Heavy quark correlations and the effective volume for quarkonia production
aa r X i v : . [ nu c l - t h ] J un Heavy quark correlations and the e ff ective volume for quarkonia production Yunpeng Liu a , Che Ming Ko a,b , Feng Li a,b a Cyclotron Institute, Texas A & M University, College Station, Texas 77843, USA b Department of Physics and Astronomy, Texas A & M University, College Station, Texas 77843, USA
Abstract
Using the Boltzmann transport approach, we study the e ff ective volume of a correlated heavy quark pair in a partonicmedium based on their collision rate. We find that the e ff ective volume is finite and depends sensitively on themomentum of the heavy quark and the temperature of the medium. Generally, it increases linearly with time t at thevery beginning and the increase then becomes slower due to multiple scattering, and finally it increases linearly withrespect to t / . We further find that the colliding heavy quark pair has an e ff ective temperature similar to that of themedium even though their initial transverse momentum spectra are far from thermal equilibrium. Keywords: heavy ion collision, quark-gluon plasma, heavy flavor
PACS: + p
1. Introduction
The conservation of Abelian charges plays an impor-tant role for particle production in heavy ion collisions.One example is the so-called canonical suppression ofstrange particle production as a result of the conserva-tion of strangeness [1–6], which requires that a hadronconsisting of a strange quark is always produced to-gether with another hadron consisting of an antistrangequark. Because of their correlated production, the twohadrons are initially close in space and their subsequentannihilation probability thus di ff ers from the case whenthey are initially randomly distributed in space. Asshown in Ref. [7], this has a significant e ff ect on theirchemical equilibration time in heavy ion collisions. Asimilar e ff ect is expected for the production of hadronsconsisting of heavy charm or bottom quarks [8, 9] due tothe conservation of charm or bottom. In particular, sincethe heavy quark and antiquark pair are produced at thesame location and their momenta are also highly cor-related according to the leading order QCD, such cor-relations , especially when there is at most one pair ofheavy quarks in an event, would then a ff ect their subse-quent collision rate and thus the rate for quarkonia pro-duction [10].The dynamics of heavy quarks in a medium has beenwidely studied using the Langevin or the Fokker-Planckequation [11–17]. These studies have shown that theobserved suppression of heavy quarks at high momen-tum implies strong interactions between heavy quarks and partons in the medium. However, it was recentlypointed out that the Boltzmann equation gives a moreaccurate description of the dynamics of heavy quarks ina medium [18]. In the present paper, we adopt the Boltz-mann equation to study the motion of a pair of heavyquarks that are initially correlated in both the coordi-nate and the momentum space, as produced in high en-ergy collisions, to study the likelihood they would scat-ter again in terms of an e ff ective volume.
2. The model
Given two particle species A and B in a volume V attime t , their collision number ∆ N AB in a small time step ∆ t can be expressed as ∆ N AB = ∆ t Z V d x Z d p A d p B (2 π ) (2 π ) f A ( x , p A , t ) · f B ( x , p B , t ) v AB σ AB . (1)In the above, the distribution functions f A and f B arefunctions of the position x and momenta p A and p B ,while v AB and σ AB are their relative velocity and totalscattering cross section.For heavy quark Q and antiquark ¯ Q , if we assume thatthe distribution functions f Q = f ¯ Q are uniform in spaceand have a Boltzmann distribution in momentum witha temperature T , then the collision number is simply Preprint submitted to Elsevier March 25, 2018 iven by ∆ N th Q ¯ Q = T V ∆ t π N Q N ¯ Q N Q N ¯ Q Z ∞√ s d √ s sp σ Q ¯ Q ( √ s ) · K ( √ s / T ) , (2)where √ s = m Q + m ¯ Q is the minimum cen-ter of mass energy √ s of the scattering pair, p = q [( s − m Q − m Q ) − m Q m Q ] / (4 s ) is their momentumin their center of mass frame, and K is the modifiedBessel function of the second kind. The number of Q in a volume V and its thermally equilibrated num-ber with vanishing chemical potential are denoted by N Q and N Q = m Q VT K ( m Q / T ) / (2 π ), respectively,while those of ¯ Q are, respectively, N ¯ Q and N ¯ Q . In thecase of a constant cross section σ Q ¯ Q and equal masses m Q = m ¯ Q = m , the above expression can be furtherexpressed as ∆ N th Q ¯ Q = N Q N ¯ Q σ Q ¯ Q ∆ tV g ( m ∗ ) (3)with g ( z ) ≡ K (2 z ) / [ zK ( z )] and m ∗ = m / T .On the other hand, if only one pair of heavy quarksis produced in a heavy ion collision, they will initiallybe at the same position and have essentially same mo-mentum but with opposite direction. As they propa-gate in the medium, they will gradually move apart andalso become thermalized. Because of the large size ofthe medium and the relatively long relaxation time forheavy quarks and antiquarks in the medium, they are notlikely to trace the whole volume of the fireball and toachieve kinetic equilibrium immediately. The deviationof the heavy quark distribution function from the uni-form and thermal distribution used in obtaining Eq. (3)for rare heavy quark events in relativistic heavy ion col-lisions (e.g. charm at SPS energy and bottom at RHICenergy) can be characterized by an e ff ective volume, V e ff ≡ lim ∆ t → ,σ Q ¯ Q → , N Q , N ¯ Q →∞ N Q N ¯ Q σ Q ¯ Q ∆ t ∆ N Q ¯ Q g ( m ∗ ) , (4)which only depends on the distribution of the heavyquark pairs in the phase space and the temperature ofthe medium. It approaches the volume of the mediumwhen the heavy quarks are thermalized.The nonequilibrium dynamics of heavy quarks canbe described by the Boltzmann equation for their phasespace distribution function f Q ( x , p , t ) ,∂ t f Q ( x , p , t ) + v Q · ∇ f Q ( x , p , t ) = C [ f Q ] , (5)where v Q is the velocity of the heavy quark, and C [ f Q ]is the collision term. For a given initial position and mo-mentum of a heavy quark, this equation can be solved using the heavy quark collision rate with thermal par-tons in the medium, given by R = mT m ∗ g N g π E Z ∞ dy e − m ∗ g cosh y cosh y r sinh y · sinh( m ∗ g sinh y r sinh y ) σ gQ ( y ) , (6)which is obtained directly from Eq. (1) by using theBoltzmann distribution for the thermal partons. In theabove, σ gQ ( y ) is the cross section for scattering betweenthe heavy quark and a parton of rapidity y in the heavyquark frame; N g is the degeneracy of the parton, whichis taken to be 16 if we include only gluons and ne-glect quarks as their scattering cross sections with heavyquarks are small [13]; y r = acosh( p · u / m ) is the rapidityof the heavy quark relative to the medium with four-velocity u µ ; m ∗ g = m g / T where m g is the mass of gluons;and E is the energy of the heavy quark. For a constantcross section and massless partons, the heavy quark col-lision rate in the rest frame of the medium can be simpli-fied to R = N g T σ gQ /π , resulting in a mean free timebetween collisions given by the inverse τ = / R . As theheavy quark traverses through the medium, the proba-bility for it to collide with a parton during a small timestep dt ≪ τ is then dt /τ . A collision occurs if a randomnumber generated between 0 and 1 is smaller than dt /τ .With the parton momentum randomly selected accord-ing to the thermal distribution, the momentum of theheavy quark after the collision can be determined fromthe energy and momentum conservations once its direc-tion is obtained from the di ff erential cross section. Forsimplicity, we take the cross section to be isotropic withthe magnitude σ gQ = Q is identical to Q , except that the directions oftheir initial momenta are opposite.For the scattering between the heavy quark and an-tiquark pair, it is treated by comparing their impactparameter with the scattering cross section and is de-scribed in detail in Appendix A. Although only onepair of heavy quarks is initially produced at the sameposition with opposite momentum in a heavy ion col-lision event, we can study their mean dynamics in thehot medium by following independently the motions ofmany similar pairs of heavy quarks and calculate the av-erage number of heavy quark scatterings as in Eq. (4).
3. Results and discussions
To illustrate the method used in our study, we firstconsider the collision dynamics of a pair of heavy2 t (fm/c) ) (f m e ff V L = 10 fmT = 0.3 GeV m = 1.25 GeV/c = 4 mb gQ σ = 5 GeV/c p Figure 1: Time evolution of the e ff ective volume V e ff of a pair of heavyquarks with mass m and back-to-back momentum p in a periodiccubic box of length L on each side, which consists of a gluonic matterat temperature T , and undergo scattering with gluons with the crosssection σ gQ . quarks in a periodic cubic box of length L =
10 fmon each side. The heavy quarks are initially locatedat same position and have opposite momentum of 5GeV / c . With the temperature of the medium taken to be0 . ff ective volume V e ff of the heavy quark pair calculated according to Eq. (4)is shown in Fig. 1. As expected, the e ff ective volume V e ff approaches the volume of the medium as time t approaches infinity, and the time scale for the heavyquarks to spread uniformly in the box is about 30 fm / c . -2 0 2 4 60.00.20.40.6 t=0.0 fm/ct=0.2 fm/ct=1.0 fm/ct=5.0 fm/cThermal (GeV/c) z p ] - [( G e V / c ) z N dpd N Figure 2: (Color on line) Longitudinal momentum p z distribution ofheavy quarks with initial momentum p = / c along the z direc-tion at di ff erent times in a finite gluonic matter of temperature T = . p z distribution calculated from Eq. (7) To see how heavy quarks approach thermal equilib-rium, we show in Fig. 2 their p z distribution at di ff erenttimes. It is seen that the initial δ -like peak at p z = / c initially moves down in p z and gradually ap-proaches a thermal distribution. During early times, the distribution deviates significantly from a Gaussianfunction, in contrast with the prediction based on theLangevin approach because of the comparable averageparton kinetic energy and heavy quark mass, which isbeyond the region where the Langevin approach is ap-plicable [19] . Since heavy quarks have a mean free time τ = .
44 fm / c , their p z distribution essentially becomesthe thermal distribution f P z ( p z ) = dNN d p z = ( m ∗ L + e − m ∗ L T m ∗ K ( m ∗ ) (7)with m ∗ L ≡ q m + p z / T at t = / c ≫ τ . In this subsection, we consider the case that the heavyquarks move in a medium of infinite volume, using thesame heavy quark mass m = .
25 GeV / c , heavy quark-gluon scattering cross section σ gQ = T = . V e ff are shown in Figs. 3, 4, and 5 for thethree time stages t ≪ τ , t ∼ τ , and t ≫ τ relative tothe mean free time τ = .
44 fm / c . In all three cases,the e ff ective volume V e ff depends sensitively on the ini-tial momentum with the larger one resulting in a largere ff ective volume. =1 GeV/c p =2 GeV/c p =3 GeV/c p =4 GeV/c p =5 GeV/c p t (fm/c) ) (f m e ff V Figure 3: Time evolution of V e ff at early times t ≪ τ for di ff erent ini-tial heavy quark momentum p in an infinite medium of temperature T = . As shown in Fig. 3, the e ff ective volume V e ff at earliertimes t ≪ τ is proportional to time, which is true ingeneral, and the proof is given in Appendix B.When the time t becomes comparable to the meanfree time τ , heavy quarks are more likely to turn aroundand collide with each other after undergoing successivecollisions. As a result, the increase of V e ff with timebecomes slower as shown in Fig. 4.3 .0 0.5 1.0 0 50100150 =1 GeV/c p =2 GeV/c p =3 GeV/c p =4 GeV/c p =5 GeV/c p t (fm/c) ) (f m e ff V Figure 4: Same as Fig. 3 for intermediate times t ∼ τ .
0 5 10 0100200 =1 GeV/c p =2 GeV/c p =3 GeV/c p =4 GeV/c p =5 GeV/c p ] [(fm/c) t ) (f m e ff V (cid:9)(cid:9)(cid:9)(cid:9)(cid:9) Figure 5: Same as Fig. 3 for long times t ≫ τ but with the horizontalaxis changed from t to t / . For time t much larger than the mean free time τ ,when heavy quarks have collided many times with par-tons in the medium, their behavior can be described byrandom walks. In this picture, the distance traveled by aheavy quark is proportional to t / , and the e ff ective vol-ume for heavy quarks to collide is thus proportional to t / as shown in Fig. 5. Although the magnitude of thee ff ective volume increases with heavy quark initial mo-mentum, as it depends on its nonequilibium dynamics,the coe ffi cient of the proportionality or the slope of thelines in the figure is independent of the initial momen-tum, since it only depends on their thermal motions.Since the initially produced charm pair are distributedin a certain volume due to their quantum nature, ourclassical calculation based on the Boltzmann equationis valid only after certain time t when the wave packetsof the heavy quark and antiquark are separated. The A detailed discussion on the value of t requires the spatial dis-tribution of the charm quark when it is produced, which is beyond thescope of study in this paper. total number of Q - ¯ Q collisions is then given by N Q ¯ Q = Z ∞ t dt σ Q ¯ Q V e ff ( t ) g ( m ∗ ) . (8)Because of the long time behavior of the heavy quarke ff ective volume, the integral in Eq. (8) converges at t = ∞ . Therefore, the heavy quark pair hardly havethe chance to collide with each other long after they areproduced even if the lifetime of the medium is infinitelylong. ff ective volume t (fm/c) ) (f m e ff V T=0.20 GeVT=0.25 GeVT=0.30 GeVT=0.35 GeVT=0.40 GeV=2 GeV/c p Figure 6: Time evolution of the e ff ective volume V e ff of heavy quarkswith initial charm momentum p = / c for di ff erent mediumtemperatures. Since both the parton density and the parton energydepend on the temperature of the medium, the heavyquark e ff ective volume also depends on the temperatureof the medium, and this is shown in Fig. 6. The e ff ec-tive volume is seen to depend sensitively on tempera-ture. For the temperature T = . ff ectivevolume already exceeds 1000 fm at t = . / c dueto the strong back to back correlation, while for T = . ff ective volume is less than 10 fm at t = / c as a result of faster thermalization of the heavyquarks. As discussed in the previous subsection, the V e ff increases with time monotonously for all temperatures. Since a heavy quark pair is produced from hard colli-sions of nucleons at high energies, their initial momenta The space dimension d = d = d =
2, the heavy quark pair will collide with each otherat a certain time since the integral in Eq. (8) diverges at t = ∞ , andthe equilibrium between the heavy quark pair and quarkonia can beestablished even if the system is infinitely large. =0.5 fm/ct=1.0 fm/ct=5.0 fm/cT=0.22 GeVT=0.25 GeVT=0.29 GeV (GeV)s ) - ( G e V S f Figure 7: Center of mass frame energy √ s distribution of collidingheavy quark pairs with initial back-to-back momentum of 5 GeV / c ina medium of temperature T = . ff erent times (shown bysymbols). Lines are collision rates from calculations based on ther-malized heavy quarks. are large and opposite in direction. As they di ff use ina medium and collide with thermal partons, their cen-ter of mass energy is a quantity of interest. Shown inFig. 7 by symbols are the distribution f √ s at di ff erenttimes for a pair of heavy quarks with an initial back-to-back momentum of 5 GeV / c in a medium of tempera-ture T = . √ s withincreasing time. Also shown in the figure by lines is thecenter of mass energy distribution of heavy quark pairsthat have a thermal distribution. According to Eqs. (2)and (3), the latter is given by f th √ S ( √ s , T ) ≡ N th Q ¯ Q dN th Q ¯ Q d √ s = s ( s − m ) K ( √ s ∗ )2 T m K (2 m ∗ ) (9)with s ∗ = s / T , and N th Q ¯ Q is the collision number be-tween thermal Q and ¯ Q within a certain time. It showsthat the distribution of √ s at t = . / c , which iscomparable to the mean free time for the heavy quarks,can be roughly described by a thermal distribution ata lower temperature T = .
22 GeV than that of themedium. This is because the heavy quarks carry verylittle momenta after reversing their direction of motionat this time. At t = . / c , the distribution of √ s can well be described by T = .
25 GeV since theheavy quarks are partially thermalized. At an even latertime t = / c , the distribution of √ s approaches theequilibrium one and can thus be described by an ef-fective temperature of T = .
29 GeV close to that ofthe medium. The approaching of the √ s distribution to the thermal distribution from a lower temperature isvery di ff erent from that of the p z distribution, which ap-proaches the thermal distribution gradually from an ini-tial hard distribution as shown in Fig. 2 .Similarly, the distribution of the total momentum | P | of colliding charm pairs, which can also be approx-imately described by a thermal distribution, is foundto have an e ff ective temperature of 0 . , .
33 and 0 . t = . , .
0, and 5 . / c , respectively.The thermal like distributions of √ s and | P | implies thatthe regeneration of heavy quarkonia from a medium isalways dominated by heavy quarks with low momen-tum when they are rare. For example, neither the high p T J /ψ at SPS nor the high p T Υ at RHIC are expectedto be produced from regeneration although the heavyquarks may not be thermalized.
4. Conclusions
Based on the Boltzmann equation, we have studiedthe e ff ective volume V e ff of a correlated classical heavyquark pair in a hot medium on their collision rate for rareheavy quark events, which is more realistic than simplyconsidering the volume of the fierball. The V e ff is finitedue to their initial spatial and momentum correlationseven though the system is an open one like in heavy ioncollisions. We have found that V e ff is proportional tothe time t when it is much shorter than their mean freetime τ between collisions with the medium partons, i.e., t ≪ τ . The increase becomes slower for t ∼ τ , and even-tually V e ff increases linearly with t / for t ≫ τ . Conse-quently, the chance for a heavy quark pair to collide witheach other per unit time increases monotonously withtime t . Also, the chance for the heavy quarks to col-lide again depends sensitively on their initial momen-tum and the temperature of the medium. Heavy quarksof lower initial momentum in a medium of higher tem-perature have a larger chance to collide. Furthermore,the distribution of heavy quark pair center of mass en-ergy corresponds to an e ff ective temperature which islower than the actual temperature of the medium. Allthese properties are important for quarkonium regener-ation in collisions where heavy quarks are rarely pro-duced. The present study was based on heavy quarkscattering cross sections with partons and among them-selves that are not from specific model calculations. Al-though the above results are not expected to qualita-tively change, a quantitative study requires more accu-rate cross sections, which we leave as a future work. The periodic condition there has no e ff ect in the momentumspace. cknowledgements We thank Ralf Rapp and Nu Xu for helpful discus-sions. This work was supported by the U.S. NationalScience Foundation under Grant No. PHY-1068572,the US Department of Energy under Contract No. DE-FG02-10ER41682, and the Welch Foundation underGrant No. A-1358.
Appendix A. Conditions for heavy quark collisions
Consider one pair of free heavy quark Q and anti-quark ¯ Q with their 4-dimensional coordinates x Q and x ¯ Q , and velocities u Q and u ¯ Q , respectively. Their tra-jectories in space are x Q ( t Q ) = x Q + u Q u Q ( t Q − t Q ) , x ¯ Q ( t ¯ Q ) = x ¯ Q + u ¯ Q u Q ( t ¯ Q − t ¯ Q ) . (A.1)Then s ( t Q , t ¯ Q ) ≡ ( x Q ( t Q ) − x ¯ Q ( t ¯ Q )) = ( t Q − t ¯ Q ) − [ x Q ( t Q ) − x ¯ Q ( t ¯ Q )] . (A.2)When the minimum distance b between Q and ¯ Q in theircenter of mass frame is reached, it is a saddle point of s ( t Q , t ¯ Q ). Requiring ∂ t Q s = ∂ t ¯ Q s = , (A.3)we find the minimum distance b and the correspondingtimes t Q and t ¯ Q to be b = (cid:16) − ( ∆ x ) − h ( ∆ x · u Q ) + ( ∆ x · u ¯ Q ) − ∆ x · u Q )( ∆ x · u ¯ Q ) u Q ¯ Q i / (cid:16) u Q ¯ Q − (cid:17)(cid:17) / , t Q = t Q + − ∆ x · u Q + ( ∆ x · u ¯ Q ) u Q ¯ Q u Q ¯ Q − u Q , t ¯ Q = t ¯ Q + ∆ x · u ¯ Q − ( ∆ x · u Q ) u Q ¯ Q u Q ¯ Q − u Q , (A.4)with ∆ x = x ¯ Q − x Q and u Q ¯ Q = u Q · u ¯ Q . During thetime interval ( t , t + ∆ t ), the two particles are regarded asundergoing a collision if and only if b ≤ p σ Q ¯ Q /π and t < ( t Q + t ¯ Q ) / < t + ∆ t are satisfied. Appendix B. Proof of the linear behavior of the ef-fective volume V e ff at short times To investigate the time dependence of V e ff , we con-sider a pair of heavy quarks in a medium and followtheir motions. When the time t is much smaller than themean free time τ , a heavy quark can have at most onecollision with the partons in the medium. Therefore, ifwe slow down the time by a factor λ to t ′ = λ t and alsostretch the coordinates by λ to l ′ = λ l , so that the veloci-ties of the particles remain unchanged, then the numberof heavy quark collisions in the original and the scaledspace-time are related by ∆ N ′ Q ¯ Q ( t ′ , ∆ t ′ , σ ′ Q ¯ Q , σ ′ gQ , f ′ g ) = ∆ N Q ¯ Q ( t , ∆ t , σ Q ¯ Q , σ gQ , f g ) (B.1)with ∆ t ′ = λ ∆ t , σ ′ Q ¯ Q = λ σ Q ¯ Q , σ ′ gQ = λ σ gQ , and f ′ g = λ − f g . On the other hand, since the number ofheavy quark collisions at a given time is, up to a con-stant, given by ∆ N Q ¯ Q ∝ ∆ t σ Q ¯ Q ( σ gQ f g ) . (B.2)we have ∆ N ′ Q ¯ Q ( t ′ , ∆ t ′ , σ ′ Q ¯ Q , σ ′ gQ , f ′ g ) = ∆ t ′ σ ′ Q ¯ Q ( σ ′ gQ f ′ g ) ∆ t σ Q ¯ Q ( σ gQ f g ) ∆ N Q ¯ Q ( t ′ , ∆ t , σ Q ¯ Q , σ gQ , f g ) = λ ∆ N Q ¯ Q ( λ t , ∆ t , σ Q ¯ Q , σ gQ , f g ) . (B.3)Combining Eqs. (B.1) and (B.3), we obtain ∆ N Q ¯ Q ( λ t , ∆ t , σ Q ¯ Q , σ gQ , f g ) = λ − ∆ N Q ¯ Q ( t , ∆ t , σ Q ¯ Q , σ gQ , f g ) , (B.4)and therefore V e ff ( λ t ) = λ V e ff ( t ) . (B.5)This proves our claim that the e ff ective volume V e ff islinearly proportional to t , as long as t is much smallerthan the mean free time τ between the collisions ofheavy quarks with medium partons. Because the scal-ing in Eq.(B.1) does not change either the velocities ofparticles or the angular distribution after their collisions,this proof is independent of the details of the cross sec-tions σ gQ or σ Q ¯ Q . References [1] R. Hagedorn, Thermal dynamics of strong interactions, CERNYellow Report 71-12 (1971) 101.
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