We discuss the calculations of quarkonium spectral functions in potential
models and their implications for the interpretation of the lattice data on
quarkonium correlators. In particular, we find that melting of different
quarkonium states does not lead to significant change in the Euclidean time
correlators. The large change of the quarkonium correlators above deconfinement
observed in the scalar and axial-vector channels appears to be due to the zero
mode contribution.Comment: 8 pages LaTeX, contribution to Lattice 2007, July 30-4 August 2007,
  Regensburg, German

Mocsy, Agnes

Petreczky, Peter

English

arXiv

We discuss the calculations of quarkonium spectral functions in potential models and their implications for the interpretation of the lattice data on quarkonium correlators. In particular, we find that melting of different quarkonium states does not lead to significant change in the Euclidean time correlators. The large change of the quarkonium correlators above deconfinement observed in the scalar and axial-vector channels appears to be due to the zero mode contribution

Mocsy, A.

Petreczky, P.

UNT (University of North Texas) Digital Library

B R O ~ ~ ~ B ~ E N  N A T I O ,  L LABORATORY RBRC 704 BNL-79724-2007-CP Quarkonium correlators at finite temperature and potential models MOCSY, A.; Petreczky, P. The XXV International Symposium on Lattice Field Theory Regensburg, Germany July 30 - August 4,2007 Physics’DepartmenVRIKEN BNL Research Center Brookhaven National Laboratory P.O. Box 5000 Upton, NY 11973-5000 www.bnl.gov Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Depamnent of Energy. The publisher by accepting the mariuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. This preprint is intended for publication in a journal or proceedings. Since changes may be made before publication, it may not be cited or reproduced without the author’s permission. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof I? 0 0 PJ - .c, i cj a a, 5: I un > vl 0 i-4 m 0 r3 r- 0 .. Quarkonium correlators at finite temperature and potential models Agnes M6csy RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973 USA, P6ter Petreczky, Department of Physics, Brookhaven National Laboratory, Upton, NY 11973 USA October 27. 2007 Abstract We discuss the calculations of quarkonium spectral functions in potential models and their implications for the interpretation of the lattice data on quarkonium cor- relators. In particular, we find that melting of different quarkonium states does not lead to significant change in the Euclidean time correlators. The large change of the quarkonium correlators above decoufinement observed in the scalar and axial-vector channels appears to be due to the zero mode contribution. 1 Introduction It was argued long ago that melting of quarkonia above the deconfinement transition can serve as a signature of quark-gluon plasma formation in heavy ion collisions [l]. The basic idea behind this proposal was that due to color screening the potential between a quark and anti-quark will not provide sufficient binding at  high temperature. This problem can be formulated more rigorously in terms of quarkonium spectral functions, which can be, in principle, extracted from Euclidean-time meson correlation functions calculated on the lattice. Attempts doing this based on the Maximum Entropy Method (MEM) have been discussed over the last few years. The initial interpretation of data led to the conclusion 1 that the 1s charmonium states survive in the deconfined medium up to temperatures of about 1.6TC, with T, being the transition temperature [2, 3, 4, 51. Recent analysis, however, has shown that, although MEM can be used to extract reliably quarkonium spectral functions at zero temperature, at finite temperature it has severe limitations [GI. At zero temperature quarkonium spectrum is well described in non-relativistic poten- tial models. Since the seminal paper by Matsui and Satz the problem of charmonium dissolution has been studied in potential models [7, 8, 9, 10, 11, 12, 13, 14, 15, 161. While the early studies used phenomenological potential, more recent studies rely on static quark anti-quark free energy calculated on the lattice. In fact significant progress has been made in understanding the in-medium modification of inter-quark forces via lattice calculations of the free energy of static quark anti-quark pair. Calculations have been done in pure gluodynamics, 3-flavor and 2-flavor QCD [17, 18, 191, and preliminary results are also available in the physically relevant case of one heavy strange quark and two light quarks [20, 211 (the light quark masses correspond to pion mass of about 220 MeV). Recently attempts to calculate quarkonium properties at finite temperature using re- summed perturbation theory have been made [22, 231. Resummed perturbation theory appears to be successful in calculations of bulk thermodynamics properties [24, 25, 261. Since the lattice calculations of spectral functions have severe limitations, in [12, 131 it has been pointed out, that comparison between the lattice data and potential models should be done in terms of the Euclidean time correlators, for which the numerical results are much more reliable. Recent studies following this line have also been presented in Refs. [14, 15, 161. In this contribution we discuss the calculation of quarkonium spectral functions in a potential model, which uses the lattice data of the free energy of a static quark anti-quark pair. Since reliable calculations of the quarkonium correlators are avail- able only in quenched approximation we consider QCD with only heavy quarks. Further details about this approach can be found in Ref. [27]. 2 Charmonium Spectral Functions in Potential Model For heavy quarks the spectral function can be related to the non-relativistic Green’s function ~ ( w )  = K-ImGn’(~,r’,E)li,j,o, - (1) 6 7r for S-wave, and P-wave charmonia, respectively. Here E = w - 2m . At leading order K = 1. Relativistic and higher order perturbative corrections will lead to a value different 2 from unity 1271. The non-relativistic Green’s function satisfies the Schrodinger equation The numerical method for solving this equation is presented in [27]. At zero temperature we use the Cornell potential V ( T )  = -a/r + UT with parameters motivated by lattice re- sults on static potential : cy = n/12 and u = (1.65 - n/12)-i2. In the actual calculations we use a potential which is screened beyond some distance T > ~ , , d  with screening length p to mimic many body effects at large energies (see Ref. [27] for further details). At finite temperature we use a potential motivated by lattice results on the singlet free energy of a static quark anti-quark pair and which is defined in section IV of Ref. [27]. At large energies, away from the threshold, the non-relativistic treatment is not applicable. The spectral function in this domain, however, can be calculated using perturbation theory. We smoothly match the non-relativistic calculation of the spectral function to the rela- tivistic perturbative result [27], Euclidean time correlators G(7,T) at some temperature T can be calculated from the spectral functions using the integral representation G(7,T) = Lmdwu(w,T)K(w,r ,T) .  (4) Here the integration kernel is coshw(T - 1/(2T)) sinh ( w / ( 2 T ) )  K(w, 7, T )  = (5) 3 Correlators at Zero Temperature In this section we discuss the comparison of the model calculations with zero temper- ature lattice data from isotropic lattices [4]. The lattice spacing has been fixed using the Sommer-scale TO = 0.5fm. Its value is slightly larger than the one used in Ref. [4], since there the string tension of fi = 420 MeV has been used to set the scale. Cal- culations have been done at the charm quark mass which corresponds to an qc mass of about 4 GeV. The renormalization constants of the lattice operators have been calculated in 1-loop tadpole improved perturbation theory (see Ref. [4] for further details). The K factors in Eq. (2) have been chosen such that at large distances the correlators calculated in potential models agree with the lattice results. In Fig. 1 we show the pseudo-scalar correlator calculated on the lattice and in the potential model for several screening param- eters, together with the corresponding spectral functions. As one can see from the figure the choice of the ad-hoc screening parameters have almost no effect on the Euclidean 3 correlator. We see a reasonably good agreement between the lattice data and potential model calculations. Also shown in the figure is the correlator corresponding only to the non-relativistic spectral function. At small Euclidean times this falls below the lattice data by more than an order of magnitude. Thus correlators calculated on the lattice are sensitive to the relativistic continuum part of the spectral functions. Similar analysis of the correlators have been done in the vector, axial vector and scalar channels. 0.2 10.' 4 6 8 10 10.2 w [Gev] 0.1 0.2 0.3 0.4 0.5 T [fml Figure 1: The pseudo-scalar charmonium correlator calculated in our model and compared to the lattice data of Ref. [4]. In the inset, the corresponding spectral functions c (w) /w2  are shown. 4 Temperature-dependence of Quarkonium Correla- tors In this section we study the temperaturedependence of quarkonium spectral functions and correlators. Since the correlators depend on the temperature through the integration kernel and the spectral functions, it is customary to study the temperature dependence of the correlators in terms of the ratio G(T,T)/G,,,(T,T), where co Grec(~ ,  T) = d w ~ ( w , T  = O)K(w, T ,T )  . (6) This way the trivial temperature dependence due to the integration kernel is taken care of. Also many uncertainties of the lattice calculations cancel out in this ratio. The finite 4 T 0 1.2T, 1.5Tc 2.0TC SO 10.975 9.541 9.462 9.384 M(1S) 9.405 9.390 9.374 9.343 Ehnd 1.570 0.151 0.088 0.041 Table 1: The mass and the binding energy of the 1s bottomonium state at different temperatures, and the continuum threshold. temperature spectral functions are shown in Fig. 2 for the pseudo-scalar channel. The 1s charmonium state is melted at  1.2TC. We see, however, a large enhancement of the spectral functions near the threshold. Note, that the height of the spectral functions near the threshold is comparable to the height of the bump in the spectral function calculated , from MEM [6]. It is therefore possible that the bump of the spectral functions calculated from lattice correlators using MEM actually corresponds to a threshold enhancement, and was mistakenly interpreted as the 1s state. In the case of bottomonium all states, except the 1s state, are dissolved above the deconfinement transition. The 1s state can survive as a resonance until temperatures of about 227,. Note, however, that the binding energy of the 1s bottomonium is significantly reduced due to  color screening, as shown in Table 1. The binding energy is defined as the distance between the continuum threshold so and the bound state peak. Due to the small binding energy the 1s state will acquire a sizable thermal width, and may not show up as the resonance in the corresponding spectral function. Therefore, the actual dissolution temperature of the 1s bottomonium will be smaller then the one estimated based on the simple potential model calculations which do not include the effect of the thermal width [28]. In the insets of Fig. 2 we also show the corresponding ratio G/G,,,. The large changes in the spectral functions are not visible in the correlation functions. These agree quite well with the lattice results. This is because even in the absence of bound states the spectral function is significantly larger than the spectral function corresponding to a freely propagating quark anti-quark pair. In the vector, scalar and axial-vector channels significant temperature dependence has been found 14, 6, 291. It has been shown that this is due to the zero mode contribution, Le. due to the wb(w)-like contribution to the quarkonium spectral functions [30]. In the vector channel the zero mode contribution corresponds to the heavy quark transport [31]. The zero mode contribution can be estimated in the free case. It can also be shown that the zero mode contribution is absent in the pseudo-scalar channel. If we add the free theory result for the zero mode contribution to the spectral function calculated in the potential model we can reproduce the temperature dependence of the scalar and axial- vector correlator both for charmonium and bottomonium. This is demonstrated by Fig. 3 where G/G,,, for the scalar and axial-vector channels is shown. 5 fie@ . . . . . . 2 3 4 5 6 7 8 9  t 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 T=O - free .._. 9 10 11 12 13 (D lGeVl Figure 2: The charmonium (left) and bottomonium (right) spectral functions at  different temperatures. For charmonium we also show the spectral functions from lattice QCD ob- tained from the MEM at  1.5'7,. The error-bars on the lattice spectral function correspond to the statistical error of the spectral function integrated in the w-interval corresponding to the horizontal error-bars. The insets show the corresponding ratio G/G,,, together with the results from anisotropic lattice calculations [6]. For charmonium, lattice calcu- lations of G/Grec are shown for T = 1.2'7, (squares), 1.5TC (circles), and 2.0'7, (triangles). For bottomonium lattice data are shown for T = 1.5'7, (circles) and 1.8'7, (triangles). 5 Conclusions We discussed the calculations of quarkonium spectral functions and the corresponding Euclidean time correlators in a potential model. We have found that all quarkonium states, 'except the 1s bottomonium state, dissolve at  temperatures smaller than about 1.2'7,. This, however, does not lead to significant change of the correlators. Zero mode contribution on the other hand could, give a large change in the correlators above the deconfinement transition. We have found that the spectral functions calculated in our model can explain quite well the temperature dependence of the quarkonium correlators obtained in lattice QCD. Acknowledgments This work has been supported by U S .  Department of Energy under Contract No. D E  AC02-98CH10886. 6 3 lattice, scalar I.-., lattice axial-veCLor ," * '' scalar - aXia,."eCtor .............. e..." lanice, scalar I-. lanice axial-vector .............. 1 1 [irn 0.05 0.1 0.15 0.2 0.05 . 0.1 0.15 0.2 Figure 3: The ratio G/G,,, in the scalar and axial-vector channel a t  T = 1.5Tc for charmonium (left) and bottomonium (right). Lattice calculation on isotropic lattices [4, 291 are shown as filled symbols. Open symbols refer to results from anisotropic lattice calculations of Ref. [ti]. References [l] T. Matsui and H. Satz, Phys. Lett. B 178 (1986) 416 [2] T. Umeda, K. Nomura and H. Matsufuru, Eur. Phys. J. C 39S1 (2005) 9 [3] M. Asakawa and T. Hatsuda, Phys. Rev. Lett. 92 (2004) 012001 [4] S. Datta, F. Karsch, P. Petreczky and I. Wetzorke, Phys. Rev. D 69 (2004)'094507 [5] G. Aarts, C. R. Allton, R. Morrin, A. P. 0. Cais, M. B. Oktay, M. J. Pear- don and J. I. Skullerud, arXiv:heplat/0610065; Nucl. Phys. A 785 (2007) 198; [arXiv:hep-lat/0608009]; arXiv:0705.2198 [hep-lat] [ti] A. JakovAc, P. Petreczky, K. Petrov and A. Velytsky, Phys. Rev. D 75 (2007) 014506 171 F. Karsch, et al., 2. Phys. C37 (1988) 617 [8] G. Ropke, D. Blaschke, H. Schulz, Phys. Rev. D 38 (1988) 3589 191 T. Hashimoto et al., Z. Phys. C 38 (1988) 251 [lo] S. Digal, et al., Phys. Lett. B 514 (2001) 57; Phys. Rev. D 64 (2001) 094015 [ll] C. Y. Wong, Phys. Rev. C 72 (2006) 034906 [12] A. M6csy and P. Petreczky, Eur. Phys. J .  C43 (2005) 77 7 [13] A. M6csy and P. Petreczky, Phys. Rev. D 73 (2006) 074007 [14] D. Cabrera, R. Rapp, arXiv:hep-ph/0610254 [15] W. M. Alberico, A. Beraudo, A. De Pace and A. Molinari, Phys. Rev. D 75 (2007) 074009 [16] C. Y. Wong and H. W. Crater, Phys Rev. D 75 (2007) 034505 [I71 0. Kaczmarek, et al., Phys. Lett. B 543 (2002) 41; Phys. Rev. D 70 (2004) 074505 [18] P. Petreczky, K. Petrov, Phys. Rev. D 70 (2004) 054503 [19] 0. Kaczmarek, F. Zantow, Phys. Rev. D 71 (2005) 114510 [20] K. Petrov [RBC-Bielefeld Coll.], arXiv:hep-lat/0610041. [21] 0. Kaczmarek, PoS CPOD07 (2007) 043 [22] M. Laine, 0. Philipsen, P. Romatschke and M. Tassler, JHEP 0703 (2007) 054 [23] M. Laine, arXiv:O704.1720 [24] F. Karsch, A. Patk6s and P. Petreczky, Phys. Lett B 401 (1997) 69 [25] J. 0. Andersen, E. Braaten and M. Strickland, Phys. Rev. Lett. 83 (1999) 2139 1261 J. P. Blaizot, E. Iancu and A. Rebhan, Phys. Rev. Lett. 83 (1999) 2906 [27] A. M6csy and P. Petreczky, arXiv:0705.2559 [hep-ph]. [28] A. M6csy and P. Petreczky, arXiv:0706.2183 [hepph], to be published in Phys. Rev. Lett. [29] S. Datta, A. JakovAc, F. Karsch and P. Petreczky, AIP Conf. PTOC. 842 (2006) 35 [30] T. Umeda, Phys. Rev. D 75 (2007) 094502; PoS (LATTICE 2007) 233 [31] P. Petreczky and D. Teaney, Phys. Rev. D 73 (2006) 014508 [32] W.M. Alberico et. al, arXiv:07062846 8 

Quarkonium correlators at finite temperature and potential models

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