Properties of J/ψ at T c : QCD second-order Stark effect
aa r X i v : . [ h e p - ph ] J a n Properties of
J/ψ at T c : QCD second-order Stark effect Su Houng Lee ∗ and Kenji Morita † Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Republic of Korea
Starting from the temperature dependencies of the energy density and pressure from lattice QCDcalculation, we extract the temperature dependencies of the electric and magnetic condensate near T c . While the magnetic condensate hardly changes across T c , we find that the electric condensateincreases abruptly above T c . This induces a small but an equally abrupt decrease in the mass of J/ψ , which can be calculated through the second-order Stark effect. Combining the present resultwith the previously determined QCD sum rule constraint, we extract the thermal width of
J/ψ above T c , which also increases fast. These changes can be identified as the critical behavior of J/ψ across T c associated with the phase transition. We find that the mass shift and width broadeningof J/ψ at 1.05 T c will be around −
100 MeV and 100 MeV respectively.
PACS numbers: 14.40.Gx,11.55.Hx,12.38.Mh,24.85.+p
Since the pioneering work by Hashimoto et al. [1] andthe seminal work by Matsui and Satz [2], many ex-perimental and theoretical works have been performedin the physics of
J/ψ suppression in heavy ion colli-sions. The subject has recently evolved into a new stageas lattice calculations using maximum entropy methods(MEM) found the peak structure to survive up to al-most 2 T c [3, 4], which was speculated before [5] as lat-tice calculations showed that non-perturbative nature ofQCD persists well above T c [6, 7]. This suggests thatthe sudden disappearance of J/ψ is not the direct sig-nature of QGP formation. Indeed, recent results fromRHIC on the suppression factors at different rapidity andhigher p T seem ever more confusing [8]. Hydrodynamiccalculations suggest that the initial temperature of theQGP formed at RHIC is in the order of 2 T c and will lastfor 3 to 4 fm/ c [9, 10]. Therefore, considering the for-mation time of charmonium states after the creation of¯ cc pair, it is crucial to know the detailed properties of J/ψ near T c to fully understand the suppression and/orenhancement of J/ψ in heavy ion collisions. Unfortu-nately, the present resolutions of the peak structure of
J/ψ from the lattice calculations based on MEM are farfrom satisfactory [11]. In fact, the peak is too broadto even discriminate between
J/ψ from ψ ′ . Moreover,the temperature region between T c to 2 T c is known tobe strongly interacting and therefore a non-perturbativemethod has to be implemented to consistently treat thecharmonium at this temperature region. In a previouswork [12], we have implemented QCD sum rules to in-vestigate the properties of J/ψ near T c . Although theresults were non-perturbative, only a constraint on thecombined mass decrease and width increase could be ob-tained. Here, we point out that the critical behavior ofQCD phase transition, could be identified with a criticalbehavior of electric condensate at T c , and then by makinguse of the QCD second order Stark effect, show that such ∗ Electronic address: [email protected] † Electronic address: [email protected] critical behavior can be translated to a sudden change inthe mass of
J/ψ across T c .We begin by characterizing the properties of thestrongly interacting quark-gluon plasma (sQGP) of thepure gluon theory across T c in terms of local opera-tors. This is accomplished by making use of the energy-momentum tensor, which has a symmetric traceless partand a trace part via the trace anomaly, T αβ = −ST ( G aαµ G aµβ ) + g αβ β ( g )2 g G aµν G aµν . (1)Here, a and α, β are the color and Lorentz indices re-spectively. The temperature dependence of the two in-dependent parts of the energy-momentum tensor can beobtained from the lattice measurement of energy densityand pressure at finite temperature. h T αβ i T = ( ε + p ) (cid:18) u α u β − g αβ (cid:19) + ( ε − p ) g αβ . (2)Here, u α is the four velocity of the heat bath. Thereforethe temperature dependencies of gluonic operators canbe identified with the pressure and energy density. Toleading order in coupling, we can identify the trace part − [ h α s π G aµν G aµν i T − h α s π G aµν G aµν i ] = M ( T ) and thenon-trace part h−ST G aαµ G aµβ ) i T = ( u α u β − g αβ ) M ( T ),where [12], M ( T ) = ( ε − p ) ,M ( T ) = ( ε + p ) , (3)and (cid:10) α s π G aµν G aµν (cid:11) ≡ G vac0 is the scalar gluon conden-sate in the vacuum. The lattice gauge theory resultfor ε and p in the pure SU(3) gauge theory was ob-tained from Ref. [13]. Figure 1 shows changes of M and M from their vacuum value scaled by their asymp-totic temperature dependence of T . One notes thatwhile M , which is also proportional to entropy densitytimes temperature, moderately reaches the asymptotictemperature dependence, M , also known as the interac-tion measure or the gluon condensate, suddenly increases T/T c (e-3p)/T (e+p)/T T/T c FIG. 1: M = ( ε − p ) and M = ( ε + p ) divided by T asfunctions of temperature. and then decreases at higher temperature. The stronglyinteracting nature of QGP is related to the large inter-action measure [15], which takes its maximum value ataround 1.1 T c . It should be noted that the temperaturedependence of the gluon condensate M extracted hereincludes both the perturbative and the non-perturbativeand thus the full temperature dependence. The suddenchange near T c is dominated by the sudden decrease ofthe non-perturbative part, which reduces to about halfof its vacuum value [7], while at higher temperatures itis dominated by the perturbative contributions [13, 16].For the heat bath at rest, one can rewrite the ther-mal expectation values of the dimension four operatorsof the energy-momentum tensor in Eq. (1) in terms ofelectric and magnetic condensate [17]. This is possibleafter making the following identification. D α s π ST ( G aαµ G aµβ ) E T ≡ α s ( T ) π D ST ( G aαµ G aµβ ) E T . (4)The scale dependence of the matrix element is transferredto the coupling constant. Therefore, we additionally needto know the temperature dependence of the coupling con-stant α s ( T ). Since we will be using the matrix element inthe operator product expansion (OPE) with the separa-tion scale relevant for the heavy bound state, we willuse the temperature dependent running coupling con-stant extracted from the lattice computation of the heavyquark free energy [14]. Then we find, D α s π E E T − D α s π E E = 211 M ( T ) + 34 α s ( T ) π M ( T ) , (5) D α s π B E T − D α s π B E = − M ( T ) + 34 α s ( T ) π M ( T ) . (6)Figure 2 shows the temperature dependence of h α s π E i T and h α s π B i T . One notes that there is a sudden -0.004-0.002 0 0.002 0.004 0.8 0.9 1 1.1 1.2 E a nd B C ond e n sa t es [ G e V ] T/T c ( α s / π )E ( α s / π )B FIG. 2: Electric and magnetic condensate near T c as functionsof temperature. increase in the electric condensate h α s π E i T , while themagnetic condensate h α s π B i T hardly changes above T c .This can be related to the fact that the area law behaviorof the space-time Wilson loop changes to the perimeterlaw above T c , while that of the space-space Wilson loopretains the area law behavior even above T c [6]. The con-nection comes in as the non perturbative behavior of arectangular Wilson loop in the S S direction can be re-lated to the non-vanishing gluon condensate h α s π G S S i via the operator product expansion [18]. Hence, one canconclude that critical behaviors of QCD phase transi-tion can be related to the sudden change in the electriccondensate. Such local changes will induce critical be-havior of a heavy quark system such as the J/ψ acrossthe phase transition, which can be obtained through theQCD second-order Stark effect.The perturbative QCD formalism for calculating theinteraction between heavy quarkonium and partons wasfirst developed by Peskin [19, 20] in the non-relativisticlimit. The formula for the mass shift reduces to thesecond-order Stark effect in QCD, which was used pre-viously to calculate the mass shift of charmonium innuclear matter [17]. The information needed from themedium is the electric field square. As the dominantchange across the phase transition is the electric conden-sate, one notes that the second-order Stark effect is themost natural formula to be used across the phase transi-tion.The second-order Stark effect for the ground statecharmonium with momentum space wave function nor-malized as R d p (2 π ) | ψ ( p ) | = 1 is as follows,∆ m J/ψ = − Z ∞ dk (cid:12)(cid:12)(cid:12)(cid:12) ∂ψ ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12) kk /m c + ǫ D α s π ∆ E E T = − π a ǫ D α s π ∆ E E T , (7) -0.3-0.25-0.2-0.15-0.1-0.05 0 0.9 0.95 1 1.05 1.1 1.15 1.2 δ m J / ψ [ G e V ] T/T c J/ ψ Stark α qq , ξ =1 α qq , ξ =2 α qq , ξ =3 FIG. 3: Mass shift from the second-order Stark effect (solidline) and the maximal mass shift obtained from QCD sumrules from Ref. [12] (points). where k = | k | and h α s π ∆ E i T denotes the value of changeof the electric condensate from its vacuum value. The sec-ond line is obtained for the Coulomb wave function. Here, ǫ is the binding energy and m c the charm quark mass.These parameters are fit to the size of the wave functionobtained in the Cornell potential model [21], and to themass of J/ψ assuming it to be a Coulombic bound statein the heavy quark limit [19]. The fit gives m c = 1704MeV, a = 0 .
271 fm and α s = 0 .
57. Few comments are inorder. The minus sign in Eq. (7) is a model independentresult and follows from the fact that the second-orderStark effect is negative for the ground state. The factorof Bohr radius square a follows from the dipole nature ofthe interaction, and the binding energy ǫ from the inversepropagator, characterizing the separation scale [19, 22].Therefore, the actual value of the mass shift does not de-pend much on the form of the wave function as long asthe size of the wave function is fixed. The Bohr radiusused in our calculation corresponds to h r i / = 0 .
47 fm,which is the size of a more realistic wave function in theCornell potential [21]. Therefore, the correction comingfrom using a more realistic wave function should be small.The solid line in Fig. 3 shows the mass shift obtainedfrom the second-order Stark effect. The second-orderStark effect formula is based on the operator productexpansion (OPE) for bound state. As mentioned before,the formalism was first established by Peskin in 1979 andthe separation scale is the binding ǫ = mg . A more sys-tematic derivation was developed recently by Brambilaet al. [23] for the bound state. Here, the relevant scalesare mv and mv , where the former is related to the po-tential 1 /r and later the kinetic energy p /m . This scale mv is the separation scale so that for effects with typ-ical momentum larger than the separation scale shouldbe taken into account through resummed perturbationby solving the Schr¨odinger equation[22], while that withsmaller scale should be taken into account through the operator product expansion. The operators contain thenon-perturbative physics of QCD, which is typically of or-der Λ QCD . Therefore the OPE for the bound state worksbest when mv ≫ Λ QCD [23]. While there are concernsthat this condition is marginal for charmonium, the ap-proach should provide a quantitative description. Nowthe question is, which approach should one take for thethermal interactions near T c . Obviously, the effects of fi-nite temperature involves new scales like the color screen-ing. However, as has been known for some time, the tem-perature region from T c to 2 T c , is known to be stronglyinteraction and can not be described by resumed pertur-bation [24]. Therefore, the approach we want to take isto calculate the non perturbative temperature effect tothe mass shift through the operator product expansion.The leading order contribution in this approach is comingfrom (cid:10) α s π ∆ E (cid:11) T as is given in Eq. (7), whose temper-ature dependence we extract directly from the lattice.The arguments for convergence of higher dimensionaloperators in our approach are twofold. First, we willrestrict to the temperature region where the change in (cid:10) α s π ∆ E (cid:11) T is smaller than the vacuum value of (cid:10) α s π E (cid:11) itself. We believe that then the OPE is under controlas has been verified by the typical QCD sum rule ap-proaches for heavy quark system in the vacuum. As canbe seen in Fig. 2, this condition restricts our applicabil-ity to 1.05 T c , as in the QCD sum rule approach at finitetemperature [12, 25]. Second, a more direct evidencecomes from the next term in the OPE correction, whichcomes from magnetic condensate. However, as can beseen from Fig. 2, the changes of (cid:10) α s π ∆ B (cid:11) T and hencethe next term in the OPE should be small up to 1.05 T c and slightly beyond. Therefore, we can conclude that thesecond-order Stark effect should be valid near T c . As canbe seen in Fig. 3, the results from second-order Stark ef-fect shows that the mass reduces abruptly above T c andbecomes smaller by about 100 MeV at 1.05 T c , reflectingthe critical behavior of the QCD phase transition.We put the present result in perspective with a non-perturbative result obtained before using the QCD sumrules [12, 25]. The points in Fig. 3 represent the max-imum mass shift obtained in Refs. [12, 25]. As canbe seen in the figure, the mass shift obtained from thesecond-order Stark effect is almost the same as the max-imum mass shift obtained in the sum rule up to T c and then becomes smaller. The mass shift at T c isabout −
50 MeV. In the QCD sum rules, only a con-straint for the combined mass shift and thermal widthof − ∆ m + Γ T ≃
80 + 17( T − T c ) MeV could be ob-tained within the temperature range from T c to 1 . T c .Therefore, the difference between the Stark effect and themaximum mass shift obtained from QCD sum rules above T c in Fig. 3 could be attributed to the non-perturbativethermal width at finite temperature. In Fig. 4, we plotthe thermal width obtained from combining the QCDsum rule constraint with the mass shift obtained fromthe QCD second-order Stark effect.[37] As can be seen inthe figure, the thermal width at 1.05 T c becomes as large Γ J / ψ [ M e V ] T/T c ξ =123 FIG. 4: Thermal width of
J/ψ obtained from the second-orderStark effect and QCD sum rule constraint. as 100 MeV. Such width slightly above T c is larger thanthat estimated from a perturbative LO and NLO QCDmethod [26, 27], but smaller than a recent phenomeno-logical estimate [28]. The mass of quarkonium at finitetemperature was also investigated in the potential mod-els [29], where the mass was found to decrease at hightemperature. However, the detailed potential has to beextracted from the lattice at each temperature and henceidentifying the critical behavior near T c will be difficult.Finally, we comment that while the present result isobtained using lattice calculation in the pure gauge the-ory, including dynamical quark will not greatly modifythe result. This follows from noting that the lattice re-sult for the temperature dependence of pressure p/T isindependent of the number of flavors if scaled to their cor-responding ideal gas limit [30]. Moreover, as was shownin Ref. [31], G ( T ) extracted from a recent full lattice cal-culation of the interaction measure [32] after subtractingthe quark contributions, and then dividing by a factor of (1+ n f ) appearing in the beta function, shows that thechange of the magnitude near T c is remarkably similar tothat of the pure gauge theory. Hence the main input forour result does not change much even in the presence ofdynamical quarks.In summary, we have shown that the sudden increasein the energy density across the phase transition, whichis a characteristic behavior of the QCD phase transitionindependent of the flavor, can be translated to a rapid in-crease in the electric condensate slightly above T c . Usingthe QCD second-order Stark effect, this translates intoan equally sudden decrease in the mass of J/ψ , whichis around −
50 MeV and −
100 MeV respectively at T c and 1.05 T c . Combining with a QCD sum rule constraint,we obtain the thermal width of J/ψ slightly above T c ,and found it to be larger than previous perturbative es-timates, and becoming as much as 100 MeV at 1.05 T c .Hence, one can conclude that the critical behavior of J/ψ at T c is not its sudden disappearance, but rather theabrupt changes of its mass and width. The mass shiftis probably too small to be detected with present reso-lutions at RHIC. However, with the expected upgradesat RHIC and plans at LHC, such direct measurementcould be possible. Indeed, the mass resolution of J/ψ for dimuon channel at LHC is 35 MeV for the CMS de-tector [33] and around 70 MeV for ALICE [34] and AT-LAS [35]. It might be better for dielectron channel. Fur-thermore, the mass shift could also influence productionrates within the statistical model [36]. The large widthof 100–150 MeV already at 1.05 T c suggest that while themaximal entropy method shows a J/ψ peak structuresurviving up to 2 T c , the actual formation at heavy ioncollisions might only be possible at lower temperatureswhere the width of the J/ψ becomes equal to its bind-ing.This work was supported by the Korean Ministry ofEducation through the BK21 Program and KRF-2006-C00011. [1] T. Hashimoto, K. Hirose, T. Kanki and O. Miyamura,Phys. Rev. Lett. , 2123 (1986).[2] T. Matsui and H. Satz, Phys. Lett. B , 416 (1986).[3] M. Asakawa and T. Hatsuda, Phys. Rev. Lett. , 012001(2004)[4] S. Datta et al. , Phys. Rev. D , 094507 (2004)[5] T. H. Hansson, S. H. Lee and I. Zahed, Phys. Rev. D ,2672 (1988).[6] E. Manousakis and J. Polonyi, Phys. Rev. Lett. , 847(1987).[7] S. H. Lee, Phys. Rev. D , 2484 (1989).[8] A. Adare et al. , (PHENIX Collaboration), Phys. Rev.Lett. , 232301 (2007).[9] P. F. Kolb and U. W. Heinz, in Quark-Gluon Plasma 3 ,edited by R. C. Hwa and X. N. Wang (World Scientific,2004), p. 643. [10] K. Morita, Braz. J. Phys. , 1039 (2007)[11] A. Jakov´ac, P. Petreczky, K. Petrov, and A. Velytsky,Phys. Rev. D , 014506 (2007).[12] K. Morita and S. H. Lee, Phys. Rev. Lett. , 022301(2008).[13] G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Leg-eland, M. L¨utgemeier, and B. Petersson, Nucl. Phys. B469 , 419 (1996).[14] O. Kaczmarek, F. Karsch, F. Zantow, and P. Petreczky,Phys. Rev. D , 074505 (2004) [Erratum-ibid. D ,059903 (2005)].[15] F. Karsch, D. Kharzeev, and K. Tuchin, Phys. Lett.B , 217 (2008).[16] D. E. Miller, Phys. Rept. , 55 (2007).[17] M. E. Luke, A. V. Manohar and M. J. Savage, Phys. Lett.B , 355 (1992). [18] M. A. Shifman, Nucl. Phys. B173 , 13 (1980).[19] M. E. Peskin, Nucl. Phys.
B156 , 365 (1979).[20] G. Bhanot and M. E. Peskin, Nucl. Phys.
B156 , 391(1979).[21] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, andT. M. Yan, Phys. Rev. D , 203 (1980).[22] Y. S. Oh, S. Kim, and S. H. Lee, Phys. Rev. C , 067901(2002).[23] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Rev.Mod. Phys. , 1423 (2005).[24] J.-P. Blaizot, E. Iancu and A. Rebhan, Phys. Lett. B , 181 (1999).[25] K. Morita and S. H. Lee, Phys. Rev. C , 064904 (2008).[26] Y. Park, K. I. Kim, T. Song, S. H. Lee, and C. Y. Wong,Phys. Rev. C , 044907 (2007).[27] T. Song, Y. Park, S. H. Lee and C. Y. Wong, Phys. Lett.B , 621 (2008).[28] ´A. M´ocsy and P. Petreczky, Phys. Rev. Lett. , 211602 (2007).[29] W. M. Alberico, A. Beraudo, A. De Pace, and A. Moli-nari, Phys. Rev. D , 074009 (2007).[30] F. Karsch, Lect. Notes Phys. , 209 (2002).[31] S. H. Lee and K. Morita, J. Phys. G: Nucl. Part. Phys. ,104024 (2008).[32] M. Cheng et al. , Phys. Rev. D , 014511 (2008).[33] D. Dutta, talk given at Quark Matter 2008.[34] B. Alessandro et al. , (ALICE Collaboration), J. Phys. G:Nucl. Part. Phys. , 1295 (2006).[35] A. Lebedev, talk given at Quark Matter 2008.[36] A. Andronic, P. Braun-Munzinger, K. Redlich, andJ. Stachel, Proc. Sci. CPOD07, 044 (2007).[37] We have improved the QCD sum rule calculation bytaking into account running mass effect properly. Thischanges the result of ξ6