Excited bottomonia in quark-gluon plasma from lattice QCD
Rasmus Larsen, Stefan Meinel, Swagato Mukherjee, Peter Petreczky
EExcited bottomonia in quark-gluon plasma from lattice QCD
Rasmus Larsen a, ∗ , Stefan Meinel b,c , Swagato Mukherjee a , Peter Petreczky a a Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA b Department of Physics, University of Arizona, Tucson, Arizona 85721, USA c RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA
Abstract
We present the first lattice QCD study of up to 3 S and 2 P bottomonia at non-zero temperatures. Correlationfunctions of bottomonia were computed using novel bottomonium operators and a variational technique, within thelattice non-relativistic QCD framework. We analyzed the bottomonium correlation functions based on simple physically-motivated spectral functions. We found evidence of sequential in-medium modifications, in accordance with the sizes ofthe bottomonium states. Keywords:
Heavy-ion collision, Quark-gluon plasma, Quarkonium, Lattice QCD
1. Introduction
Quarkonium suppression has been proposed as a sig-nature of quark-gluon plasma (QGP) formation in heavy-ion collisions [1]. The main idea behind this proposalwas the observation that color screening within a decon-fined medium can make the interaction between the heavyquark and anti-quark short ranged, leading to the dissolu-tion of quarkonia in QGP. At a given temperature, differ-ent quarkonium states are expected to be affected differ-ently by QGP— a more tightly bound quarkonia having asmaller size is less influenced by the medium than a rela-tively loosely bound, larger one. Therefore, following thehierarchy of their binding energy and sizes, a sequentialpattern of in-medium modification is expected [2, 3]. Someevidence of sequential in-medium modification of quarko-nia comes from lattice QCD studies of S-wave and P-wavequarkonium correlators along the temporal [4–8] and spa-tial [9, 10] directions. Recent studies have revealed thatinclusion of dissipative effects lead to a more complex the-oretical picture of in-medium heavy quark and anti-quarkinteractions [11, 12]. However, the main conclusion, i.e.,that quarkonium dissolve in QGP when the temperatureis large enough compared to its inverse size and bindingenergy, have remained unchanged [13–17]. Hints for se-quential in-medium modification of bottomonia have alsobeen observed in heavy-ion collision experiments [18–22].While the connection between the observed hierarchy ofthe Υ( nS ) yields in heavy-ion collisions and the expectedsequential melting of these states in QGP is complicated ∗ Corresponding author
Email addresses: [email protected] (Rasmus Larsen ), [email protected] (Stefan Meinel ), [email protected] (Swagato Mukherjee), [email protected] (PeterPetreczky) by dynamical effects, such a link is expected to exist [23–25]. For this reason, the study of sequential in-mediumquarkonium modifications in heavy-ion collisions is a sub-ject of extensive experimental and theoretical efforts; forrecent reviews see Refs. [26, 27].There have been many attempts to study in-mediumproperties of charmonium [28–36] and bottomonium [4–8, 33, 37, 38] in lattice QCD, almost entirely focused onin-medium modifications of ground states of S- and P-wavequarkonium. Previous lattice QCD studies of in-mediumquarkonium used point meson operators, i.e., operatorswith quark and anti-quark fields located in the same spa-tial point, which are known to have non-optimal overlapwith the quarkonium wave-functions, especially, with theexcited states. As a result, these correlators are largelydominated by the vacuum continuum parts of the spectralfunction, and isolating the contributions of in-medium bot-tomonium becomes quite difficult [16, 17, 39]. Recently,we explored the possibility of studying in-medium bot-tomonium properties using correlators of extended mesonoperators [40]. We found that such operators have verygood overlap with the lowest S- and P-state bottomonia,thereby allowing us to cleanly isolate the vacuum contin-uum contributions to the bottomonium correlators. Weshowed that these correlators are more sensitive to the in-medium bottomonium properties than the ones with pointsources. Analyzing these correlators, we found evidence forthermal broadening of the 1 S - and 1 P -state bottomonia,however, excited bottomonium states remained elusive. Inthis letter we introduce novel extended meson operatorswithin the lattice non-relativistic QCD (NRQCD) formal-ism, which, for the first time, allow us to probe in-mediummodifications of up to 3 S and 2 P bottomonium. Preprint submitted to Elsevier June 1, 2020 a r X i v : . [ h e p - l a t ] M a y r (fm) -1-0.5 0 0.5 1 r (fm) 0 0.1 0.2 0.3 0.4 Ψ (GeV ) -1 -0.5 0 0.5 1 r (fm) -1-0.5 0 0.5 1 r (fm) 0 0.1 0.2 0.3 0.4 Ψ (GeV ) -1 -0.5 0 0.5 1 r (fm) -1-0.5 0 0.5 1 r (fm) 0 0.1 0.2 0.3 0.4 Ψ (GeV ) Figure 1: The shape-functions Ψ S (left), Ψ S (middle), Ψ S (right) used to calculate η b and Υ correlators for lattices with a = 0 .
2. Methodology
The lattice NRQCD Lagrangian employed in this studyis exactly the same as in Refs. [40, 41]— tree-level tadpoleimproved, accurate up to order v , but also includes v spin dependent terms. For the background gauge fields weused 2 + 1-flavor gauge configurations on 48 ×
12 latticeswith bare gauge couplings β = 6 .
74, 6 .
88, 7 .
03, 7 .
28 and7 . a = 0 . . . . . T = 151, 173,199, 251 and 334 MeV, respectively. For each gauge cou-pling we also carried out the corresponding vacuum T = 0calculations. All gauge configurations were generated bythe HotQCD collaboration [42, 43], with the physical valueof the strange quark mass, and up/down quark masses cor-responding to the pion mass of 160 MeV in the continuumlimit. The lattice spacings, a , were determined using the r scale from the static quark anti-quark potential and thevalue r = 0 . η b meson, described in detail in Ref. [40].To calculate bottomonium correlators we used novelextended meson operators in Coulomb gauge of the form O i ( x , t ) = (cid:88) r Ψ i ( r )¯ q ( x + r , t )Γ q ( x , t ) . (1)The different choices of Γ used in this work can be foundin Table 1 of Ref. [40]. The index i refers to the dif-ferent states in a given channel, e.g., 1S, 2S, 3S etc .The shape-functions Ψ i were obtained by solving the dis-cretized Schrodinger equation with a Cornell potential on a3-dimensional lattice having a lattice spacing and a volumeexactly the same as that of the corresponding QCD back-ground. Spin interactions were neglected. For the latticeSchrodinger equation we used an O ( a )-improved Laplaceoperator. For the Cornell potential the string tension waschosen to be (468 MeV) , and the Coulomb part was com-puted at tree-level in lattice perturbation theory for theSymanzik-improved lattice gluon action with a fixed strongcoupling constant α S = 0 .
24. The bottom-quark mass wasset to m b = 4 .
676 GeV. More details can be found in theAppendix D of Ref. [41]. The Ψ’s used for calculations of η b and Υ correlators for lattices with a = 0 . i is a good approximation for the wave-functionof the i th vacuum bottomonium, as expected, the corre-sponding operator O i was found to have a good overlapwith the i th state. However, the off-diagonal correlators, G ij ( t ) = (cid:104) O i ( t ) O † j (0) (cid:105) for i (cid:54) = j , were found to be non-zero, though small. Thus, we resorted to the variationalanalysis by considering linear combinations ˜ O α = Ω αj O j such that (cid:104) ˜ O α ( t ) ˜ O † β (0) (cid:105) ∝ δ α,β . The matrices Ω αj wereobtained using the generalized eigenvalue problem [45–49] G ij ( t )Ω αj = λ α ( t, t ) G ij ( t )Ω αj . We calculated the cor-relators of optimized operators C α ( t ) = (cid:104) ˜ O α ( t ) ˜ O † α (0) (cid:105) for1S, 1P, 2S, 2P and 3S. When calculating Ω αj , the value of t was chosen such that it corresponds to physical extent τ = t · a (cid:39) . t was chosen to be τ = 0 fm.Choosing t /a to be 1 or 2 does not change the resultssignificantly.
3. Results
In Fig. 2 we show some examples of the effectivemasses, aM eff ( t ) = ln[ C α ( t ) /C α ( t + a )], at T = 0. Thosewere found to reach a plateau for τ (cid:39) . τ ofaround 1 . E S = ( E η b + 3 E Υ ) /
4, to be the zero of the mass/energy,and quote masses/energies of the rest of the bottomoniumstates with respect to these baselines for each lattice spac-ing. (Note that, in Ref. [40] this baseline was set by the η b energy level.) The reason being ¯ E S remains unaffectedby the spin-spin interaction, which is difficult to repro-duce accurately using tree-level NRQCD Lagrangian usedin this study. The energy differences show a very mild de-pendence on the lattice spacing, and in most cases can befitted with a constant to obtain our final estimate for theenergy differences. The only exceptions are the energy of2 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ▲ Υ ( ) ● Υ ( ) ■ Υ ( ) τ ( fm ) M e ff ( G e V ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲● ● ● ● ● ● ● ● ● ●● ▲ χ b0 ( ) ● χ b0 ( ) τ ( fm ) M e ff ( G e V ) Figure 2: The effective masses, M eff , at T = 0 for different Υ (left) and χ b (right) states calculated on the lattice with a = 0 .
109 fm. χ b and the 2S hyperfine splitting, where the a -dependencecannot be neglected. We also fitted the resulting energydifferences with a constant plus a term proportional to a to remove the remaining discretization effects, this re-sulted in small differences in the central value but largerstatistical errors for the final estimate. We used this proce-dure to compare with the experimental results, except for3S state, where it gives too large statistical errors. Thecomparison of the zero temperature results on the massdifferences with the experimental results from the ParticleData Group (PDG) [50] is shown in Table 1. We see verygood agreement between our lattice NRQCD calculationsand the experimental results within the estimated errorsin most cases. The only exception is the χ b (1 P ) state,where a small tension between our result and the PDGvalue is observed. Using the result for the 3S hyperfinesplitting we can also predict the mass of the η b (3 S ) stateto be 10341 . ± . In NRQCD the spectral function, ρ ( ω, T ), is related tothe Euclidean time correlation function: C α ( τ, T ) = (cid:90) ∞−∞ dωρ α ( ω, T ) e − ωτ . (2)Here, α labels the bottomonium operator of interest. Bot-tomonium states correspond to peaks in the spectral func-tion having some in-medium width. At large ω many statescontribute to the spectral function, forming a continuum.Therefore, we can write the spectral function as ρ α ( ω, T ) = ρ med α ( ω, T ) + ρ high α ( ω ) , (3)with the second term parameterizing the continuum partof the spectral function. In the zero-temperature limit ρ med α ( ω, T ) = A α δ ( ω − M α ), M α being the mass of thecorresponding bottomonium state. Here we note that theuse of extended operators reduces the relative contributionof ρ high α [51, 52]. For this reason the effective masses in Fig.2 approach a plateau at relatively small τ . The continuum state ∆ M [MeV] ∆ M ( P DG ) [MeV]Υ(3 S ) 906.0(25.0)(5.2) 910.3(0.7) h b (2 P ) 804.4(35.8)(4.7) 814.9(1.3) χ b (2 P ) 809.2(36.2)(4.7) 823.8(0.9) χ b (2 P ) 802.2(34.9)(4.7) 810.6(0.7) χ b (2 P ) 786.8(32.7)(4.6) 787.6(0.8)Υ(2 S ) 582.7(9.8)(3.4) 578.4(0.6) h b (1 P ) 454.5(4.7)(2.6) 454.4(0.9) χ b (1 P ) 463.3(4.8)(2.7) 467.3(0.6) χ b (1 P ) 448.9(4.6)(2.6) 447.9(0.6) χ b (1 P ) 421.3(4.7)(2.4) 414.5(0.7)hyperfine(3S) 13.4(6.2)(0.1) NAhyperfine(2S) 24.1(1.0)(0.1) 24.5(4.5) Table 1: Comparisons of mass differences ∆ M (in units of MeV)of various bottomonium states with respect to the 1S spin-averagedmass obtained from our lattice calculations with that from PDG [50].The last two rows show the 2S and 3S hyperfine splitting. The seconderror in our results corresponds to the uncertainty of the r scale. part, ρ high α ( ω ), is expected to be temperature independent.This was seen to fit with our calculation, as the temper-ature dependence of C α ( τ, T ) for τ (cid:46) . f m fm was verysmall, with the small difference being in agreement withchanges due to the medium. Thus, following Ref. [40],for each lattice spacing we can identify the contribution of ρ high α ( ω ) to the correlator, C high α ( τ ), as C α ( τ, T = 0) = A α e − M α τ + C high α ( τ ) . (4)Here, A α and M α are the amplitude and mass of the corre-sponding bottomonium state, and C high α ( τ ) is the Laplacetransform of ρ high α . Using the single-exponential fits to thevacuum correlators for τ (cid:38) . C α ( τ, T = 0) we isolated C high α ( τ ) foreach value of β . Further, following Ref. [40], for each tem-perature we then defined the continuum-subtracted corre-lator as C sub α ( τ, T ) = C α ( τ, T ) − C high α ( τ ) . (5)3 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■▲ Υ ( ) ● Υ ( ) ■ Υ ( ) τ ( fm ) M e ff s ub ( G e V ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲● ● ● ● ● ● ● ● ● ● ●▲ χ b0 ( ) ● χ b0 ( ) - τ ( fm ) M e ff s ub ( G e V ) Figure 3: Continuum-subtracted effective masses, M subeff , of Υ states at T = 251 MeV (left) and χ b states at T = 199 MeV (right). ● ● ● ● ● ● ● ● ● ● ●▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲● Υ ( ) - - - τ ( fm ) M e ff s ub ( G e V ) Figure 4: Comparison of the continuum-subtracted effective mass, M subeff , of Υ(1 S ) at T = 251 MeV obtained in this study (circle) andusing Gaussian extended source (triangle) of Ref. [40]. Thus, the continuum-subtracted correlator, C sub α ( τ, T ), ismostly sensitive to ρ med α ( ω, T ), encoding the in-mediumbottomonium properties. We then studied the in-mediumbottomonium properties using the continuum-subtractedeffective masses, aM subeff ( τ, T ) = ln (cid:0) C sub α ( τ, T ) /C sub α ( τ + a, T ) (cid:1) . (6)In Fig. 3 we show typical examples of M subeff as a func-tion of τ — for the Υ states at T = 251 MeV and for the χ b states at T = 199 MeV. At small τ , M subeff are ap-proximately equal to the vacuum masses. As τ increases,we see an approximately linear decrease of M subeff . Finally,for τ (cid:39) /T we see a rapid drop-off. Similar behaviors of M subeff for the ground states were also observed in the pre-vious study using Gaussian smeared meson operators [40].As discussed in Ref. [40], the slope of the linear decreaseof M sub eff can be understood in terms of a thermal width.We see that the slope is larger for higher excited bot-tomonium states, i.e., the thermal width of different bot-tomonium states follows the expected hierarchy of theirsizes. Higher excited states have larger size and there-fore are more affected by the medium, leading to largerwidth. The behavior of the effective masses at τ (cid:39) /T is related to the tail of the spectral function at small ω , and may depend on the choice of the meson operator [40].Therefore, it is important to compare the results on thesubtracted effective masses obtained with different mesonoperators. In Fig. 4 we compare M subeff of subtracted Υ(1 S )at T = 251 MeV with the corresponding results obtainedwith Gaussian smeared sources of Ref. [40]. Good agree-ment was found between the present results and those ob-tained of Ref. [40], especially for τ (cid:28) /T . Therefore, ourconclusion regarding the in-medium modification of thespectral functions is not affected by the choices of mesonoperators. For τ (cid:39) /T we see a smaller drop-off in M subeff compared to that observed in Ref. [40]. Thus, the small- ω tail of the spectral functions plays a less prominent rolehere. We found that the behaviors of M subeff for η b ( nS )are very similar to that of Υ( nS ), and that of χ b ( nP ), χ b ( nP ) and h b ( nP ) are very similar to that of χ b ( nP ).As introduced in Ref. [40], the simplest theoreticallymotivated parameterization of the in-medium spectral func-tion that can describe the generic behavior of M subeff ob-served here is as follows ρ med α ( ω, T ) = A cut α ( T ) δ (cid:0) ω − ω cut α ( T ) (cid:1) + A α ( T ) exp (cid:32) − [ ω − M α ( T )] α ( T ) (cid:33) . (7)The first term in the above equation provides a simple pa-rameterization of the low- ω tail of the spectral function.As explained in Ref. [40], this tail is important for un-derstanding the behavior of the effective masses around τ (cid:39) /T . The second term gives rise to the linear be-havior in τ of M subeff , with the slope given by Γ α . Foreach temperature, we fitted M subeff ( τ ) with the Ansatz givenby Eq. (7), and using Eqs. (2), (6), to determine the in-medium masses, M α ( T ), and width, Γ α ( T ), of differentbottomonium states. Since the tail of the spectral func-tion plays a less prominent role in the present study, for T ≤
173 MeV and all temperatures for Υ 1S, we per-formed fits, setting A cut α = 0, and omitting 1-3 data pointsfor the largest values of τ . Only for higher temperatureswas the term proportional to A cut α included. We generallyfind good fits with χ divided by degrees of freedom being4round 0 .
5. In some cases the data points fluctuate morethan the size of the estimated errors. Examples of suchcases include Υ(1 S ) and also χ b at 199 MeV, as can beseen on the right in Fig. 3. In these cases we found that χ divided by degrees of freedom was around 2. The fitstill seem to work nicely, so it is most likely the errors thatwere a bit too small.The change of the in-medium mass parameter com-pared to the vacuum mass ( M α ), ∆ M α ( T ) = M α ( T ) − M α ,and width parameter, Γ α ( T ), are shown in Figs. 5 and 6,respectively. The in-medium masses of different states ob-tained from the fits turned out to be very similar to thevacuum masses. In fact, we do not see any statistically sig-nificant deviations from the T = 0 results. On the otherhand, Γ α ( T ) shows a clear increase with increasing tem-perature. For large enough temperatures, Γ α ( T ) appearsto approximately rise linearly with T . Nearly for the en-tire T -range, the in-medium width was found to follow thesequential hierarchical pattern according to the increas-ing sizes of the bottomonium states: Γ S ( T ) < Γ P ( T ) < Γ S ( T ) < Γ P ( T ) < Γ S ( T ). Moreover, Γ S (cid:38) M S − M S and Γ P (cid:38) M P − M P for T (cid:38)
200 MeV. As a result, atthese temperatures 2 S and 3 S , as well as the 1 P and 2 P states will together appear as broad structures in theirrespective spectral functions. These observations lead usto conclude that, similar to what have been observed inthe experiments [18, 19], for T (cid:38)
200 MeV it will becomedifficult to individually identify the 2 S , 3 S , 1 P and 2 P states within the experimentally measured line shapes ofthe invariant-mass distributions.Lastly, we address the question to what extent the es-timated thermal widths of bottomonium states depend onthe model for the spectral function used to interpret ourlattice QCD results presented here. Since M subeff ( τ ) show alinear in τ behavior and we do not observe any significantthermal mass shift, following Ref. [40], we also used thefollowing model for the spectral function: ρ med α ( ω, T ) = A cut α ( T ) δ ( ω − ω cut α ( T ))+ δ ( ω − M α + ∆ α ( T ))+ δ ( ω − M α )+ δ ( ω − M α − ∆ α ( T )) . (8)Here, M α is the vacuum bottomonium mass, and the pa-rameters A cut α and ω cut α describe the low- ω tail of the spec-tral function. For T ≤
173 MeV again we used A cut α = 0.The equivalent thermal width in this case is Γ α ( T ) = (cid:112) / α ( T ). Carrying out fits with the above Ansatz weobtained thermal widths that, as shown in Fig. 6, withinerrors, agreed with the ones obtained by using the Gaus-sian Ansatz. Therefore, our estimates of thermal width donot depend very much on the precise functional form ofthe fit Ansatz.
4. Conclusion
For the very first time, we studied in-medium prop-erties up to 3 S and 2 P excited bottomonium states us-ing lattice QCD at temperatures T (cid:39) −
350 MeV.This lattice QCD study was made possible through theintroduction of novel bottomonium operators within thelattice NRQCD framework, and implementation of a vari-ational analysis based on these novel operators. We foundthat the effective masses constructed out of the continuum-subtracted bottomonium correlation functions drop off lin-early in Euclidean time. We argued that the behaviors ofthe continuum-subtracted effective masses can be under-stood in terms of a couple of theoretically-motivated, sim-ple models of the bottomonium spectral functions. For allof the models considered, we found indications of thermalbroadening of bottomonium states in QGP. For the entiretemperature range, the magnitudes of the thermal broad-ening were found to follow the expected sequential hierar-chical pattern according to the increasing sizes of the bot-tomonium states. Further, we found that for T (cid:38)
200 MeVthe thermal broadening of the 2 S , 3 S , 1 P and 2 P statesbecomes large enough that it would be difficult to identifythese states separately within the corresponding spectralfunctions. Acknowledgments
This material is based upon work supported by the U.S.Department of Energy, Office of Science, Office of NuclearPhysics: (i) Through the Contract No. DE-SC0012704;(ii) Through the Scientific Discovery through Advance Com-puting (SciDAC) award Computing the Properties of Mat-ter with Leadership Computing Resources. (iii) StefanMeinel acknowledges support by the U.S. Department ofEnergy, Office of Science, Office of High Energy Physicsunder Award Number DE-SC0009913.This research used awards of computer time: (i) Pro-vided by the USQCD consortium at its Fermi NationalLaboratory, Brookhaven National Laboratory and Jeffer-son Laboratory computing facilities; (ii) Provided by theINCITE program at Argonne Leadership Computing Fa-cility, a U.S. Department of Energy Office of Science UserFacility operated under Contract No. DE-AC02-06CH11357;(ii) Provided by the ALCC program at National EnergyResearch Scientific Computing Center, a U.S. Departmentof Energy Office of Science User Facility operated underContract No. DE-AC02-05CH11231; (iii) Provided by theINCITE programs at Oak Ridge Leadership ComputingFacility, a DOE Office of Science User Facility operatedunder Contract No. DE-AC05-00OR22725.
References [1] T. Matsui, H. Satz,
J/ψ
Suppression by Quark-Gluon PlasmaFormation, Phys. Lett. B178 (1986) 416–422. doi:10.1016/0370-2693(86)91404-8 . ■ ■ ■ ■■ Υ ( ) - ● ● ● ● ●● Υ ( ) - Δ M α ( M e V ) ▲ ▲ ▲ ▲ ▲▲ Υ ( )
150 200 250 300 - - ( MeV ) ● ● ● ● ●● χ b0 ( ) - - Δ M α ( M e V ) ▲ ▲ ▲ ▲ ▲▲ χ b0 ( )
150 200 250 300 - - ( MeV ) Figure 5: The change of the in-medium mass compared to the vacuum mass, ∆ M α = M α ( T ) − M α , for the Υ (left) and χ b states (right) asfunction of the temperature. ▲ ▲ ▲ ▲ ▲● ● ● ● ●■ ■ ■ ■ ■△ △ △ △ △○ ○ ○ ○ ○□ □ □ □ □▲ Υ ( ) ● Υ ( ) ■ Υ ( )
150 200 250 3000200400600800 T ( MeV ) Γ α ( M e V ) ▲ ▲ ▲ ▲ ▲● ● ● ● ●△ △ △ △ △○ ○ ○ ○ ○▲ χ b0 ( ) ● χ b0 ( )
150 200 250 3000100200300400500600 T ( MeV ) Γ α ( M e V ) Figure 6: Thermal width, Γ α , of Υ (left) and χ b states (right) as function of the temperature using Eq. (7) (filled) and using Eq. (8)(empty). Fit with Eq. (8) assumes M α ( T ) = M α and error bars are thus slightly smaller.[2] F. Karsch, M. T. Mehr, H. Satz, Color Screening and Deconfine-ment for Bound States of Heavy Quarks, Z. Phys. C37 (1988)617. doi:10.1007/BF01549722 .[3] S. Digal, P. Petreczky, H. Satz, Quarkonium feed down andsequential suppression, Phys. Rev. D64 (2001) 094015. arXiv:hep-ph/0106017 , doi:10.1103/PhysRevD.64.094015 .[4] G. Aarts, S. Kim, M. P. Lombardo, M. B. Oktay, S. M. Ryan,D. K. Sinclair, J. I. Skullerud, Bottomonium above decon-finement in lattice nonrelativistic QCD, Phys. Rev. Lett. 106(2011) 061602. arXiv:1010.3725 , doi:10.1103/PhysRevLett.106.061602 .[5] G. Aarts, C. Allton, S. Kim, M. P. Lombardo, S. M. Ryan,J. I. Skullerud, Melting of P wave bottomonium states in thequark-gluon plasma from lattice NRQCD, JHEP 12 (2013) 064. arXiv:1310.5467 , doi:10.1007/JHEP12(2013)064 .[6] G. Aarts, C. Allton, T. Harris, S. Kim, M. P. Lombardo,S. Ryan, J.-I. Skullerud, The bottomonium spectrum at finitetemperature from N f = 2 + 1 lattice QCD, JHEP 07 (2014)097. arXiv:1402.6210 , doi:10.1007/JHEP07(2014)097 .[7] S. Kim, P. Petreczky, A. Rothkopf, Quarkonium in-mediumproperties from realistic lattice NRQCD, JHEP 11 (2018) 088. arXiv:1808.08781 , doi:10.1007/JHEP11(2018)088 .[8] S. Kim, P. Petreczky, A. Rothkopf, Lattice NRQCD study ofS- and P-wave bottomonium states in a thermal medium with N f = 2 + 1 light flavors, Phys. Rev. D91 (2015) 054511. arXiv:1409.3630 , doi:10.1103/PhysRevD.91.054511 .[9] A. Bazavov, F. Karsch, Y. Maezawa, S. Mukherjee, P. Pe-treczky, In-medium modifications of open and hidden strange-charm mesons from spatial correlation functions, Phys. Rev.D91 (5) (2015) 054503. arXiv:1411.3018 , doi:10.1103/PhysRevD.91.054503 . [10] F. Karsch, E. Laermann, S. Mukherjee, P. Petreczky, Signa-tures of charmonium modification in spatial correlation func-tions, Phys. Rev. D85 (2012) 114501. arXiv:1203.3770 , doi:10.1103/PhysRevD.85.114501 .[11] M. Laine, O. Philipsen, P. Romatschke, M. Tassler, Real-timestatic potential in hot QCD, JHEP 03 (2007) 054. arXiv:hep-ph/0611300 , doi:10.1088/1126-6708/2007/03/054 .[12] N. Brambilla, J. Ghiglieri, A. Vairo, P. Petreczky, Static quark-antiquark pairs at finite temperature, Phys. Rev. D78 (2008)014017. arXiv:0804.0993 , doi:10.1103/PhysRevD.78.014017 .[13] M. Laine, A Resummed perturbative estimate for the quarko-nium spectral function in hot QCD, JHEP 05 (2007) 028. arXiv:0704.1720 , doi:10.1088/1126-6708/2007/05/028 .[14] Y. Burnier, M. Laine, M. Vepsalainen, Heavy quarkonium inany channel in resummed hot QCD, JHEP 01 (2008) 043. arXiv:0711.1743 , doi:10.1088/1126-6708/2008/01/043 .[15] A. Beraudo, J. P. Blaizot, C. Ratti, Real and imaginary-timeQ anti-Q correlators in a thermal medium, Nucl. Phys. A806(2008) 312–338. arXiv:0712.4394 , doi:10.1016/j.nuclphysa.2008.03.001 .[16] P. Petreczky, C. Miao, A. Mocsy, Quarkonium spectral func-tions with complex potential, Nucl. Phys. A855 (2011) 125–132. arXiv:1012.4433 , doi:10.1016/j.nuclphysa.2011.02.028 .[17] Y. Burnier, O. Kaczmarek, A. Rothkopf, Quarkonium at finitetemperature: Towards realistic phenomenology from first prin-ciples, JHEP 12 (2015) 101. arXiv:1509.07366 , doi:10.1007/JHEP12(2015)101 .[18] S. Chatrchyan, et al., Indications of suppression of excitedΥ states in PbPb collisions at √ S NN = 2.76 TeV, Phys.Rev. Lett. 107 (2011) 052302. arXiv:1105.4894 , doi:10.1103/PhysRevLett.107.052302 .
19] S. Chatrchyan, et al., Observation of Sequential Upsilon Sup-pression in PbPb Collisions, Phys. Rev. Lett. 109 (2012)222301, [Erratum: Phys. Rev. Lett.120,no.19,199903(2018)]. arXiv:1208.2826 , doi:10.1103/PhysRevLett.109.222301,10.1103/PhysRevLett.120.199903 .[20] V. Khachatryan, et al., Suppression of Υ(1 S ) , Υ(2 S ) and Υ(3 S )production in PbPb collisions at √ s NN = 2.76 TeV, Phys.Lett. B770 (2017) 357–379. arXiv:1611.01510 , doi:10.1016/j.physletb.2017.04.031 .[21] A. M. Sirunyan, et al., Suppression of Excited Υ States Relativeto the Ground State in Pb-Pb Collisions at √ s NN =5.02 TeV,Phys. Rev. Lett. 120 (14) (2018) 142301. arXiv:1706.05984 , doi:10.1103/PhysRevLett.120.142301 .[22] P. Wang, Γ measurements in Au+Au collisions at √ s NN =200GeV with the STAR experiment, Nucl. Phys. A982 (2019)723–726. doi:10.1016/j.nuclphysa.2018.09.025 .[23] B. Krouppa, A. Rothkopf, M. Strickland, Bottomonium sup-pression using a lattice QCD vetted potential, Phys. Rev.D97 (1) (2018) 016017. arXiv:1710.02319 , doi:10.1103/PhysRevD.97.016017 .[24] X. Yao, M. Berndt, Approach to equilibrium of quarkoniumin quark-gluon plasma, Phys. Rev. C97 (1) (2018) 014908,[Erratum: Phys. Rev.C97,no.4,049903(2018)]. arXiv:1709.03529 , doi:10.1103/PhysRevC.97.049903,10.1103/PhysRevC.97.014908 .[25] P. Petreczky, C. Young, Sequential bottomonium production athigh temperatures, Few Body Syst. 58 (2) (2017) 61. arXiv:1606.08421 , doi:10.1007/s00601-016-1188-8 .[26] A. Mocsy, P. Petreczky, M. Strickland, Quarkonia in the QuarkGluon Plasma, Int. J. Mod. Phys. A28 (2013) 1340012. arXiv:1302.2180 , doi:10.1142/S0217751X13400125 .[27] G. Aarts, et al., Heavy-flavor production and medium proper-ties in high-energy nuclear collisions - What next?, Eur. Phys.J. A53 (5) (2017) 93. arXiv:1612.08032 , doi:10.1140/epja/i2017-12282-9 .[28] T. Umeda, K. Nomura, H. Matsufuru, Charmonium at finitetemperature in quenched lattice QCD, Eur. Phys. J. C39S1(2005) 9–26. arXiv:hep-lat/0211003 , doi:10.1140/epjcd/s2004-01-002-1 .[29] S. Datta, F. Karsch, P. Petreczky, I. Wetzorke, A Study ofcharmonium systems across the deconfinement transition, Nucl.Phys. Proc. Suppl. 119 (2003) 487–489, [,487(2002)]. arXiv:hep-lat/0208012 , doi:10.1016/S0920-5632(03)01591-3 .[30] F. Karsch, S. Datta, E. Laermann, P. Petreczky, S. Stickan,I. Wetzorke, Nucl. Phys. A715 (2003) 701–704. arXiv:hep-ph/0209028 , doi:10.1016/S0375-9474(02)01470-7 .[31] S. Datta, F. Karsch, P. Petreczky, I. Wetzorke, Behavior of char-monium systems after deconfinement, Phys. Rev. D69 (2004)094507. arXiv:hep-lat/0312037 , doi:10.1103/PhysRevD.69.094507 .[32] M. Asakawa, T. Hatsuda, J / psi and eta(c) in the decon-fined plasma from lattice QCD, Phys. Rev. Lett. 92 (2004)012001. arXiv:hep-lat/0308034 , doi:10.1103/PhysRevLett.92.012001 .[33] A. Jakovac, P. Petreczky, K. Petrov, A. Velytsky, Quarkoniumcorrelators and spectral functions at zero and finite tempera-ture, Phys. Rev. D75 (2007) 014506. arXiv:hep-lat/0611017 , doi:10.1103/PhysRevD.75.014506 .[34] H. Ohno, S. Aoki, S. Ejiri, K. Kanaya, Y. Maezawa, H. Saito,T. Umeda, Charmonium spectral functions with the variationalmethod in zero and finite temperature lattice QCD, Phys. Rev.D84 (2011) 094504. arXiv:1104.3384 , doi:10.1103/PhysRevD.84.094504 .[35] H. T. Ding, A. Francis, O. Kaczmarek, F. Karsch, H. Satz,W. Soeldner, Charmonium properties in hot quenched latticeQCD, Phys. Rev. D86 (2012) 014509. arXiv:1204.4945 , doi:10.1103/PhysRevD.86.014509 .[36] H.-T. Ding, O. Kaczmarek, S. Mukherjee, H. Ohno, H. T. Shu,Stochastic reconstructions of spectral functions: Application tolattice QCD, Phys. Rev. D97 (9) (2018) 094503. arXiv:1712.03341 , doi:10.1103/PhysRevD.97.094503 . [37] G. Aarts, C. Allton, S. Kim, M. P. Lombardo, M. B. Oktay,S. M. Ryan, D. K. Sinclair, J. I. Skullerud, What happens tothe Υ and η b in the quark-gluon plasma? Bottomonium spectralfunctions from lattice QCD, JHEP 11 (2011) 103. arXiv:1109.4496 , doi:10.1007/JHEP11(2011)103 .[38] G. Aarts, C. Allton, S. Kim, M. P. Lombardo, M. B. Ok-tay, S. M. Ryan, D. K. Sinclair, J.-I. Skullerud, S wave bot-tomonium states moving in a quark-gluon plasma from lat-tice NRQCD, JHEP 03 (2013) 084. arXiv:1210.2903 , doi:10.1007/JHEP03(2013)084 .[39] A. Mocsy, P. Petreczky, Can quarkonia survive deconfinement?,Phys. Rev. D77 (2008) 014501. arXiv:0705.2559 , doi:10.1103/PhysRevD.77.014501 .[40] R. Larsen, S. Meinel, S. Mukherjee, P. Petreczky, Thermalbroadening of bottomonia: Lattice nonrelativistic QCD withextended operators, Phys. Rev. D100 (2019) 074506. arXiv:1908.08437 , doi:10.1103/PhysRevD.100.074506 .[41] S. Meinel, Bottomonium spectrum at order v from domain-walllattice QCD: Precise results for hyperfine splittings, Phys. Rev.D82 (2010) 114502. arXiv:1007.3966 , doi:10.1103/PhysRevD.82.114502 .[42] A. Bazavov, et al., The chiral and deconfinement aspects ofthe QCD transition, Phys. Rev. D85 (2012) 054503. arXiv:1111.1710 , doi:10.1103/PhysRevD.85.054503 .[43] A. Bazavov, et al., Equation of state in ( 2+1 )-flavor QCD,Phys. Rev. D90 (2014) 094503. arXiv:1407.6387 , doi:10.1103/PhysRevD.90.094503 .[44] A. Bazavov, et al., Results for light pseudoscalar mesons, PoSLATTICE2010 (2010) 074. arXiv:1012.0868 .[45] K. Nochi, T. Kawanai, S. Sasaki, Bethe-Salpeter wave func-tions of η c (2 S ) and ψ (2 S ) states from full lattice QCD, Phys.Rev. D94 (11) (2016) 114514. arXiv:1608.02340 , doi:10.1103/PhysRevD.94.114514 .[46] C. Michael, Adjoint Sources in Lattice Gauge Theory, Nucl.Phys. B259 (1985) 58–76. doi:10.1016/0550-3213(85)90297-4 .[47] M. Luscher, U. Wolff, How to Calculate the Elastic ScatteringMatrix in Two-dimensional Quantum Field Theories by Nu-merical Simulation, Nucl. Phys. B339 (1990) 222–252. doi:10.1016/0550-3213(90)90540-T .[48] B. Blossier, M. Della Morte, G. von Hippel, T. Mendes, R. Som-mer, On the generalized eigenvalue method for energies andmatrix elements in lattice field theory, JHEP 04 (2009) 094. arXiv:0902.1265 , doi:10.1088/1126-6708/2009/04/094 .[49] K. Orginos, D. Richards, Improved methods for the study ofhadronic physics from lattice QCD, J. Phys. G42 (3) (2015)034011. doi:10.1088/0954-3899/42/3/034011 .[50] M. Tanabashi, et al., Phys. Rev. D 98 (2018) 030001. doi:10.1103/PhysRevD.98.030001 , [link].URL https://link.aps.org/doi/10.1103/PhysRevD.98.030001 [51] I. Wetzorke, F. Karsch, E. Laermann, P. Petreczky, S. Stickan,Meson spectral functions at finite temperature, Nucl. Phys.Proc. Suppl. 106 (2002) 510–512. arXiv:hep-lat/0110132 , doi:10.1016/S0920-5632(01)01763-7 .[52] S. Stickan, F. Karsch, E. Laermann, P. Petreczky, Free mesonspectral functions on the lattice, Nucl. Phys. Proc. Suppl. 129(2004) 599–601, [,599(2003)]. arXiv:hep-lat/0309191 , doi:10.1016/S0920-5632(03)02654-9 ..