We report on a continuum extrapolation of the vector current correlation
function for light valence quarks in the deconfined phase of quenched QCD. This
is achieved by performing a systematic analysis of the influence of cut-off
effects on light quark meson correlators at $T\simeq 1.45 T_c$ using clover
improved Wilson fermions. We discuss resulting constraints on the electrical
conductivity and the thermal dilepton rate in a quark gluon plasma. In addition
new results at 1.2 and 3.0 $T_c$ will be presented.Comment: 4 pages, 6 eps figures, to appear in the proceedings of Quark Matter
  2011, 23-28 May 2011, Annecy, Franc

A Francis

Arnold P B

Aurenche P

O Kaczmarek

Zhao J

English

arXiv

Kaczmarek O, Francis A. Electrical conductivity and thermal dilepton rate from quenched lattice QCD. J.Phys.G. 2011;38(12): 124178.We report on a continuum extrapolation of the vector current correlation function for light valence quarks in the deconfined phase of quenched QCD. This is achieved by performing a systematic analysis of the influence of cut-off effects on light quark meson correlators at $T\simeq 1.45 T_c$ using clover improved Wilson fermions. We discuss resulting constraints on the electrical conductivity and the thermal dilepton rate in a quark gluon plasma. In addition new results at 1.2 and 3.0 $T_c$ will be presented

Kaczmarek, Olaf 

Francis, Anthony

Publications at Bielefeld University

arXiv:1109.4054v1  [hep-lat]  19 Sep 2011Electrical conductivity and thermal dilepton ratefrom quenched lattice QCDO. Kaczmarek and A. FrancisFakulta¨t fu¨r Physik, Universita¨t Bielefeld, D-33615 Bielefeld, GermanyE-mail: okacz@physik.uni-bielefeld.deAbstract. We report on a continuum extrapolation of the vector current correlationfunction for light valence quarks in the deconfined phase of quenched QCD. This isachieved by performing a systematic analysis of the influence of cut-off effects on lightquark meson correlators at T ≃ 1.45Tc using clover improved Wilson fermions [1]. Wediscuss resulting constraints on the electrical conductivity and the thermal dilepton ratein a quark gluon plasma. In addition new results at 1.2 and 3.0 Tc will be presented.1. IntroductionThe measured dilepton rates in heavy ion experiments show an enhancement in thelow invariant mass regime of a few hundred MeV and are getting sensitive to transportproperties which are relevant in the hydrodynamic regime of small invariant masses[2, 3]. The spectral representation of the correlation functions of the vector currentdirectly relates to the invariant mass spectrum of dileptons and photons and in thelimit of small frequencies determines a transport coefficient, in the case of the vectorcorrelation function of light quarks, the electical conductivity.At temperatures relevant for current heavy ion experiments, non-perturbative techniquesare mandatory for the determination of those quantities. Perturbative studies of thevector spectral functions [4, 5] and also the inclusion of nonperturbative aspects throughthe hard thermal loop resummation scheme [6] break down, especially in the lowinvariant mass region indicated by an infrared divergent Euclidean correlator [7] leadingto an infinite electrical conductivity. Instead it was demonstrated that the spectralfunction at low invariant masses will increase linearly resulting in a finite electricalconductivity of the quark gluon plasma [8, 9].In [1] we have analyzed the behaviour of the vector correlation function at T ≃ 1.45Tcand performed its extrapolation to the continuum limit based on precise data at variouslattice sizes, corresponding to different lattice cutoffs. While only small finite volumeeffects were observed in this study, large cutoff effects in the correlation functions requiresmall lattice spacings and a proper continuum extrapolation to obtain reliable resultsfor the determination of the spectral properties and the extraction of the dilepton ratesand transport coefficients.Electrical conductivity and thermal dilepton rate from quenched lattice QCD 2 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 0  0.1  0.2  0.3  0.4  0.5τTT2GV(τT)/[χqGVfree(τT)]1283x241283x321283x48contfit [0.2:0.5] 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 0  0.05  0.1  0.15  0.2  0.25  0.3τ[fm]GV(τT)/GVfree(τT)1283x40,1.2Tc1283x32,1.5Tc1283x48,1.5Tc1283x16,3.0TcFigure 1. The vector correlation function GV (τT ) calculated at T ≃ 1.45Tc (left)normalized by the continuum version of the corresponding free vector correlationfunction together with the continuum extrapolation [1]. The same quantity on thefinest lattices for three different temperatures (right).2. Thermal vector correlation and spectral functionIn fig. 1 (left) our results for the vector correlation function normalized by thecorresponding continuum free correlator and the quark number susceptibilty χq areshown for various lattice sizes N3σ × Nτ together with the continuum extrapolation [1].The strong cutoff effects at small separations τT on the lattices with small temporalextend Nτ clearly show the necessity for the continuum extrapolation. Only on the finestlattice and in the extrapolation the relevant physical behaviour of the vector correlationfunction becomes apparent and a reliable continuum extrapolation at distance τT>∼0.2could be performed.We used an Ansatz for the vector spectral function,ρ00(ω) = − 2πχqωδ(ω) , (1)ρii(ω) = 2χqcBWωΓ/2ω2 + (Γ/2)2+32π(1 + k) ω2 tanh(ω/4T ) , (2)that depends on four temperature dependent parameters; the quark numbersusceptibility χq(T ), the strength (cBW (T )) and width (Γ(T )) of the Breit-Wigner peakand the parameter k(T ) that parametrizes deviations from a free spectral function atlarge energies. At high temperature and for large energies, ω/T ≫ 1, we expect to findk(T ) ≃ αs/π.The parameters are determined by a fit to the continuum extrapolated vector correlationfunction. In order to analyze the influence of the low energy structure of the spectralfunction and to analyze the systematic uncertainties of the Ansatz, we have smoothlytruncated the continuum contribution at some energy ω0 by multiplying the second termin (2) with Θ(ω0,∆ω) =(1 + e(ω20−ω2)/ω∆ω)−1. Details of the fit procedure includingadditional information obtained from thermal moments of the spectral functions arediscussed in [1]. The resulting spectral functions are shown in Fig. 2 for different valuesElectrical conductivity and thermal dilepton rate from quenched lattice QCD 3 0 1 2 3 4 5 6 0  2  4  6  8  10ρii(ω)/ωTω/T∆ω/T=0.5ω0/T=00.51.01.75cont 0 1 2 3 4 5 6 0  2  4  6  8  10ρii(ω)/ωTω/Tω0/T=1.5∆ω/T=00.10.5contFigure 2. Spectral functions obtained from fits to the vector correlation function [1].For comparison we also show only the continuum part of the spectral function. Theleft hand figure shows results for different values of the cut-off (ω0) and fixed width(∆ω). The right hand figure shows results for fixed ω0/T = 1.5 and several values of∆ω. The curve labeled ’cont’ is the continuum contribution to the fit described in (2).of ω0 and ∆ω that lead to χ2/d.o.f smaller than unity.3. Electrical conductivity and thermal dilepton rateIn Fig. 3 we show the thermal dilepton ratedNl+l−dωd3p= Cemα2em6π3ρV (ω, ~p, T )(ω2 − ~p2)(eω/T − 1), (3)for two massless (u, d) flavors. The results are compared to a dilepton spectrumcalculated within the hard thermal loop approximation [6] using a thermal quark massmT/T = 1. For ω/T ≥ 2 the results are in good agreement and for 1<∼ω/T<∼2 differencesbetween the HTL spectral function and our numerical results are about a factor two,which is the intrinsic uncertainty in our spectral analysis.While for energies ω/T<∼1 the HTL results grow too rapidly, from our numerical resultswe obtain a finite electrical conductivity,1/3 <∼1CemσT<∼ 1 at T ≃ 1.45 Tc . (4)4. ConclusionsWe have performed a detailed analysis of the vector correlation function at a fixedtemperature of 1.45Tc in the high temperature phase of quenched QCD. The resultsallowed for a determination of the spectral properties and the resulting dilepton rate aswell as an estimate for the electrical conductivity of the QGP.First results at different temperatures, 1.2 and 3.0Tc (Fig. 1 (right)), show a qualitativelysimilar behaviour for the vector correlation function. Even at the smallest temperatureElectrical conductivity and thermal dilepton rate from quenched lattice QCD 4 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0  2  4  6  8  10dNl+l-/dω d3pω/Tω0/T=0.0, ∆ω/T=0.0ω0/T=1.5, ∆ω/T=0.5HTLBorn 0 1 2 3 4 5 0  2  4  6  8  10ω0/T=0.0, ∆ω/T=0.0ω0/T=1.5, ∆ω/T=0.5HTLfreeρii(ω)/ωTω/TFigure 3. Thermal dilepton rate in 2-flavor QCD (left). Shown are results from fitswithout a cut-off on the continuum contribution (ω0/T = 0) and with the largest cut-off tolerable in our fit ansatz (ω0/T = 1.5). The HTL curve is for a thermal quarkmass mT /T = 1 and the Born rate is obtained by using the free spectral function. Theright hand part of the figure shows the spectral functions that entered the calculationof the dilepton rate.of 1.2Tc no signals for a sizeable contribution of a ρ-resonance are visible. Howevera detailed analysis of the spectral properties at temperatures close to the critical oneremains to be performed in future.Furthermore in order to analyze to what extent the low mass enhancement observedin our estimate for the dilepton rate can account for the experimentally observedenhancement of dilepton rates at low energies [2, 3, 10] results over the wholetemperature region probed experimentally as well as knowledge on its momentumdependence is needed together with a realistic model for the hydrodynamic expansionof dense matter created in heavy ion collisions [11].References[1] H. T. Ding et al., Phys. Rev. D 83, 034504 (2011).[2] A. Adare et al. (PHENIX Collaboration), Phys. Rev. Lett. 104, 132301 (2010) and Phys. Rev. C81, 034911 (2010).[3] J. Zhao (for the STAR collaboration), these proceedings, arXiv:1106.6146.[4] for an overview and further references, see for instance: F. Gelis, Nucl. Phys. A 715, 329 (2003).[5] J. P. Blaizot and F. Gelis, Eur. Phys. J. C 43, 375 (2005).[6] E. Braaten and R.D. Pisarski, Nucl. Phys. B 337, 569 (1990).[7] F. Karsch, M. G. Mustafa and M. H. Thoma, Phys. Lett. B 497, 249 (2001).[8] P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP 0011, 001 (2000); JHEP 0112, 009 (2001) andJHEP 0305, 051 (2003).[9] P. Aurenche, F. Gelis, G. D. Moore and H. Zaraket, JHEP 0212, 006 (2002).[10] D. Adamova et al., Phys. Lett. B 666, 425 (2008).[11] H. van Hees and R. Rapp, Nucl. Phys. A 806, 339 (2008).

Electrical conductivity and thermal dilepton rate from quenched lattice QCD

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