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Heavy quark potential in lattice QCD at finite temperature

ITEP-LAT/2002–31 KANAZAWA 02–40

HEAVY QUARK POTENTIAL IN LATTICE QCD AT FINITE TEMPERATURE

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arXiv:hep-lat/0301002v1 5 Jan 2003

V. BORNYAKOV a,b,c , M. CHERNODUB a,b , Y. KOMA a , Y. MORI a , Y. NAKAMURA a , M. POLIKARPOV b, G. SCHIERHOLZ d , D. SIGAEV b , ¨ A. SLAVNOV e , H. STUBEN f , T. SUZUKI a , P. UVAROV b,e , A. VESELOV b ITP, Kanazawa University, Kanazawa, 920-1192, Japan ITEP, B. Cheremushkinskaya 25, Moscow, 117259, Russia c IHEP, Protvino, 124280, Russia d NIC/DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany and Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany e Steklov Mathematical Institute, Vavilova 42, 117333 Moscow, Russia f ZIB, D-14195 Berlin, Germany
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Abstract Results of the study of lattice QCD with two ?avors of nonperturbatively improved Wilson fermions at ?nite temperature are presented. The transition temperature for mπ ? 0.8 and lattice spacing a ? 0.12 fm is mρ determined. A two-exponent ansatz is successfully applied to describe the heavy quark potential in the con?nement phase.

Studies of Nf = 2 lattice QCD at ?nite temperature with improved actions have provided consistent estimates of Tc [1, 2]. Still there are many sources of systematic uncertainties and new computations of Tc with di?erent actions are useful as an additional check. To make such check we performed ?rst large scale simulations of the nonperturbatively O(a) improved Wilson fermion action at ?nite temperature. Other goals of our work were to study the heavy quark potential and the vacuum structure of the full QCD at T > 0. We employ Wilson gauge ?eld action and fermionic action of the same form as used by UKQCD and QCDSF collaborations [3] in T = 0 studies. To ?x the physical scale and mπ ratio we use their results. Our simulations were performed mρ on 163 8 lattices for two values of the lattice gauge coupling β = 5.2, 5.25.
given by V. Bornyakov at “Con?nement V”, Gargano, Italy, 10-14 Sep. 2002. work is partially supported by grants INTAS-00-00111, RFBR 02-02-17308, 01-0217456, 00-15-96-786 and CRDF RPI-2364-MO-02. M.Ch. is supported by JSPS Fellowship No. P01023. P.U. is supported by Kanazawa foundation.
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As numerical results show [1] both Polyakov loop and chiral condensate susceptibilities can be used to locate the transition point. We use only Polyakov loop susceptibility. We found critical temperature Tc = 213(10) and 222(10)MeV at mπ /mρ = 0.78, 0.82, respectively. These values are in good agreement with previous results [1] at comparable mπ /mρ . To test ?nite size e?ects simulations on 243 · 8 lattice for T /Tc = 0.94 have been made. We found that results for all our observables agree with our smaller volume results within error bars. Thus ?nite size e?ects do not introduce strong systematic uncertainties in our results. The heavy quark potential V (r, T ) in full QCD at non-zero temperature has ? been studied in [1]. It is given by Lx Ly /9 =e?V (r,T )/T , where Lx is Polyakov
? loop. In the limit |x ? y| → ∞, Lx Ly approaches the cluster value | L |2 , where | L |2 = 0 because the global Z3 symmetry is broken by the fermions. The spectral representation for the Polyakov loop correlator is [4] ∞ ? Lx Ly = n=0

wn e?En (r)/T .

At T = 0 one gets V (r, T = 0) = E0 (r). In contrast, V (r, T ) at T > 0 gets contributions from all possible states. We assume that in the con?nement phase, at temperatures below Tc , the Polyakov loop correlator can be described with the help of two states, namely string state and broken string (two static-light meson) state: 1 ? Lx Ly = e?(V0 +Vstr (r,T ))/T + e?2E(T )/T , (1) 9 1 π xT 1 T Vstr (r, T ) = arctanx ? + σ(T )r + arctan + ln 1 + x2 , (2) 6r 12r 3 x 2 E(T ) = V0 /2 + m(T ) , (3) where m(T ) is the e?ective quark mass at ?nite temperature, x = 2rT . The T = 0 string potential (2) was derived in [5]. The alternative ?t of our data can be done using the ?nite temperature QCD static potential [6]: VKMS (r, T ) = α σ ? (1 ? e??r ) ? e??r , ? r (4)

where σ , ? and α are parameters. We used function (4) to ?t the data. ? In computation of the Polyakov loops correlator to reduce statistical errors hypercubic blocking [7] has been employed. Details of this computation were reported in [8]. Parameters of the ?t (1)-(3) are presented in Fig. 1. The values for the ratio σ(T )/σ(0) are higher than those obtained in quenched QCD [9], especially close to Tc . The values for m(T ) are also 20-30 % higher than those obtained in [10]. Using parameters of the potential we calculate the string breaking distance rsb from relation Vstr (rsb , T ) = 2m(T ). In Fig. 1 one can see that rsb decreases down to values ? 0.3 fm when temperature approaches critical 2

1

σ(T)/σ(0)

1.25

m(T)r0
0.5 β=5.2 β=5.25 0 0.85 0.9 0.95 T/Tc 0.75

rbr/r0

2 1.5 1

0.25 0.85

β=5.2 β=5.25 0.9 0.95 T/Tc

β=5.2 β=5.25 0.85 0.9 0.95 T/Tc

0.5 0

Figure 1: Best ?t parameters for ?t eq.(1) as functions of temperature. Solid line on the left-hand ?gure show quenched results9 . Dashed horizontal lines show T = 0 results. r0 = 0.5 fm. value. Our ?t using Vstr (r, T ), eq.(2), is probably not valid when rsb becomes so small. It still provides reasonable values for string tension and e?ective mass for T /Tc < 0.95 when rsb > 0.5f m. The comparison of two ?ts, eqs.(1–3) and eq.(4) showed that both ?ts are equally good within our error bars. There is an indication that with more precise data one can discriminate between these two ?ts at low temperatures. We found also that parameters of the ?t eq.(4) are in a clear disagreement with parameters suggested in [11].

References
[1] F. Karsch, E. Laermann and A. Peikert, Nucl. Phys. B 605 (2001) 579. [2] A. Ali Khan et al., Phys. Rev. D 63 (2001) 034502. [3] S. Booth et al., Phys. Lett. B 519 (2001) 229. [4] M. L¨ scher, P. Weisz, JHEP 0207 (2002) 049. u [5] M. Gao, Phys. Rev. D 40 (1989) 2708. [6] F. Karsch, M. T. Mehr and H. Satz, Z. Phys. C 37 (1988) 617. [7] A. Hasenfratz and F. Knechtli, Phys. Rev. D 64 (2001) 034504. [8] V. Bornyakov et al., arXiv:hep-lat/0209157. [9] O. Kaczmarek et al., Phys. Rev. D 62 (2000) 034021. [10] S. Digal, P. Petreczky and H. Satz, Phys. Lett. B 514 (2001) 57. [11] C. Y. Wong, Phys. Rev. C 65 (2002) 034902 [arXiv:nucl-th/0110004].

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