Finite temperature charmonium spectral functions in the pseudoscalar(PS) and
vector(V) channels are studied in lattice QCD with 2+1 flavours of dynamical
Wilson quarks, on fine isotropic lattices (with a lattice spacing of 0.057fm),
with a non-physical pion mass of 545MeV. The highest temperature studied is
approximately 1.4Tc. Up to this temperature no significant variation of the
spectral function is seen in the PS channel. The V channel shows some
temperature dependence, which seems to be consistent with a temperature
dependent low frequency peak related to heavy quark transport, plus a
temperature independent term at omega > 0. These results are in accord with
previous calculations using the quenched approximation.Comment: Conference proceedings: The 32nd International Symposium on Lattice
  Field Theory - Lattice 2014 June 23-28, 2014 Columbia University, New York,
  New York This conference contribution draws heavily from the paper:
  arXiv:1401.5940 [hep-lat

Borsanyi, Szabolcs

Dürr, Stephan

Fodor, Zoltán

Hoelbling, Christian

Katz, Sándor D.

Krieg, Stefan

Mages, Simon

Nógrádi, Dániel

Pásztor, Attila

Schäfer, Andreas

Szabó, Kálmán K.

Trombitás, Norbert

Tóth, Bálint C.

English

arXiv

Finite temperature charmonium spectral functions in the pseudoscalar(PS) and vector(V) channels are studied in lattice QCD with 2+1 flavours of dynamical Wilson quarks, on fine isotropic lattices (with a lattice spacing of 0.057fm), with a non-physical pion mass of 545MeV. The highest temperature studied is approximately 1.4Tc. Up to this temperature no significant variation of the spectral function is seen in the PS channel. The V channel shows some temperature dependence, which seems to be consistent with a temperature dependent low frequency peak related to heavy quark transport, plus a temperature independent term at omega > 0. These results are in accord with previous calculations using the quenched approximation

Szabo, Kalman

Juelich Shared Electronic Resources

PoS(LATTICE2014)218Spectral functions of charmonium with 2+1 flavoursof dynamical quarksSzabolcs Borsányia, Stephan Dürrab, Zoltán Fodorabc, Christian Hoelblinga, SándorD. Katzcd , Stefan Kriegab, Simon Magese, Dániel Nógrádicd , Attila Pásztor∗cd ,Andreas Schäfere, Kálmán K. Szabóab, Bálint C. Tótha, Norbert TrombitáscdaUniversity of Wuppertal, Department of Physics, Wuppertal D-42097, GermanybJülich Supercomputing Center, Jülich D-52425, GermanycEötvös University, Budapest 1117, HungarydMTA-ELTE Lendület Lattice Gauge Theory Research GroupeUniversity of Regensburg, Regensburg D-93053, GermanyFinite temperature charmonium spectral functions in the pseudoscalar(PS) and vector(V) channelsare studied in lattice QCD with 2+1 flavours of dynamical Wilson quarks, on fine isotropic lattices(with a lattice spacing of 0.057fm), with a non-physical pion mass of mpi ≈ 545MeV. The highesttemperature studied is approximately 1.4Tc. Up to this temperature no significant variation of thespectral function is seen in the PS channel. The V channel shows some temperature dependence,which seems to be consistent with a temperature dependent low frequency peak related to heavyquark transport, plus a temperature independent term at ω > 0. These results are in accord withprevious calculations using the quenched approximation. The details of the study can be found in[1].The 32nd International Symposium on Lattice Field Theory - Lattice 2014June 23-28, 2014Columbia University, New York, New York∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LATTICE2014)218Spectral functions of charmonium with 2+1 flavours of dynamical quarks Attila Pásztor1. Introduction1.1 Mesonic spectral functionsThe charmonium systems at finite T have been under heavy investigation, since the suppressionof J/Ψ is regarded as an important experimental signal for the formation of the plasma state [2].Charmonium states are expected to dissolve somewhat above the transition temperature, becauseof the screening of the inter-quark potential and collisions with the hot medium. In this paper, weinvestigate the in medium properties of the J/Ψ and ηc mesons from a lattice QCD perspective.The spectral function (SF) of a correlator of self-adjoint operators is the imaginary part of theFourier-transform of the real time retarded correlator. In this article, we will deal with correlatorsbetween local charmonium meson currents, and the corresponding SFs. It can be shown, that theSF is related to the Euclidean correlator by an integral transformG(τ,~p) =∫ ∞0dωA(ω,~p)K(ω,τ), (1.1)whereK(ω,τ) =cosh(ω(τ−1/2T ))sinh(ω/2T )(1.2)is the integral kernel, and G(τ,~p) is the Euclidean correlator (at zero chemical potential). Knowl-edge of the SFs is of great importance. In the SF a stable particle gives a δ like peak, while anunstable particle in matter gives a smeared peak. Also, the Kubo-formula states that the heavyquark diffusion constant D is related to the V spectral function as D = 16χ limω→0∑3i=1Aii(ω,T )ω ,where χ is the (heavy) quark number susceptibility and Aii is the spectral function correspondingto the V channel. If the transport coefficient is non vanishing, we expect some finite value of ρ/ωfor small ω . This implies the presence of a transport peak. We will investigate the anticipatedmelting of the heavy meson states J/Ψ and ηc in the quark gluon plasma.1.2 The Maximum Entropy MethodTo get the SFs from a lattice study one has to invert equation (1.1). This inversion howeveris ill-defined, since the number of frequencies for which one wants to reconstruct the SF is higherthan the number of data points. In this case a χ2 fit on the shape of the SF discretized to Nω pointsis degenerate. One has to regularize the problem in some way. 1 The determination of hadronic SFsvia the Maximum Entropy Method (MEM) was first suggested in [4]. Here, one has to maximize:Q = αS− 12χ2, (1.3)where as usualχ2 =Ndata∑i, j=1(Gfiti −Gdatai )C−1i j (Gfitj −Gdataj ), (1.4)1Note that the number of data points vs. parameters is not the only problem when trying to reconstruct SFs. Even ifone has a large number of data points (e.g. on time anisotropic lattices), the Euclidean correlator is rather insensitive tofine details of the SF. Therefore the inversion introduces large uncertainties [3].2PoS(LATTICE2014)218Spectral functions of charmonium with 2+1 flavours of dynamical quarks Attila Pásztorwith Ci j being the covariance matrix of the data (in Euclidean time), and the Shannon–Jaynesentropy isS =∫dω(A(ω)−m(ω)−A(ω) log(A(ω)m(ω))), (1.5)where the so called prior function m(ω) is supposed to summarize our prior knowledge on the shapeof the SF (such as the leading perturbation theory behaviour). After equation (1.3) is maximized ata given value of α , and the optimal Aα is obtained, the regularization parameter α has to be aver-aged over, with the conditional probability from [4]. It can then be shown that the maximum of Q ata given α lies in an Ndata dimensional subspace of the Nω dimensional space of possible A(ωi) vec-tors, that can be parametrized as A(ω) =m(ω)exp(∑Ndatai=1 si fi(ω)). The particular parametrizationof the subspace we use is fi(ω) = K(ω,τi) and was introduced by Ref. [5]. In our experience thisproved to be numerically more stable than the more widely known Bryan method[6]. The shapeof the subspace is of course strongly dependent on the choice of the prior function. This is thesource of a systematic uncertainty, that has to be considered. We don’t carry out an error analysisof the full spectral function, since with the current statistics that would give huge errors. Instead,we only give errors to some physically interesting quantities related to the spectral function, thatare more stable. Systematic errors have been estimated with varying the prior function shapes:m0/a2, m0ω/a, m0ω2, a−3/(m0/a+ω) and the constant m0 = 0.001,0.01,0.1,1.0. Stastical errorshave been estimated with 10 jackknife samples. We use a modified version of the kernel for thereconstruction [7, 8]: Kˆ (τ,ω) = tanh(ω/2)K (τ,ω). This “cures" the low frequency divergent1/ω behaviour of the kernel, without spoiling the high ω behaviour. We also mention here, thatthe reliable use of the MEM, even with this method, which does not contain a SVD, still requiredarbitrary precision arithmetics in the implementation.Lattice studies of charmonium SFs using the MEM have been carried out on numerous occa-sions, but so far not in 2+1 flavour QCD. A recent, detailed study of charmonium SFs in quenchedQCD can be found in [9]. Results regarding spectral functions with 2 flavours of dynamical quarkscan be found in Ref. [10].1.3 Lattice configurationsWe use the same lattice configurations as in [11], details can be found there. The gauge actionused for the calculations was the Symanzik tree level improved gauge action, and for the fermionicsector the clover improved Wilson action. Six steps of stout smearing with smearing parameterρ = 0.11 were used. The clover coefficient was set to its tree level value, cSW = 1.0, which, forthis type of smeared fermions, essentially leads to an O(a) improved action. The full hadronspectrum using this action was determined in Ref. [12]. The bare masses of the u and d quarkswere taken to be degenerate, therefore the configurations were generated using an N f = 2+ 1flavor algorithm. The u and d quarks were implemented via the Hybrid Monte Carlo (HMC)algorithm [13], whereas the strange quark was implemented using the Rational Hybrid MonteCarlo (RHMC) algorithm [14]. From the study in [11], we only used the finest lattices, withgauge coupling β = 3.85, corresponding to a lattice spacing of a = 0.057(1)fm. The bare lightquark masses where choosen to be amud = −0.00336 and ams = 0.0050, which, when fixing thescale with a physical Ω baryon mass, corresponds to a pion mass mpi = 545MeV. The lattices3PoS(LATTICE2014)218Spectral functions of charmonium with 2+1 flavours of dynamical quarks Attila Pásztorhad spatial extent N3s = 643 and temporal extent Nt = 28,20,18,16,14,12. Nt = 28 correspondsto T = 123MeV, Nt = 12 to T = 288MeV. As for the charm mass tuning, from ref. [15] theratio mc/ms = 11.85. Since with Wilson fermions, there is an additive renormalization, it is notpossible to use this ratio directly in setting the charm mass. However, we know that for ud and sthe masses used in the simulation correspond to a mass ratio of 1.5 [16], from this we get (mc−ms)/(ms−mud) = 35.55 which gives the estimate for the charm mass that was used. To check ifthis is approximately the correct charm mass, we checked the masses of the different mesons statescontaining s and c quarks, and they were indeed in the right ballpark.2. Results2.1 Zero temperature analysis: stability testSince the temperature is T = 1/(Nta), as the temperature increases we have less and less datapoints for our reconstruction of the SFs. That means that the reliability of the method decreaseswith increasing temperature. To the highest temperature, where the MEM results are still likelyto be trusted, we drop points (starting from t = 0) from the lowest temperature correlators and doa MEM reconstruction with these limited number of points. We say that the reconstruction is nolonger reliable when we can not reconstruct the first peak. The result is that Nt = 14 seams reliablestill, but Nt = 12 does not.2.2 MEM reconstructed spectral functionsSo as far as our direct MEM analysis can tell, the PS SF is temperature independent in thegiven range. See Figures 1. The situation is a little bit more complicated in the V channel. Here,reconstruction of the SF shows some temperature dependence. The first peak appears to go downto lower energies at the highest temperature. Due to some properties of the analysis (i.e. possiblemerging of adjacent peaks and problems with the resolution of the transport peak) using MEMalone one cannot draw any firm conclusions about the nature of the change in the SF - at least atthe current level of statistical errors.2.3 The ratio G/GrecAn alternative aproach to study spectral functions was suggested in [5]. The ratio:G(t,T )Grec (t,T )=G(t,T )∫A(ω,Tref)K(ω, t,T )dω(2.1)has a few advantages: MEM reconstruction is only needed at Tref, where we have the most datapoints, and so a more reliable reconstruction. We use Nt = 28 as reference temperature. We cancalculate this ratio even at high temperatures, where the MEM reconstruction is already unreliable.Also, if the spectral function is temperature independent, then the trivial temperature dependenceof the correlators, coming from the integral kernel will drop out, and the ratio will be G/Grec = 1.It is also useful to study the same ratio with mid-point subtracted correlators [17].G−G−rec=G(t,T )−G(Nt/2,T )Grec (t,T )−Grec (Nt/2,T ) =G(t,T )−G(Nt/2,T )∫A(ω,Tref) [K(ω, t,T )−K(ω,Nt/2,T )]dω (2.2)4PoS(LATTICE2014)218Spectral functions of charmonium with 2+1 flavours of dynamical quarks Attila Pásztor 0 0.5 1 1.5 2 2.5 0.5  1  1.5  2  2.5  3  3.5A (ω) / ω2ωaPSEUDOSCALAR CHANNEL, Nt=20, T/Tc≈0.9m(ω)=0.01/a2m(ω)=0.1/a2m(ω)=1.0/a2m(ω)=10.0/a2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.5  1  1.5  2  2.5  3  3.5A (ω) / ω2ωaPSEUDOSCALAR CHANNEL, Nt=20, T/Tc≈0.9m(ω)=0.01/a2m(ω)=0.1ω2m(ω)=1.0/a2/(10.0+ωa) 0 0.05 0.1 0.15 0.2 0.25 0  0.5  1  1.5  2  2.5  3  3.5A (ω) / ω2aωVECTOR CHANNEL, m(ω) = 0.1 ω2Nt=14, T/Tc ≈ 1.30Nt=16, T/Tc ≈ 1.14Nt=18, T/Tc ≈ 1.01Nt=20, T/Tc ≈ 0.91 0 0.05 0.1 0.15 0.2 0.25 0  0.5  1  1.5  2  2.5  3  3.5A (ω) / ω2aωPSEUDOSCALAR CHANNEL, m(ω) = 0.1 ω2Nt=14, T/Tc ≈ 1.30Nt=16, T/Tc ≈ 1.14Nt=18, T/Tc ≈ 1.01Nt=20, T/Tc ≈ 0.91 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075P os it i on  o f f ir st  pe ak  i n l at t ic e un it saT=1/Nt 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075P os it i on  o f f ir st  pe ak  i n l at t ic e un it saT=1/NtFigure 1: The sensitivity of the reconstruction on the prior function(top). Only PS channel is includedin the figure, but the V channel looks similar. The temperature dependence of the reconstucted spectralfunctions(middle). The position of the first PS(left) and V(right) peak as a function of temperature(bottom).This way, one can drop the zero-mode (constant) contribution to the correlators. These have to dowith transport coefficients, or other low frequency (ω  T ) features of the spectral functions. Ifthe ratio of G/Grec is different from one, but the ratio with the middle-point substracted correlatorsis not, that means that the temperature dependence of the SFs should be well described by just azero-mode contribution f (T ) ·ωδ (ω − 0+). The results of such an analysis can be seen in Figs.2. As one can see, the results in the PS channel are consistent with a temperature independent SF,while the results in the V channel show a temperature dependent zero mode/low frequency con-tribution in the SF. We can also try to extract the zero mode contribution itself by considering thedifference G−Grec. This is only plotted in the V channel, in Fig. 3 (in the PS channel it is alwaysconsistent with zero). The difference has big errors, but on the two highest temperatures it is nonzero within 1σ . At every temperature it is consistent with a time separation independent constant.With the ansatz A(ω,T ) = f (T )ωδ (ω−0+)+A(ω,T0) we get f (T )T ≈ (3±1.5) ·10−5 at 1.4Tcin lattice units 2. This ansatz, taken strictly, would imply a diverging diffusion constant. However,2Since we are not using the conserved current on the lattice, but a local current, this will have a finite, latticespacing dependent renormalization constant of O(1). We neglect this fact, since we don’t do a continuum limit, and the5PoS(LATTICE2014)218Spectral functions of charmonium with 2+1 flavours of dynamical quarks Attila Pásztor 0.94 0.96 0.98 1 1.02 1.04 1.06 2  4  6  8  10G/ Gr ect/aNt=12Nt=14Nt=16Nt=18Nt=20 0.9 0.95 1 1.05 1.1 1.15 2  4  6  8  10G/ Gr ect/aNt=12Nt=14Nt=16Nt=18Nt=20 0.9 0.95 1 1.05 1.1 2  4  6  8  10G-/ G-r ect/aNt=12Nt=14Nt=16Nt=18Nt=20 0.9 0.95 1 1.05 1.1 2  4  6  8  10G-/ G-r ect/aNt=12Nt=14Nt=16Nt=18Nt=20Figure 2: The ratio G/Grec in the PS(left) and V(right) channels(). Also, the ratio G−/G−rec(bottom).-1e-05-5e-06 0 5e-06 1e-05 2  4  6  8  10G- Gr ect/aNt=16Nt=14Nt=12Figure 3: The difference G−Grec in the V channel at the 3 highest temperatures.the data do not restrict the shape of the transport peak, they are only sensitive to the area. By usingthis ansatz, we do not mean to say that the diffusion constant diverges, we simply extract that areaof the transport peak. To get a diffusion constant additional information is needed. (The width orthe height of the peak, which is too narrow to resolve at this point.) The survival of J/Ψ up tosuch high temperatures is consistent with previous results in quenched and 2 flavour QCD (see eg.[9, 10]).3. SummaryWe have performed a lattice study of charmonium spectral functions with 2+1 dynamical Wil-son quarks. The MEM reconstruction of the spectral functions is hampered by the limited numberrenormalization is temperature independent.6PoS(LATTICE2014)218Spectral functions of charmonium with 2+1 flavours of dynamical quarks Attila Pásztorof data points at higher temperatures, so the highest temperature we used for MEM reconstructionwas approximately 1.3Tc. The PS spectral functions did not show any noticable temperature vari-ation. The V spectral functions showed a temperature variation, which, according to the analysisof the ratio G/Grec is consistent with a temperature dependent zero mode, and a temperature inde-pendent non-zero part in the SFs. In conclusion, we can say that we observed no melting of the ηcand J/Ψ mesons up to temperature of 1.4Tc, and we observe no variations in the spectral functionsof ηc whatsoever.AcknowledgmentComputations were carried out on GPU clusters at the Universities of Wuppertal and Budapestas well as on supercomputers in Forschungszentrum Juelich. This work was supported by the EUFramework Programme 7 grant (FP7/2007-2013)/ERC No 208740, by the Deutsche Forschungsge-meinschaft grants FO 502/2, SFB-TR 55 and by Hungarian Scientific Research Fund grant OTKA-NF-104034..References[1] S. Borsanyi, S. Dürr, Z. Fodor et al., JHEP 1404, 132 (2014) [arXiv:1401.5940 [hep-lat]].[2] T. Matsui, H. Satz, Phys. Lett. B178, 416 (1986).[3] G. Aarts and J. M. Martinez Resco, JHEP 0204 (2002) 053 [hep-ph/0203177].[4] M. Asakawa, T. Hatsuda and Y. Nakahara, Prog. Part. Nucl. Phys. 46, 459 (2001) [hep-lat/0011040].[5] A. Jakovác, P. Petreczky, K. Petrov, A. Velytsky, Phys. Rev. D 75, 014506 (2007) [hep-lat/0611017].[6] R. K. Bryan, Eur. Biophys J. 18, 165 (1990)[7] G. Aarts, C. Allton, J. Foley, S. Hands, S. Kim, Phys. Rev. Lett. 99 (2007) 022002 [hep-lat/0703008].[8] J. Engels, O. Vogt, Nucl. Phys. B832, 538-566 (2010). [arXiv:0911.1939 [hep-lat]].[9] H. T. Ding, A. Francis et al. Phys. Rev. D 86, 014509 (2012) [arXiv:1204.4945 [hep-lat]].[10] A. Kelly, J. -I. Skullerud, C. Allton, D. Mehta and M. B. Oktay, arXiv:1312.0791 [hep-lat].[11] S. Borsányi, S. Dürr, Z. Fodor et al., JHEP 1208, 126 (2012) [arXiv:1205.0440 [hep-lat]].[12] S. Dürr, Z. Fodor et al., Science 322 (2008) 1224 [arXiv:0906.3599 [hep-lat]].[13] S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Phys. Lett. B 195 (1987) 216.[14] M. A. Clark and A. D. Kennedy, Phys. Rev. Lett. 98 (2007) 051601 [arXiv:hep-lat/0608015].[15] C. T. H. Davies, C. McNeil et al., Phys. Rev. Lett. 104, 132003 (2010) [arXiv:0910.3102 [hep-ph]].[16] S. Dürr, Z. Fodor, C. Hoelbling et al., Phys. Lett. B 701 (2011) 265 [arXiv:1011.2403 [hep-lat]].[17] T. Umeda, Phys. Rev. D 75, 094502 (2007) [hep-lat/0701005].7

Spectral functions of charmonium with 2+1 flavours of dynamical quarks

Spectral functions of charmonium with 2+1 flavours of dynamical quarks

Abstract

Similar works

Full text

Available Versions

Juelich Shared Electronic Resources