We compute correlators and spectral functions for J/psi and eta_c mesons at
nonzero momentum on anisotropic lattices with Nf=2. We find no evidence of
significant momentum dependence at the current level of precision. In the
pseudoscalar channel, the ground state appears to survive up to T~450MeV or
2.1T_c. In the vector channel, medium modifications may occur at lower
temperatures.Comment: 5 pages, 12 figure

Oktay, M. B.

Skullerud, J. I.

English

arXiv

We compute correlators and spectral functions for J/  and c mesons at nonzero momentum

on anisotropic lattices with Nf = 2. We find no evidence of significant momentum dependence

at the current level of precision. In the pseudoscalar channel, the ground state appears to survive

up to T  450MeV or 2.1Tc. In the vector channel, medium modifications may occur at lower

temperatures

Bugrahan Oktay, Mehmet

Skullerud, Jon-Ivar

Maynooth EPrints and ETheses Archive

Momentum-dependence of charmonium spectral functions from lattice QCD

NUI Maynooth Eprint Archive

MURAL - Maynooth University Research Archive Library

arXiv:1005.1209v1  [hep-lat]  7 May 2010Momentum-dependence of charmonium spectral functions from lattice QCDMehmet Bugˇrahan Oktay1 and Jon-Ivar Skullerud21Department of Physics, University of Utah, 115 S 1400 E, Salt Lake City, UT 84112-0830 USA.2Department of Mathematical Physics, NUI Maynooth, County Kildare, Ireland.(Dated: May 10, 2010)We compute correlators and spectral functions for J/ψ and ηc mesons at nonzero momentumon anisotropic lattices with Nf = 2. We find no evidence of significant momentum dependenceat the current level of precision. In the pseudoscalar channel, the ground state appears to surviveup to T ≈ 450MeV or 2.1Tc. In the vector channel, medium modifications may occur at lowertemperatures.I. INTRODUCTIONThe survival and properties of charmonium boundstates in the quark–gluon plasma is a hot topic in theinterpretation of experimental results from relativisticheavy-ion collisions. A variety of models have been putforward to explain the observed pattern of suppressionof the J/ψ yield, including sequential suppression [1],charmonium regeneration through coalescence of uncor-related cc¯ pairs [2–4], and combinations of these twomechanisms [5].Results from lattice simulations [6–9] suggest (thoughfar from decisively [10]) that S-wave ground states (J/ψand ηc) survive up to temperatures close to twice thepseudocritical temperature Tc, while excited states (ψ′)and P-waves (χc) melt at temperatures close to Tc, inrough agreement with the sequential suppression sce-nario. These results are however not sufficient to rulein favour of one or other scenario.An additional handle on the problem may be gainedby considering charmonium states which are moving withrespect to the thermal medium. The two main scenarios(regeneration and sequential suppression) predict differ-ent transverse momentum and rapidity dependence of thecharmonium yield. This is primarily due to the momen-tum dependence of the reaction rates included in thesescenarios, but in order to complete the story it will alsobe necessary to investigate any momentum dependenceof the baseline survival probability of charmonium boundstates. This paper provides the first steps in this direc-tion.The analytical structure of QCD correlators impliesthat both spatial and temporal correlators are governedby the same spectral function. However, in order to de-scribe spatial correlators and screening masses, it is nec-essary to also include the spatial-momentum dependenceof the spectral function.Finally, computing spectral functions at nonzero mo-mentum may also facilitate the determination of trans-port coefficients such as the heavy quark diffusion con-stant [11], as the zero-frequency transport peak is broad-ened and shifts to nonzero frequency.We will work on a spatially isotropic lattice, whichmeans that the anisotropy of the plasma created inheavy-ion collisions is not taken into account. Our mo-mentum is best interpreted as the transverse momentum,and we will therefore not be able to say anything at thispoint about the rapidity dependence of any observables.II. FORMULATIONWe have simulated two degenerate flavours of O(a)-improved Wilson fermions, with mpi/mρ = 0.54 (corre-sponding approximately to the strange quark mass), andwith spatial lattice spacing as = 0.162fm and anisotropyξ = as/aτ = 6. The lattice action is described in moredetail in Ref. [12], and the high-temperature simulationparameters are discussed in Ref. [13]. Our parametershere correspond to Run 7 in Ref. [13]. The same fermionaction is used for both light sea and heavy valence quarks.The lattices, corresponding temperatures and number ofconfigurationsNcfg used in this study are given in Table I.The main difference compared to Ref [13] is that all ourdata are now obtain using the tuned (Run 7) parametersand on a spatial volume of 123 in lattice units.Ns Nτ T (MeV) T/Tc Ncfg8 80 92 0.42 25012 80 92 0.42 25012 32 230 1.05 100012 28 263 1.20 100012 24 306 1.40 50012 20 368 1.68 100012 18 408 1.86 100012 16 459 2.09 1000TABLE I: Lattice volumes, temperatures and number of con-figurations used in this simulation. The separation betweenconfigurations is 10 HMC trajectories, except for the Nτ = 80runs where configurations were separated by 5 trajectories.The lattice spacings are as ∼ 0.162fm, aτ ' 7.35 ± 0.03GeV.Spectral functions were computed using the maxi-mum entropy method as described in [13], with themodified kernel introduced in Ref. [14]. The charmo-nium correlators were computed using all-to-all propaga-tors with dilution in time, color and varying levels inspace and spin. The momentum values are given by200.511.522.5m-m(ηc) [GeV]0-+ 1-- 1+- 0++ 1++ 2++ 2-+ 2-- 3--DD-m(ηc)mexp-m(ηc)expJPCFIG. 1: The mass splittings of the low-lying charmoniumspectrum at zero temperature, with the known experimentalvalues and the DD¯ threshold also indicated.p2 =(2piasNs)2n2, with n2 = 0, 1, 2, 3, 4, correspondingto p = 0, 0.66, 0.93, 1.14, 1.32 GeV. We consider only thepseudoscalar (ηc) and vector (J/ψ) channels. Our corre-lators have not been renormalised, so the vertical scaleon the figures presented is arbitrary, but the shape of thecurves is not.III. RESULTSA. Zero-temperature spectrumIn order to establish a baseline for our high-temperature results, we first present results for the spec-trum and dispersion relation at zero temperature. Takingadvantage of the all-to-all propagators, we used the varia-tional method [15, 16] with a large basis of interpolatingoperators to compute the low-lying spectrum of S-, P-and D-waves. This is shown in Fig. 1, along with the ex-perimentally known masses. The operator basis used inthe spectrum study is summarized in Table II. Takingthe difference between the spin-averaged 1S masses andthe 1P1(hc) mass to set the scale, we find the inverse lat-tice temporal lattice spacing to be a−1τ = 7.35(3)GeV[19].In Figure 1 the highest lying radial excitations in eachchannel are colored red to indicate that they are contam-inated by higher excited states. The difference betweenour results and experiment is largely due to the fairlycoarse spatial lattice spacing used. Including a cloverterm in the action has been found to improve the agree-ment with experiment [17].Figure 2 shows the ground-state pseudoscalar (ηc) dis-persion relation. The bare anisotropy for the valencecharm quarks is ξc0 = 5.90, which on the 83× 80 latticehad given a renormalised anisotropy of 6. However, onthe larger (123×80) lattice we find that the dispersion re-lation gives us a renormalised anisotropy of 5.829±0.05,giving rise to a 3% systematic error in our final results.JPC 2S+1LJ State Operators0−+ 1S0 ηc, η′c γ5,γ5∑isi1−− 3S1 J/Ψ,Ψ(2S) γj ,γj∑isi1+− 1P1 hc,h′c γiγj ,γ5pj0++ 3P0 χc0 ,χ′c01, ~γ · ~p1++ 3P1 χc1 ,χ′c1γ5γi, ~γ × ~p2++ 3P2 χc2 ,χ′c2~γ × ~p, γ1p1 − γ2p22γ3p3 − γ1p1 − γ2p22−+ 1D2 11D2 γ5(s1 − s2),γ5(2s3 − s1 − s2)2−− 3D2 13D2 γj(si − sk), γ1t1 − γ2t22γ3t3 − γ1t1 − γ2t23−− 3D3 13D3 ~γ · ~tTABLE II: Operator basis used to obtain the T = 0 charmo-nium spectrum. The definitions of si,pi and ti are given inRef. [18].0 0.5 1 1.5as2p20.190.20.210.220.230.24aτ2 E2ηc, ξq0 = 5.9ξqR = 5.829(5)FIG. 2: The ηc dispersion relation.B. High-temperature spectral functionsIn order to facilitate comparisons between spectralfunctions at different temperatures and momenta, wehave determined the spectral functions ρ(ω, ~p;T ) at allT and ~p using the same default model, m(ω) = m0ω2with aτm0 = 6 and aτm0 = 4 for the pseudoscalar andvector channel respectively. These values are close tothe ‘best’ values determined from a one-parameter fit ofthe correlator for each temperature and momentum con-sidered. We have repeated the analysis for a range ofdifferent default models to study the robustness of ourresults.As in Ref. [13], the first two timeslices were discardedin the analysis since these will be dominated by latticeartefacts. The effect of varying the time range has alsobeen investigated.Figure 3 shows the spectral function in both channelsat zero momentum for the various temperatures. Theseresults may be compared to Figs. 5 and 6 in Ref. [13].In the pseudoscalar channel, our results suggest that the32 4 6 8 10ω (GeV)01020304050ρ(ω)/ω2T = 230MeVT = 263MeVT = 306MeVT = 368MeVT = 408MeVT = 459MeVT = 0ηcp = 0.00GeV2 4 6 8 10ω (GeV)01020304050ρ(ω)/ω2T = 230MeVT = 263MeVT = 306MeVT = 368MeVT = 408MeVT = 459MeVT = 0J/ψp = 0.00GeVFIG. 3: The pseudoscalar (top) and vector(bottom) spectralfunction at zero momentum for various temperatures.ground state (ηc) survives up to the highest tempera-tures accessible in our simulations, while the J/ψ (vec-tor state) appears to dissolve for T >∼ 370MeV or 1.7Tc.It must be noted, however, that the uncertainty in theMEM procedure increases as the temperature increases,and it is therefore not possible at present to say for cer-tain whether the pattern observed at the highest tem-peratures is genuine or merely reflects the inability ofthe MEM algorithm to determine the spectral functiongiven the available data. This remains the case for allmomenta and in all channels.Figure 4 shows the pseudoscalar spectral function fordifferent momenta at T = 230 MeV. Using τmin = 3(lower panel), we see a clear structure of peaks orderedby momentum, with the separation between the peakscorresponding reasonably well to the zero-temperatureenergy levels. The peak positions, however, appear tobe shifted compared to the zero-temperature energies.This is most likely due to the maximum entropy methodnot being able to resolve the full detail of the spectralfunction for the available data. This is supported by theupper panel of Fig. 4, showing the spectral functions de-termined using τmin = 2. Here the peak position for thelowest non-zero momentum corresponds precisely to thezero-temperature energy, while for the other momenta2 3 4 5 6ω (GeV)01020304050ρ(ω)/ω2p = 0p = 0.66GeVp = 0.93GeVp = 1.14GeVp = 1.32GeVηcT=226MeV2 3 4 5 6ω (GeV)01020304050ρ(ω)/ω2p = 0p = 0.66GeVp = 0.93GeVp = 1.14GeVp = 1.32GeVηcT=226MeVFIG. 4: The pseudoscalar spectral function at nonzero mo-mentum for T = 230 MeV, using timeslices 2–16 (top) and3–16 (bottom). The zero-temperature energy levels are alsoshown for comparison.the peaks are smeared out and shifted to higher energies.Figure 5 shows the pseudoscalar spectral function fordifferent momenta at two higher temperatures. There isstill some evidence of a surviving ground state peak at allmomenta, but uncertainties in the MEM reconstructionmeans that we are no longer able to resolve the orderingof the states given our current precision.At nonzero momentum, the vector meson correlatoris decomposed into transverse and longitudinal polarisa-tions,Vij(τ, ~p) =(δij −pipjp2)VT (τ, ~p) +pipjp2VL(τ, ~p) . (1)The transversely and longitudinally polarised J/ψ mayin principle behave differently in the medium. We havetherefore analysed the two separately. Figures 6 and7 show the transverse and longitudinal spectral func-tions for our largest and smallest momentum, respec-tively. The longitudinal spectral function for the lowestmomentum (p = 0.66GeV) shows clear evidence of a sur-viving peak corresponding to the J/ψ ground state atthe two lowest temperatures, but there are indicationsof medium modifications already at T = 300MeV. The42 3 4 5 6ω (GeV)01020304050ρ(ω)/ω2p = 0p = 0.66GeVp = 0.93GeVp = 1.14GeVp = 1.32GeVηcT=362MeV2 3 4 5 6ω (GeV)01020304050ρ(ω)/ω2p = 0p = 0.66GeVp = 0.93GeVp = 1.14GeVp = 1.32GeVηcT=451MeVFIG. 5: The pseudoscalar spectral function at nonzero mo-mentum for T = 368 MeV (top) and T = 459 MeV (bottom).The zero-temperature energy levels are also shown for com-parison.transverse spectral function, on the other hand, appearsto be subject to medium modifications (indicated by thesofter ground state peak) already at the lowest tempera-ture, just above Tc.It is worth emphasising that the transverse spectralfunction at T = 230MeV, p = 0.66GeV displays littleor no sensitivity to the choice of model function, so weconsider our result in this case to be robust. At highermomenta and temperatures, the MEM reconstructionof spectral functions is no longer stable with regard tovariations in the model function and time range used.Our results for both transverse and longitudinal J/ψ atp = 1.32GeV should therefore be taken as no more thanindicative, even at the lowest temperature. With thisproviso, the indications are that the longitudinal J/ψsurvives at least up to 265MeV (1.2Tc) and is most likelymelted by T = 400MeV, while the transversely polarisedJ/ψ may experience medium modifications already closeto Tc.IV. DISCUSSIONWe have presented first results for charmonium spec-tral functions at nonzero momentum. Our results suggestthat the 1S pseudoscalar meson (ηc) survives up to tem-peratures close to twice the pseudocritical temperatureof QCD, for all momenta. No substantial momentumdependence was found.In the vector channel, there appears to be a distinctionbetween the transverse and longitudinal channels, withthe longitudinally polarised J/ψ experiencing smallermedium modifications. Again, no substantial momen-tum dependence was found, although the reconstructionof the spectral function became progressively more un-certain with increasing momentum. Therefore, some ad-ditional momentum dependence can not be ruled out.Whether the difference between transverse and longi-tudinal polarisations is real or merely a reflection of theuncertainty in the MEM procedure given the data usedin this study, still needs to be determined.We are in the process of generating configurations withsmaller lattice spacing. This will provide greater tem-poral resolution, leading to a more reliable determina-tion of spectral functions from imaginary-time correla-tors. The finer lattice is also expected to bring the zero-temperature spectrum in closer agreement with exper-iment, and allow a clearer separation between physicalfeatures and lattice artefacts in our spectral functions. Itwill also allow us to access higher temperatures, whereall model studies up to now have found the charmoniumground states to be dissolved.We are also computing correlators of the conserved vec-tor current, which will provide the correct nonperturba-tive renormalisation of the vector operator used in thisstudy, and permit a quantitative determination of thecharm quark diffusion rate and the charmonium contri-bution to the dilepton rate.AcknowledgmentsThis work was supported by SFI grant RFP-08-PHY1462, U. S. Department of energy under grant num-ber DE-FC06-ER41446 and by the U. S. National Sci-ence Foundation under grant numbers PHY05-55243 andPHY09-03571. We are grateful to the Trinity Centre forHigh-Performance Computing for their support. We havebenefited greatly from numerous discussions with Sine´adRyan and Mike Peardon.[1] F. Karsch, D. Kharzeev and H. Satz, Phys. Lett. B637,75 (2006), [hep-ph/0512239].[2] P. Braun-Munzinger and J. Stachel, Phys. Lett. B490,196 (2000), [nucl-th/0007059].52 3 4 5 6ω (GeV)0102030ρ(ω)/ω2T = 230MeVT = 263MeVT = 306MeVT = 368MeVT = 408MeVT = 459MeVT = 0J/ψ (T)p = 0.63GeV2 3 4 5 6ω (GeV)0102030ρ(ω)/ω2T = 230MeVT = 263MeVT = 306MeVT = 368MeVT = 408MeVT = 459MeVT = 0J/ψ (L)p = 0.63GeVFIG. 6: The transverse (left) and longitudinal (right) vector meson spectral function at p = 0.66GeV for various temperatures.2 3 4 5 6ω (GeV)01020304050ρ(ω)/ω2T = 230MeVT = 263MeVT = 306MeVT = 368MeVT = 408MeVT = 459MeVT = 0J/ψ (T)p = 1.26GeV2 3 4 5 6ω (GeV)01020304050ρ(ω)/ω2T = 230MeVT = 263MeVT = 306MeVT = 368MeVT = 408MeVT = 459MeVT = 0J/ψ (L)p = 1.26GeVFIG. 7: The transverse (left) and longitudinal (right) vector meson spectral function at p = 1.32GeV for various temperatures.[3] R. L. Thews, M. Schroedter and J. Rafelski, Phys. Rev.C63, 054905 (2001), [hep-ph/0007323].[4] R. L. Thews and M. L. Mangano, Phys. Rev. C73,014904 (2006), [nucl-th/0505055].[5] L. Grandchamp and R. Rapp, Nucl. Phys. A709, 415(2002), [hep-ph/0205305].[6] T. Umeda, K. Nomura and H. Matsufuru, Eur. Phys. J.C39S1, 9 (2005), [hep-lat/0211003].[7] M. Asakawa and T. Hatsuda, Phys. Rev. Lett. 92, 012001(2004), [hep-lat/0308034].[8] S. Datta, F. Karsch, P. Petreczky and I. Wetzorke, Phys.Rev. D69, 094507 (2004), [hep-lat/0312037].[9] A. Jakovac, P. Petreczky, K. Petrov and A. Velytsky,Phys. Rev. D75, 014506 (2007), [hep-lat/0611017].[10] A. Mocsy and P. Petreczky, arXiv:0705.2559 [hep-ph].[11] P. Petreczky and D. Teaney, Phys. Rev. D73, 014508(2006), [hep-ph/0507318].[12] R. Morrin, A. O´ Cais, M. Peardon, S. M. Ryanand J.-I. Skullerud, Phys. Rev. D74, 014505 (2006),[hep-lat/0604021].[13] G. Aarts, C. Allton, M. B. Oktay, M. Peardon andJ.-I. Skullerud, Phys. Rev. D76, 094513 (2007),[arXiv:0705.2198 [hep-lat]].[14] G. Aarts, C. Allton, J. Foley, S. Hands and S. Kim, Phys.Rev. Lett. 99, 022002 (2007), [hep-lat/0703008].[15] M. Lu¨scher and U. Wolff, Nucl. Phys. B339, 222 (1990).[16] C. Michael, Nucl. Phys. B259, 58 (1985).[17] S. M. Ryan, private communication.[18] UKQCD, P. Lacock, C. Michael, P. Boyle and P. Row-land, Phys. Rev. D54, 6997 (1996), [hep-lat/9605025].[19] This replaces the value 7.23GeV reported in Ref. [13],which was obtained using preliminary values for the 1Sand 1P masses. Our final values for the ηc and J/ψ massesare also slightly too high as a result.

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Momentum-dependence of charmonium spectral functions from lattice QCD

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