The spectral functions of QCD can give us insight into properties of hadrons,
and they are useful in probing the QCD vacuum. I will discuss the correlators
and spectral functions of charmonium in high temperature two flavour QCD. The
spectral functions have been obtained using the Maximum Entropy Method from
anisotropic lattice data using the conserved vector current. This work has been
done as part of the FASTSUM collaboration. We find that the spectral functions
for zero momentum are stable. At non-zero momentum the spectral functions are
less stable but still produce resonance and transport peaks. This work is part
of our programme to calculate the heavy quark diffusion constant.Comment: 7 pages, 5 figures, 31st International Symposium on Lattice Field
  Theory LATTICE 201

Allton, Chris

Kelly, Aoife

Mehta, Dhagash

Oktay, Mehmet B.

Skullerud, Jon-Ivar

English

arXiv

The spectral functions of QCD can give us insight into properties of hadrons, and they are useful

in probing the QCD vacuum. I will discuss the correlators and spectral functions of charmonium

in high temperature two flavour QCD. The spectral functions have been obtained using the Maximum

Entropy Method from anisotropic lattice data using the conserved vector current. This work

has been done as part of the FASTSUM collaboration. We find that the spectral functions for zero

momentum are stable. At non-zero momentum the spectral functions are less stable but still produce

resonance and transport peaks. This work is part of our programme to calculate the heavy

quark diffusion constant

Maynooth EPrints and ETheses Archive

Spectral functions of charmonium from 2 flavour

anisotropic lattice data

Maynooth University ePrints and eTheses Archive

MURAL - Maynooth University Research Archive Library

PoS(LATTICE 2013)170Spectral functions of charmonium from 2 flavouranisotropic lattice dataAoife Kelly∗Department of Mathematical Physics, National University of Ireland Maynooth, Maynooth, Co.Kildare, Ireland.E-mail: aoifek@thphys.nuim.ieJon-Ivar SkullerudDepartment of Mathematical Physics, National University of Ireland Maynooth, Maynooth, Co.Kildare, Ireland.E-mail: jonivar@thphys.nuim.ieChris AlltonDepartment of Physics, Swansea University, Swansea, SA2 8PP Wales, UKE-mail: c.allton@swan.ac.ukDhagash MehtaDepartment of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USAE-mail: dbmehta@ncsu.eduMehmet B. OktayDepartment of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242 USAEmail: boktay@ucair.med.utah.eduThe spectral functions of QCD can give us insight into properties of hadrons, and they are usefulin probing the QCD vacuum. I will discuss the correlators and spectral functions of charmoniumin high temperature two flavour QCD. The spectral functions have been obtained using the Maxi-mum Entropy Method from anisotropic lattice data using the conserved vector current. This workhas been done as part of the FASTSUM collaboration. We find that the spectral functions for zeromomentum are stable. At non-zero momentum the spectral functions are less stable but still pro-duce resonance and transport peaks. This work is part of our programme to calculate the heavyquark diffusion constant.31st International Symposium on Lattice field theory LATTICE 2013July 29 August 3, 2013Mainz, Germany∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LATTICE 2013)170Charmonium spectral functions Aoife Kelly1. IntroductionExperimental observations have been made of the quark-gluon plasma at both RHIC and theLHC. As yet, the theory explaining this phase of matter has not been fully developed. The charmquarks were seen to thermalise as effectively as the light quarks. This motivates the study ofcharm diffusion [1] and J/ψ suppression [2]. There have been several previous lattice studies ofcharmonium at high temperature [3, 4]. Hydrodynamical transport coefficients are related to thelow frequency range of the spectral function by Kubo formulae [5, 6]. The heavy quark diffusioncan be related to the behaviour of vector current correlators at large times. In order to calculatethis diffusion, the correlator is decomposed into its transverse and longitudinal components [1].The heavy quark diffusion constant D is related to the spatial components of the vector spectralfunction,D =16χ00limω→03∑i=1ρVii (ω,~p = 0,T )ω, (1.1)where χ00 is the quark number susceptibility which is defined through the zeroth component of thetemporal correlator in the vector channel [7]. The density-density retarded correlator χNN(~k,ω) isalso related to the heavy quark diffusion constant [1].χNN(ω,~p) =χs(~p)Dp2−iω+ p2D −χs(~p)Dp2−iω+η (1.2)Here χs(~p) is the static susceptibility and η is the drag coefficient. Additional information maythus be obtained by studying the spectral functions of charmonium for non-zero momentum. Forfrequencies ω ∼ Dp2 the first term dominates and this represents the solution to the diffusionequation.Different methods have been used to compute the spectral functions of charmonium. One suchmethod includes the use of the Maximum Entropy Method (MEM) [8] and phenomenologicallymotivated fit functions [7]. Our approach is to use MEM analysis to obtain the spectral functionsfor charmonium in the vector channel, J/ψ and to compute the heavy quark diffusion constant.We use the updated Bryan’s method for this [9]. This work has been performed with anisotropiclattices with two dynamical flavours of fermions.2. Lattice ParametersThe lattices used were generated [10] with two dynamical flavours of light quarks with aniso-tropy ξ = as/aτ = 6. The spatial and temporal lattice spacings are as ' 0.162fm and a−1τ '7.35GeV. The temperature, T = 1/(aτNτ), ranges from the confined phase up to ∼ 2Tc, whereTc is the deconfining transition. The fermion action is a Hamber-Wu action in the spatial directionsand a Wilson action in the temporal direction. A two-plaquette Symanzik-improved gauge actionwas employed. The lattices, their corresponding temperatures and the number of configurations arelisted in Table 1.2PoS(LATTICE 2013)170Charmonium spectral functions Aoife KellyNs Nτ T (MeV) T /Tc Ncfgs12 16 459 2.09 100012 18 408 1.86 70012 20 368 1.68 100012 24 306 1.40 50012 28 263 1.20 100012 32 230 1.05 875Table 1: Lattice sizes N3s ×Nτ , temperatures T and number of configurations Ncfgs.3. Spectral Functions from MEMThe Maximum Entropy Method (MEM) was implemented in order to find the spectral func-tions for the J/ψ particle. This is a well-established method [8] of determining the spectral functionρ(ω) from the imaginary time correlator G(τ),G(τ) =∞∫0dω2piρ(ω)cosh[ω(τ− β2 )]sinh(βω2 ), (3.1)for β = 1/T . MEM is contingent on Bayes’ Theorem and produces the most probable spectralfunction given the Monte Carlo data.The resulting spectral functions rely on certain input information known as the default modelsm(ω). The inclusion of this prior information leads to a unique solution. However, while it isnecessary to include some prior information in these m(ω), the ensuing spectral function shouldhave no dependence on them. We have studied the default models listed in equations (3.2)–(3.4),where m0 is a free parameter and m1 is scaled with temperature,m(ω) = m0ω2, (3.2)m(ω) = m0, (3.3)m(ω) = m0ω(m1+ω). (3.4)The parameter m0 was chosen to be the same for each temperature T , close to the value givinga best fit to the correlator. It was then varied by factors of 10 and 1/10 in order to examine thedependence of m(ω) on it. The parameter m1 was set to 1/T and then varied to 2piT and T/2pi .The analysis in the variation of m1 is ongoing. Most of the results shown are using (3.4).4. Conserved Vector CurrentThe focus of this study is the vector channel, J/ψ . The conserved current was used to createthis state and has the advantage that it requires no renormalisation. The spatial component of theconserved current for the Hamber-Wu action is given in equation (4.1). Here rA = 6s= 6/8, where3PoS(LATTICE 2013)170Charmonium spectral functions Aoife Kellys = 1/8 is the spatial Wilson parameter. The temporal component is the usual conserved Wilsoncurrent, which is described in equation (4.2), where r = 1 is the usual Wilson parameter.V Ai (x) =−23ψ¯(x)(rA− γi)Ui(x)ψ(x+ ıˆ)+ 23 ψ¯(x+ ıˆ)(rA+ γi)U†i (x)ψ(x)+112us[ψ¯(x)(2rA− γi)Ui(x)Ui(x+ ıˆ)ψ(x+2ıˆ)+(x→ x− ıˆ)]− 112us[ψ¯(x+ ıˆ)(2rA+ γi)U†i (x)U†i (x− ıˆ)ψ(x− ıˆ)+(x→ x+ ıˆ)].(4.1)Vt(x) =12[ψ¯(x+ tˆ)(r+ γ0)U†t (x)ψ(x)− ψ¯(x)(r− γ0)Ut(x)ψ(x+ tˆ)]. (4.2)In order to calculate the heavy quark diffusion constant [1], it is necessary to decompose theconserved vector current into its longitudinal (VL) and transverse (VT ) polarisations, as in equation(4.3).Vi j(τ,~p) =(δi j− pi p jp2)VT (τ,~p)+pi p jp2VL(τ,~p). (4.3)It is the longitudinal component which will yield the charm diffusion. The momentum values hereare given by p2 =(2pin2asNs)2, with n2 = 0, 1, 2, 3, 4, corresponding to p = 0, 0.66, 0.93, 1.14, 1.32GeV respectively.5. Zero Momentum ResultsFigure 1 shows the spectral functions obtained for momentum p2 = 0 for each temperature.The default model used in these calculations is given in equation (3.4). A peak structure is evident(in particular for Nτ = 28, 32) close to the experimental value of the J/ψ mass. For the highertemperatures, we see no evidence of a peak. The secondary peak can be attributed to lattice artifacts.Figure 1: Spectral functions for each temperature with p2 = 0. The vertical line represents theexperimental value for the J/ψ mass.4PoS(LATTICE 2013)170Charmonium spectral functions Aoife Kelly6. Non-zero Momentum ResultsIn studying the non-zero momenta it was noted that the high temperature lattices did not yieldgood results. The systematic errors were quite large. This was observed in [10]. As such, only the123×28 and 123×32 lattices are shown here. Figure 2 shows the spectral functions obtained forthe longitudinal polarisation of momenta p2 = 1 and p2 = 3. At p2 = 1 there is a peak structuredetectable, which is non-existent at p2 = 3. However, we cannot definitively argue that this is dueto a melted state. Further analysis is required.Figure 2: Spectral functions for the longitudi-nal polarisation of the conserved vector currentat non-zero momenta.Figure 3: Spectral functions for the transversepolarisation of the conserved vector current atnon-zero momenta.In figure 3 we present the plots for the spectral functions of the transverse component of theconserved vector current for momenta p2 = 1 and p2 = 2. At these low momenta we see evidenceof a stable peak in the region of the J/ψ mass, with a secondary peak due to lattice artifacts.When using MEM it is prudent to check the dependence of the spectral function on the defaultmodel. Thus, in our studies of the charmonium spectral functions three default models were used,see equations (3.2) – (3.4). The dependence is shown in figure 4 on the 123× 28 lattice for thetransverse polarisation of the conserved current with p2 = 1. As the spectral functions have similarshapes with peak positions very close to one another, we can be reasonably confident in our results.7. Low Frequency RegionOur objective is to calculate the diffusion transport coefficient and therefore we are particularlyinterested in the low frequency region of our spectral functions. Figure 5 shows the low frequencyzone for the spectral function the 123× 32 lattice with p2 = 0 for each default model. There arediscrepancies between the heights of these transport peaks, however the range is small. It is thesepeaks which we will use to find the coeffiecient of charm diffusion. As ω→ 0 the default model inequation (3.2) (represented on the plot by the black line) will converge to 0, and the default modelin equation (3.3) (represented by the red line) will diverge. These therefore will give extreme upperand lower bounds. Our main result is given by the blue line. A variation in values for m0 and m15PoS(LATTICE 2013)170Charmonium spectral functions Aoife KellyFigure 4: Dependence of ρ(ω) on m(ω) for p2 = 0 on the 123×32 lattice.has also been studied, and the dependence of the spectral functions on these parameters is work inprogress.Figure 5: Low frequency region of spectral function from 12s×32 lattice with zero momentum.8. Conclusions and OutlookWe have computed the spectral functions in the vector channel (corresponding to J/ψ) fromLattice QCD using anisotropic lattices with the conserved vector current. At zero momentum, thecalculations agree quite well and confirm previous studies suggesting that J/ψ survives up to atleast 1.2Tc. At non-zero momentum, the errors are larger and we are less confident with our results.We are particularly interested in the low frequency region of the spectral functions. In ourfuture work the transport coefficients will be evaluated, including the heavy quark diffusion and6PoS(LATTICE 2013)170Charmonium spectral functions Aoife Kellythe susceptibility. Lattices with 2+1 dynamical flavours of quarks and a finer spatial lattice spacingwill also be examined.AcknowledgementsThis work was supported by the Irish Research Council and Science Foundation Ireland grant08-RFP-PHY1462. We acknowledge the use of computing resources from the Irish Centre forHigh End Computing, Fermilab Lattice QCD Clusters, the Trinity Centre for High PerformanceComputing and the IITAC project funded by the HEA under the Program for Research in ThirdLevel Institutes (PRTLI) co-funded by the Irish Government and the European Union. DM wassupported by a DARPA Young Faculty Award and the U.S. Department of Energy under contractno. DE-FG02-85ER40237.References[1] P. Petreczky and D. Teaney, Heavy quark diffusion from the lattice, Phys. Rev. D 73 (2006) 014508[arXiv:0507318 [hep-lat]].[2] T. Matsui and H. Satz, J/ψ suppression by quark-gluon plasma formation, Phys. Lett. B 178 (1986)416.[3] G. Aarts, C. Allton, M. B. Oktay, M. Peardon and J.-I. Skullerud, Charmonium at high temperature in2 flavour QCD, Phys. Rev. D 76 (2007) 094513 [arXiv:0705.2198 [hep-lat]].[4] M. Asakawa and T. Hatsuda, Charmonia above the deconfinement phase transition [arXiv:0309001[hep-lat]][5] D. Forster, Hydrodynamics, fluctuations, broken symmetry and correlation functions, Perseus Books(1990).[6] J.P. Boon and S. Yip, Molecular Hydrodynamics, McGraw-Hill (1980).[7] H.-T. Ding, A. Francis, O. Kaczmarek, F. Karsch, H. Satz and W. Soeldner, Charmonium propertiesin hot quenched lattice QCD, Phys. Rev. D 86 (2012) 014509.[8] M. Asakawa, Y. Nakahara and T. Hatsuda, Maximum entropy analysis of the spectral functions inlattice QCD, Prog. Part. Nucl. Phys. 46 (2001) 459 [arXiv:0011040 [hep-lat]].[9] G. Aarts, C. Allton, J. Foley and S. Hands, Spectral functions at small energies and the electricalconductivity in hot, quenched lattice QCD, Phys. Rev. Lett. 99 (2007) 022002 [arXiv:0703008[hep-lat]][10] M. B. Oktay and J.-I. Skullerud, Momentum dependence of charmonium spectral functions fromlattice QCD, [arXiv:1005.1209 [hep-lat]].7

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Spectral functions of charmonium from 2 flavour anisotropic lattice data

Spectral functions of charmonium from 2 flavour anisotropic lattice data

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