Fast Heating Dissociation of Upsilon(1S) in Heavy Ion Collisions at RHIC
aa r X i v : . [ nu c l - t h ] J u l Fast Heating Dissociation of
Υ(1 S ) in Heavy Ion Collisions at RHIC Yunpeng Liu and Baoyi Chen ∗ Department of Applied Physics, School of Science, Tianjin University, Tianjin, 300072, China (Dated: July 10, 2020)With the adiabatic assumption in the cooling process, we discussed a new mechanism on Υ(1 S )suppression that is due to the fast heating process at the early stage of the fireball instead of itsfinite decay width in finite temperature medium produced in the heavy ion collisions. We calculatedthe transition probability after the fast heating dissociation as a function of the temperature of themedium and the nuclear modification factor in central collisions, and found that the suppression isnot negligible at RHIC, even if the width of Υ(1 S ) vanishes. The phase diagram of quark matter is one of the most interesting topics in high energy nuclear physics. At hightemperature and/or high baryon density, the state of the quark-gluon plasma (QGP) has been predicted and widelystudied both in theory [1–7] and in experiments [8–12]. Lots of synchrotrons and colliders are built for heavy-ioncollisions to produce the QGP. However, it is difficult to measure the temperature of the fireball directly, especially atits very early stage, because of its small size and short lifetime. From the statistical model [13], one can extract thetemperature from the spectrum or the yield of light hadrons. It seems that strange particles freeze-out earlier thanpions [14], which means that they carries information of the fireball at earlier time. Heavy quarkonia may even survivethe QGP due to their large binding energies and carry information from the early stage of the fireball. However, theycan hardly be thermalized, either in kinetics or in chemistry. Anomalous suppression of
J/ψ was suggested as a signalof the formation of QGP very early [15], which actually contains different processes such as color screening [16], gluonscattering [17–20], and quasi-free scattering from quarks [20, 21]. It is found that the inverse process, that is theregeneration of
J/ψ from charm quarks in QGP also plays an important role in relativistic heavy collisions at RHICand higher colliding energies [21–24], which makes the story more complicated.Although it is difficult to measure the temperature of the fireball at early time from the momentum distribution ofheavy quarkonia, the sequential dissociation model [16] offers another way to take heavy quarkonia as thermometersof the fireball, which assumes that a heavy quarkonium survives if and only if the temperature of the fireball is abovethe dissociation temperature T d of the quarkonium. Most of other dynamic studies focus on the dissociation rate (orthe width) of a heavy quarkonium at a certain temperature T [21–23, 25–30]. As a matter of fact, the sequentialdissociation model can also be regarded as a special case of the dynamic models with a dissociation rate that isinfinitely large above T d and zero below T d .However, even if the dissoicaiton rate of a heavy quarkonium vanishes at finite temperature, quarkonium suppressioncan still happen in heavy-ion collisions because of the fast heating process at the very early stage of the bulk medium,which is the main effect we try to discuss in this paper. In relativistic heavy ion collisions, the fireball reaches itshighest temperature within 1 fm/c, and cools down for much longer time to freeze-out finally. For simplicity, wetreat the initial heating process as a sudden process, and the cooling process as a very slow process. According tothe adiabatic theorem, the yield of heavy quarkonia keeps constant during the cooling process if the width of thequarkonia is negligible. Therefore the suppression is mainly due to the transition from initial heavy quarkonia toheavy quarkonia in the hot medium. In this case, we can map the nuclear modification factor of heavy quarkonia tothe temperature of the fireball after the fast heating process directly. As discussed in the previous paragraph, J/ψ at RHIC and at colliders with higher beam energies does not fit such a model, since both the scattering dissociationand the regeneration are very important. Therefore we will consider Υ(1 S ) instead in the following discussion andcalculations, since both its dissociation rate and its regeneration rate in medium are small at RHIC energy. [31, 32]The evolution of the wave function ψ ( r , t ) of Υ(1 S ) at time t and the relative radius r between the bottom quarkand the anti-bottom quark can be described by the Schroedinger equation i∂ t ψ ( r , t ) = (cid:20) − m b ∇ + V ( r, T ( t )) − i Γ( r , T ) (cid:21) ψ ( r , t ) , (1)where m b is the mass of a bottom quark, and we have taken ~ = 1. In the above the thermal fluctuation [33] isneglected. The corresponding stationary radial Schroedinger equation for a give temperature T writes (cid:20) − m b r d d r r + l ( l + 1) m b r + V ( r, T ) − i Γ( r, T ) (cid:21) ψ r ( r ) = Eψ r ( r ) , (2) ∗ Email: [email protected] where ψ r ( r ) is the radial wave function of Υ(1 S ), and E is the eigen energy of Υ(1 S ). For Υ(1 S ), the azimuthalquantum number is l = 0. To focus on the new mechanism, we neglect the particle scattering process and take thein-medium width Γ = 0. The potential V is taken in the form of a screened Cornell potential [34] V ( r ) = − αr e − µr − σ / Γ(3 / (cid:18) rµ (cid:19) / K / [( µr ) ] , (3)witr α = π , σ = 0 . [34]. The Γ and K above are the gamma function and the modified Bessel function,respectively. We have dropped the constant term that does not vanish at infinity for simplicity in V ( r ). Thetemperature T dependence comes from the screening mass µ . We fit the free energy of heavy quarks by the latticeQCD [34, 35], and parameterize the screening mass µ (scaled by √ σ ) as µ ( ¯ T ) √ σ = s ¯ T + aσ t r π (cid:20) erf (cid:18) b √ σ t (cid:19) − erf (cid:18) b − ¯ T √ σ t (cid:19)(cid:21) , (4)with ¯ T = T /T c , s = 0 . a = 2 . b = 1 . σ t = 0 . z ) = √ π R z e − x d x . Here T c is the critical temperature of the phase transition. =0.0T=1.0T=1.5T=2.0T=2.5T r r ψ (cid:9) (cid:9)(cid:9) (cid:9)(cid:9) FIG. 1: (Color Online)Scaled radial wave functions ¯ ψ r = m − b ψ of Υ(1 S ) as a function of scaled radius ¯ r = m b r , at differentscaled temperature ¯ T = T /T c . The radial eigen wave function is shown in Fig. 1. To be dimensionless, we scaled the radius and the wave functionas ¯ r = m b r , and ¯ ψ r = m − / b ψ r , respectively, resulting in R (cid:12)(cid:12) ¯ ψ r (cid:12)(cid:12) ¯ r d¯ r = 1. It can be seen that the wave function ofΥ(1 S ) at T = T c is similar to that at T = 0, while it becomes more and more broad at higher and higher temperature.The dissociation temperature is T d ≈ T c .The transition probability from a Υ(1 S ) at zero temperature to that at T is P ( ¯ T ) = |h ψ ( T ) | ψ (0) i| , (5)which is shown as a function of ¯ T = T /T c in Fig. 2. It decreases with ¯ T monototically, since the overlap between thewave function at finite temperature and that at zero temperature becomes small when ¯ T increases. It is very close tounit at ¯ T = 1 as already indicated by Fig. 1, and it vanishes at T d ≈ T c .Now we check the adiabatic approximation. At RHIC energy, the highest temperature of the fireball is around 2 T c when the system reaches local thermal equilibrium. We suppose that the temperature decreases with time linearlyfrom 2 T c to T c , and evolve the wave function ψ ( r , t ) of a Υ(1 S ) by Eq. (1) with its initial condition as an eigen Υ(1 S )at 2 T c . The survival probability as a function of time is shown in Fig. 3. The typical time for the fireball to cooldown to T c is 5 ∼
10 fm/ c . As one can see from the figure, the survival probability is about 0 .
98 when the evolutiontime is 10 fm/ c , which implies that the adiabatic approximation is very good in such a case. Even if we take a lowervalue of 5 fm/ c , the survival probability 0 .
93 is obviously larger than P (2 .
0) = 0 .
76 shown in Fig. 2. This result isqualitatively consistent with the result in Ref. [36], where the adiabatic approximation is examinded for Υ(1 S ) atLHC energy with a finite dissociation rate.Now we include the spacial distribution of temperature. In practice the temperature is not uniform in space. Thetemperature is high in the center of the fireball, while it is low in peripheral regions. Therefore the survival probability T)TP( (cid:9)
FIG. 2: Transition probability P [defined in Eq. (5)] of a Υ(1 S ) from temperature 0 to temperature T as a function of scaledtemperature ¯ T = T /T c . c T / T s u r v i va l p r ob a b ili t y time t(fm/c) FIG. 3: (Color Online) Upper panel: Different cooling systems with the medium temperature decreasing linearly with time.Lower panel: Time evolution of survival probability of Υ(1 S ) at different cooling speed calculated by Schroedinger equationwith an initial Υ(1 S ) at its eigen state at the initial temperature. for Υ(1 S ) is an average of all produced Υ(1 S )s. Since the production of Υ(1 S ) is a hard process, we assume that thedensity of produced Υ(1 S ) is proportional to the number density of binary collisions n c ( x T ) at transverse coordinate x T . Therefore we have R AA = R P ( ¯ T ( x T ))d N Υ(1 S ) R d N Υ(1 S ) = R P ( ¯ T ( x T )) n c ( x T )d x T R n c ( x T )d x T . (6)We assume that the entropy density s is proportional to the density of the number of participants n p , and regard thehot medium as ideal gas, so that the entropy density is also proportional to T . As a result, the spacial distributionof temperature is ¯ T ( x T ) = ¯ T ( ) (cid:18) n p ( x T ) n p ( ) (cid:19) / , (7)where ¯ T ( x T ) is the scaled local temperature T /T c at x T , and ¯ T ( ) is the scaled local temperature at x T = . Incentral collisions, the number density of participants n p and number density of binary collisions n c are n p ( x T ) = 2 T ( x T ) h − e − σ NN T ( x T ) i , (8) n c ( x T ) = σ NN T ( x T ) , (9)where σ NN is the inelastic cross section of nucleons, and T ( x T ) is the thickness function of a gold nucleus. Forsimplicity, we take a sharp-cut-off thickness function T ( x T ) = 3 A p R − x T πR , (10)where R and A are the radius and mass number of the nucleus, respectively. Substitute Eq. (7-10) to Eq. (6), weobtain the nuclear modification factor in central collisions R AA = 4 Z P ¯ T (0) s x − e − N m x − e − N m x d x, (11)with N m = σ NN T ( x T = 0) = 3 σ NN A/ (2 πR ), and P ( ¯ T ) = , ¯ T > T d /T c , |h ψ ( T ) | ψ (0) i| , < ¯ T < T d /T c , , ¯ T < . (12)where we have taken P = 1 below T c as an approximation. We take R = 6 .
38 fm and A = 197 for gold [37], and σ NN = 41 mb at RHIC energy [25]. The R AA as a function of ¯ T ( ) is shown in Fig. 4. It can be seen that the R AA is above 0 . T (0) is lower than 1 . T c , while it is below 0 . T (0) is higher than 2 . T c .One can expect that this effect is not negligible at the RHIC and is remarkable at the LHC. The factor x in Eq. (11)comes from two facts: 1) more Υ(1 S )s are produced in central of the fireball, and 2) the thickness changes slow withradius in central of the fireball. As a result, the R AA relies more on the survival probability P in the center of thefireball, that is at x T = 0. Therefore the qualitative behavior of the R AA in Fig. 4 is similar to the P in Fig. 2, andthey are quantitatively similar when P ( ¯ T ( )) is large. (0)T AA R (cid:9) FIG. 4: The nuclear modification factor R AA in central Au+Au collisions due to the heating dissociation effect as a functionof the scaled temperature ¯ T ( ) at the central of the fireball. We have two remarks on this result. 1) Even if the width (or dissociation rate) Γ vanishes at finite temperature,there is a fast heating dissociation effect for Υ(1 S ) suppression, which is not carefully considered before. 2) If the widthof Υ(1 S ) is negligible as in some calculations, then the heating dissociation of Υ(1 S ) can be used as a thermometer todetect the temperature of the fireball at early time, and it is not sensitive to the temperature later on. It is necessaryto clarify that such a temperature measured by Υ(1 S ) should never be interpreted as the highest temperature of thefireball, but the temperature a Υ(1 S ) feels. As a matter of fact, the highest temperature at very early time is not welldefined and the change of the temperature at very early time is so quick that the adiabatic theorem breaks, whichmeans the Υ(1 S ) may not feel the temperature before the temperature drops down relatively slowly. Actually themost interesting temperature is not the high and short-lived temperature at the very beginning, but the temperaturethat can be felt by particles. In this sense, the Υ(1 S )-felt temperature of the medium is more meaningful.In summary we discussed a new mechanism on Υ(1 S ) dissociation which is due to the fast heating process atthe early stage of the fireball instead of a non-zero width in a steady hot medium. Because of such a fast heatingdissociaiton, the suppression of Υ(1 S ) is observable at RHIC energy even if the width of Υ(1 S ) at finite temperatureis zero, and such a mechanism may be used as a measure of the temperature of the fireball at early time.Acknowlegements: the work is supported by the NSFC under the Grant No.s 11547043, 11705125 and by the“Qinggu” project of Tianjin University. [1] R. Hagedorn, Nuovo Cimento, Suppl. , 147 (1965), URL http://cds.cern.ch/record/346206 .[2] F. Karsch, E. Laermann, and A. Peikert, Nucl. Phys. B605 , 579 (2001), hep-lat/0012023.[3] H.-T. Ding, F. Karsch, and S. Mukherjee, Int. J. Mod. Phys.
E24 , 1530007 (2015), 1504.05274.[4] R. D. Pisarski and F. Wilczek, Phys. Rev.
D29 , 338 (1984).[5] K. Fukushima and T. Hatsuda, Rept. Prog. Phys. , 014001 (2011), 1005.4814.[6] M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Phys. Rev. Lett. , 4816 (1998), hep-ph/9806219.[7] D. Kharzeev and A. Zhitnitsky, Nucl. Phys. A797 , 67 (2007), 0706.1026.[8] M. Gonin et al. (NA50), Nucl. Phys.
A610 , 404c (1996).[9] X. Luo and N. Xu, Nucl. Sci. Tech. , 112 (2017), 1701.02105.[10] K. Adcox et al. (PHENIX), Phys. Rev. Lett. , 022301 (2002), nucl-ex/0109003.[11] A. Adare et al. (PHENIX), Phys. Rev. Lett. , 162301 (2007), nucl-ex/0608033.[12] S. Chatrchyan et al. (CMS), Phys. Rev. C84 , 024906 (2011), 1102.1957.[13] A. Andronic, P. Braun-Munzinger, and J. Stachel, Acta Phys. Polon.
B40 , 1005 (2009), 0901.2909.[14] R. Bellwied, J. Noronha-Hostler, P. Parotto, I. P. Vazquez, C. Ratti, and J. Stafford,
Extracting the strangeness freeze-outtemperature from net-kaon data at rhic (2019), 1904.12711.[15] T. Matsui and H. Satz, Phys. Lett.
B178 , 416 (1986).[16] F. Karsch, D. Kharzeev, and H. Satz, Phys. Lett.
B637 , 75 (2006), hep-ph/0512239.[17] M. E. Peskin, Nucl. Phys.
B156 , 365 (1979).[18] X.-M. Xu, D. Kharzeev, H. Satz, and X.-N. Wang, Phys. Rev.
C53 , 3051 (1996), hep-ph/9511331.[19] S. Chen and M. He, Phys. Lett.
B786 , 260 (2018), 1805.06346.[20] X. Yao and B. Muller, Phys. Rev.
D100 , 014008 (2019), 1811.09644.[21] L. Grandchamp and R. Rapp, Phys. Lett.
B523 , 60 (2001), hep-ph/0103124.[22] R. L. Thews, M. Schroedter, and J. Rafelski, Phys. Rev.
C63 , 054905 (2001), hep-ph/0007323.[23] L. Yan, P. Zhuang, and N. Xu, Phys. Rev. Lett. , 232301 (2006), nucl-th/0608010.[24] X. Yao and T. Mehen, Phys. Rev. D99 , 096028 (2019), 1811.07027.[25] X. Zhu, P. Zhuang, and N. Xu, Phys. Lett.
B607 , 107 (2005), nucl-th/0411093.[26] T. Song, Y. Park, S. H. Lee, and C.-Y. Wong, Phys. Lett.
B659 , 621 (2008), 0709.0794.[27] X. Zhao and R. Rapp, Nucl. Phys.
A859 , 114 (2011), 1102.2194.[28] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Nucl. Phys.
B566 , 275 (2000), hep-ph/9907240.[29] M. A. Escobedo, F. Giannuzzi, M. Mannarelli, and J. Soto, Phys. Rev.
D87 , 114005 (2013), 1304.4087.[30] A. Mishra, A. Jahan CS, S. Kesarwani, H. Raval, S. Kumar, and J. Meena, Eur. Phys. J.
A55 , 99 (2019), 1812.07397.[31] H. Liu (STAR), Nucl. Phys.
A830 , 235c (2009), 0907.4538.[32] X. Du, M. He, and R. Rapp, Physical Review
C96 (2017), ISSN 2469-9993, URL http://dx.doi.org/10.1103/PhysRevC.96.054901 .[33] R. Katz and P. B. Gossiaux, Annals Phys. , 267 (2016), 1504.08087.[34] H. Satz, J. Phys.
G32 , R25 (2006), hep-ph/0512217.[35] O. Kaczmarek, F. Karsch, P. Petreczky, and F. Zantow, Phys. Lett.
B543 , 41 (2002), hep-lat/0207002.[36] J. Boyd, T. Cook, A. Islam, and M. Strickland, Phys. Rev.
D100 , 076019 (2019), 1905.05676.[37] H. De Vries, C. De Jager, and C. De Vries, Atom. Data Nucl. Data Tabl.36