Static potentials for quarkonia at finite temperatures
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Static potentials for quarkonia at finite temperatures
Owe Philipsen
Institut f¨ur Theoretische Physik, Universit¨at M¨unster, 48149 M¨unster, Germany
Abstract
We review non-perturbative static potentials commonly used in potential models for quarkoniaat finite T . Potentials derived from Polyakov loop correlators are shown to be inappropriate forthis purpose. The q ¯ q free energy is physical but has the wrong spatial decay and perturbativelimit. The so-called singlet free energy is gauge dependent and unphysical. An appropriatestatic real time potential can be defined through a generalisation of pNRQCD to finite T . Inperturbation theory, its real part reproduces the Debye-screened potential, its imaginary partaccounts for Landau damping. Possibilities for its non-perturbative evaluation are discussed. Key words:
Thermal field theory, Lattice gauge theory, Quark gluon plasma, Quarkonia
PACS:
1. Introduction
The properties of quarkonia are believed to provide a useful probe of the QCD plasmaat high temperatures, in particular for the quark-hadron transition. This expectation wasoriginally based on a potential model [1], in which the linearly confining potential forzero temperature gets replaced by a Debye-screened potential at high T .Potential models have a long history for the description of quarkonia at zero tempera-ture. The basic idea is that for heavy quarks of mass M , which move non-relativistically,the binding energy ( E − M ) is small compared to M and can be obtained by solving astatic Schr¨odinger equation (cid:18) ∇ M + V ( r ) (cid:19) ψ = ( E − M ) ψ. (1) V ( r ) is the (radially symmetric) potential between the static quark anti-quark pair sep-arated by a distance r . Initially V ( r ) was modelled by the Cornell potential (Coulombplus linear), more recently non-perturbative lattice data are used as input. The crucialobservation is that the Schr¨odinger equation follows from an effective theory approach. Preprint submitted to Elsevier 6 November 2018 tarting from QCD, one can use of the scale separation between the heavy quark mass M and the binding energy E − M , to obtain an effective theory, pNRQCD [2], forthe low energy dynamics in the confining potential. In this framework, the static po-tential appears as a perturbative matching coefficient of the effective theory. Hence, theSchr¨odinger equation can be improved systematically by computing higher order termsin the scale hierarchy. Note, that a very successful spectroscopy with ∼
1% accuracy isobtained in this way.It is tempting to employ this approach also at finite T . Matsui and Satz heuristicallyused the same equation, but with a Debye-screened potential from perturbation theory, V ( r, T ) ≈ − g C F π e − m D ( T ) r r . (2)However, there are a number of problems. Firstly, it is not clear if the bound stateSchr¨odinger equation can be translated to a finite T many body situation, in a waythat temperature effects show up only in the potential. Secondly, at finite T there existsa variety of non-perturbative potentials, and it is not clear which one constitutes thenon-perturbative generalisation of Eq. (2).
2. Static potentials from the lattice at zero and finite T At T = 0, the static potential can be defined non-perturbatively on a euclidean L × N τ space time lattice. Consider a meson correlation function with an interpolating operator¯ ψ ( x ) U ( x , y ) ψ ( y ), where U denotes a straight line gauge string between the quarks. Inthe limit M → ∞ the heavy quarks can be integrated out, taking the correlator to theeuclidean Wilson loop, h ¯ ψ ( x , τ ) U ( x , y ; τ ) ψ ( y , τ ) ¯ ψ ( x , U ( x , y ; 0) ψ ( y , i −→ e − Mτ W E ( | x − y | ) . (3)Inserting a complete set of eigenstates of the Kogut-Susskind Hamiltonian (in temporalgauge), the Wilson loop evaluates to ( r = | x − y | , U r ≡ U ( x , y ; 0)) W E ( r, τ ) = 1 Z X n,m |h n | U r | m i| e − E n N τ e − ( E m ( r ) − E n ) τ (4) N τ →∞ −→ X m |h | U r | m i| e − ( E m ( r ) − E ) τ τ →∞ −→ |h | U r | i| e − ( E ( r ) − E ) τ , (5)with E m ( r ) eigenvalues in the sector with sources, and E n in the sector without. Onthe lattice, T = 1 / ( aN τ ), hence T = 0 implies N τ → ∞ in the second line. Takingfurthermore the limit τ → ∞ , the sum is dominated by the lowest energy state. The staticpotential is defined to be the lowest energy of the static quark anti-quark configurationat a given separation, V ( r ) ≡ E ( r ) − E . Note that the matrix element with the stringoperator is of no interest here.The generalisation to finite T is difficult to interpret because of the finite and shorttemporal extent, N τ = 1 / ( aT ). Thus, we have to deal with the full superposition Eq. (4),to which now also the matrix elements contribute, and the result still depends on τ .A different definition of the static potential which does generalise to finite T is basedon the Polyakov loop L ( x ) = Q τ =1 ,N τ U ( x , τ ), i.e. a static quark sitting at x and2 V(R,T)/T R T β R TV(R)/T β -200 0 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2F i (r,T) [MeV] r [fm]F qq T/T c =0.91F T/T c =0.91F T/T c =0.91F qq T/T c =1.24F T/T c =1.24F T/T c =1.24 Fig. 1. Static quark anti-quark free energy/potential, Eq. (6), for
T < T c (left) and T > T c (middle) [5].Right: Free energies for the three channels Eq. (9). The solid line is the zero temperature potential [8]. propagating in euclidean time through the periodic boundary. It transforms in the adjoint,so its trace is gauge invariant. By spectral analysis one establishes that the Polyakov loopcorrelator represents the free energy of a static quark anti-quark pair separated by r [3],e − F ¯ qq ( r,T ) /T = 1 N c h Tr L † ( x ) Tr L ( y ) i = 1 ZN c X n e − E n ( r ) /T . (6)The energy levels entering this Boltzmann sum are identically the same as the E m ( r )from the Wilson loop, Eq. (4). Hence, for T → V ( r ), cf. [4]. The free energyis thus often called a T -dependent potential, V ( r, T ) ≡ F ¯ qq ( r, T ). The Polyakov loopcorrelator is readily simulated, with results as in Fig. 1. It gives a linear potential in theconfined phase, whose string tension reduces with temperature, while in the deconfinedphase the potential is screened, Fig. 1. Unfortunately, this is not the Debye-screenedpotential we want, as becomes apparent when considering its spatial decay at high T .Fitting to F q ¯ q T = − c ( T )( rT ) d e − m ( T ) r , (7)gives d ≈ . m = M +++ , i.e. the screening mass corresponds to the lightest, gauge-invariant glueball channel [6]. This can already be seen in perturbation theory, where theleading term is by two-gluon exchange and thus m = 2 m D [7].It was thus suggested to decompose the Polyakov loop correlator into channels withrelative colour singlet and octet orientations of the quark anti-quark pair [3], e − F q ¯ q ( r,T ) /T = 19 e − F ( r,T ) /T + 89 e − F ( r,T ) /T , (8)e − F ( r,T ) /T = 13 h Tr L † ( x ) L ( y ) i , e − F ( r,T ) /T = 18 h Tr L † ( x ) Tr L ( y ) i − h Tr L † ( x ) L ( y ) i . (9)Note that the correlators in the singlet and octet channels are gauge dependent, and thecolour decomposition only holds perturbatively in a fixed gauge. However, in perturba-tion theory the singlet channel indeed displays the expected Debye-screened behaviour, F ( T, r ) ∼ e − m D ( T ) r / πr . This has motivated lattice simulations of these correlatorsin fixed Coulomb gauge, with results as in Fig. 1 (right). The three different channels3how different r -dependence, and hence lead to different binding energies when used inSchr¨odinger equations. There is a vast literature employing F or the corresponding in-ternal energy U = F + T S , and from the solutions trying to reconstruct lattice mesoncorrelation functions to check which fits better [9].However, both options are unphysical at a non-perturbative level. To understand this,let us start from something physical and consider a meson operator in an octet state, O a = ¯ ψ ( x ) U ( x , x ) T a U ( x , y ) ψ ( y ), with x the meson’s center of mass. In the plasmathe colour charge can always be neutralised by a gluon. In the correlators for the singletand octet operators, we integrate out the heavy quarks, replacing them by Wilson lines, h O ( x , y ; 0) O † ( x , y ; N τ ) i ∝ h Tr L † ( x ) U ( x , y ; 0) L ( y ) U † ( x , y ; N τ ) i , h O a ( x , y ; 0) O a † ( x , y ; N τ ) i ∝ (cid:20) N c − h Tr L † ( x ) Tr L ( y ) i (10) − N c ( N c − h Tr L † ( x ) U ( x , y ; 0) L ( y ) U † ( x , y ; N τ ) i (cid:21) . We have now arrived at gauge invariant expressions, because we used a gauge stringbetween the sources. The singlet correlator corresponds to a periodic Wilson loop whichwraps around the boundary. The connection to the gauge fixed correlators is readilyestablished, replacing the gauge string by gauge fixing functions, U ( x , y ) = g − ( x ) g ( y ).Thus, in axial gauge U ( x , y ) = 1 (and only there) the gauge fixed correlators are identicalto the gauge invariant ones.Next, let us perform the spectral analysis. While indeed the energy eigenvalues in thespectral sum are independent of the operators [10], the full correlators take the form [11]e − F ( r,T ) /T = 1 ZN c X n h n δγ | U γδ ( x , y ) U † αβ ( x , y ) | n βα i e − E n ( r ) /T , e − F ( r,T ) /T = 1 ZN c X n h n δγ | U aγδ ( x , y ) U † aαβ ( x , y ) | n βα i e − E n ( r ) /T . (11)The energy levels in the exponents are identically the same in Eqs. (6,11) and correspondto the familiar gauge invariant static potential at zero temperature and its excitations.However, while Eq. (6) is purely a sum of exponentials and thus a true free energy, thesinglet and octet correlators contain matrix elements which do depend on the operatorsused, thus giving a path/gauge dependent weight to the exponentials contributing to F , F . This is illustrated numerically in Fig. 2 in the low temperature limit, where theground state potential dominates and one can cleanly separate the exponential and thematrix elements. The r -dependent structure is entirely in the matrix elements, whichdepend on operators and/or the gauge.I do not see how this is evaded by applying smearing techniques, as recently suggestedin [12]. These authors replace the spatial string swith a smeared object in order toincrease the overlap with the ground state, i.e. to get the ground state matrix elementclose to one. However, most smearing changes the expectation values of correlators, thusdestroying their mutual relations, Eq. (9). Secondly, smearing increases the weight ofthe lower energy states at the cost of the higher ones, and thus undoes the effect offinite temperature in a procedure dependent way. Finally, if one could get all matrix4 N t "average"singlet"adjoint" r/a c "average"singlet"adjoint" Fig. 2. Left: Polyakov loop correlators, Eq. (9), for r/a = 1 in the case of 3d SU(2) in the low temperaturelimit. All decay with the same ground state exponential. Right: The corresponding matrix elements inthe three channels introduces operator dependent r -dependence, except for the average channel [11]. elements equal to one, the different channels would simply be equal, up to the trivialcolour coefficients, with no additional information.To summarise, since the spectral information contained in the average and gauge fixedsinglet and octet channels is the same, we must conclude that any difference betweenthose correlators is entirely gauge dependent and thus unphysical, and so are all bindingenergies calculated in F or U .
3. A real time static potential for finite T quarkonia
Progress was made recently by generalising the effective theory approach quarkoniumphysics at T = 0, namely pNRQCD, to finite temperatures [13,14,15], as reviewed at thisconference [16]. The analysis is performed in a perturbative setting in Minkowski time.Just as at zero temperature, the static potential then appears as a matching coefficientin the effective theory after the heavy modes have been integrated out. The relevantcorrelation function is the quarkonium correlator in real time, but evaluated as a thermalexpectation value. Not surprisingly, after integrating out the static quarks, the correlatoris proportional to a Wilson loop in Minkowski time, W E ( it, r ). Of course, the expectationvalue implied in W E is now a thermal one, i.e. N τ is finite for fixed lattice spacing. Hencewe need the analytic continuation of the double spectral sum in Eq. (4). From the effectivetheory it is easy to see that this correlator obeys a real time evolution equation[ i∂ t − V > ( t, r )] W E ( it, r ) = 0 . (12)This represents the desired Schr¨odinger equation for quarkonia in the plasma, and definesthe relevant real time dependent potential. The required scale hierarchy for this equationto be valid is g M < T < gM . Furthermore, for non-relativistic bound states p ≪ E ,hence we need t ≫ r , i.e. the static pontential is obtained in the long time limit V ( ∞ , r ).Eq. (12) may be also be viewed as a non-perturbative definition of the potential of in-terest via a correlation function, just as was the case for the zero temperature potential.Unfortunately, this one is defined for Minkowski time and thus requires analytic contin-uation, i.e. it cannot be evaluated directly from euclidean lattice simulations. However,a first impression about this object can be gained form HTL-resummed perturbationtheory, for which the leading order result is5 > ( ∞ , r ) = − g C F π (cid:20) m D + exp( − m D r ) r (cid:21) − ig T C F π φ ( m D r ) , with φ ( x ) = 2 ∞ Z d z z ( z + 1) (cid:20) − sin( zx ) zx (cid:21) . (13)The most striking feature of this potential is that it is complex, contrary to the freeenergies discussed before. The real part features the expected Debye-screened potential.The imaginary part is due to Landau damping and must necessarily be there for a correcteffective description of the plasma dynamics. Its derivation and properties are discussedin more detail in [13,14,16].What are the corrections to this potential? Firstly, there are corrections from HTL-resummed perturbation theory of the order g T /
Λ, where Λ is the UV cut-off, with acalculable coefficient. Here, we are interested in the non-perturbative corrections frominfrared modes ∼ g T /m mag . These are due to the soft colour magnetic modes m mag ∼ g T , and thus cannot be calculated in perturbation theory.However, one can calculate these non-perturbative corrections by classical lattice sim-ulations in that sector of the theory, which has high occupation numbers and is wellrepresented by a classical approximation. To identify this sector it is instructive to takethe limit ~ → ~ needs to bereinstated by the replacements g → g ~ , 1 /T → ~ /T , leading tolim ~ → V > ( ∞ , r ) = − ig T C F π φ ( m D r ) . (14)Thus, only the imaginary part survives in the continuum limit. This is easy to understandsince the long range physics of Landau damping is dominated by classical fields, e.g. inscalar field dynamics, whereas the binding is a generic quantum effect, cf. the hydrogenproblem. Thus, we can evaluate the non-perturbative infrared effects for the imaginarypart of the potential.
4. Imaginary part from classical lattice simulations
This has been done in [17], following the technical setup that was also used for theevaluation of the sphaleron rate in the electroweak theory [18]. In order to performreal time simulations one reformulates the theory in a Hamiltonian approach. Fixingtemporal gauge U = 0, the conjugate field operators are the links and the electric fieldsdefined by ˙ U ( x , t ) = iE i ( x , t ) U i ( x , t ). Full gauge invariance is restored by imposing theGauss constraint G ( x ) ≡ P i h E i ( x ) − U − i ( x ) E i ( x − ˆ i ) U †− i ( x ) i − j ( x ) = 0. A thermaldistribution at some initial time is generated by the partition function Z = Z D U i D E i δ ( G ) e − βH , H = 1 N c X x (cid:20)X i 5. Non-perturbative real time potential? It is clearly desirable to go beyond the classical approximation and construct an opera-tor from which the whole quantum potential, including its real part, can be extracted. Ofcourse, a full quantum computation of the real time Wilson loop is impossible, just as thecalculation of any other real time correlation function. The whole point of the potentialapproach is to bypass the need for such correlators. In our case, what is needed is thestatic potential in the infinite time limit, which is clearly less information than havingto know the full time dependence. In terms of correlators, the information we need islim t →∞ W E ( it, r ) , lim t →∞ ∂ t W E ( it, r ) . (18)7t least in principle, these limits ought tho be representable by Euclidean operators, thechallenge is to construct those in practice. 6. Conclusions We have argued that many potential models used for the description of quarkoniaat finite T have significant flaws. The connection between the Schr¨odinger equation tothe underlying quantum field theory is unclear, and lattice potentials extracted fromPolyakov loop correlators, which are typically used as input for those models, are thewrong quantities for this purpose. The average free energy is gauge invariant and welldefined, but in its perturbative limit does not reduce to the Debye-screened potential.The so-called singlet potential is gauge dependent and therefore unphysical.These problems can be overcome by using an effective field theory obtained by inte-grating out the heavy quarks, which is pNRQCD generalised to finite temperatures. Theresulting Schr¨odinger equation is the real time evolution equation for a quarkonium cor-relator, and the static potential in this equation is a matching coefficient in the effectivetheory. To leading order in HTL-resummed perturbation theory, this potential is complex,its real part showing the correct Debye-screened behaviour and its imaginary part reflect-ing Landau damping. In the classical limit, only the imaginary part survives. This partcan be calculated non-perturbatively with classical lattice simulations in real time. Theresult agrees in all qualitative features with the HTL result with slightly strengtheneddamping. It is now important to search for a lattice operator that represents the real part. Acknowledgements: Some of the work presented here is supported by the BMBFproject Hot Nuclear Matter from Heavy Ion Collisions and its Understanding from QCD. References [1] T. Matsui and H. Satz, Phys. Lett. B (1986) 416.[2] N. Brambilla, A. Pineda, J. Soto and A. Vairo, Nucl. Phys. B (2000) 275.[3] L. D. McLerran and B. Svetitsky, Phys. Rev. D (1981) 450.[4] M. L¨uscher and P. Weisz, JHEP (2002) 049 and references therein.[5] O. Kaczmarek, F. Karsch, E. Laermann and M. Lutgemeier, Phys. Rev. D (2000) 034021.[6] A. Hart, M. Laine and O. Philipsen, Nucl. Phys. B (2000) 443,S. Datta and S. Gupta, Phys. Rev. D (2003) 054503.[7] S. Nadkarni, Phys. Rev. D (1986) 3738; Phys. Rev. D (1986) 3904.[8] O. Kaczmarek and F. Zantow, Phys. Rev. D (2005) 114510.[9] A. Mocsy and P. Petreczky, Phys. Rev. D (2008) 014501.[10] O. Philipsen, Phys. Lett. B (2002) 138.[11] O. Jahn and O. Philipsen, Phys. Rev. D (2004) 074504.[12] A. Bazavov, P. Petreczky and A. Velytsky, arXiv:0809.2062 [hep-lat].[13] M. Laine, O. Philipsen, P. Romatschke and M. Tassler, JHEP (2007) 054.[14] N. Brambilla, J. Ghiglieri, A. Vairo and P. Petreczky, Phys. Rev. D (2008) 014017.[15] M. A. Escobedo and J. Soto, arXiv:0804.0691 [hep-ph].[16] M. Laine, arXiv:0810.1112 [hep-ph].[17] M. Laine, O. Philipsen and M. Tassler, JHEP (2007) 066.[18] D. B¨odeker, G. D. Moore and K. Rummukainen, Phys. Rev. D (2000) 056003.(2000) 056003.