BNL- 73704 -2005-CP Heavy Quark Potentials and Quarkonia Binding Peter Petreczky Presented at Hard Probes 2004 Ericeira, Portugal November 4-1 0,2004 Physics Department Nuclear Theory Group Brookhaven National Laboratory P.O. Box 5000 Upton, NY 1 1973-5000 www.bnI.gov Managed by ' Brookhaven Science Associates, LLC for the United States Department of Energy under Contract No. DE-AC02-98CH10886 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. - EPJ manuscript No. (will be inserted by the editor) Heavy Quark Potentials and Quarkonia Binding PBter Petreczky Nuclear Theory Group, Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-500, USA Abstract. I review recent progress in studying in-medium modification of inter-quark forces at finite tem- perature in lattice QCD. Some applications to the problem of quarkonium binding in potential models is also discussed. PACS. 1 1.15.Ha1ll.lO.Wx, 12.38.Mh, 25.75.Nq 1 Introduction is the temporal Wilson line. L(z)= TrW(z) is also known as Polyakov loop and in the case of pure gauge theory it The study of in-medium modifications of inter-quark forces is an order parameter of the deconfinement transition. As at high temperatures is important for detailed theoretical the QQ pair can be either in color singlet or octet state understanding of the properties of Quark Gluon Plasma one should separate these irreducible contributions to the as well to detect its formation in heavy ion collisions. In partition function. This can be done using the projection particular, it was suggested by Matsui and Satz that color operators PI and P8 onto color singlet and octet states screening at high temperature will result in dissolution introduced in Refs. [7,8]. Applying PI and p8 to ZQQ(T,T) of quarkonium state and the corresponding quarkonium we get the following expression for the singlet and octet suppression could be a signal of Quark Gluon Plasma for- free energies of the static QQ pair mation [I]. Usually the problem of in-medium modification of inter- quark forces is studied in terms of so-called finite tem- $erature heavy quark potentials, which are, in fact, the --3 'Tr(W(T)W+(O)) (3) differences in the free energies of the system with static quark anti-quark pair and the same system without static charges. Alternatively this problem can be studied in terms of finite temperature quarkonium spectral functions [2, 3,4] which were also discussed during this conference by 1 1 =-(TrW(r)TrWt(O)) - -Tr(W(r)Wt(O)). (4) Karsch, Hatsuda and Petrov 151, Recently substantial prog- 8 24 ress has been made in studying the free energy of static Although usually FI,~is referred to as the free energy quark anti-quark pair which I am going to review in the of the static QQ pair, it is important to keep in mind present paper. An important question is what can we learn that it refers to the difference between the free energy of about the quarkonium properties from the free energy of the system with static quark anti-quark pair and the free static charges which will be discussed at the end of the energy of the system without static charges. paper. As W(z)is a not gauge invariant operator we have to fix a gauge in order to define F1 and Fs. As we want that F1 and F8 have a meaningful zero temperature limit we 2 The free energy of static charges better to iix the Coulomb gauge because in this gauge a transfer matrix can be defined and the free energy differ- Following McLerran and Svetitsky the partition function ence can be related to the interaction energy of a static of the system with static quark anti-quark (QQ) pair at QQ pair at zero temperature (T = 0). Another possibil- finite temperature T can be written as ity discussed in Ref. [9] is to replace the Wilson line by a gauge invariant Wilson line using the eigenvector of the zQa(r, T, = (W(r)Wt(0))Z(T), (1) spatial covariant Laplacian [9]. For the singlet free energy with Z(T),being the partition function of the system with- both methods were tested and they were shown to give out static charges and numerically indistinguishable results, which in the zero temperature limit are the same as the canonical results ob- 1/T tained from Wilson loops. The interpretation of the color W(o)= Pexp(igh d.rAo(.r,z)) (2) octet free energy at small temperatures is less obvious and 2 PBter Petreczky: Heavy Quark Potentials and Quarkonia Binding 1.1 s ' QuknchidQCO 2 1 N,=3 QCD I--+-- 1.5 1 0.5 0 -0.5 0.3l " " " " I 0 0.5 1 1.5 2 2.5 3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Fig. 2. The effective screening radius versus T/Tc[14] . 1.5 T [MeV] 138 -E- 150 -*- la rc 3 Free energy of a static QQ pair and 176 -*--I 1 190 -+- 205 - screening of inter-quark forces at high P temperatures 2 0.5 Perturbatively the quark anti-quark potential can be re- 0 lated to the scattering amplitude corresponding to one gluon exchange and in the non-relativistic limit it is given -0.5 f. , r[pl . 0.2 0.4 0.6 0.8 1 1.2 1.4 bY Fig. 1. The color singlet free energy in quenched [10,23] (top) and three flavor [14] (bottom) QCD. The solid black line is the T = 0 singlet potential. Here .&o(rC) is the temporal part of the Coulomb gauge gluon propagator and in general it has the form will be discussed separately. One can also define the color &o(k) = (k2+ 1700(k))-1. averaged free energy Furthermore the averaging over color gives (T"Tb)= -4/3 for the color singlet and (TaTb)= +1/6 for the color octet case. At zero temperature the polarization operator IT00 gives rise only to running of the coupling constant g = g(r) =; (TrW(T)TrW+(0)), (5) (recall that a, = g2/(4n)).But at finite temperature T it has a non-trivial infrared limit I70o(Ic 4 0) = m; = which is expressed entirely in terms of gauge invariant gT.. Therefore at distances r >> 1/T the Polyakov loops. This is the reason why it was extensively potential has the form I studied on lattice during the last two decades. The color b g2 averaged free energy is a thermal average over the free V(r,T)= (TaT )- exp(-mor). (10) energies in color singlet and color octet states 4nr The singlet and octet free energies defined in the pre- eXP(-&v(r, T)/T)= vious section can also be easily calculated in leading order perturbation theory. Again because of lIOo(k: = 0) = rng exp(-4 (r,T)/T) exp( -F8 (r,T)/T). 9 + 6 (6) one has Therefore it gives less direct information about medium 4 1 g2 modification of inter-quark forces. Given the partition func- Fl,s(r,T)= (--, -)-exp(-mor). (11) ' 3 6 4x7- tion ZQ,(r,T) we can calculate not only the free energy At leading order the singlet free energy has exactly the but also the entropy as well as the internal energy of the same form as the potential and has no entropy contribu- static charges tion. This is the reason why the free energies of static QQ pair were (mis)interpreted as potentials. At next to lead- ing order which is 0(g3)the free energies have the form 4 1 g2 g2mD Fl,s(r,T)= (--,-)-exp(-mDr) - -3n 3 6 47rr ' (12) and the entropy contribution -TS appears (recall Eq. 7). = Fi(r,T) + TSi(r,T), (8) For the singlet case the entropy has the form i = 1,8,ab. n g"mD Sl(r,T)= -(l - exp(--mDr)). 3nT (13) P&er Petreczky: Heavy Quark Potentials and Quarkonia Binding 3 2 1.5 12 - 124T -A- 1:WJT: 1 . 221T - 11 1’48T’ - 2:95Tz - 10 . * - 5.97-r’ c* . 9- 7- 0.2 0.4 0.6 0.8 1 1.2 Fig. 3. The color octet free energy in quenched QCD [23]. The Fig. 4. The ratio of color singlet and color octet free energies solid black line is the zero temperature singlet potential. ~31. It has the asymptotic value of at large distances and below the transition temperature T, N 270MeV the free vanishes for rT << 1. Similarly one can calculate the color energy rises linearly with the distance T signaling confine- octet entropy to be ment. Above deconfinement T > T, the free energy has a finite value at infinite separation indicating screening.