Bottomonium in the plasma: lattice results
G. Aarts, C. Allton, W. Evans, P. Giudice, T. Harris, A. Kelly, S. Kim, M.P. Lombardo, S. Ryan, J-I Skullerud
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Owned by the authors, published by EDP Sciences, 2018
Bottomonium in the plasma
Lattice results
Gert Aarts , Chris Allton , Wynne Evans , Pietro Giudice , Tim Harris , Aoife Kelly ,Seyong Kim , Maria Paola Lombardo , Sinead Ryan , and Jon-Ivar Skullerud Department of Physics, College of Science, Swansea University, Swansea SA2 8PP, U.K. Institut für Theoretische Physik, Universität Münster, D-48149 Münster, Germany School of Mathematics, Trinity College, Dublin 2, Ireland Department of Mathematical Physics, Maynooth University, Maynooth, Co.Kildare, Ireland Department of Physics, Sejong University, Seoul 143-747, Korea INFN, Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
Abstract.
We present results on the heavy quarkonium spectrum and spectral functionsobtained by performing large-scale simulations of QCD for temperatures ranging fromabout 100 to 500 MeV, in the same range as those explored by LHC experiments. Wediscuss our method and perspectives for further improvements towards the goal of fullcontrol over the many systematic uncertainties of these studies.
Most ordinary hadrons can only exist up to temperatures of about 150–170 MeV. Beyond that, chiralsymmetry is restored and confinement is lost. We know that this hot state of matter — the Quark-Gluon Plasma (QGP) — existed in the early universe: the transition from the QGP to the hadronicworld is the latest cosmological transition. The QGP can be re-created in accelerators: the talk byRoberta Arnaldi [1] provides an excellent introduction into the status of this rich experimental pro-gram.At low temperature the thermal medium consists of a gas of light pions, while towards infinitetemperature quarks and gluons become free, with a corresponding increase of the pressure. Aftera debate lasting several years a consensus has been reached on how to interpolate between thesetwo di ff erent, limiting regimes [2]. It turns out that there is a large intermediate temperature rangewhich is not amenable to any analytic approaches, even when the most sophisticated high temperatureexpansions and model analyses are being used. This is the region explored by experiments, and thisis where our lattice simulations are being performed.Hadrons are of course dramatically a ff ected in the QGP: the light quarks lose their dynamicalmasses and chiral partners approach degeneracy. Quark–antiquark states bound by long-distance,confining forces dissolve. It is very remarkable that heavy quarkonia behave very di ff erently in thisrespect, as their fundamental states might well persist into the plasma: indeed heavy quarks andantiquarks are bound by short range Coulombic interactions which are not immediately a ff ected bytemperatures of the order of 200 MeV. Experimental evidence has been reviewed at this meeting [3]and our motivation is to provide a solid theoretical baseline for these studies. a r X i v : . [ h e p - l a t ] N ov CD@Work 2014
A comprehensive review has recently appeared [4], and we concentrate here on our own recentwork [5–8]. The next section is an introduction into spectral functions and related methodology. Thenwe give an overview of quarkonia in the Quark-Gluon Plasma. The following section is devoted to amore detailed presentation of the bottomonium results. We close with a brief discussion.
Spectral functions play an important role in understanding how elementary excitations are modifiedin a thermal medium. In a relativistic field theory approach the temperature T is realized through(anti)periodic boundary conditions in the Euclidean time direction and the spectral decomposition ofa zero-momentum Euclidean propagator G ( τ ) at finite temperature is given by G ( τ ) = (cid:90) ∞ d ω π K ( τ, ω ) ρ ( ω ) , ≤ τ < T , (1)where ρ ( ω ) is the spectral function and the kernel K is given by K ( τ, ω ) = (cid:16) e − ωτ + e − ω (1 / T − τ ) (cid:17) − e − ω/ T . (2)The τ dependence of the kernel reflects the periodicity of the relativistic propagator in imaginarytime, as well as its T symmetry. The Bose–Einstein distribution, intuitively, describes the wrappingaround the periodic box which becomes increasingly important at higher temperatures. When thesignificant ω range greatly exceeds the temperature, K ( τ, ω ) (cid:39) (cid:16) e − ωτ + e − ω (1 / T − τ ) (cid:17) : backwards andforwards propagations are decoupled and the spectral relation reduces to G ( τ ) = (cid:90) ∞ ω d ω (cid:48) π exp( − ω (cid:48) τ ) ρ ( ω (cid:48) ) . (3)This approximation holds true in NRQCD: the interesting physics takes place around the two-quarkthreshold, ω ∼ M ∼ b quarks, which is still much larger than our temperatures T < . ω = M + ω (cid:48) .Turning to the actual computational methodology, the calculation of the spectral functions usingEuclidean propagators as an input is a di ffi cult, ill-defined problem. We will tackle it using the Maxi-mum Entropy Method (MEM) [10], which has proven successful in a variety of applications. We havestudied the systematics carefully, including the dependence on the set of lattice data points in time,and on the default model m ( ω ) which enters in the parametrisation of the spectral function, ρ ( ω ) = m ( ω ) exp (cid:88) k c k u k ( ω ) , (4)where u k ( ω ) are basis functions fixed by the kernel K ( τ, ω ) and the number of time slices, while thecoe ffi cients c k are to be determined by the MEM analysis [10]. We find that the results are insensitiveto the choice of default model, provided that it is a smooth function of ω , and we will provide someexamples in the next section.Recently, an alternative Bayesian reconstruction of the spectral functions has been proposed in ref.[11, 12], and applied to the analysis of HotQCD configurations [13]. Some preliminary results for thebottomonium spectral functions obtained using this new reconstruction on our ensembles becameavailable after the QCD@work meeting and have been presented at recent conferences [5, 6]. CD@Work 2014
Figure 1.
The spectral function for the charmonium states η c (S-wave) and χ c (P-wave) for varying temperatures,using gauge field ensembles with dynamical u, d and s quarks. The results from two di ff erent default models areshown in the upper and lower diagrams respectively to demonstrate the stability of the analysis. The results on bottomonium presented in this note should be framed in the broader context of studiesof quarkonia as QCD thermometers, either from lattice first principles simulations, or from a lattice-informed potential model approach. Our most recent results for bottomonium [7] have been obtainedby analysing gauge field configurations with two active light quarks and one heavier quark. The lighterquarks are still heavier than the physical up and down quarks as at T = m π / m ρ (cid:39) .
4, while themass of the heavier quark is close to the strange mass. These results can be contrasted with earlierones obtained with an infinite ‘strange’ mass (two active flavours) [14, 15]: in brief summary, wehave found that the results from the di ff erent ensembles are broadly consistent, and we defer a moredetailed comparison to future work.The spectral functions of the charmonium states have been studied as a function of both tempera-ture and momentum, using as input relativistic propagators with two light quarks [16, 17] and, morerecently, including the strange quark. These most recent results are shown in fig. 1, for temperaturesranging between 0 . T c and 1 . T c . The sequential dissolution of the peaks corresponding to the S-and P-wave states is clearly seen. Transport coe ffi cients can be obtained from the low frequency do-main, and this is an important aspect of our research [17, 18]. Furthermore, the inter-quark potentialin charmonium was calculated using the HAL-QCD method, originally developed for the study of thenucleon–nucleon potential [19]. At low temperatures, we observe agreement with the Cornell poten-tial, and the potential flattens (weakens), as expected, when the temperature increases. This is the firstab initio calculation of force between relativistic quarks as a function of temperature. The results areconsistent with the expectation that charmonium melts at high temperature.Bottomonium mesons have been studied using the NRQCD approximation for the bottom quark[20]. We defer the discussion of details to the next section, and here we focus on the main results CD@Work 2014
FASTSUM ρ ( ω ) / m b ω (GeV) 9 10 11 12 13 14 15 Υ T /T c = 0 . .
84 0 . .
95 0 . . . .
27 1 . .
52 1 . . Figure 2.
The spectral functions for the Υ at di ff erent temperatures, obtained using the maximum entropy method. — the spectral functions. Note that in this case the low frequency limit is excluded: transport peaksare sensitive to long-distance, nearly constant modes and do not develop when winding along theEuclidean time direction is suppressed. The results [7] for the Upsilon shown in fig. 2 clearly demon-strate the persistence of the fundamental state above T c as well as the suppression of the excited states.These patterns should be contrasted, for instance, with the one observed by the CMS experiment: foran estimated temperature of about 420 MeV the excited peaks of the invariant mass distribution aresuppressed. Consider now the rate of production of muon pairs dN µ ¯ µ d xd q = F ( q , T , ... ) ρ ( ω ). The connec-tion between the invariant mass distribution and the spectral function is clear, although the dynamicalfactor F is largely unknown. Understanding in detail this connection is an important aspect of on-going research [21]. In the following we will limit ourselves to the presentation and discussion ofour spectral functions. The comparison of important features of our results — masses, as seen in thecentral peak positions, and associated widths — with e ff ective models is satisfactory, and gives usfurther confidence in our analysis.Our results for the P -wave χ b [7] are shown in fig. 3. Here checks of the systematic errors are stillin progress, and in particular we would like to assess the fate of the fundamental state at T c , possiblybefore experimental results — which are still lacking in this sector — appear! We discuss here in more detail our results for the Upsilon and the χ b . The analysis starts with thecomputation of the correlators in Euclidean time within the NRQCD formalism, which in turn areinput to the spectral functions presented above.NRQCD is an e ff ective field theory with power counting in the heavy quark velocity in the bot-tomonium rest frame. The heavy quark and anti-quark fields decouple and their numbers are sepa-rately conserved. Their propagators, S ( x ), solve an initial-value problem whose discretization leads CD@Work 2014
FASTSUM ρ ( ω ) / m b ω + E (GeV) 9 10 11 12 13 14 15 χ b T /T c = 0 . .
84 0 . .
95 0 . . . .
27 1 . .
52 1 . . Figure 3.
The spectral functions for the χ b at di ff erent temperatures, obtained using the maximum entropymethod. to the following choice for the evolution equation S ( x + a τ e τ ) = (cid:32) − a τ H | τ + a τ k (cid:33) k U † τ ( x ) (cid:32) − a τ H | τ k (cid:33) k (1 − a τ δ H ) S ( x ) , (5)where U τ ( x ) is the temporal gauge link at site x and e τ the temporal unit vector. The leading orderHamiltonian is defined by H = − ∆ (2) m b , with ∆ (2 n ) = (cid:80) i = (cid:16) ∇ + i ∇ − i (cid:17) n . The higher order covariant finitedi ff erences are written in terms of the components of the usual forward and backward first order ones.Further details can be found e.g. in ref. [7]. Only energy di ff erences are physically significant inNRQCD because the rest-mass energy can be removed from the heavy quark dispersion relation byperforming a field transformation. Since there is no rest mass term in the NRQCD action one candispense with the demanding constraint a (cid:28) / m b .In our most recent work [7] we have tuned the heavy quark mass by requiring the spin-averaged1S kinetic mass, M (1S) = ( M ( η b ) + M ( Υ )) /
4, to be equal to its experimental value. The tunedvalue of the heavy quark mass corresponds to M (1S) = M expt (1S) = . ρ free ( ω ) ∝ ( ω − ω ) α Θ ( ω − ω ) , where α = / , S wave;3 / , P wave . (6) We have included a threshold, ω , to account for the additive shift in the quarkonium energies. For free quarks the thresholdoccurs at 2 m b , which within NRQCD corresponds to ω = CD@Work 2014 G ( τ ; T ) / G ( τ ; T ≈ ) τ/a τ Υ .
90 =
T/T c . . . . . .
76 11.041.081.121.161.2 0 5 10 15 20 25 30 35 40 G ( τ ; T ) / G ( τ ; T ≈ ) τ/a τ χ b T/T c = 1 . . . . . . . Figure 4.
Thermal modification, G ( τ ; T ) / G ( τ ; T ≈ Υ (left) and χ b (right)channels. a τ M e ff ( τ ) τ/a τ Υ T/T c = 0 . . . . . . .
90 0.30.40.50.6 5 10 15 20 25 30 35 40 a τ M e ff ( τ ) τ/a τ χ b T/T c = 0 . . . . . . . Figure 5.
Temperature dependence of the e ff ective mass in the Υ (left) and the χ b (right) channels. The correlation functions then have the following behaviour G free ( τ ) ∝ e − ω τ τ α + . (7)To show their temperature dependence we consider the ratios of the correlation functions at finite tem-perature to those at zero temperature. They are shown in fig. 4. We see that the thermal modificationsare much larger in the P-wave than in the S-wave channel. A useful numerical tool is the so-callede ff ective mass M e ff ( τ ): M e ff ( τ ) ≡ − G ( τ ) dG ( τ ) d τ G = G free −→ ω + α + τ . (8)The results are shown in fig. 5. The S-wave e ff ective mass displays little temperature dependence(left) but a clear e ff ect is seen in the P-wave channel e ff ective mass (right). In ref. [20] it was alsoobserved that the S-wave e ff ective mass showed little variation with temperature while the temperaturedependence in the P-wave channel e ff ective mass was even more pronounced than visible here.These results clearly show a temperature dependence, but it is not easy to assess with confidencethe fate of bound states. While in real time the information on the long term dynamics is fully acces-sible, in imaginary time all the information is squeezed within the periodicity τ P = / T . One would CD@Work 2014 G ( τ ; T = . T c ) / G ( τ ; T ≈ ) τ/a τ Υ a s m b = 1 . . . . .
32 0.9511.051.11.151.21.251.31.351.4 0 2 4 6 8 10 12 14 16 G ( τ ; T = . T c ) / G ( τ ; T ≈ ) τ/a τ χ b a s m b = 1 . . . . . Figure 6.
Dependence on the heavy quark mass of the modification in the correlators at the highest accessibletemperature, T / T c = .
90, in the Υ (left) and χ b (right) channels. need an extremely high accuracy on extremely fine lattices to make quantitative statements from thecorrelators alone. This further motivates an analysis in terms of spectral functions. In summary, we have a coherent scenario for the Υ : the fundamental state survives up to at leasttwice the critical temperature, while the excited states dissolve. With the caveats mentioned above,this is consistent with the observations of CMS, ALICE and PHENIX. The fundamental state hassome modifications whose basic features can be captured by e ff ective field theories. However, at atemperature of about 420 MeV ALICE results [3] indicate that the suppression of Υ and J /ψ as afunction of the number of participants is comparable, within the present uncertainties. This can beexplained by the J /ψ being more suppressed, but also more sensitive to regeneration, the two e ff ectscompeting in such a way that the resulting R AA is similar to that of the Υ . When comparing RHICand LHC results, it is found that the nuclear modification factor at RHIC is smaller than at the LHC— the so called quarkonium suppression puzzle. New theoretical ideas have been put forward tointerpret this behaviour [22]. All this confirms the interest in ab initio lattice studies of charmonia andbottomonia in hot matter with full control of systematical errors. On the lattice we might also takeadvantage of the freedom to simulate arbitrary masses: some preliminary results were presented inref. [23] and the most recent ones [7] are shown in fig. 6. Guided by these analysis we might be ableto locate a melting line in the temperature–mass plane which passes through the individual meltingtemperatures observed in di ff erent channels. These studies might help unravel general features of thedissolutions of heavy states and their interrelation with gauge dynamics.One important next step is a full control over matter content in our lattice simulations. The sim-ulations reported here have been performed with m π m ρ (cid:39) .
4, and with m s either set to infinity or toits physical value. We aim at physical m u , d , s masses which should correspond to the correct mattercontent in the range T ≤
400 MeV. Above 400 MeV a dynamical charm quark might become relevantas well.We have already mentioned the subtleties related with the reconstruction of the spectral functions.To gain confidence in our analysis we will continue cross checking MEM results with those based onthe novel Bayesian approach [5, 6, 12]; applications of a generalised integral transform might ease theinversion task [24]; and model calculations will provide very useful testbeds for these new techiques[25, 26].
CD@Work 2014
Acknowledgements
It is a pleasure to thank Yannis Burnier and Alexander Rothkopf for many usefuldiscussions. MpL wishes to thank the organisers of QCD@Work 2014 for their very nice hospital-ity and a most interesting meeting. AK acknowledges financial support through the Irish ResearchCouncil.
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