The charmonium states at non-zero temperature are studied on anisotropic lattices with 2 dynamical

quark flavours. Non-local operators are used to determine the Nambu-Bethe-Salpeter (NBS)

wavefunctions via both conventional fitting methods and the Maximum Entropy Method. The

interquark potential is determined from the solution of the Schrodinger equation, given the NBS

wavefunction as input following the HAL QCD method. We observe a temperature dependent

potential which becomes steeper as the temperature decreases

Allton, Chris

Evans, Wynne

Skullerud, Jon-Ivar

English

MURAL - Maynooth University Research Archive Library

PoS(Lattice 2012)082Charmonium Potentials at Finite TemperatureChris Allton∗Swansea UniversityE-mail: c.allton@swan.ac.ukWynne EvansSwansea UniversityE-mail: pyevans@swan.ac.ukJon-Ivar SkullerudNational University of Ireland MaynoothE-mail: jonivar@thphys.nuim.ieThe charmonium states at non-zero temperature are studied on anisotropic lattices with 2 dynam-ical quark flavours. Non-local operators are used to determine the Nambu-Bethe-Salpeter (NBS)wavefunctions via both conventional fitting methods and the Maximum Entropy Method. Theinterquark potential is determined from the solution of the Schrodinger equation, given the NBSwavefunction as input following the HAL QCD method. We observe a temperature dependentpotential which becomes steeper as the temperature decreases.The 30th International Symposium on Lattice Field Theory - Lattice 2012June 24-29, 2012Cairns, Australia∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(Lattice 2012)082Charmonium Potentials at Finite Temperature Chris Allton1. IntroductionThe quark-gluon plasma phase of QCD has been studied experimentally at both RHIC and theLHC. However, a full theoretical understanding of this phase is still being developed. One quantitywhich can aid this understanding is the potential between quarks as a function of temperature. Dueto the widely cited J/ψ suppression [1], it is natural to consider the interquark potential of thecharmonium system.There has been theoretical work studying the interquark potential in quarkonia as a functionof temperature from early models [2] to perturbative QCD calculations [3]. Furthermore, therehave been some recent non-perturbative QCD studies (i.e. using lattice simulations) of interquarkpotentials which is relevant to the work presented here. These fall into two categories: (i) non-zero temperature studies of the static quark potential [4]; and (ii) zero temperature studies of thepotential between quarks with finite masses [5]. This work presents a study of the interquarkpotential of charmonium using physical charm quark masses at finite temperature and uses two-flavours of dynamical quarks. A particular feature of our work is that our lattices are anisotropicwhich has the significant advantage in that our correlation function data is determined at a largenumber of temporal points.Our method follows the HAL QCD method [6] of determining the potential in lattice simu-lations.1 This is based on using the Nambu-Bethe-Salpeter (NBS) wavefunction as input into theSchrödinger equation and solving for the potential.In this work we use two methods to determine the wavefunction from our lattice simulations,conventional exponential fits and the Maximum Entropy Method. We find that both approaches givequalitatively similar results. Our main conclusion is that we observe a temperature dependence inthe charmonium potential which is consistent with expectations, i.e. the potential is steepest forlow temperatures, T , and shows signs of flattening at large distances as T increases.We are in the process of extending our work by using the “time-dependent” approach of HALQCD [8] which determines the potential directly from the hadron correlation functions [9].2. Schrödinger Equation ApproachWe follow the HAL QCD method to determine the potential [6]. We begin by determining theNBS wavefunction of charmonium ψ(r∼) from the (t → ∞ behaviour of the) correlators of point-split operators, J(x, r∼) = q(x)ΓU(x,x+ r∼)q(x+ r∼),C(r∼, t) = ∑x∼< J(0; r∼) J(x; r∼)> −→ |ψ(r∼)|2 e−Et . (2.1)The NBS wavefunction is determined from the matrix element of the point-split operator, ψ(r∼) ∼〈0|J(r∼)|gnd〉. Once the NBS wavefunctions and energies, E , are extracted from the correlators (seesection 4) we use the Schrödinger equation to solve for the potential V (r),1The original HAL QCD programme studied internucleon rather than interquark potentials from lattice QCD sim-ulations.2PoS(Lattice 2012)082Charmonium Potentials at Finite Temperature Chris Allton(−∇22µ +V(r))ψ(r) = E ψ(r). (2.2)Here, µ = 12mq '14MJ/ψ , is the reduced mass. We note that this is the opposite approach nor-mally associated with the Schrödinger equation, i.e. the potential is “reverse engineered” from theinputted wavefunction.3. Lattice Parameters and CorrelatorsOur lattices are generated with two dynamical flavours of light quarks using a Wilson-typeaction with anisotropy of ξ = as/aτ = 6 with as ' 0.162fm and a−1τ ' 7.35GeV [7]. Table 3 liststhe lattice parameters used. We note that the range of temperatures, T = 1/(aτ Nτ), is from theconfined phase up to ∼ 2Tc where Tc is the deconfining transition. The charm quark is simulatedwith the (anisotropic) clover action and its mass is set by matching the experimental ηc mass atzero temperature.Ns Nτ T (MeV) T/Tc Ncfg12 80 90 0.42 25012 32 230 1.05 100012 28 263 1.20 100012 24 306 1.40 50012 20 368 1.68 100012 16 458 2.09 1000Table 1: Lattice parameters used, including spatial and temporal dimension, Ns and Nτ .In Fig.1 we plot the pseudoscalar charmonium correlation functions for quark separations r atvarious temperatures, see table 3. Only on-axis separations, r∼, were studied in this work.4. Wavefunctions and Potentials from Exponential FitsWe extract the NBS wavefunction, ψ(r), discussed in section 2 using a standard exponentialfit of the point-split correlators at large t, C(r, t) = |ψ(r)|2 e−Et (see eq.(2.1)). The normalisedwavefunctions are plotted in Fig.2 for both the pseudoscalar (ηc) and vector (J/ψ) channels. Ascan be seen, the expected behaviour for s-wave ground states is observed with the maximum ofψ(r) at r = 0, and ψ(r)→ 0 as r → 0.We use eq.(2.2) to determine the potential for both the pseudoscalar and vector channels.Figure 3 shows the spin-independent potential, Vqq(r), definedVqq(r) =14[VPS(r)+3VV (r)]. (4.1)As can be seen from Fig. 3, there is evidence of a temperature dependency in the potential: thepotential flattens as the temperature increases. This is in accord with expectations. Note however,that the Nt = 16 correlator fits cannot be made over a large time range due to the brevity of the3PoS(Lattice 2012)082Charmonium Potentials at Finite Temperature Chris Allton0 2 4 6 8t-10-8-6-4-202log(C(t))r=0r=1r=2r=3r=4r=5r=6Nt = 160 2 4 6 8 10 12t-14-12-10-8-6-4-202log(C(t))r=0r=1r=2r=3r=4r=5r=6Nt = 240 2 4 6 8 10 12 14 16t-20-15-10-50log(C(t))r=0r=1r=2r=3r=4r=5r=6Nt = 320 10 20 30 40t-30-25-20-15-10-50log(C(t))r=0r=1r=2r=3r=4r=5r=6Nt = 80Figure 1: Pseudoscalar charmonium correlators of point-split operators for various temperatures.lattice in the temporal direction. For this reason we associate some uncertainty with this ensemble(and these points are depicted with open symbols in Fig. 3).5. Wavefunctions from MEMThe Maximum Entropy Method (MEM) has been used by many lattice studies to extract spec-tral information from correlators [10]. The fundamental equation isC(r, t) =∫ρ(r,ω) K(t,ω) dω , (5.1)where ρ is the spectral function and the lattice kernel isK(t,ω) =cosh[ω(t−Nt/2)]sinh[ωNτ/2]. (5.2)Figure 4 shows the spectral function obtained via MEM from the correlators of point-split opera-tors, C(r, t), at the highest temperature. The vertical bands in Fig. 4 are placed at the ground andfirst excited states’ masses as obtained from an MEM analysis at lower temperatures. We estimatethe NBS wavefunction from the integral of ρ(ω) over this energy interval. Figure 2 includes thewavefunctions obtained in this way. As can be seen there is reasonable agreement between theMEM determination and the exponential fit approach from section 4.4PoS(Lattice 2012)082Charmonium Potentials at Finite Temperature Chris Allton0 0.2 0.4 0.6 0.8 1r [fm]00.20.40.60.8|ψ(r)|2PS expV expPS MEMV MEMNt = 160 0.2 0.4 0.6 0.8 1r [fm]00.20.40.60.8|ψ(r)|2PS expV expPS MEMV MEMNt = 240 0.2 0.4 0.6 0.8 1r [fm]00.20.40.60.8|ψ(r)|2PS expV expPS MEMV MEMNt = 320 0.2 0.4 0.6 0.8 1r [fm]00.20.40.60.8|ψ(r)|2PS expV exp Nt = 80Figure 2: Pseudoscalar (PS) and vector (V) normalised wavefunctions, ψ(r), for various temperatures. Boththe results from the exponential and MEM fits are shown.0 0.2 0.4 0.6 0.8 1r [fm]00.511.52Vqqbar(r) [GeV]803228242016Figure 3: The spin-independent potential, Vqq, for various temperatures. Statistical errors only are shown,and the points are shifted horizontally for clarity. A vertical constant was added to each temperature’spotential to align the first point, i.e. Vqq(r/as = 1) is defined to be zero. The Nt = 16 points are depictedwith open symbols due to the concerns about their fits as discussed in the text.5PoS(Lattice 2012)082Charmonium Potentials at Finite Temperature Chris Allton0 1 2 3ω01020304050ρ(ω)/ωr=0r=1r=2r=3r=4r=5r=6Nt = 16Figure 4: The spectral function obtained from MEM for the Nτ = 16 case for various separations r. Thevertical bands are placed at the position of the ground and excited states.Note that the estimate of first excited state’s wavefunction from the second vertical band inFig. 4 decreases for small separations, r, as r → 0, as expected for an excited state.6. Conclusions and OutlookThere is a significant body of theoretical work studying the interquark potential in charmoniumat non-zero temperature using model, perturbative and lattice (non-perturbative) approaches. Thiswork improves upon earlier lattice simulations by considering quarks with finite mass, and thusrepresents a first-principle calculation of the charmonium potential of QCD at finite temperature.The method we use is based on the HAL QCD approach which obtains the potential from correla-tors of point-split operators [6]. Our determination of the potential shows the expected flatteningas the temperature increases. This work adds to previous charmonium studies performed by ourcollaboration with the same lattice parameters [11].We’ve used two different methods of fitting the point-split correlators: conventional fits toexponentials and the MEM. There is qualitative agreement between the wavefunctions determinedfrom both. However, it is clear from Fig 4 that the (high temperature) correlators, are not a simplesum of discrete exponentials, i.e. each spectral feature has a finite width. For this reason, fittingthe correlator to an exponential (as in section 4) is an approximation. A better approach would beto determine the wavefunction from MEM as described in section 5. However, it is necessary tohave symmetric correlators2 when using MEM to maintain positive semi-definite spectral weights,ρ . Since the operators in question in this study are point-split and therefore are inherently noisy,demanding symmetric correlators increases the noise substantially compared to non-symmetriccorrelators.In forthcoming work [9] we will use the HAL QCD “time dependent” approach [8] whichallows us to calculate the potential directly from the correlators, thus circumventing the problemsdiscussed in both the exponential and MEM fitting procedures. We also will be studying signif-2i.e. correlators of the same operator at the source and sink6PoS(Lattice 2012)082Charmonium Potentials at Finite Temperature Chris Alltonicantly larger lattices (with a spatial volume of 323) with 2+1 flavours of dynamical quarks andhope to extend our work to determine the potential between NRQCD quarks [12].7. AcknowledgementsWe acknowledge the support and infrastructure provided by the Trinity Centre for High Per-formance Computing and the IITAC project funded by the HEA under the Program for Research inThird Level Institutes (PRTLI) co-funded by the Irish Government and the European Union. Thework of CA and GA is carried as part of the UKQCD collaboration and the DiRAC Facility jointlyfunded by STFC, the Large Facilities Capital Fund of BIS and Swansea University. WE and CAare supported by STFC. CRA thanks the Galileo Galilei Institute for Theoretical Physics for hos-pitality and the INFN for support during the writing up of this work. We are very grateful to SinyaAoki, Balint Joó and Robert Edwards for useful discussions.References[1] T. Matsui and H. Satz, Phys. Lett. B 178 (1986) 416.[2] F. Karsch, M. T. Mehr and H. Satz, Z. Phys. C 37 (1988) 617.[3] Y. Burnier, M. Laine and M. Vepsalainen, JHEP 0801 (2008) 043 [arXiv:0711.1743 [hep-ph]],N. Brambilla, J. Ghiglieri, A. Vairo and P. Petreczky, Phys. Rev. D 78 (2008) 014017[arXiv:0804.0993 [hep-ph]], A. Dumitru, Y. Guo, A. Mocsy and M. Strickland, Phys. Rev. D 79(2009) 054019 [arXiv:0901.1998 [hep-ph]].[4] A. Rothkopf, T. Hatsuda and S. Sasaki, PoS LAT 2009 (2009) 162 [arXiv:0910.2321 [hep-lat]], Phys.Rev. Lett. 108 (2012) 162001 [arXiv:1108.1579 [hep-lat]], A. Rothkopf, arXiv:1207.5486 [hep-ph],Y. Burnier and A. Rothkopf, Phys. Rev. D 86 (2012) 051503 [arXiv:1208.1899 [hep-ph]]. A. Bazavovand P. Petreczky, arXiv:1210.6314 [hep-lat], P. Petreczky, C. Miao and A. Mocsy, Nucl. Phys. A 855(2011) 125 [arXiv:1012.4433 [hep-ph]].[5] T. Kawanai and S. Sasaki, Phys. Rev. D 85 (2012) 091503 [arXiv:1110.0888 [hep-lat]], PoSLATTICE 2011 (2011) 126 [arXiv:1111.0256 [hep-lat]], Phys. Rev. Lett. 107 (2011) 091601[arXiv:1102.3246 [hep-lat]], Y. Ikeda and H. Iida, PoS LATTICE 2010 (2010) 143 [arXiv:1011.2866[hep-lat]], PoS LATTICE 2011 (2011) 195.[6] N. Ishii, S. Aoki and T. Hatsuda, Phys. Rev. Lett. 99 (2007) 022001 [nucl-th/0611096], S. Aoki,T. Hatsuda and N. Ishii, Prog. Theor. Phys. 123 (2010) 89 [arXiv:0909.5585 [hep-lat]], S. Aoki et al.[HAL QCD Collaboration], arXiv:1206.5088 [hep-lat].[7] M. B. Oktay and J. -I. Skullerud, arXiv:1005.1209 [hep-lat].[8] N. Ishii [HAL QCD Collaboration], PoS LATTICE 2011 (2011) 160.[9] W. Evans, C.R. Allton, J.-I. Skullerud, in preparation.[10] M. Asakawa, T. Hatsuda and Y. Nakahara, Prog. Part. Nucl. Phys. 46 (2001) 459 [hep-lat/0011040].[11] G. Aarts, C. Allton, M. B. Oktay, M. Peardon and J. -I. Skullerud, Phys. Rev. D 76 (2007) 094513[arXiv:0705.2198 [hep-lat]].[12] G. Aarts, C. Allton, S. Kim, M. P. Lombardo, M. B. Oktay, S. M. Ryan, D. K. Sinclair andJ. -I. Skullerud, arXiv:1210.2903 [hep-lat].7

Charmonium Potentials at Finite Temperature

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Charmonium Potentials at Finite Temperature

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