- Email: [email protected]
Charmonium production from hot hadronic matter
24 December 1998
Physics Letters B 444 Ž1998. 237–244
Charmonium production from hot hadronic matter C.M. Ko
a,1
, X.N. Wang
b,2
, B. Zhang
a,3
, X.F. Zhang
a,4
a
b
Cyclotron Institute and Physics Department, Texas A & M UniÕersity, College Station, TX 77843, USA Nuclear Science DiÕision, Lawrence Berkeley National Laboratory, UniÕersity of California, Berkeley, CA 94720, USA Received 11 August 1998; revised 16 September 1998 Editor: W. Haxton
Abstract Charmonium production from DD annihilation in the hadron gas formed in ultra-relativistic heavy-ion collisions is studied. With initial conditions taken from the HIJING parton model, we have estimated the number of Jrc produced from an expanding hadron gas, using the Jrc absorption and production cross sections obtained from either an effective Lagrangian or the quark-exchange model. We find that Jrc production is negligible in heavy-ion collisions at the Relativistic Heavy Ion Collider ŽRHIC. but may be important at the Large Hadron Collider ŽLHC., where more charm mesons are produced. Similar results are obtained for c X production from the hadron gas at RHIC and LHC. q 1998 Elsevier Science B.V. All rights reserved. PACS: 25.75.-q; 24.10.Lx
1. Introduction One of the signals proposed for identifying the existence of a quark-gluon plasma in ultra-relativistic heavy-ion collisions is the suppression of Jrc production compared to that expected from the superposition of nucleon-nucleon collisions w1x. According to Matsui and Satz, if a quark-gluon plasma is created in heavy-ion collisions, Jrc ’s produced from initial nucleon-nucleon interactions will dissociate as a result of Debye screening and vanishing string tension between c and c. Recent experimental results from 1
E-mail: [email protected] E-mail: [email protected] 3 E-mail address: [email protected] 4 Present address: Department of Physics and Astronomy, Iowa State University, Ames, IA 50011. 2
heavy-ion collisions at CERN Super Proton Synchrotron ŽSPS. energies w2–4x have indeed shown a reduction of Jrc production. Part of the effects can be attributed to absorption by nucleons as it is more likely that Jrc is first produced as a pre-resonance in a color octet state and thus has a larger interaction cross section with nucleon w5–8x. Such an effect is needed to account for the observed Jrc suppression in proton-nucleus collisions w9x, where one does not expect the formation of a quark-gluon plasma. Also, Jrc absorption by comovers, which are most pions and rho mesons produced in the collisions, has been suggested w10–13x. In collisions of light ions such as S q U at 200 GeVrnucleon, these absorption mechanisms are sufficient to explain the experimental data. However, for central collisions of heavy nuclei such as Pb q Pb at 160 GeVrnucleon, the anomalous large suppression of Jrc observed in the ex-
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 3 9 0 - 2
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C.M. Ko et al.r Physics Letters B 444 (1998) 237–244
periments has led to the suggestions that a quarkgluon plasma is formed in these collisions, and the Jrc yield is reduced due to dissociation either by Debye screening w14–16x or collisions with gluons w17x. If the quark-gluon plasma has already been formed in heavy-ion collisions at SPS energies, it is almost certain that it will also be formed in heavy-ion collisions at RHIC and LHC, where energies are much higher. Therefore, Jrc suppression at RHIC and LHC is considered as one of the most prominent signals for the quark-gluon plasma. But will Jrc production at RHIC and LHC be really reduced? This depends on whether Jrc can be regenerated from the hadronic matter after the phase transition of the quark-gluon plasma. Such an effect has been shown to be important for hadrons made of strange quarks w18x. In hadronic matter, Jrc can be produced from the interactions of charm mesons in reactions such as DD ) , D ) D, D ) D ) ™ Jrcp and DD, DD ) , D ) D, D ) D ) ™ Jrcr . These reactions are apparently unimportant at SPS energies but may become significant at RHIC and LHC since charm production in nucleon-nucleon interaction increases with center-of-mass energy while the Jrc to cc ratio remains essentially at a constant value of ; 2.5 = 10y2 w19x. Furthermore, most these reactions are exothermic as 2 m D ; 3.73 GeVrc 2 , m D q m )D ; 3.87 GeVrc 2 and m D ) q m D ) ; 4.01 GeVrc 2 while m Jr c q mp ; 3.15 GeVrc 2 and m Jr c q mr ; 3.87 GeVrc 2 , it is thus more likely that Jrc ’s produced from the hadronic matter will survive and be detected in experiments. If this is the case, then the final number of Jrc may be comparable to or even larger than that of primary Jrc ’s which are dissociated in the quark-gluon plasma, leading instead to an absence of Jrc suppression or even an enhanced Jrc production. In the following, we shall make an estimate of Jrc production from the hadron gas formed in heavy-ion collisions at RHIC and LHC energies.
2. Dynamics of heavy ion collisions To model heavy-ion collisions at such high energies, we use the results from the HIJING calculation w20x, which takes into account parton production from semihard scatterings. It shows that at an initial
proper time t 0 a thermally equilibrated although chemically non-equilibrated quark-gluon plasma of temperature T0 is formed. For Au q Au collisions, HIJING predicts that t 0 ; 0.7 and 0.5 fmrc, T0 ; 0.57 and 0.83 GeV at RHIC and LHC energies, respectively. The quark-gluon plasma then cools due to expansion and production of additional partons. It reaches the critical temperature Tc ; 200 MeV at about tc ; 3 fmrc at RHIC and 6 fmrc at LHC, when the quark-gluon plasma starts to make a transition to a hadron gas, consisting mostly of pions and rho mesons. Since neither the partons have reached chemical equilibrium nor the order of phase transition is definitely known, the time for the system to remain at Tc can not be properly determined. We shall assume that the proper time at which the quark-gluon plasma is completely converted to a hadron gas is t h ; 2tc and will study how the results depend on the value of t h . Since transverse expansion is not expected to be appreciable in this early phase, the initial volume of the hadronic matter at midrapidity is simply Vh ; p R 02t h if one uses the boost-invariant model of Bjorken w21x. In the above, R 0 is the radius of the colliding nuclei. Assuming that the hadron gas expands isentropically, its temperature then decreases according to the inverse of the cubic root of volume. To include the transverse expansion of the hadron gas, we introduce an acceleration a. Then its transverse radius increases with time according to RŽt . s R 0 q aŽt y t h . 2r2 until the velocity reaches the velocity of light, when the transverse radius increases linearly with time. From entropy conservation, the temperature of the hadron gas then decreases as T Žt . s Žt hrt .1r3 Ž R 0rRŽt .. 2r3 Tc . Final hadron yields and spectra are determined at freeze out with a temperature Tf ; 120 MeV. We note that for hadron gas in equilibrium the pion and rho meson densities are approximately given by np ; 0.285ŽTr200 MeV. 3.3 fmy3 and nr ; 0.15ŽTr200 MeV. 6.5 fmy3 , respectively. The time evolution of the temperature and volume of the hadron gas for Au q Au collisions at both RHIC and LHC is shown in Fig. 1 for a s 0.1 c 2rfm. The transverse expansion velocity at freeze out is about 0.75c and 0.9c at RHIC and LHC, respectively. The initial number of charm quarks at midrapidity produced in central heavy ion collisions can be
C.M. Ko et al.r Physics Letters B 444 (1998) 237–244
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both Jrc ’s and D’s are expected to be dissociated due to Debye screening. As shown in Ref. w20x, thermal charm production from the quark-gluon plasma is small, it is thus reasonable to assume that the c and c quark numbers remain constant during the quark-gluon phase. After phase transition to a hadron gas, these charm quarks are more likely to form D and D mesons and their excited states due to the large abundance of light quarks in the plasma. Conservation of charm then requires that the number of charm mesons at the beginning of the hadron phase is the same as the initial one.
3. J r c production cross section and rate Fig. 1. Time evolution of the temperature and volume of hadron gas in AuqAu collisions at RHIC and LHC.
estimated from their production cross section d sccN N rdy < ys 0 in nucleon-nucleon interaction, d NccAB dy
sTA B Ž 0 . ys 0
d sccN N dy
,
Ž 1.
ys0
where TA B Ž0. is the thickness function at zero impact parameter and has a value f 30 mby1 for Au q Au collisions. The cross section d sccN N rdy < ys 0 has been evaluated using the perturbative QCD w20x, and it has a value 0.055 mb and 1.77 mb per unit rapidity in nucleon-nucleon interaction at RHIC and LHC energies, respectively. For Au q Au collisions, the initial number of cc Ži.e., DD . pairs at midrapidity is thus about 1.65 at RHIC and 53 at LHC. The initial Jrc number can be similarly estimated using its production cross section per unit rapidity in the nucleon-nucleon collision, i.e., 6.3 = 10y4 mb and 1.6 = 10y2 mb per unit rapidity at RHIC and LHC, respectively. This gives at midrapidity 0.0189 Jrc at RHIC and 0.48 Jrc at LHC. The ratio of the Jrc number to that of cc with invariant mass below the DD threshold, which is about 1r2 at RHIC and LHC, is thus similar to that observed in nucleon-nucleon interaction at centerof-mass energy below 40 GeV w19x. Since a quark-gluon plasma is almost certain to be created in heavy-ion collisions at RHIC and LHC,
The Jrc production cross section from the reaction DD ™ Jrcp can be related to the absorption cross section sJr cp ™ D D via the detailed balance relation 2
sD D ™ Jr cp s d Ž k Jr cprk D D . sJr cp ™ D D ,
Ž 2.
where the degeneracy factor d is 3r4 for DD ) and D ) D annihilation, and 1r4 for D ) D ) annihilation; k Jr cp and k D D are, respectively, the relative momenta of Jrcp and DD. The magnitude of Jrc absorption cross section by pion, sJr cp ™ D D , is still under debate. In the comover model for Jrc suppression in heavy-ion collisions at SPS energies, it is taken to be about 3 mb. Both the perturbative QCD estimate w22x and the effective Lagrangian calculation based on the exchange of a D meson w23x give values which are about a factor of 10 smaller. On the other hand, studies including nonperturbative effects via quark-exchange model w24x give a peak value of about 6 mb. In the following, we shall use both this cross section and that from the effective Lagrangian. The cross section from the quark-exchange model has been parameterized as s0
2
ž ž //
sJr cp ™ D D Ž s . f s 0 1 y
s
=exp ya 's y s1
ž
( /
u Ž s y s0 . ,
Ž 3.
C.M. Ko et al.r Physics Letters B 444 (1998) 237–244
240
where s is the square of the Jrcp center-of-mass energy and s0 is the square of the threshold energy. For the final states DD ) and DD ) , the parameters are s 0 s 2.5 mb, s0 s 15.05 GeV 2 , s1 s 17.6 GeV 2 , and a s 11 GeVy1 ; while for the final state D ) D ) , they are s 0 s 3.4 mb, s0 s 16.2 GeV 2 , s1 s 19.0 GeV 2 , and a s 11 GeVy1 . Similarly, the Jrc production cross section from the reaction DD ™ Jrcr is related to the absorption cross section sJr cr ™ D D by a similar detailed balance relation as Eq. Ž2.. In this case, the factor d is 27r4 for DD annihilation, 9r4 for DD ) and D ) D annihilation, and 3r4 for D ) D ) annihilation. In the perturbative QCD and effective Lagrangian calculations, the Jrc absorption cross section sJr cr ™ D D also has a small value as sJr cp ™ D D . However, calculations have not been done for this cross section in the quark-exchange model. The quantity needed for estimating Jrc production from DD interaction in a hadron gas is the thermal average of sD D ™ Jr c M Õ, where Õ is the relative velocity between D and D and M denotes either a pion or rho meson, i.e., ² sD D ™ Jr c M Õ : 2
2
s 4 Ž m D 1rT . Ž m D 2rT . K 2 Ž m D 1rT . =K 2 Ž m D 2rT . x
y1
`
Hz dz 0
= z 2 y Ž m D 1rT q m D 2rT .
2
= z 2 y Ž m D 1rT y m D 2rT .
2
=K 1 Ž z . sD D ™ Jr c M .
Ž 4.
In the above, m D 1 and m D 2 are the masses of the two charm mesons; z 0 s maxwŽ m D 1 q m D 2 .rT,Ž m Jr c q m M .rT x with m Jr c and m M the Jrc and meson masses, respectively; and K 1 and K 2 are, respectively, the modified Bessel function of the first and second kind. We first show in Fig. 2 the temperature dependence of the thermally averaged Jrc production and absorption cross sections from the effective Lagrangian model of Ref. w23x. We see that although the Jrc absorption cross section by pion and rho
Fig. 2. Temperature dependence of the thermally averaged Jrc absorption and production cross sections from the effective Lagrangian model w23x.
meson has similar magnitude, the thermally averaged cross section for rho meson ² sD D ™ Jr cr Õ : is much larger than that for pion ² sD D ) ™ Jr cp Õ :, defined by the sum of ² sD D ) ™ Jr cp Õ : and ² sD ) D ™ Jr cp Õ :, as a result of threshold effects in the latter so only pions with sufficient energy can destroy a Jrc . We note that the value for ² sJr cr ™ D D Õ : is consistent with, while that for ² sJr cp ™ D D ) Õ : is smaller than, that inferred from Jrc suppression data in heavy-ion collisions w11x. The thermally averaged Jrc production cross sections are both larger than the absorption ones due to the large degeneracy factor in the case of rho meson and the fact that the one involving pion is exothermic. Similarly, we show in Fig. 3 the thermally averaged Jrc production and absorption cross sections from the quark-exchange model w24x. Again, the Jrc production cross section is larger than its absorption cross section as it is exothermic. The value for ² s D D ) ™ J r cp Õ : is larger than that for ² sD ) D ) ™ Jr cp Õ : as a result of both a larger degeneracy factor and the smaller cross section of the latter reaction. Furthermore, ² sD D ) ™ Jr cp Õ : is much larger than ² sJr cp ™ D D Õ : at all temperatures. The latter has values ranging between 0.25 to 1.0 mb, which is much smaller than the cross section itself
C.M. Ko et al.r Physics Letters B 444 (1998) 237–244
Fig. 3. Temperature dependence of the thermally averaged Jrc absorption and production cross sections from the quark-exchange model w24x.
due to threshold effects and is consistent with the available experimental constraint w11x. The rate of Jrc production from the hadron gas is given by dR dt
241
find that at RHIC the number of Jrc produced from DD annihilation is only 3.0 = 10y4 and is two orders of magnitude smaller than the number of primary Jrc . This is different at LHC where the number of DD present in the hadron gas is much larger. In Fig. 4, we show by the solid curve the Jrc number as a function of time in Au q Au collisions at LHC. It is seen that the Jrc number increases with time and at freeze out is 0.20, which is only a factor of two smaller than the number of primary Jrc , shown by the arrow on the left. To see the dependence of our results on the Jrc absorption cross section, we have also used the cross sections given by the quark-exchange model w24x. This increase the Jrc number at RHIC to 9.6 = 10y4 , which is, however, still much smaller than the number of primary Jrc . At LHC, the Jrc number is increased to 0.60, shown in Fig. 4 by the dashed curve, and is now larger than the primary Jrc number. We have also studied the dependence of our results on the values of the parameters in the model. The results given below are obtained by changing only one parameter a time and using default values
s ² sD D ™ Jr cr Õ : n2D q ² sD D ) ™ Jr cp Õ : n D n D ) q ² sD ) D ) ™ Jr cp Õ : n2D ) y ² sJr cp ™ D D Õ : =n Jr c np y ² sJr cr ™ D D Õ : n Jr c nr ,
Ž 5.
where n D and n D ) are the densities of D and D ) , respectively, and their relative value is determined by assuming that they are in chemical equilibrium. In Eq. Ž5., we have used the fact that the densities of charm and anti-charm mesons are the same. The Jrc density is denoted by n Jr c .
4. J r c production from heavy ion collisions The total number of produced Jrc from heavy ion collisions is obtained by multiplying the rate, Eq. Ž5., by the volume of the hadron gas and then integrating over time. Using the default parameters of Section 2 for the collision dynamics and the cross sections from the effective Lagrangian model, we
Fig. 4. Time evolution of the abundance of Jrc from AuqAu collisions at LHC using cross sections from the effective Lagrangian w24x and the quark-exchange model w23x and the default parameters for the collision dynamics. The initial Jrc number is denoted by the arrow on the left.
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default parameters to 0.48 for a s 0.2 c 2rfm, 0.76 for a s 0.05 c 2rfm, 0.57 for mp s 130 MeV, 0.35 for t h s 4tc , 0.98 for t h s tc , and 0.56 for Tc s 150 MeV.
5. c X production in heavy ion collisions
X
Fig. 5. Temperature dependence of the thermally averaged c absorption and production cross sections from the effective Lagrangian model w23x.
for other parameters and the cross sections from the effective Lagrangian model. For larger values of transverse acceleration a, the hadron gas expands faster so its temperature drops more appreciably, leading to a slight reduction of the number of produced Jrc to 0.16 for a s 0.2 c 2rfm. However, the transverse expansion velocity in this case reaches the velocity of light after a few fmrc, which seems unreasonable. On the other hand, with smaller values of a, the final Jrc number is increased to 0.25 for a s 0.05 c 2rfm. If pions and rho mesons are out of chemical equilibrium w25x, Jrc absorption may be enhanced. Using a pion chemical potential of 130 MeV and a rho meson chemical potential twice as large, the final Jrc number is only slightly reduced to 0.19. The lifetime of the phase transition also affects the result. If it is doubled, then the number of produced Jrc is reduced to about 0.12, while it is increased to 0.31 if the phase transition is instantaneous. Reducing the value of the temperature at which the phase transition occurs affects also the number of Jrc produced from the hadron gas. Using Tc s 150 MeV, the Jrc number is reduced to 0.12. In the case cross sections from the quark-exchange model are used, calculations similar to the above change the Jrc number from 0.60 using the
The above analysis can be generalized to c X production in ultra-relativistic heavy-ion collisions. The primary c X number is about a factor of 5 smaller than the Jrc number, i.e., about 3.8 = 10y3 and 0.096 at RHIC and LHC, respectively. Since the radius of c X is about twice that of c , its absorption cross section by meson is expected to be larger. Indeed, if we assume that the coupling constant gc X D D has a similar value as gc D D f 7.7 used in the effective Lagrangian model of Ref. w23x, then the c X absorption cross section by pion and rho meson are both a few times larger than those for Jrc . The thermally averaged c X absorption and production cross sections from this model are shown in Fig. 5. Because of the larger c X mass than that of Jrc , the thermally averaged c X absorption cross section turns out to be order of magnitude larger than that for Jrc .
X
Fig. 6. Temperature dependence of the thermally averaged c absorption and production cross sections from the quark-exchange model w24x.
C.M. Ko et al.r Physics Letters B 444 (1998) 237–244
The c X absorption cross section has not been evaluated in the quark-exchange model. To make an estimate, we assume that the same parameterization shown in Eq. Ž3. holds but the threshold energy s0 is replaced by the appropriate value, i.e., 3.83 GeV for c Xp ™ DD ) and c Xp ™ DD ) , while it remains the same for c Xp ™ D ) D ) . Furthermore, s 0 is increased by a factor of four to account for the larger size of c X than Jrc . The thermally averaged cross sections are shown in Fig. 6 and are about an order of magnitude larger than those from the effective Lagrangian model. With the cross sections from the effective Lagrangian model, we find that the number of c X produced from the hadronic matter in Au q Au collisions at RHIC is about 8.0 = 10y5 and is negligible compared with that from primary nucleon-nucleon collisions. For the same reaction at LHC, the produced c X from the hadronic matter is about 0.052 as shown by the solid curve in Fig. 7 and is more than 1r2 of the primary ones shown by the arrow on the left of the figure. Using the cross sections from the quark exchange model leads to an increase of the number of c X produced from the hadronic matter in Au q Au collisions at RHIC to 3.4 = 10y4 , which is
X
Fig. 7. Time evolution of the abundance of c from AuqAu collisions at LHC using cross sections from the effective Lagrangian w24x and the quark-exchange model w23x and the default X parameters for the collision dynamics. The initial c number is denoted by the arrow on the left.
243
still an order of magnitude smaller than the number of primary c X . For LHC, we obtain 0.19 c X from the hadronic matter, shown by the dashed curve in Fig. 7, which is about twice the primary c X number.
6. Summary To summarize, we have estimated the number of Jrc produced from the reaction DD ™ Jrc M in the hadron gas formed in heavy-ion collisions at RHIC and LHC using the initial conditions determined from the HIJING parton model and the cross section given by either the effective Lagrangian or the quark-exchange model. Although this is a negligible effect at RHIC, it becomes important at LHC as a result of the appreciable number of D and D mesons in the hadron gas. We have found that the number of Jrc produced from this reaction at LHC may be comparable to that produced from initial primary collisions, which are either absorbed by nucleons or dissociated in the quark-gluon plasma formed in the collisions. A similar estimate has been made for c X production by assuming in the effective Lagrangian model that the c X DD coupling is the same as the c DD coupling and in the quark-exchange model that the c X absorption cross sections have similar energy dependence as those for Jrc except a different threshold and four times larger magnitude due to the larger size of c X . As in the case of Jrc , c X production from the hadron gas is found insignificant in heavy ion collisions at RHIC but important at LHC. To use the Jrc and c X yields as signals for the quark-gluon plasma at LHC thus requires a good understanding of their production from the hadron gas. For a more reliable evaluation of this effect, a better determination of the Jrc production and absorption cross sections from both experiments and theoretical models is needed. Also, a comprehensive transport model for describing the heavy ion collision dynamics at RHIC and LHC is needed. In the present study based on the schematic model described in Section 2, the final pion number at midrapidity is only about 600 per unit rapidity at LHC if the default parameters are used. This number is about a factor of three smaller than that predicted by the HIJING model, which,
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however, does not include final hadron rescattering. The final pion number in our model can be increased if we introduce one or more of the following mechanisms; a longer proper time t h for hadronization, a finite chemical potential mp for pions, and the production of entropy in the hadron gas. As shown in our schematic study, increasing t h reduces the number of produced Jrc and c X as a result of a decreased charm meson density in the hadron gas, but increasing mp does not affect much the final Jrc and c X numbers. Allowing entropy production during the expansion of the hadronic matter, which would lead to a slower decrease of its temperature, also makes it possible to increase the final pion number without reducing the Jrc and c X numbers as the latter are mostly produced during the initial high density hadronic matter. At present, it is not clear which of the above mechanisms will be more important in heavy ion collisions at ultrarelativistic energies.
Acknowledgements This work was started while the authors were visiting the Institute for Nuclear Theory at University of Washington for the program on Probes of Dense Matter in Ultrarelativistic Heavy Ion Collisions, and they thank Y. Asakawa, C. Gale, J. Kapusta, V. Koch, and C. Y. Wong for useful discussions. We are also grateful to D. Blaschke for communications regarding the quark-exchange model. The work of CMK, BZ, and XFZ was supported in part by the National Science Foundation under Grant No. PHY-9509266 and PHY-9870038, the Welch Foundation under Grant No. A-1358, and the Texas Advanced Research Program. The work of XNW was supported by the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High energy and Nuclear Physics of the U.S.
Department of energy under Contract No. DE-AC0376SF00098.
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Physics Letters B 444 Ž1998. 237–244
Charmonium production from hot hadronic matter C.M. Ko
a,1
, X.N. Wang
b,2
, B. Zhang
a,3
, X.F. Zhang
a,4
a
b
Cyclotron Institute and Physics Department, Texas A & M UniÕersity, College Station, TX 77843, USA Nuclear Science DiÕision, Lawrence Berkeley National Laboratory, UniÕersity of California, Berkeley, CA 94720, USA Received 11 August 1998; revised 16 September 1998 Editor: W. Haxton
Abstract Charmonium production from DD annihilation in the hadron gas formed in ultra-relativistic heavy-ion collisions is studied. With initial conditions taken from the HIJING parton model, we have estimated the number of Jrc produced from an expanding hadron gas, using the Jrc absorption and production cross sections obtained from either an effective Lagrangian or the quark-exchange model. We find that Jrc production is negligible in heavy-ion collisions at the Relativistic Heavy Ion Collider ŽRHIC. but may be important at the Large Hadron Collider ŽLHC., where more charm mesons are produced. Similar results are obtained for c X production from the hadron gas at RHIC and LHC. q 1998 Elsevier Science B.V. All rights reserved. PACS: 25.75.-q; 24.10.Lx
1. Introduction One of the signals proposed for identifying the existence of a quark-gluon plasma in ultra-relativistic heavy-ion collisions is the suppression of Jrc production compared to that expected from the superposition of nucleon-nucleon collisions w1x. According to Matsui and Satz, if a quark-gluon plasma is created in heavy-ion collisions, Jrc ’s produced from initial nucleon-nucleon interactions will dissociate as a result of Debye screening and vanishing string tension between c and c. Recent experimental results from 1
E-mail: [email protected] E-mail: [email protected] 3 E-mail address: [email protected] 4 Present address: Department of Physics and Astronomy, Iowa State University, Ames, IA 50011. 2
heavy-ion collisions at CERN Super Proton Synchrotron ŽSPS. energies w2–4x have indeed shown a reduction of Jrc production. Part of the effects can be attributed to absorption by nucleons as it is more likely that Jrc is first produced as a pre-resonance in a color octet state and thus has a larger interaction cross section with nucleon w5–8x. Such an effect is needed to account for the observed Jrc suppression in proton-nucleus collisions w9x, where one does not expect the formation of a quark-gluon plasma. Also, Jrc absorption by comovers, which are most pions and rho mesons produced in the collisions, has been suggested w10–13x. In collisions of light ions such as S q U at 200 GeVrnucleon, these absorption mechanisms are sufficient to explain the experimental data. However, for central collisions of heavy nuclei such as Pb q Pb at 160 GeVrnucleon, the anomalous large suppression of Jrc observed in the ex-
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periments has led to the suggestions that a quarkgluon plasma is formed in these collisions, and the Jrc yield is reduced due to dissociation either by Debye screening w14–16x or collisions with gluons w17x. If the quark-gluon plasma has already been formed in heavy-ion collisions at SPS energies, it is almost certain that it will also be formed in heavy-ion collisions at RHIC and LHC, where energies are much higher. Therefore, Jrc suppression at RHIC and LHC is considered as one of the most prominent signals for the quark-gluon plasma. But will Jrc production at RHIC and LHC be really reduced? This depends on whether Jrc can be regenerated from the hadronic matter after the phase transition of the quark-gluon plasma. Such an effect has been shown to be important for hadrons made of strange quarks w18x. In hadronic matter, Jrc can be produced from the interactions of charm mesons in reactions such as DD ) , D ) D, D ) D ) ™ Jrcp and DD, DD ) , D ) D, D ) D ) ™ Jrcr . These reactions are apparently unimportant at SPS energies but may become significant at RHIC and LHC since charm production in nucleon-nucleon interaction increases with center-of-mass energy while the Jrc to cc ratio remains essentially at a constant value of ; 2.5 = 10y2 w19x. Furthermore, most these reactions are exothermic as 2 m D ; 3.73 GeVrc 2 , m D q m )D ; 3.87 GeVrc 2 and m D ) q m D ) ; 4.01 GeVrc 2 while m Jr c q mp ; 3.15 GeVrc 2 and m Jr c q mr ; 3.87 GeVrc 2 , it is thus more likely that Jrc ’s produced from the hadronic matter will survive and be detected in experiments. If this is the case, then the final number of Jrc may be comparable to or even larger than that of primary Jrc ’s which are dissociated in the quark-gluon plasma, leading instead to an absence of Jrc suppression or even an enhanced Jrc production. In the following, we shall make an estimate of Jrc production from the hadron gas formed in heavy-ion collisions at RHIC and LHC energies.
2. Dynamics of heavy ion collisions To model heavy-ion collisions at such high energies, we use the results from the HIJING calculation w20x, which takes into account parton production from semihard scatterings. It shows that at an initial
proper time t 0 a thermally equilibrated although chemically non-equilibrated quark-gluon plasma of temperature T0 is formed. For Au q Au collisions, HIJING predicts that t 0 ; 0.7 and 0.5 fmrc, T0 ; 0.57 and 0.83 GeV at RHIC and LHC energies, respectively. The quark-gluon plasma then cools due to expansion and production of additional partons. It reaches the critical temperature Tc ; 200 MeV at about tc ; 3 fmrc at RHIC and 6 fmrc at LHC, when the quark-gluon plasma starts to make a transition to a hadron gas, consisting mostly of pions and rho mesons. Since neither the partons have reached chemical equilibrium nor the order of phase transition is definitely known, the time for the system to remain at Tc can not be properly determined. We shall assume that the proper time at which the quark-gluon plasma is completely converted to a hadron gas is t h ; 2tc and will study how the results depend on the value of t h . Since transverse expansion is not expected to be appreciable in this early phase, the initial volume of the hadronic matter at midrapidity is simply Vh ; p R 02t h if one uses the boost-invariant model of Bjorken w21x. In the above, R 0 is the radius of the colliding nuclei. Assuming that the hadron gas expands isentropically, its temperature then decreases according to the inverse of the cubic root of volume. To include the transverse expansion of the hadron gas, we introduce an acceleration a. Then its transverse radius increases with time according to RŽt . s R 0 q aŽt y t h . 2r2 until the velocity reaches the velocity of light, when the transverse radius increases linearly with time. From entropy conservation, the temperature of the hadron gas then decreases as T Žt . s Žt hrt .1r3 Ž R 0rRŽt .. 2r3 Tc . Final hadron yields and spectra are determined at freeze out with a temperature Tf ; 120 MeV. We note that for hadron gas in equilibrium the pion and rho meson densities are approximately given by np ; 0.285ŽTr200 MeV. 3.3 fmy3 and nr ; 0.15ŽTr200 MeV. 6.5 fmy3 , respectively. The time evolution of the temperature and volume of the hadron gas for Au q Au collisions at both RHIC and LHC is shown in Fig. 1 for a s 0.1 c 2rfm. The transverse expansion velocity at freeze out is about 0.75c and 0.9c at RHIC and LHC, respectively. The initial number of charm quarks at midrapidity produced in central heavy ion collisions can be
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both Jrc ’s and D’s are expected to be dissociated due to Debye screening. As shown in Ref. w20x, thermal charm production from the quark-gluon plasma is small, it is thus reasonable to assume that the c and c quark numbers remain constant during the quark-gluon phase. After phase transition to a hadron gas, these charm quarks are more likely to form D and D mesons and their excited states due to the large abundance of light quarks in the plasma. Conservation of charm then requires that the number of charm mesons at the beginning of the hadron phase is the same as the initial one.
3. J r c production cross section and rate Fig. 1. Time evolution of the temperature and volume of hadron gas in AuqAu collisions at RHIC and LHC.
estimated from their production cross section d sccN N rdy < ys 0 in nucleon-nucleon interaction, d NccAB dy
sTA B Ž 0 . ys 0
d sccN N dy
,
Ž 1.
ys0
where TA B Ž0. is the thickness function at zero impact parameter and has a value f 30 mby1 for Au q Au collisions. The cross section d sccN N rdy < ys 0 has been evaluated using the perturbative QCD w20x, and it has a value 0.055 mb and 1.77 mb per unit rapidity in nucleon-nucleon interaction at RHIC and LHC energies, respectively. For Au q Au collisions, the initial number of cc Ži.e., DD . pairs at midrapidity is thus about 1.65 at RHIC and 53 at LHC. The initial Jrc number can be similarly estimated using its production cross section per unit rapidity in the nucleon-nucleon collision, i.e., 6.3 = 10y4 mb and 1.6 = 10y2 mb per unit rapidity at RHIC and LHC, respectively. This gives at midrapidity 0.0189 Jrc at RHIC and 0.48 Jrc at LHC. The ratio of the Jrc number to that of cc with invariant mass below the DD threshold, which is about 1r2 at RHIC and LHC, is thus similar to that observed in nucleon-nucleon interaction at centerof-mass energy below 40 GeV w19x. Since a quark-gluon plasma is almost certain to be created in heavy-ion collisions at RHIC and LHC,
The Jrc production cross section from the reaction DD ™ Jrcp can be related to the absorption cross section sJr cp ™ D D via the detailed balance relation 2
sD D ™ Jr cp s d Ž k Jr cprk D D . sJr cp ™ D D ,
Ž 2.
where the degeneracy factor d is 3r4 for DD ) and D ) D annihilation, and 1r4 for D ) D ) annihilation; k Jr cp and k D D are, respectively, the relative momenta of Jrcp and DD. The magnitude of Jrc absorption cross section by pion, sJr cp ™ D D , is still under debate. In the comover model for Jrc suppression in heavy-ion collisions at SPS energies, it is taken to be about 3 mb. Both the perturbative QCD estimate w22x and the effective Lagrangian calculation based on the exchange of a D meson w23x give values which are about a factor of 10 smaller. On the other hand, studies including nonperturbative effects via quark-exchange model w24x give a peak value of about 6 mb. In the following, we shall use both this cross section and that from the effective Lagrangian. The cross section from the quark-exchange model has been parameterized as s0
2
ž ž //
sJr cp ™ D D Ž s . f s 0 1 y
s
=exp ya 's y s1
ž
( /
u Ž s y s0 . ,
Ž 3.
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where s is the square of the Jrcp center-of-mass energy and s0 is the square of the threshold energy. For the final states DD ) and DD ) , the parameters are s 0 s 2.5 mb, s0 s 15.05 GeV 2 , s1 s 17.6 GeV 2 , and a s 11 GeVy1 ; while for the final state D ) D ) , they are s 0 s 3.4 mb, s0 s 16.2 GeV 2 , s1 s 19.0 GeV 2 , and a s 11 GeVy1 . Similarly, the Jrc production cross section from the reaction DD ™ Jrcr is related to the absorption cross section sJr cr ™ D D by a similar detailed balance relation as Eq. Ž2.. In this case, the factor d is 27r4 for DD annihilation, 9r4 for DD ) and D ) D annihilation, and 3r4 for D ) D ) annihilation. In the perturbative QCD and effective Lagrangian calculations, the Jrc absorption cross section sJr cr ™ D D also has a small value as sJr cp ™ D D . However, calculations have not been done for this cross section in the quark-exchange model. The quantity needed for estimating Jrc production from DD interaction in a hadron gas is the thermal average of sD D ™ Jr c M Õ, where Õ is the relative velocity between D and D and M denotes either a pion or rho meson, i.e., ² sD D ™ Jr c M Õ : 2
2
s 4 Ž m D 1rT . Ž m D 2rT . K 2 Ž m D 1rT . =K 2 Ž m D 2rT . x
y1
`
Hz dz 0
= z 2 y Ž m D 1rT q m D 2rT .
2
= z 2 y Ž m D 1rT y m D 2rT .
2
=K 1 Ž z . sD D ™ Jr c M .
Ž 4.
In the above, m D 1 and m D 2 are the masses of the two charm mesons; z 0 s maxwŽ m D 1 q m D 2 .rT,Ž m Jr c q m M .rT x with m Jr c and m M the Jrc and meson masses, respectively; and K 1 and K 2 are, respectively, the modified Bessel function of the first and second kind. We first show in Fig. 2 the temperature dependence of the thermally averaged Jrc production and absorption cross sections from the effective Lagrangian model of Ref. w23x. We see that although the Jrc absorption cross section by pion and rho
Fig. 2. Temperature dependence of the thermally averaged Jrc absorption and production cross sections from the effective Lagrangian model w23x.
meson has similar magnitude, the thermally averaged cross section for rho meson ² sD D ™ Jr cr Õ : is much larger than that for pion ² sD D ) ™ Jr cp Õ :, defined by the sum of ² sD D ) ™ Jr cp Õ : and ² sD ) D ™ Jr cp Õ :, as a result of threshold effects in the latter so only pions with sufficient energy can destroy a Jrc . We note that the value for ² sJr cr ™ D D Õ : is consistent with, while that for ² sJr cp ™ D D ) Õ : is smaller than, that inferred from Jrc suppression data in heavy-ion collisions w11x. The thermally averaged Jrc production cross sections are both larger than the absorption ones due to the large degeneracy factor in the case of rho meson and the fact that the one involving pion is exothermic. Similarly, we show in Fig. 3 the thermally averaged Jrc production and absorption cross sections from the quark-exchange model w24x. Again, the Jrc production cross section is larger than its absorption cross section as it is exothermic. The value for ² s D D ) ™ J r cp Õ : is larger than that for ² sD ) D ) ™ Jr cp Õ : as a result of both a larger degeneracy factor and the smaller cross section of the latter reaction. Furthermore, ² sD D ) ™ Jr cp Õ : is much larger than ² sJr cp ™ D D Õ : at all temperatures. The latter has values ranging between 0.25 to 1.0 mb, which is much smaller than the cross section itself
C.M. Ko et al.r Physics Letters B 444 (1998) 237–244
Fig. 3. Temperature dependence of the thermally averaged Jrc absorption and production cross sections from the quark-exchange model w24x.
due to threshold effects and is consistent with the available experimental constraint w11x. The rate of Jrc production from the hadron gas is given by dR dt
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find that at RHIC the number of Jrc produced from DD annihilation is only 3.0 = 10y4 and is two orders of magnitude smaller than the number of primary Jrc . This is different at LHC where the number of DD present in the hadron gas is much larger. In Fig. 4, we show by the solid curve the Jrc number as a function of time in Au q Au collisions at LHC. It is seen that the Jrc number increases with time and at freeze out is 0.20, which is only a factor of two smaller than the number of primary Jrc , shown by the arrow on the left. To see the dependence of our results on the Jrc absorption cross section, we have also used the cross sections given by the quark-exchange model w24x. This increase the Jrc number at RHIC to 9.6 = 10y4 , which is, however, still much smaller than the number of primary Jrc . At LHC, the Jrc number is increased to 0.60, shown in Fig. 4 by the dashed curve, and is now larger than the primary Jrc number. We have also studied the dependence of our results on the values of the parameters in the model. The results given below are obtained by changing only one parameter a time and using default values
s ² sD D ™ Jr cr Õ : n2D q ² sD D ) ™ Jr cp Õ : n D n D ) q ² sD ) D ) ™ Jr cp Õ : n2D ) y ² sJr cp ™ D D Õ : =n Jr c np y ² sJr cr ™ D D Õ : n Jr c nr ,
Ž 5.
where n D and n D ) are the densities of D and D ) , respectively, and their relative value is determined by assuming that they are in chemical equilibrium. In Eq. Ž5., we have used the fact that the densities of charm and anti-charm mesons are the same. The Jrc density is denoted by n Jr c .
4. J r c production from heavy ion collisions The total number of produced Jrc from heavy ion collisions is obtained by multiplying the rate, Eq. Ž5., by the volume of the hadron gas and then integrating over time. Using the default parameters of Section 2 for the collision dynamics and the cross sections from the effective Lagrangian model, we
Fig. 4. Time evolution of the abundance of Jrc from AuqAu collisions at LHC using cross sections from the effective Lagrangian w24x and the quark-exchange model w23x and the default parameters for the collision dynamics. The initial Jrc number is denoted by the arrow on the left.
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default parameters to 0.48 for a s 0.2 c 2rfm, 0.76 for a s 0.05 c 2rfm, 0.57 for mp s 130 MeV, 0.35 for t h s 4tc , 0.98 for t h s tc , and 0.56 for Tc s 150 MeV.
5. c X production in heavy ion collisions
X
Fig. 5. Temperature dependence of the thermally averaged c absorption and production cross sections from the effective Lagrangian model w23x.
for other parameters and the cross sections from the effective Lagrangian model. For larger values of transverse acceleration a, the hadron gas expands faster so its temperature drops more appreciably, leading to a slight reduction of the number of produced Jrc to 0.16 for a s 0.2 c 2rfm. However, the transverse expansion velocity in this case reaches the velocity of light after a few fmrc, which seems unreasonable. On the other hand, with smaller values of a, the final Jrc number is increased to 0.25 for a s 0.05 c 2rfm. If pions and rho mesons are out of chemical equilibrium w25x, Jrc absorption may be enhanced. Using a pion chemical potential of 130 MeV and a rho meson chemical potential twice as large, the final Jrc number is only slightly reduced to 0.19. The lifetime of the phase transition also affects the result. If it is doubled, then the number of produced Jrc is reduced to about 0.12, while it is increased to 0.31 if the phase transition is instantaneous. Reducing the value of the temperature at which the phase transition occurs affects also the number of Jrc produced from the hadron gas. Using Tc s 150 MeV, the Jrc number is reduced to 0.12. In the case cross sections from the quark-exchange model are used, calculations similar to the above change the Jrc number from 0.60 using the
The above analysis can be generalized to c X production in ultra-relativistic heavy-ion collisions. The primary c X number is about a factor of 5 smaller than the Jrc number, i.e., about 3.8 = 10y3 and 0.096 at RHIC and LHC, respectively. Since the radius of c X is about twice that of c , its absorption cross section by meson is expected to be larger. Indeed, if we assume that the coupling constant gc X D D has a similar value as gc D D f 7.7 used in the effective Lagrangian model of Ref. w23x, then the c X absorption cross section by pion and rho meson are both a few times larger than those for Jrc . The thermally averaged c X absorption and production cross sections from this model are shown in Fig. 5. Because of the larger c X mass than that of Jrc , the thermally averaged c X absorption cross section turns out to be order of magnitude larger than that for Jrc .
X
Fig. 6. Temperature dependence of the thermally averaged c absorption and production cross sections from the quark-exchange model w24x.
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The c X absorption cross section has not been evaluated in the quark-exchange model. To make an estimate, we assume that the same parameterization shown in Eq. Ž3. holds but the threshold energy s0 is replaced by the appropriate value, i.e., 3.83 GeV for c Xp ™ DD ) and c Xp ™ DD ) , while it remains the same for c Xp ™ D ) D ) . Furthermore, s 0 is increased by a factor of four to account for the larger size of c X than Jrc . The thermally averaged cross sections are shown in Fig. 6 and are about an order of magnitude larger than those from the effective Lagrangian model. With the cross sections from the effective Lagrangian model, we find that the number of c X produced from the hadronic matter in Au q Au collisions at RHIC is about 8.0 = 10y5 and is negligible compared with that from primary nucleon-nucleon collisions. For the same reaction at LHC, the produced c X from the hadronic matter is about 0.052 as shown by the solid curve in Fig. 7 and is more than 1r2 of the primary ones shown by the arrow on the left of the figure. Using the cross sections from the quark exchange model leads to an increase of the number of c X produced from the hadronic matter in Au q Au collisions at RHIC to 3.4 = 10y4 , which is
X
Fig. 7. Time evolution of the abundance of c from AuqAu collisions at LHC using cross sections from the effective Lagrangian w24x and the quark-exchange model w23x and the default X parameters for the collision dynamics. The initial c number is denoted by the arrow on the left.
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still an order of magnitude smaller than the number of primary c X . For LHC, we obtain 0.19 c X from the hadronic matter, shown by the dashed curve in Fig. 7, which is about twice the primary c X number.
6. Summary To summarize, we have estimated the number of Jrc produced from the reaction DD ™ Jrc M in the hadron gas formed in heavy-ion collisions at RHIC and LHC using the initial conditions determined from the HIJING parton model and the cross section given by either the effective Lagrangian or the quark-exchange model. Although this is a negligible effect at RHIC, it becomes important at LHC as a result of the appreciable number of D and D mesons in the hadron gas. We have found that the number of Jrc produced from this reaction at LHC may be comparable to that produced from initial primary collisions, which are either absorbed by nucleons or dissociated in the quark-gluon plasma formed in the collisions. A similar estimate has been made for c X production by assuming in the effective Lagrangian model that the c X DD coupling is the same as the c DD coupling and in the quark-exchange model that the c X absorption cross sections have similar energy dependence as those for Jrc except a different threshold and four times larger magnitude due to the larger size of c X . As in the case of Jrc , c X production from the hadron gas is found insignificant in heavy ion collisions at RHIC but important at LHC. To use the Jrc and c X yields as signals for the quark-gluon plasma at LHC thus requires a good understanding of their production from the hadron gas. For a more reliable evaluation of this effect, a better determination of the Jrc production and absorption cross sections from both experiments and theoretical models is needed. Also, a comprehensive transport model for describing the heavy ion collision dynamics at RHIC and LHC is needed. In the present study based on the schematic model described in Section 2, the final pion number at midrapidity is only about 600 per unit rapidity at LHC if the default parameters are used. This number is about a factor of three smaller than that predicted by the HIJING model, which,
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however, does not include final hadron rescattering. The final pion number in our model can be increased if we introduce one or more of the following mechanisms; a longer proper time t h for hadronization, a finite chemical potential mp for pions, and the production of entropy in the hadron gas. As shown in our schematic study, increasing t h reduces the number of produced Jrc and c X as a result of a decreased charm meson density in the hadron gas, but increasing mp does not affect much the final Jrc and c X numbers. Allowing entropy production during the expansion of the hadronic matter, which would lead to a slower decrease of its temperature, also makes it possible to increase the final pion number without reducing the Jrc and c X numbers as the latter are mostly produced during the initial high density hadronic matter. At present, it is not clear which of the above mechanisms will be more important in heavy ion collisions at ultrarelativistic energies.
Acknowledgements This work was started while the authors were visiting the Institute for Nuclear Theory at University of Washington for the program on Probes of Dense Matter in Ultrarelativistic Heavy Ion Collisions, and they thank Y. Asakawa, C. Gale, J. Kapusta, V. Koch, and C. Y. Wong for useful discussions. We are also grateful to D. Blaschke for communications regarding the quark-exchange model. The work of CMK, BZ, and XFZ was supported in part by the National Science Foundation under Grant No. PHY-9509266 and PHY-9870038, the Welch Foundation under Grant No. A-1358, and the Texas Advanced Research Program. The work of XNW was supported by the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High energy and Nuclear Physics of the U.S.
Department of energy under Contract No. DE-AC0376SF00098.
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