entropy term stabilizes the problem and guarantees an unique solution for the coefficients of the eigenfunctions NAH99 . In our analysis, we use only the basis functions for which θi>10−12×θ1 holds, where θi is i-th largest singular value. This restriction has no significant effect on the result.
iv.3 χ2 fit with ansatz for the spectral function
The standard χ2 fit method minimizes the likelihood function L, Eq. (29), with an assumed form for the spectral function. A most simple form for fit function is the delta function:
Apole(ω;r,m)=rδ(ω−m). (33)
This form is referred to as pole ansatz in the following. At T=0, a sum of several pole terms should describe well the correlators. For the correlators below Tc, where narrow thermal widths are expected, the multi-pole form is also convenient to test this assumption.
As noted in Sec. II, to describe a peak structure with finite width we adopt the relativistic Breit-Wigner form (referred to as BW form),
ABW(ω;r,m,γ)=ω2rmγ(ω2−m2)2+m2γ2. (34)
This is the same form as Eq. (13), with slightly modified notation for convenience. This ansatz neglects the ω dependence of r, and hence is valid only for the case that the width of the peak, γ, is sufficiently small compared with the change of the smearing function in the region of interest.
Combining these two forms, we fit the numerical data of correlators to the following ansätze:
2-pole form: this form is suitable for description of correlators at T=0. It is also expected to be a good representation of correlators at T<Tc;
A(ω)=Apole(ω;r0,m0)+Apole(ω;r1,m1). (35)
1-BW form: if the contribution from the high frequency part of the spectral function is negligible, a collective excitation at low energy is expected to be well represented by a single Breit-Wigner type ansatz:
A(ω)=ABW(ω;r0,m0,γ0). (36)
This form is also a good representation for the spectral function of the smeared correlator composed of free quarks.
BW+pole form: although the lowest peak is well represented with the Breit-Wigner type function, there may exist contribution from high frequency region. Since the smearing of operator suppress such contribution, remaining effects of this region may be represented as a single pole-like term. This is the reason that we fit the data to the form
A(ω)=ABW(ω;r0,m0,γ0)+Apole(ω;r1,m1). (37)
This is the most general ansatz for χ2 fit in this paper.
These ansätze form are the basis for the analysis of the lowest peak structure, which corresponds to the ground state at T<Tc. The structure in the high frequency region is out of the scope of this paper.
iv.4 Analysis of correlators composed of free quarks
Figure 2: The result of MEM analysis for the spectral function from the smeared PS correlator composed of free quarks at T=0.
Figure 3: The result of χ2 fit analysis for the spectral functions from the PS correlators composed of free quarks at Nt=26.
To learn how to distinguish physical features of the correlators from artifacts due to the smearing we apply MEM and the χ2 fit analyses to the correlators composed of free quarks on our lattice, which were discussed in Sec. III.3.
For the MEM analysis, the fluctuations of the correlators are given by hand in the same manner as the mock data analyses in Refs. NAH99 ; CPPACS01a . The deviations of the correlators are less than those of the data from the Monte Carlo simulation. Figure 2 shows the result of MEM applied to the smeared correlator in pseudoscalar channel composed of free quarks at T=0. The upper bound of t region used for the analysis, tmax, is varied to observe how the result of MEM depends on the t-region used. With decreasing tmax, the resolution of the spectral function becomes worse, and for tmax≤16 the spectral function extracted with MEM displays just one peak. Therefore for the circumstances specific at T>0 MEM does not have sufficient resolution to distinguish individual states if they are closer than about 0.05a−1τ. If the correlator is composed of almost free quarks, the width extracted with MEM may be of order of 0.05–0.1 in temporal lattice units.
We also apply the χ2 fit analysis to the correlators composed of free quarks. In this case the errors of the same size as in Monte Carlo simulation are just put on the correlators without fluctuating them. Figure 3 shows the result of the χ2 fit of the correlators composed of free quarks at Nt=26 using the 1-BW ansatz. The tmin dependence of mass and width (with fixed tmax=13) indicates that the single BW ansatz describes rather well the correlators. As is observed in Figure 3, if the correlator is composed of free quarks, the χ2 fit gives sizable difference in mass and width parameters for the smeared and half-smeared correlators. This dependence is in agreement with the fact that the propagator should only show the two free quark cut and no particle-like excitations and indicates that by testing the dependence of the result on the smearing function we can distinguish physical effects from artifacts due to smearing.
V Setup of numerical simulation
v.1 Lattice setup
The zero temperature lattice used in this paper is the third one of Ref. Aniso01b : a quenched lattice of size 203×160, generated with the standard plaquette action with (β,γG)=(6.10,3.2108). These coupling and bare anisotropy correspond to the renormalized anisotropy ξ=aσ/aτ=4 within 1% accuracy Kla98 , and the spatial lattice cutoff a−1σ=2.030(13) GeV set by the hadronic radius r0 Som94 . At T=0, 500 configurations are generated with the pseudo-heat-bath update algorithm, each separated by 2000 sweeps after 20000 sweeps for thermalization. The mean-field values are defined as the average values of link variables in the Landau gauge, and obtained as uσ=0.8059(1) and uτ=0.9901.
To determine the critical temperature, we measure the Polyakov loop susceptibility at Nt=27, 28, and 29 at β=6.10, and in addition, at several values of β (with corresponding values of γG) around β=6.10 at fixed Nt=28. At β=6.05 the lattice scale set by r0 is a−1σ=1.892(10) GeV, which together with a−1σ at β=6.10 determine the scales at the other values of β by linear interpolation. The susceptibility peaks at about β=6.10 and Nt=28. The critical temperature is obtained as Tc≃290 MeV with 10 MeV of roughly estimated uncertainty.
The charmonium correlators at T>0 are measured for two values of temporal lattice extent, Nt=32 and 26. Corresponding temperatures are 0.88Tc for Nt=32, and 1.08Tc for Nt=26. For brevity, these temperatures are hereafter referred to as 0.9Tc and 1.1Tc, respectively. Thus the temperatures treated in this paper are in the vicinity of the transition. At each of these two Nt's, we generate 1000 configurations each separated by 500 pseudo-heat-bath sweeps after 20000 sweeps for thermalization.
At Nt=26, we find that the configurations almost stay in a single Polyakov loop sector during the whole updating history. Transitions to other sectors occurred only once after more than 470k sweeps. In Ref. TARO01 it was reported that the light mesonic correlators behave differently on configurations in different sectors. We regard quenched lattices as approximations to lattices with dynamical quarks, on which Z3 center symmetry is explicitly broken and Polyakov loop prefers to stay on the real axis. Therefore we transform all the configurations to the the real sector of the Polyakov loop.
v.2 Charmonium correlators
As mentioned in Sec. III.2, we use (κ,γF)=(0.112,4.00), which correspond to mQ≃0.98 GeV, roughly the charm quark mass. The statistical errors for the results are estimated with the jackknife method with appropriate binning.
In the next section, we start our analysis at T=0 with the examination of the point correlators. On each configuration, we calculate the point correlators four times with four different source points: t=0 and 80 at spatial site →x=(0,0,0) and t=40 and 160 with →x=(10,10,10). Then these correlators are averaged with appropriate shifts in t-direction. At T=0 the standard χ2 fit of data to a multipole form works well. To fix the general picture we anticipate on the analysis of the next sections and list in Table 1 the result of two-pole fits of the point correlators in PS and V channels. The fit ranges [tmin,tmax] are determined by varying tmin with fixed tmax=80 and observing the stability of the fit parameters. The results of the ground state masses are very close to the experimentally observed charmonium masses. On the other hand, the hyperfine splitting is with about 74 MeV smaller than the experimental value, mJ/ψ−mηc≃117 MeV. This is a well-known feature of the quenched lattice simulations and is considered mainly due to quenching, although lattice artifacts in the charmonium system can also play an important role TARO02 .
0.36835(37) 0.37748(49)
first exc.
0.449(31) 0.463(42)
fit range
0.36856(9) 0.37769(12)
Table 1: The spectrum at zero temperature determined from the point and the smeared correlators. The masses in PS and V channels are determined with the two-pole fit.
The smeared correlators are the main target of this paper for reasons we already explained. The smearing parameters are described in Sec. III.2. The result of a fit to 2-pole form is also shown in Table 1. It is well-known that the correlators with smeared sink are quite noisy. To reduce the noise, we measure sixteen correlators with different source points on each configuration, and average them. We change the spatial center of the smearing function, as well as the time slice, to reduce the correlation as much as possible. In contrast to a naive expectation, we find this procedure efficient. In fact, at T=0, the statistical errors are reduced by a factor of about 0.3 in the region t=8–16, which is the most important region for the present analyses, in both the PS and V channels. This reduction of errors corresponds to an increase in statistics of about 10 times. Almost the same amount of reduction of errors is observed at Nt=32. At Nt=26, the errors are reduced by factors of 0.40–0.45 in the range t=8–13. This way of reducing statistical errors will be advantageous in the case of dynamical simulations.
For the present kind of analysis, whether the analysis is efficient or not may significantly depends on the statistics. In the temporal region t=8–16, the statistical errors of point correlators at T=0 are about 0.12–0.18% and 0.08–0.15% for PS and V channels, respectively. In the same region, the smeared correlators have statistical errors of 0.14–0.17% and 0.15–0.20%. Therefore at T=0 the errors are of the same order for the point and smeared correlators. At finite temperature, with 1000 configurations, the statistical errors are of roughly the same size as at T=0. We also note that the correlation between the correlators at neighboring time slices is stronger for the point correlators than for the smeared ones.
Figure 4: The effective masses for the smeared correlators in PS and V channels at three temperatures.
To obtain an impression of the physics to be expected at various temperatures we show in Figure 4 the effective mass plot for the smeared correlators, with meff(t) defined through
C(t)C(t+1)=cosh[meff(t)(Nt/2−t)]cosh[meff(t)(Nt/2−t−1)]. (38)
If the correlator is dominated by a single (stable) state, meff(t) shows a plateau. In Fig. 4, the effective masses at T=0 show plateaus beyond t≃16. The effective masses at Nt=32 (T≃0.9Tc) show almost the same behaviors as at T=0. This implies that the charmonium states are almost unchanged at this temperature, compared to those at T=0. In contrast, meff(t)'s at Nt=26 (T≃1.1Tc) show quite different behavior: they decrease rapidly compared to those at T<Tc, and in the large t region (t≥10) the pseudoscalar and vector channels are almost degenerate. Therefore a qualitatively different behavior of the spectral function is expected below and above Tc. These features are consistent with observations in earlier work Ume01 .
Vi Analysis at zero temperature
vi.1 Result of MEM for point correlators
We start with MEM analysis of the point correlators at zero temperature.
First we discuss the default model function. Following previous applications of MEM to lattice data NAH99 ; CPPACS01a , we adopt as mDM in Eq. (32) the asymptotic values of correlators in the perturbative QCD,
1ω2Acont(ω≫ΛQCD)=r14π(1+r2αsπ), (39)
where r1=3/2 and r2=11/3 for PS channel, and r1=1 and r2=1 for V channel. As value of the strong coupling constant αs, we use αs=0.2 at 8 GeV from Ref. PDG00 as a typical value at the temporal cutoff of the present lattice. The matching of the lattice theory with the continuum theory is performed through the tadpole-improved tree-level, and hence the renormalization of the quark field is represented only by the KLM normalization factor (22). These settings give for the parameter of the default model function mDM=4.1 for PS channel, and 2.4 for V channel. If MEM works as a method to extract the spectral function without assuming a specific form, the result should not be sensitive to the choice of this parameter. In addition, the observed asymptotic behavior of the spectral function from lattice data around the cutoff is actually different from that of Eq. (39). We therefore observe the dependence of the result on the value of mDM by changing mDM by factors of 10 and 0.1, to verify the insensitivity of the result to mDM.
The further setting of MEM is as follows. In all cases, the correlator at t=0 is not used for the analysis. The minimum of t used depends on the type of analysis, in most cases tmin=1 is adopted. The frequency ω is discretized with Δω=0.002, in the region [ωmin,ωmax]=[0.001,4.0]. The dependence of the result on the parameters for ω is sufficiently small around the adopted values.
Figure 5: The result of MEM analysis of the point correlator in PS channel at T=0. The figure shows the tmax dependence of the resultant spectral function.
Figure 6: The result of MEM analysis of the point correlator in PS channel at T=0. This figure shows the tsep dependence of the result of the spectral function with fixed tmax=48.
We first examine how the restriction of degrees of freedom of the correlator affects the reconstructed spectral function. Three parameters tmin, tmax and tsep are introduced, that is MEM is applied for the correlators at times tmin≤t≤tmax in every tsep time slices. We examine two types of restrictions of the numbers of degrees of freedom:
(I).
tmax is varied as 48, 24, 16, and 12, with fixed tmin=1 and tsep=1. The corresponding numbers of data points are the same as tmax.
(II).
tsep is varied as 1, 2, 4, and 8, with fixed tmax=48. tmin is varied accordingly as tmin=tsep. Correspondingly, the numbers of data points are 48, 24, 12, and 6, respectively.
From the point of view of the basis functions of the singular value decomposition in the spectral function space, case (I) reduces the number of basis functions while keeping the functions unchanged. On the other hand, case (II) dilate the basis functions by a factor of tsep simultaneously reducing their number. Reliability of the result at finite temperature requires the obtained spectral function to be stable under the type (I) restriction. Although larger values for tmax are possible, tmax=48 is sufficiently large and has appropriate physical range (tmaxaτ≃1.2 fm) for the present purpose.
The result is shown in Figs. 5 and 6 for the pseudoscalar channel. For the vector correlator a similar result is obtained. In the case of tmax=48 and tsep=1, i.e. without restriction, the reconstruction of the spectral functions seems successful, and their fundamental features appear to be the same as in previous works NAH99 ; CPPACS01a . The peak positions of the ground and first excited states are consistent with the result of χ2 fit summarized in Table 1. Here we do not discuss the higher excited states because of uncertainties not only in MEM but also in the χ2 fit analysis.
Figure 5 displays the result under restriction conditions (I). Decreasing tmax, the spectral function becomes increasingly different from that with tmax=48. For tmax≤24, the first excited state peak does not appear at the correct place, and for tmax≤16 even the ground state peak is located at an incorrect place and has broad width. These lowest peaks for different tmax's are all significant in the sense of the error analysis of MEM NAH99 . Therefore tmax≤16 is not acceptable for extracting reliable information of the ground state from the point correlators. Since at finite temperature we are restricted to tmax≤16 because of short temporal extent this is the reason why we abandon using the point correlators and apply the smearing technique.
Figure 6 shows the result with restriction condition (II). In this case, the reconstructed spectral functions are rather stable, while the sharpness of peaks is lost by increasing tsep. Even with 6 degrees of freedom (as a linear combination of 6 functions), the reconstructed spectral function at least exhibits the ground state peak at the correct position. To summarize, if one has a temporal region of the correlator of the order of 1 fm, the spectral function can be rather nicely reconstructed even with less than 10 degrees of freedom.
From these observation, we conclude that with O(10) degrees of freedom the crucial condition is determined by the region of t where the correlators are measured. tmaxaτ of the order of 1 fm seems necessary for a reliable extraction of the spectral function from the point correlator. Increasing the number of degrees of freedom above 10 with fixed physical range of [tmim,tmax] improves the situation only slightly.
Now we consider the effect of the statistics. We apply the same analysis of the case (I) restriction to the correlators averaged over first 100 configurations. The observed features of tmax dependence are essentially the same as with 500 configurations. Therefore at the present level of statistics (order of several hundreds), increasing statistics does not seem to improve drastically the situation of the tmax dependence. We also compare the correlators averaged over 100 configurations randomly selected from the total of 500 configurations. A comparison of the results with tmax=48 from five such correlators shows that the first excited state does not always appear at the same place as with 500 configurations, but deviates in a range of ω=0.44–0.52. Thus with less statistics only the ground state peak can be trusted even with tmax=48. These observations indicate the required statistics for each situation.
Finally we briefly discuss the dependence on the default model. We compare the results obtained with mDM in Eq. (32) multiplied by 10 and 0.1 with otherwise fixed parameters, tmax=48 and tsep=1. These results of the spectral function display the peaks at the same places as with original mDM, while increasing mDM the peaks become sharper. Although present analysis is quite simple, we conclude that the default model dependence of the result of MEM is not severe, at least for a rough estimate of the shape of the spectral function.
The most important conclusion in the analysis of the point correlator is that we need the order of 1 fm for the range of t where the correlators are measured, for reliable extraction of the spectral function from the correlator with present level of statistics. This requirement cannot be fulfilled at finite temperature, and a brute force application of MEM to the point correlators at T>0 would produce an unreliable result. This result warns us against physical significance of the results in Refs. Bielefeld02a ; Bielefeld02b ; Bielefeld02c ; AHN02 . This difficulty may originate in that the point correlator at short distance contains the contribution from wide range of frequency up to the lattice cutoff. We therefore give up to analyze further the point correlators, and concentrate our attention on the smeared correlators.
vi.2 Result of MEM for smeared correlators
Figure 7: The spectral functions at Nt=160 determined with MEM. The top and bottom panels show the tmax dependences of obtained spectral functions in PS and V channels, respectively.
Figure 8: The default model dependence of the extracted spectral function with MEM from the smeared correlators in PS channel. The figure shows the result of MEM with three values of mDM, the parameter of the default model function Eq. (32).
We now turn to MEM analysis of the smeared correlators. Since the smearing technique reduces the high frequency part, there is no a priori choice for the form of the default model. As described in Sec. IV.2, we use the same default model function as for the point correlator, with the normalization of the smeared correlator which gives the same overlap with the ground state as the point correlator. The default model dependence is examined in the same manner as for the point correlators. Other parameters are the same as for the point correlator, except that ωmax=2.0 is adopted, since the high frequency region is sufficiently suppressed.
First we examine the dependence of the result on the numbers of degrees of freedom, using the same restriction conditions as for the point correlators. Since the smearing technique enhances the low energy part, we only consider whether the ground state peak is correctly reproduced or not. The result for the restriction condition (I) is displayed in Fig. 7. In the both cases of PS and V channels the place of the ground state peak is stable with decreasing tmax, although the width is gradually broadened. Since the ground state peak evidently has no physical width at T=0, this fictitious width is a systematic uncertainty of MEM due to the insufficient number of basis functions in the spectral function space. Therefore, in particular at finite temperature, an estimate of the physical width requires a careful analysis combined with other procedures. The instability of the first excited state peak under changing tmax is explained by the smallness of the contribution to the correlators from energies above the ground state. In fact, the two-pole fit analysis indicates that the overlap of the first excited state with the smeared correlator is about 7% (6%) of that of the ground state for the PS (V) channel. Thus the instability of the excited state peaks has no significance for the following analysis. We conclude that for the smeared correlators, MEM with tmax≃12 would work for a rough estimate of the structure of the spectral function in the low energy part, such as examination of whether strong ground state peak exist, while a quantitative evaluation of width, for example, is difficult. In the case of the restriction condition (II), a similar tendency as for the point correlators is observed: the position of the ground state peak is stable with increasing tsep, while the sharpness of the peak is gradually lost.
The mDM dependence of the result is examined in two cases of tmax, tmax=48 and 12, by using rescaling factors 10 and 0.1. In the case of tmax=48, the ground state peak is strong, and changing mDM does not cause a large effect other than the decrease of the peak height with slight increase of the width when decreasing mDM. Fig. 8 shows the case of tmax=12 for the PS channel. Although the peak shape is broadened with decreasing mDM, the essential features of the result are stable. A similar result is observed for the V channel.
Let us summarize the MEM analysis of the smeared correlators at T=0. In contrast to the case of the point correlators, MEM with restricted numbers of degrees of freedom works up to tmax=12, i.e. tmaxaτ≃0.3 fm, at least for a rough estimate of the shape of the spectral function in the low energy part. For this purpose, systematic uncertainties in the default model parameter do not appear significant. This is an encouraging result for an application of MEM to finite temperature as a precedent analysis to the χ2 fit.
vi.3 Result of χ2 fit analysis
Figure 9: The result of the χ2 fit analysis at Nt=160. The top and bottom panels show the results for mass and width of the ground state peak, respectively.
The results with 2-pole ansatz have been given in Table 1. For completeness, we also apply the χ2 fits to the smeared correlators with forms other than the 2-pole fit. The result is shown in Fig. 9. As fit range [tmin,tmax] we fix tmax=13, considering the severest case at T>0, and observe the tmin dependence of the result. The top and bottom panels of Fig. 9 show the results of mass and width of the ground state peak, respectively.
As is expected, the 2-pole form describes well the correlators, and the other two fit forms are consistent with the 2-pole fit. In the cases of 1-BW and BW+pole fits the tmin dependences of the widths are significant. In the bottom panel the results for the width decreases as tmin increases, and seem to approach zero. Therefore no indication of a finite width is found at T=0, as it should be.
Vii Analysis at T<Tc
Now the two analysis procedures are applied to the correlators at T≃0.9Tc (Nt=32). In this and the next sections we no longer discuss the point correlators and focus only on the smeared ones.
vii.1 Result of MEM analysis
Figure 10: Spectral functions at Nt=32 determined with MEM.
We begin with the MEM analysis, in accordance with our strategy. The parameters concerning MEM are almost the same as for the smeared correlators at T=0. The range of t is fixed to tmin=1, tsep=1, and tmax=16.
The result is displayed in Fig. 10 for three values of mDM. For the reasons already mentioned we focus on the ground state peak and do not discuss in detail the high frequency part of the spectral function. As apparent in the figure a ground state peak appears in both the PS and V channels. Although with decreasing mDM the widths of the ground state peaks increase, the positions of the peaks are stable and almost the same as at T=0. In the present case, since there is no intrinsically advisable value for mDM beyond an order estimate, this ambiguity of the width of the peak should be considered as an uncertainty of MEM applied to the smeared correlators. We also perform the same analysis with less statistics, 500 configurations, to see how this result depends on the statistics. The result is essentially the same, and thus statistically stable.
These results support the assumption that the mesonic ground states are persistent up to this temperature, with almost the same masses as at T=0. The width of the ground state peaks is small while finite, although it strongly depends on the value of mDM. Whether the width is physically finite or not should therefore be examined with the χ2 fit analysis.
vii.2 Result of χ2 fit analysis
Figure 11: The result of χ2 fit analysis at Nt=32. The top and bottom panels show the results of the mass and width parameters for the ground state peak, respectively.
According to the result of MEM we can assume that the spectral function of the smeared correlator at this temperature has a peak structure at almost the same mass as at T=0 and with small width. Therefore it is expected that the χ2 fits with the three types of fit forms, 2-pole, 1-BW, and BW+pole forms can clarify the low energy structure of the spectral function.
The results are summarized in Fig. 11. The top panel displays the dependence of the mass parameters for the ground state peak on the lower bound of the fit range, tmin. The upper bound is fixed to tmax=16. It is apparent that the fit to the 2-pole form exhibits stable results both for the PS and V channels beyond tmin=3. The result of fit to 1-BW form exhibits no plateau, indicating that this form does not explain the whole structure of the correlators. However, the values seem to approach to the corresponding results of the 2-pole fits. The result of fit to the BW+pole form is consistent with those of 2-pole fit at tmin≤2.
The bottom panel of Fig. 11 shows the result for the width parameter. In the case of the fit to the 1-BW form the value of width gradually decreases with increasing tmin. This behavior is consistent with a vanishing width. The fit to BW+pole form also indicate that the width is consistent with zero. Therefore, there is no indication of a finite width for the spectral function at this temperature.
All the results of fits to the three forms indicate that the ground state peak is well described by a strong peak with vanishing width, i.e. a pole-like structure, and the associated mass is almost the same as at zero temperature.
Viii Analysis at T>Tc
viii.1 Result of MEM analysis
Figure 12: The spectral function at Nt=26 determined with MEM (smeared correlators).
Figure 13: The results of MEM for the spectral function of the half-smeared correlators at Nt=26.
In this section, we analyze the correlators at T≃1.1Tc (Nt=26).
The setup of MEM analysis is the same as at T≃0.9Tc, except for tmax=13. Figure 12 shows the result. In both the PS and V channels there appear strong peak structures around ω≃0.4. This peak, hereafter called "the ground state peak" for simplicity, appears at almost the same position as at T≃0.9 but with larger width. Although, as a common tendency, the peak becomes sharper as the default model parameter mDM increases, the position of the peak is unchanged. It is also verified that the result is essentially the same with less statistics, namely 500 configurations. Therefore it is presumable that the spectral function still has a peak structure at almost the same position as below Tc.
This result should be compared with the case where the correlators are composed of free quarks, considered in Sec. III.3. In the latter case, as shown in Sec. IV.4, MEM with the present number of degrees of freedom may produce the same feature of the spectral function but with a lower peak position. The above result therefore does not exclude the possibility that the correlators are composed of two weakly interacting quarks with rather large effective mass.
To judge between almost free quarks and genuine bound-state-like structure we analyze the half-smeared correlator. We repeat the same analysis as applied to the smeared correlators for the half-smeared correlators and verify that MEM works up to tmax=12 on almost the same level as for the smeared ones. As discussed in Sec. IV.4, if the correlator is composed of two almost free quarks the extracted spectral function should exhibit now a wider peak at a higher position. Figure 13 shows the result of MEM analysis for the half-smeared correlators. The analysis is performed in the same manner as for the smeared correlators. Comparing with the result for the smeared correlator, the peak position for the half-smeared one is almost unshifted, while the width of the peak tends to broaden slightly. The latter effect , however, can also be easily explained as an effect of default model uncertainty. This result contradicts therefore the assumption that the peak structure is an artifact of the smearing. Hence the MEM analysis supports a bound-state-like structure at this temperature.
viii.2 Result of χ2 fit analysis
Figure 14: The result of χ2 fit analysis for the smeared correlators at Nt=26.
Figure 15: The result of χ2 fit analysis for the half-smeared correlators at Nt=26.
Figure 16: The spectral functions obtained with χ2 fit analysis at Nt=26. For the correlators composed of free quarks only the result of BW fit is displayed, while a BW+pole fit is used for the correlators measured in the simulation.
According to the result of MEM, it is reasonable to assume that the spectral function at this temperature also have a peak structure, similarly to the T<Tc case. Therefore the three forms for the χ2 fit analysis are still reasonable assumptions to investigate the low energy structures of the correlators.
Figure 14 shows the result of the χ2 fit analysis for the smeared correlators. The top panel shows the tmin dependence of the mass parameters. In contrast to the case below Tc, the masses from 2-pole fit no longer approach those of the other two fit forms but fall monotonously. The masses from 1-BW and BW+pole fits exhibit consistent behaviors, indicating that these fits represent better the correlators. Actually the values of χ2/NDF are consistently fluctuating around unity at tmin>8 for 1-BW fit, and in the whole range of tmin for BW+pole fit. In the BW+pole fit, the mass parameters for the second peaks take values around 0.9 and 1.0 for PS and V channels, respectively, which is roughly consistent with the result of MEM. The consistency of 1-BW and BW+pole fits also holds for the width parameters, as displayed in the bottom panel of Figure 14. These results indicate therefore that the widths associated with the ground state peaks are finite for both the correlators in PS and V channels.
We now analyze the half-smeared correlators to examine whether this observation of peak structure is an artifact of the smearing. The analysis is performed in the same manner as for the (fully) smeared correlators. The result is displayed in Figure 15. The mass parameters for the half-smeared correlators (top panel) show a similar tendency as for the smeared correlators, although the result of 1-BW and BW+pole fits approach slightly larger values than those of smeared correlators. The shifts of the masses are at most 5% and can be explained with the frequency dependence of the overlap R(ω) in Eq. (13). A similar tendency is also found for the width parameters in the bottom panel of Fig. 15, which appear even more consistent with those of smeared correlators than the mass parameters. Therefore we find no significant shift of the peak position or broadening of the peak width, such as expected for the case with almost free quarks discussed in Sec. III.3.
Figure 16 illustratively compares these spectral functions extracted with the χ2 fits from four kinds of correlators: the smeared and half-smeared correlators, numerically measured and composed of free quarks. The results of the numerical simulation are of the BW+pole fit with tmin=4. For the free quark case the results of 1-BW fit with tmin=8 is displayed. We note that the normalizations of the spectral functions are not significant in the present analysis. In both the PS and V channels the spectral functions for the numerically observed correlators strongly peak near ω=0.4 with small widths. The effect of the smearing seems not significant. In contrast, the spectral functions for the correlators composed of free quarks are strongly dependent on the applied smearing. Both a sizable shift of mass and a broadening of width are observed in this case. It is difficult to exclude completely the possibility that the correlators are composed of almost free quarks with rather large effective mass and nontrivial dispersion relation. However, it is quite suggestive that a nontrivial structure indicating the existence of bound-state-like structure in the low energy region of spectral function subsists at this value of T.
Summarizing the results of MEM and χ2 fit analysis we conclude that the charmonium correlators in PS and V channels at T≃1.1Tc possess a nontrivial peak structure in the low energy region. As representative values of m0 and γ0, we quote the results of the BW+pole fit analysis with tmin=4 for the smeared correlators:
PS: m0=3.06(2) GeV, γ0=0.12(3) GeV, (40)
V: m0=3.13(2) GeV, γ0=0.21(2) GeV. (41)
The quoted errors in m0 and γ0 are only the statistical ones and do not include the error in a−1τ. As obvious from the discussions in this sections, these results should contain systematic uncertainties of the order of 5% due to the analysis procedures, apart from other uncertainties such as finite lattice artifacts and quenching effects. Compared with the result at T≃0.9Tc, the widths at T≃1.1Tc is sizable, indicating a genuine temperature effect in the deconfined phase. On the other hand, the masses are almost unchanged. This result is contradictory to the absence of bound states argued from the potential model approach MS86 ; Potential_model .
Ix Conclusion
The main goals of this paper were (1) to elucidate the technical problems in extraction of the spectral function from lattice data of t-correlators, and (2) to investigate the temperature effect on the spectral function near the deconfining phase transition. In the following, we summarize and discuss the results obtained in this paper.
As techniques to extract the information on the spectral function we examined maximum entropy method (MEM) and the standard χ2 fit assuming suitable forms of the spectral function. It is essential to check the reliability of the applied methods to the systems in question. Our condition for reliability at finite temperature is that the methods reproduce the correct form of the spectral function at T=0 when the t-interval is restricted to the one forced on us at T>0. We examined MEM by applying it to the point and smeared correlators. We find that MEM does not meet this requirement for the former, while it does for the latter. This is understandable, since the smearing enhances the low energy part of the spectral function, which is what we are interested in, while a much wide energy region contributes to the point correlators. We note that whether MEM correctly works or not depends primarily on the extension of the physical t-region. In particular, for the point correlators a region of t of O(1fm) is necessary. For the smeared correlator, this condition is much relaxed. Therefore only the smeared correlators were analyzed at finite temperatures.
Since MEM is ambiguous concerning the quantitative detail of the spectral function, the latter should be also analyzed with other procedures. As such a procedure, the χ2 fit is a reasonable candidate, since MEM already gives a hint for a suitable ansatz for the spectral function. With several assumed forms and examining the fit range dependence of the resultant values for the parameters, this procedure gives us more quantitative information on the properties of spectral function. We emphasize that MEM and χ2 fit analysis as used here are complementary to each other. This two-step approach actually worked for analyses of the correlators at finite temperature, as well as at T=0.
Now we discuss the physical implications of our results below and above the critical temperature. We remind that our numerical simulation was performed without dynamical quark effects.
Below the deconfining temperature, at T≃0.9Tc, the reconstructed spectral function has a strong peak corresponding to the ground state, with almost the same mass as at T=0 and narrow width consistent with zero. In contrast to the potential model analysis Has86 , the charmonium mass is not changed up to this temperature. Similar tendencies have been reported in previous lattice QCD calculations for the mesonic channels TARO01 ; Ume01 , while sizable reduction of mass has been observed for glueballs ISM02 . Considering the rather quantitative success of the potential model approach for the charmonium systems at T=0 Cornel , it is important to explain this discrepancy.
At T≃1.1Tc we observed an indication that the spectral functions still has strong peaks at almost the same positions as T<Tc, and with widths of about 0.12 and 0.21 GeV for PS and V channels, respectively. This result presumably indicates the existence of quasi-stable bound-state-like structures persistent up to this temperature. The possibility of observing correlators composed of almost free quarks (but of large effective masses) is, however, not completely excluded. Finally, also above Tc the observations are not in accord with the expectation from the potential model approach Potential_model . This result implies that the plasma phase has a nontrivial structure at least near the critical temperature.
For a more definite understanding of hadronic correlators at T>0 more studies containing dynamical simulations are necessary, as well as calculations in wider range of temperatures. The techniques adopted in this paper should be applicable to these situations. It is also important to repeat the same sort of analysis in the light hadron sectors, for which a nontrivial structure of correlators above Tc was also reported TARO01 .
We would like to express our sincere gratitude to Professor Osamu Miyamura, who tragically passed away in July 2001, for his inspiring us with this work and for valuable discussions. We thank N. Ishii, T. Kunihiro, A. Nakamura, T. Onogi, I.-O. Stamatescu, and T. Yamazaki for useful discussions. We are also grateful to the members of QCD-TARO Collaboration; this work was done partly in accordance with their long standing physical goals. The simulation has been done on NEC SX-5 at Research Center for Nuclear Physics, Osaka University and Hitachi SR8000 at KEK (High Energy Accelerator Research Organization). T. U. is supported by the center-of-excellence (COE) program at CCP, University of Tsukuba. H. M. is supported by Japan Society for the Promotion of Science for Young Scientists.
(1) T. Hashimoto, O. Miyamura, K. Hirose and T. Kanki, Phys. Rev. Lett. 57, 2123 (1986).
(2) T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986).
(3) NA50 Collaboration, M. C. Abreu et al., Phys. Lett. B 477, 28 (2000).
(4) For a review, H. Satz, Nucl. Phys. B (Proc. Suppl.) 94, 204 (2001).
(5) A.A. Abrikosov, L.P.Gor'kov, and Dzyaloshinskii, Sov. Phys. JETP 36(9), 636 (1959); E.S. Fradkin, Sov. Phys. JETP 36(9), 912 (1959).
(6) T. Hashimoto, A. Nakamura and I. O. Stamatescu, Nucl. Phys. B400, 267 (1993); Nucl. Phys. B406, 325 (1993).
(7) QCD-TARO Collaboration, Ph. de Forcrand et al., Phys. Rev. D 63, 054501 (2001).
(8) T. Umeda, R. Katayama, O. Miyamura and H. Matsufuru, Int. J. Mod. Phys. A 16, 2215 (2001).
(9) Y. Nakahara, M. Asakawa and T. Hatsuda, Phys. Rev. D 60, 091503 (1999); M. Asakawa, T. Hatsuda and Y. Nakahara, Prog. Part. Nucl. Phys. 46, 459 (2001).
(10) For pioneering works, QCD-TARO Collaboration, Ph. de Forcrand et al., Nucl. Phys. B (Proc. Suppl.) 63, 460 (1998); E. G. Klepfish, C. E. Creffield, E. R. Pike, ibid., 655 (1998).
(11) I. Wetzorke and F. Karsch, hep-lat/0008008.
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(15) Difficulty in extracting transport coefficients from lattice data has been pointed out in: G. Aarts and J. M. Martinez Resco, JHEP 0204, 053 (2002); hep-lat/0209033.
(16) F. Karsch, E. Laermann, P. Petreczky, S. Stickan and I. Wetzorke, Phys. Lett. B530, 147 (2002); I. Wetzorke et al., Nucl. Phys. B (Proc. Suppl.) 106, 510 (2002).
(17) P. Petreczky, F. Karsch, E. Laermann, S. Stickan and I. Wetzorke, Nucl. Phys. B (Proc. Suppl.) 106, 513 (2002).
(18) S. Datta, F. Karsch, P. Petreczky and I. Wetzorke, hep-lat/0208012.
(19) M. Asakawa, T. Hatsuda and Y. Nakahara, hep-lat/0208059.
(20) N. Ishii, H. Suganuma and H. Matsufuru, Phys. Rev. D 66, 014507 (2002); hep-lat/0206020, to appear in Phys. Rev. D.
(21) K. Nomura, O. Miyamura, T. Umeda and H. Matsufuru, hep-lat/0209139.
(22) F. Karsch, Nucl. Phys. B205, 285 (1982).
(23) J. Harada, A. S. Kronfeld, H. Matsufuru, N. Nakajima and T. Onogi, Phys. Rev. D 64, 074501 (2001).
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text:
Jets QB Sam Darnold (shoulder) doubtful to play Monday vs. Patriots
Published: Nov 07, 2020 at 02:27 PM
Adam Maya
NFL.com Digital Content Producer
Sam Darnold is down again, and he might soon be out.
The Jets quarterback didn't practice Saturday and is listed as doubtful to play Monday night against the Patriots. The third-year signal-caller reaggravated the AC joint in his throwing shoulder last week, causing him to sit out practice Thursday and participate in a limited capacity Friday. Now he's expected to miss a third game this season.
Darnold first injured his arm in a Week 4 loss to the Broncos. He sat out the ensuing two games, while New York managed to score just 10 points combined. Thus, another absence would be a notable blow, even for the league's only winless team.
Joe Flacco claimed the previous two vacant starts and completed just 51.9% of his throws with one touchdown and one interception. If Darnold can't go, it's possible that the NFL's worst offense will be even worse in prime time.

redpajama_set_name: RedPajamaCommonCrawl


Hi, in my first post- need help or advice for designing audio amplifier circuit.
I have to design an audio amplifier circuit. We are told the impedance of the headphones and the minimum voltage they require and also the voltage source. After researching I have found that the LM386N-4 is a popular choice. (According to a datasheet of LM386N-4 - https://www.icrfq.com/part/2377557-LM386N_4.html ). I've seen many circuit diagrams but what I don't understand is how to pick the voltage gain needed for the amplifier.
I originally thought that need to know the input audio signal voltage and when I multiplied that by the gain that would need to give the minimum voltage required by the headphones, but it appears that what matters is the current which I don't get.
And are you able to design such a circuit without knowing about the audio input signal? If you do need to know about it what is it you need to know about it and why?
I would greatly appreciate any help as I am very lost.
Have you calculated what your output voltage swing needs to be to drive the headphones?
Thanks a lot, for your reply and sharing a valuable information to me. It might be helpful for me. I had bookmarked this information for my future.
No, I didn't calculate the output voltage but as my research maybe it would be 9V. Let you know my update very soon.


text:
Man Who Defiles His Underage Daughter Charged
by EDITOR January 25, 2023 January 25, 2023 045
In Lagos State, a 35-year-old father named Temitayo Bamiro has been detained and charged with defiling his 13-year-old daughter through anal intercourse.
They live in Lagos' Iba New Site neighbourhood of Ojo, where the incident took place.
He was accused of defiling a woman in front of the Ojo Magistrate's Court in Lagos State, which carries a sentence of up to life in prison if he is proved guilty.
According to the prosecution, Simon Uche, the defendant, Bamiro, committed the crime on January 7, 2023, and it was not the first time he had done so because Bamiro often violated his daughter's anus without her permission.
In light of the defendant's plea of not guilty, the prosecutor argued that the accused offence violated Lagos State criminal laws and requested that the court consult with the DPP.
The prosecutor was instructed to duplicate the file and transmit it to the DPP by the presiding magistrate, Mr. L.J.K. Layeni, who also ordered his remand in custody at the Badagry Town Correctional Centre until the conclusion of the DPP's advice.
The matter was postponed until February 22, 2023, for mention while the DPP's report, which will determine whether or not the matter is referred to the High Court over jurisdiction, is awaited.
UNICEF says 75% of Nigerian Children Can't Read
Islamic teacher sentenced to death for defiling eight students
Development Webinars: Focus on Upstream Oil and Strategic Mineral Ores
EDITOR July 8, 2021 July 8, 2021
Police Rescue 4 Kidnapped Victims From Den
World Leaders Ridicule Buhari, Give Appointment to Jonathan
EDITOR October 8, 2019 October 8, 2019


Schedule via and cross-referenced with. If you spot any errors, let us know below. Conferences reflect home teams. Ranked teams bold; rankings from the USA Today Coaches Poll. Times (EST).
A busy day of football awaits, especially in the Big 10 and SEC; if you want to see how a pro does it, check out the viewing schedule of our own Tim Burke.


Heliozid Océ operates directly through its branch offices in UAE, Saudi Arabia and Lebanon. All other locations have appointed distributors to cater to their designated markets. We provide regular product and technical trainings to all our sales and service staff so that they can understand your printing requirements and suggest the best solution to meet with your business needs.
To find details of one of our sales or services team member operational in your region, please contact us or leave a short message below and we will get back to you at the earliest.


text:
Wires Home
Pardon sought for ex-CIA officer in leak case
Published: 13:00 EST, 15 October 2015 | Updated: 13:00 EST, 15 October 2015
The wife of an ex-CIA officer convicted of leaking classified information called Thursday for a presidential pardon, saying the case was part of a stealth war on the news media.
Holly Sterling, wife of former CIA officer Jeffrey Sterling, made the appeal to President Barack Obama at a Washington news conference, flanked by members of activist groups supporting press freedom and government "whistleblowers."
Sterling was sentenced in May to 3.5 years in prison after a jury found him guilty of leaking classified information to a New York Times journalist.
Jeffrey Sterling was sentenced in May to 3.5 years in prison after a jury found him guilty of leaking classified information to a New York Times journalist ©Saul Loeb (AFP/File)
Holly Sterling said "the true story has not been out there" and that her husband, despite the case, "loved his job at the CIA and loved his country."
In an open letter to Obama, she said that the government was seeking "to punish a man who had the audacity to ... stand up for his constitutional rights."
The case centered around a botched US attempt during president Bill Clinton's administration to sabotage Iran's nuclear program that instead may have accelerated it, according to New York Times journalist James Risen, who detailed the covert operation in his 2006 book "State of War."
Holly Sterling said her husband has maintained his innocence in the case, and that his conviction was based on "metadata" which indicated he had been in contact with Risen by telephone and email.
While the trial produced evidence that Jeffrey Sterling had been in contact with Risen -- who was covering a discrimination lawsuit Sterling had filed against the CIA -- his wife said that dozens of people at the agency could have leaked the information about "Operation Merlin."
His conviction marked a victory for the US administration in its crackdown on leaks. The case had dragged on in court for years as prosecutors pressed Risen to reveal his sources. He refused to do so.
Jesselyn Radack, a lawyer at the whistleblower defense group Expose Facts, called the case "a gross miscarriage of justice" and said the Obama administration was seeking to intimidate anyone from releasing information that could be embarrassing, even if it were in the public interest.
- Punishing journalists by proxy -
She said the prosecution relied on the little-used Espionage Act to prosecute Sterling, calling it a "bludgeon to silence, coerce and imprison whistleblowers."
She added that "punishing whistleblowers is a backdoor way of punishing journalists" because it chills any potential attempt at leaking important information.
Delphine Halgand, US director at the media rights group Reporters Without Borders, said the case set a troubling precedent by convicting someone on the basis of metadata showing Sterling had been in contact with a reporter.
"We are extremely concerned by this precedent," she told the news conference, adding that Sterling "was convicted on circumstantial evidence. Only the metadata has been made public."
She added that the Justice Department "chose to make an example to warn people against talking to journalists."
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The Department of Economics was established in 1968 with B.A. Economics and it was upgraded as PG Department in 1986 with the introduction of M.A. Economics. In the year 2016, B.A. Economics was started in the self-financed stream. The department has 12 faculty members out of which 7 faculty members are in the aided stream and remaining faculty members are in the self-supporting stream. The Department of Economics was elevated into a full-fledged research department in the year 2014 with part-time and full-time scholars.


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Q: SQL Server Management Studio for sql server 2008 Is there any other version of SQL Server Management Studio except EXPRESS one?
I need one for MS SQL Server 2008 STD version.
I've been trying to search almost everywhere, but it seems there is not.
Can someone agree?
If there is any STANDARD version of this app, i would be very grateful for your help.

A: Microsoft SQL Server 2008 Standard shipss with the SSMS as part of the installation package. I think that you cannot find the SSMS, other than Express editions, as a separate download. If you already have SQL Server 2008 Standard, just add the SSMS via the SQL Server 2008 R2 installer

redpajama_set_name: RedPajamaStackExchange


Branding is more than just a logo... It's your Identity.
What is branding & why does it matter?
Branding is more than just a logo - it's your promise to your customers & the "Big Picture" of your company. Consumers are buying more than just a product, they want to buy a lifestyle; an ideal; a dream. They want to buy "you". Who you present your company as should directly reflect your target customers.
Want to see this idea in action? Look at Lululemon - you might think they just sell yoga pants, but to their loyal customers they are selling the idea that they, too, can be a yogi who takes months off to retreat in the Himalayas & find their inner peace & these yoga pants will help them do it.
Be more than a just product or a brand. Be something your customers are inspired by & aspire to.
To start our work together, I will ask you to fill out a brief questionnaire so that I can get an initial idea of what your brand is/does, where you see your brand in the future & your aesthetic preferences. Next, I study your brand & your competitors to get a better sense whats been done & how we can make your brand stand out from the rest.
Contact us for a free, no-strings-attached consultation!
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With other developers, you have to explain every little detail & lets be honest, most of them just know how to code, but know very little when it comes to design or pleasing aesthetics. With my design background, you're getting professional design advice & direction. If we don't think it looks good or is cohesive with your brands message, it doesn't go on your site.
We also work hard to create an easy flow through your site. Wether you have a lot of information you want to convey or you have a lot of products to display, we will organize your menus & pages in such a way that does not confuse or overwhelm your customers. Reducing the amount of time your customer spends searching for the information or products they need will increase your sales & return customers.
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Ok, sounds good... But why Squarespace instead of Wix or Wordpress?
We have experience with several other platforms, but simply put, Squarespace is the best. It's intuitive dashboard & content editing system makes it easy for our clients to edit their content on their own after their site is done being built, relieving them of the need to continue to pay a third party for site updates if they don't want to. On the other hand, if they don't have time to edit themselves, it makes it easy for us to go in there & edit without having to share passwords or change user settings. As a Thesis client, you are never tied to using us to update or change your site in the future if you don't want to. We give you the tools you need to do it yourself & Squarespace is the most user friendly platform with 24 hour customer support.
Another reason we love Squarespace is that it has built-in responsive design - meaning, your site looks beautiful no matter what device it is being viewed on. These days, web traffic is not only coming from desktop servers, but more & more from mobile devices & tablets. For a lot of customers, if you don't have an easy to navigate mobile site, you won't earn their business. With Squarespace, you don't have to worry about that. It's already included.
Squarespace also has whats known as "Developer Mode". If we feel your site needs something extra that isn't already integrated into your template, we can add it on the back end using CSS (coding). This keeps your site from looking like a "cookie cutter" template like the other guys & gives us more freedom when it comes to site features & functions.
So, now you have a beautiful Brand Identity system... How can you use it?
Let us apply it to every aspect of your business via your business cards, letterhead, envelopes, flyers/mailers, etc. With our print materials services, we can help you put your best foot forward always, wether you're networking, sending official correspondence, advertising or invoicing your customers. Make sure your company always looks professional with beautiful, cohesive design across all materials.
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Is your logo outdated, irrelevant or just not resonating with your target customers? Did you use a DIY cookie cutter website template just to get up & running years ago, but want something better?
If your brand identity & website weren't designed in this decade, or even if they were but you're not 100% happy, we can re-brand & re-design, all while paying homage to what helped you get where you are today! We will go over whatever elements you would like to transfer to your new brand identity & site. If you want to change your domain name but all of your advertising & print materials have your old domain, we can redirect it to your new domain & site so you won't lose any customers or clients during this transition.
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If you don't have time to update your website, we do! Let me know what you need ones & I will quote a flat fee for the project. Wether it's adding information, new pages or products - we can get it done for you - FAST!


This is a placeholder page for Esparza Amy, which means this person is not currently on this site. We do suggest using the tools below to find Esparza Amy.
You are visiting the placeholder page for Esparza Amy. This page is here because someone used our placeholder utility to look for Esparza Amy. We created this page automatically in hopes Esparza Amy would find it. If you are not Esparza Amy, but are an alumni of Homer Hanna High School, register on this site for free now.


text:
Sounders FC has the week off of training before returning on June 14 to prepare for the 2014 Lamar Hunt U.S. Open Cup. Seattle will host PSA Elite (USASA) in the fourth round of the tournament on June 18. Kickoff is set for 7 p.m. PT at Starfire Stadium in Tukwila, Wash., with a live stream on SoundersFC.com.
After resuming MLS play against D.C. United, Sounders FC will stay on the road for the July 5 match against the Vancouver Whitecaps FC. The next home match is set for July 13 against the Portland Timbers on the Xbox Pitch at CenturyLink Field.
Tickets for Sounders FC home matches, including the Tottenham Hotspur friendly, are available now at www.SoundersFC.com and 1-877-MLS-GOAL. The 5-Match ticket package is also available now with expanded seating to include the upper bowl at CenturyLink Field.


text:
module TripToPrint {
    export class CommandZoomIn extends BaseCommand {
        getTitle() { return "Increase size"; }

        getImageName() { return "ZoomIn.png"; }
    }
}

redpajama_set_name: RedPajamaGithub


Malawi has always relied on paper registration for voters, but electoral authorities say that hasn't worked so well. "We used to have a lot of problems in the past" with the passports and driver's licenses used for registration "because photographs may fall off" or names may get misspelled, said Yahya Mmadi, a member of the Malawi Electoral Commission. But the southeast African country's recently unveiled biometric system, being put in place before the 2019 general elections, "will be 100 percent correct," he said. It relies on unique markers such as fingerprints.
Full Article: Malawi Pushes New Biometric Voter Registration, Despite Doubts.


Check out our VIP Prediction for Wisla Plock vs Gornik Zabrze!
Wisla Plock and Gornik Zabrze will be face to face this Saturday in Ekstraklasa at 14:30 so try not to miss it.
Wisla Plock is ranked on the 8th place in the standings with 39 points, 42 goals scored and 44 goals received in the last 30 leagueevents. They have failed to win 11 of their last 30 matches in Ekstraklasa.In their last match they draw against Arka Gdynia. Wisla Plock lost 2 of their last 5 matches before this event with Gornik Zabrze.
The attacking skills of Wisla Plock are mild, they won 2events from the last 5 played.
On this page you can find our predictions & bettings tips for the match between Wisla Plock and Gornik Zabrze. Our predictions for Wisla Plock vs Gornik Zabrze are based on a large number of factors such as direct meetings, Wisla Plock last matches, Gornik Zabrze last matches, Wisla Plock lineup, Gornik Zabrze lineup, standings, insider informations and so on. By visiting our website daily you will increase your winning chances. All the predictions posted here have a huge rate of success!


We are a random and motley collection of pilgrims and searchers, generally aligned with progressive Christianity, meeting monthly in and around Tunbridge Wells in Kent.
Tunbridge Wells Radical Pilgrims continue to meet monthly generally on the last Tuesday of the month. We are currently meeting at Christ Church rooms, on High Street. Numbers vary between twelve and up to twenty. The group is ecumenical but also with members who no longer belong to any church.
Please contact me by phone or e-mail for further information.
We meet at Christ Church, Tunbridge Wells, generally on the last Tuesday of the month, but this can change to tie up with school terms.
Radical Pilgrims was formed about ten years ago. During that time we have met regularly, held a few day conferences and travelled individually to good number in various locations.


Over the past 15 years, no American unionist has organized as many workers, or won them raises as substantial, as Rolf. Which makes it all the more telling that Rolf believes the American labor movement, as we know it, is on its deathbed, and that labor should focus its remaining energies on bequeathing its resources to start-up projects that may find more effective ways to advance workers' interests than today's embattled unions can.
Most labor leaders and activists concur that the union movement is in something close to mortal peril. With Rolf, they believe that the inadequacies of the 1935 National Labor Relations Act (the Wagner Act), which union-busting employers are able to violate with impunity, have made it almost impossible to organize private-sector workers. Unable to grow, labor has also seen its ranks diminished by the offshoring of millions of jobs and the relegation of millions more to the ranks of contingent labor or ostensible self-employment. Today, the percentage of private-sector workers in unions has dropped to a bare 6.7 percent—roughly its level at the beginning of the 20th century, before the advent of a sizable middle class.
This October, with funding from his local, from the national SEIU, and from several liberal foundations, Rolf will unveil The Workers Lab, housed at the Roosevelt Institute in New York. The center will study and, in time, invest in organizations that, in Rolf's words, "have the potential to build economic power for workers, at scale, and to sustain themselves financially." Whatever those organizations may be, they won't be unions—at least, not unions as they currently exist.


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Bill Burr Is the New Louis C.K.
Meet the undisputed heavyweight champ of rage-fueled humor
Carina Chocano
Carina Chocano's Most Recent Stories
The Weaponized Comedy of Hannah Gadsby
'Killing Eve': How Jodie Comer Turned a Pink-Dressed Psychopath Into a Breakout Role
Why The Harvey Weinstein Allegations Could Change Our Culture
Rob Kim/Getty Images
You might have seen comedian Bill Burr on Breaking Bad, playing Saul Goodman's redheaded henchman, Kuby. Or during one of his regular appearances on Letterman, Conan and Fallon. But to really get a sense of what Burr's about, you have to see him live – raging about everything from overfed children to racism – during one of his near-endless string of big-venue dates. "I'm an unbelievably flawed human being," says the 45-year-old stand-up. Or, as he put it in a classic bit, "Like, I went to this place and I'm ordering food. The guy behind the counter asks me if I want a cookie. All of a sudden I have this unbelievable urge to blast this guy right in the face. I'm not trying to be paranoid, but what kind of a man asks another man if he wants a cookie?"
See Who Made Our List of the 50 Funniest People Now
Burr released his third hourlong special, You People Are All the Same, last year, selling it for $5 on his website. He's been working in comedy since the early 1990s; his first high-profile gig was on Chappelle's Show. Since 2007, he's hosted the popular "Monday Morning Podcast," and he recently launched the podcast network All Things Comedy with The Daily Show's Al Madrigal because, Burr says, "Artists start scenes, and then the businessmen come in and own everything." But Breaking Bad was a whole new level. "It was a huge moment for me," he says. "It was like getting sucked into your TV."
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As an Eddie Murphy-obsessed kid growing up in Canton, Massachusetts, where success was measured mostly via sports ("You either sucked in gym class or you were good"), Burr never dreamed of a career in comedy. But working in a warehouse after college, he met a guy who loved stand-up as much as he did. "He said, 'One of these nights I'm gonna take a shot of Jack Daniel's and go up onstage,'" Burr recalls. "I said if he can do it, I'm gonna try it. And thank God I did. Because I sucked at everything else."
This story is from the November 7th, 2013 issue of Rolling Stone.
In This Article: Bill Burr,	Comedy,	Dave Chappelle,	The Daily Show


We have listed all of the Intercession City food pantries that we have found. With help from users like you we have compiled a list of some. If you know of a food pantry that is not included here, please submit new food pantries to our database by going to the ADD A LISTING link on the header of our website.
Provides a food pantry. Serves Orange, Osceola and Polk Counties.Pantry Hours: Monday - Saturday7:30am - 10:00amFor more information, please call.
Due to the low number of listings in Intercession City, we have added area listings below.