$\Omega_{ccc}$ Production in High Energy Nuclear Collisions

We investigate the production of $\Omega_{ccc}$ baryon in high energy nuclear collisions via quark coalescence mechanism. The wave function of $\Omega_{ccc}$ is solved from the Schr\"odinger equation for the bound state of three charm quarks by using the hyperspherical method. The production cross section of $\Omega_{ccc}$ per binary collision in a central Pb+Pb collision at $\sqrt{s_{NN}}=2.76$ TeV reaches 9 nb, which is at least two orders of magnitude larger than that in a p+p collision at the same energy. Therefore, it is most probable to discover $\Omega_{ccc}$ in heavy ion collisions at LHC, and the observation will be a clear signature of the quark-gluon plasma formation.

From Quantum Chromodynamics (QCD) at finite temperature, it is widely accepted that there exists a deconfinement phase transition from hadron gas to a quarkgluon plasma (QGP) at a critical temperature T c ∼ 155 MeV [1].Such a phase transition is expected to be realized in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC).Since the fireball formed in a heavy ion collision expands rapidly, one cannot observe directly the QGP in the final state and needs probes to signal the QGP formation in the early stage of the fireball evolution.The quarkonium suppression is considered as such a sensitive probe [2].The measured J/ψ nuclear modification factor and especially the transverse momentum distributions at RHIC [3][4][5][6] and LHC [7][8][9] show a strong hot medium effect.
In this paper we investigate Ω ccc production as an alternative probe of the QGP formation in heavy ion collisions at LHC.The existence of Ω ccc baryon, the ground bound state of three charm quarks, is a direct result of the quark model.In p+p collisions at LHC energy, the Ω ccc production is difficult, since it requires at least three pairs of charm quarks with small relative momenta in an event.In relativistic heavy ion collisions, however, there are plenty of off-diagonal charm quarks in the fireball, and the Ω ccc production becomes much easier.The coalescence mechanism [10] has been successfully used to describe the light hadron production, especially the quark number scaling of the elliptic flow [11] and the enhancement of the baryon to meson ratio [12][13][14].Taking into account this mechanism, the yield of Ω ccc is proportional to the cube of the charm quark number, N Ωccc ∼ N 3 c , at given temperature and volume of the fireball.For central Pb+Pb collisions at LHC energy, the Ω ccc production becomes significant and may play an important role in the probe of QGP.The coalescence mechanism [15] or statistical emission [16][17][18] or regeneration [19][20][21] for quarkonium production is widely discussed in heavy ion collisions and successfully explains the J/ψ yield and momentum distributions.Recently it was suggested that B c mesons can be observed at RHIC and LHC due to the regeneration mechanism [22,23].The production of par-ticles with double, triple and hidden charm in heavy ion collisions was also studied in the framework of a statistical coalescence model [24], and the symmetries of the three-heavy-quark system was also investigated within the effective field theory framework of potential nonrelativistic QCD [25].
In coalescence models the change in the constituent distribution before and after the coalescence process is required to be small, namely the number of constituents involved in the coalescence must be small compared with the total constituent number of the system.In this sense, the coalescence mechanism is more suitable for the production of rare particles like Ω ccc .The coalescence probability in phase space, namely the Wigner function, is usually parameterized as a Gaussian distribution [15,41] and the width is fixed by fitting the data in heavy ion collisions.For Ω ccc , there are currently no data, and an adjustable coalescence probability will lose the prediction power of the calculation.Fortunately, for charmed hadrons like J/ψ and Ω ccc we can calculate their wave function and in turn the Wigner function by solving the Schrödinger equation with the help of the lattice simulated heavy quark potential at finite temperature [27].
In the following we first solve the three-body Schrödinger equation via hyperspherical method to get the wave function in coordinate space and the Wigner function in phase space of Ω ccc , and then fix the coalescence hypersurface and derive the Ω ccc momentum distribution via the coalescence mechanism.We will numerically calculate the Ω ccc production in Pb+Pb collisions at LHC energy and summarize the results and physics in the end.
Since charm quarks are so heavy, we can employ the non-relativistic Schrödinger equation in the coordinate representation to describe the bound states of three charm quarks, with charm quark mass m c and total energy E T .As a usually used approximation [28], we neglect the threebody interaction and express the potential as a sum of pair interactions, According to the leading order QCD, the diquark potential is only one half of the quark-antiquark potential, We assume that such a relation still holds in the case of strong coupling and take the Cornell potential where r ij = r i − r j is the relative distance between the two quarks i and j, and α = π/12 and σ = 0.2 GeV 2 are coupling parameters of the potential which together with the charm quark mass m c = 1.25 GeV reproduce well the J/ψ and Υ masses [29] in vacuum.In hot and dense medium, the strength of the interaction between two quarks should decrease with temperature.However, from the lattice calculation [30], the J/ψ spectral function is clearly broadened only at T > T c .Therefore, we still take the Cornell potential between a pair of charm quarks at the coalescence which happens at T c .It is hard to solve a three-body problem exactly, and one usually take some approximations to simplify the problem.One of the most popular and effective approaches is the hyperspherical method [28].Its main idea is to change a low dimensional three-body problem to a high dimensional one-body problem with the assumption of hyperspherical symmetry for the potential [31].Since the potential (2) is only related to the relative coordinates r i − r j , the motion of the three-quark bound state can be factorized into the motion of the baryon and the relative motion among the quarks, by making the transformation between r 1 , r 2 , r 3 and the baryon coordinate R and relative coordinates r x , r y , (R, r x , r y ) = (r 1 , r 2 , r 3 )M T with the transformation matrix Then by rewriting r x and r y in terms of their azimuthal angles θ x , φ x , θ y , φ y and the hyperradius r = r 2 x + r 2 y = (r 2 12 + r 2 23 + r 2 31 ) /3 and hyperpolar angle α = arctan(|r y |/|r x |), the volume element in hyper coordinates is represented as (5) and the kinetic energy in center of mass frame becomes where L is the hyper angular momentum and lx and ly are the normal angular momenta.Since the potential V (|r i − r j |) depends on both the radius and the 5 angles, one can not directly separate the relative motion into a radial part and an angular part.The approximation [31] we take here is to average the potential over all the angles, With this homogeneous potential, the equation of relative motion can now be factorized into the radial equation (for the ground state with L = 0) and the angular equation where ϕ(r) is the radial wave function, Y (Ω) the eigenstate of the hyper angular momentum operator L2 with Ω representing all the angle variables {α, θ x , φ x , θ y , φ y }, L the corresponding angular momentum number, and E the relative energy.The radial wave function ϕ(r) is normalized as where P (r) = |ϕ(r)| 2 r 5 is the probability to find the three charm quarks in the ground bound state in a hyper spherical shell of unit thickness at radius r.By solving the radial equation for Ω ccc , we obtain its mass m Ω = 4.7 GeV and binding energy ǫ Ω = 900 MeV.From the radial probability shown in Fig. 1, Ω ccc is a tightly bound state of three charm quarks with average radius r ∼ 0.5 fm which is almost the same as J/ψ.
We now construct the Wigner function in the center of mass frame of Ω ccc , where p = (p x , p y ) is the 6D relative momentum corresponding to r = (r x , r y ), and the 3D relative momenta p x , p y and the Ω ccc momentum P, corresponding to r x , r y and R, are associated with the three quark momenta p 1 , p 2 , p 3 via the transformation (P, p x , p y ) = (p 1 , p 2 , p 3 )M −1 .Using the above obtained relative wave function ψ(r) = ϕ(r)Y (Ω) in the approximation of hyperspherical symmetry and taking the first axis of the vector y in the direction of p and the second axis on the plane constructed by p and r, the Wigner function for the ground bound state Ω ccc (Y (Ω) = π −3/2 ) is simplified as Note that the vectors r and p in the Wigner function are correlated with each other through the angle θ between them.By integrating out the angle we obtain the probability to find the three charm quarks in the ground bound state in a hyperspherical shell in coordinate space at radius r and in a hyperspherical shell in momentum space at radius p, which satisfies the normalization Fig. 2 shows the probability P(r, p).The most probable position in the phase space is located at (r, p) ∼ (0.5 fm, 1 GeV), leading to r • p ≈ 2.5, which is near to the result r 2 p 2 = 3 from the uncertainty relation for a Gaussian Wigner function.
The observed momentum distribution of Ω ccc via coalescence mechanism can be calculated from the Wigner function [32,33], (15) where F is the distribution function of the three charm quarks in phase space, C the factor to count the intrinsic symmetry.For Ω ccc , it is a color singlet and carries spin 3/2.Since there is only one color singlet state in the 3 × 3 × 3 = 27 possible color states and 4 spin 3/2 states in the 2 × 2 × 2 = 8 possible spin states, we obtain C = 1/27 × 4/8 = 1/54.In relativistic heavy ion collisions, the hadronization of the parton system happens on the hypersurface of confinement phase transition.The 4D coordinates R µ = (t, R) on the hypersurface is constrained by the hydronization condition, which leads to t = t(T c , R), where T c is the critical temperature of the confinement phase transition, and the local temperature T (R µ ) and fluid velocity u µ (R µ ) (which will be used in the charm quark distribution) are determined by hydrodynamic equations with T µν = (ǫ + p)u µ u ν − g µν p being the energy momentum tensor and ǫ and p the energy density and pressure.To close the hydrodynamical equations one needs to know the equation of state of the medium.We follow Ref. [34] where the deconfined phase at high temperature is an ideal gas of gluons and massless u and d quarks plus 150 MeV massed s quarks, and the hadron phase at low temperature is an ideal gas of all known hadrons and resonances with mass up to 2 GeV [35].There is a first order phase transition between these two phases with the critical temperature T c = 165 MeV.For the initialization of the hot medium, we take the same treatment as in Ref. [36].The maximum temperature of the medium at the starting time τ 0 = 0.6 fm/c is T 0 = 484 MeV for central 2.76 TeV Pb+Pb collisions at LHC. Changing the volume integral d 3 R to the covariant integral on the hypersurface Σ, the Ω ccc distribution is rewritten as where P T , η and P 0 = P 2 + m 2 Ω are, respectively, the Ω ccc transverse momentum, rapidity and energy.Remember that the Wigner function obtained above is derived in the center of mass frame of Ω ccc and the Ω ccc moves with 4-velocity v µ = P µ /m Ω in the laboratory frame, the coordinates r 1 , r 2 , r 3 or r x , r y in the parton distribution function F should be replaced by rµ x = L µ ν r ν x , rµ y = L µ ν r ν y with the boost matrix elements , and ξ = 1/(1 + v 0 ).Since in the center of mass frame of the three charm quarks the coalescence happens at the same time, there is r 0 x = r 0 y = 0. Similarly, the momenta p i (i = 1, 2, 3) in the charm quark distribution F are replaced by pµ i = L µ ν p ν i with p 0 = p 2 i + m 2 c .We now consider the integral element dσ µ (R) over the coalescence hypersurface Σ.In the framework of Bjorken hydrodynamics [37], we take the rapidity η = (1/2) ln[(t+ and the azimuth angle φ = arctan(R y /R x ) as independent variables instead of R and regard the proper time τ = t 2 − R 2 z as a function of η, R T , φ through the coalescence condition ( 16), the hypersurface element can be expressed as The three quark distribution F (r 1 , r 2 , r 3 , p 1 , p 2 , p 3 ) can be factorized as where S counts the symmetry of the same specie of quarks.For Ω ccc we simply take S = 1/3!= 1/6, since the number of charm quarks in an event is much larger than 3 at LHC energy.The single charm quark distribution f is in principle between the pQCD and equilibrium distributions.From the experimental data at LHC [38,39], the observed large quench factor and elliptic flow for charmed mesons indicate that the charm quarks interact strongly with the medium.Therefore, one can take, as a good approximation, a kinetically thermalized phase space distribution for charm quarks, where the local temperature T (r i ) and fluid 4-velocity u µ (r i ) of the medium are determined by the hydrodynamics, and is the normalization factor.The number density ρ(r i ) is controlled by the charm conservation equation The charm quark number density at initial time τ 0 = 0.6 fm/c is fixed by the colliding energy and nuclear geometry, where T A and T B are the thickness functions of the lead nuclei with nuclear matter density following the Woods-Saxon distribution, dσ cc pp /dη is the rapidity distribution of charm quark cross section in p + p collisions, and b is the impact parameter.We now apply the above coalescence approach to Ω ccc production in relativistic heavy ion collisions.The yield at middle rapidity in Pb+Pb collisions at colliding energy √ s N N = 2.76 TeV is shown in Fig. 3 as a function of the number of binary collisions N coll .The charm production cross section is taken as dσ cc /dη = 0.7 mb [40].The yield increases almost linearly with N coll and reaches 5 × 10 −4 in the most central collisions.If we consider a homogeneous fireball with volume V at the coalescence time, the yield of Ω ccc can be estimated as where N c is the charm quark yield.Supposing both N c and V are proportional to N coll , the yield of Ω ccc is then proportional to N coll , which approximately explains the linear increase in Fig. 3. From the Ω ccc yield we can define an effective cross section per binary collision, With the inelastic proton cross section σ pp = 62 mb, the rapidity range ∆η = 1.8, and N coll = 2000 for the most central collision, we have σ Ω = 9 nb, which is much larger than the cross section 0.06-0.13nb at 7 TeV and 0.1-0.2nb at 14 TeV in p+p collisions at mid rapidity |η| < 2.5 [41].It is necessary to point out that the tightly bound states of heavy quarks are in principle continuously produced in the medium above T c and suffer from dissociation due to the interaction with the medium [42].Therefore, the above obtained Ω ccc yield from the sudden coalescence approach without considering dissociation is more like the upper limit of the production.We also calculated the wave function and in turn the Wigner function for J/ψ where the interaction between the c and c is exactly the Cornell potential.The calculated nuclear modification factor R AA = N AA /(N pp N coll ) for J/ψ in Pb+Pb collisions at LHC energy is shown in Fig. 4 as a function of the number of participant nucleons N part , where N AA and N pp are, respectively, the J/ψ yield in Pb+Pb and p+p collisions.The model calculation with charm cross section dσ/dη = 0.7 mb, corresponding to the upper limit of the theoretical band, is clearly overestimated, in comparison with the experimental data [43].This is probably due to the lack of J/ψ dissociation in the hot medium and the large charm cross section.Note that the J/ψ production in heavy ion collisions is more complicated than Ω ccc .J/ψs can be produced via both initial p+p collisions and later coalescence, while the coalescence is the only way for Ω ccc production at LHC energy.A good description of the experimental J/ψ data needs dσ/dη = 0.5 mb in our calculation, see the lower limit of the theoretical band in Fig. 4. is due to the uncertainty in the charm cross section, the upper and lower limit of the band correspond to dσ/dη = 0.7 and 0.5 mb.The data are from the ALICE collaboration [43].
In heavy ion collisions, transverse motion is developed during the dynamical evolution of the system.The mi-croscopically high particle density and multiple scatterings are essential for the finally observed transverse momentum distributions.The distributions are therefore sensitive to the medium properties, like the equation of state.In order to understand the Ω ccc production mechanism and extract the properties of the medium, we calculated the transverse momentum distributions of Ω ccc and J/ψ, shown in Fig. 5 with the assumption of thermalized charm quark distribution.In both cases the distribution drops down monotonously with transverse momentum p T .For Ω ccc it is about one order of magnitude smaller at p T = 4 GeV than that at p T = 0.As a characteristic of the coalescence mechanism [12][13][14], the decreasing of J/ψ becomes faster than Ω ccc at high p T .In summary, we investigated the production of Ω ccc baryon via coalescence mechanism in relativistic heavy ion collisions.We solved the Schrödinger equation for the ground bound state of three charm quarks by the hyperspherical method and derived the radial wave function of Ω ccc .With the obtained Wigner function as the coalescence probability and thermalized charm quark distribution, we calculated the Ω ccc yield and transverse momentum spectrum in Pb+Pb collisions at colliding energy √ s N N = 2.76 TeV.The obtained production cross section per binary collision is at least two orders of magnitude larger than that in a p+p collision at LHC energy.Therefore, from the weak decay modes like the non-leptonic decay channel Ω ccc → Ω sss + 3π + [41], it becomes most probable to observe Ω ccc in heavy ion collisions at LHC, and its observation is a clear signature of the quark-gluon plasma formation.

FIG. 1 .
FIG.1.The radial probability P (r) to find the three charm quarks in the ground bound state in a hyper spherical shell of unit thickness at radius r.

9 FIG. 2 .
FIG.2.The probability P(r, p) to find the three charm quarks in the ground bound state in a hyper spherical shell in coordinate space at radius r and in a hyper spherical shell in momentum space at radius p.

9 FIG. 3 .
FIG. 3. The Ωccc yield as a function of the number of binary collisions N coll in Pb+Pb collisions at middle rapidity and colliding energy √ sNN = 2.76 TeV.

9 FIG. 4 .
FIG. 4. The J/ψ nuclear modification factor RAA at middle rapidity as a function of the number of participant Npart in Pb+Pb collisions at √ sNN = 2.76 TeV.The theoretical band

9 FIG. 5 .
FIG. 5.The transverse momentum distribution of Ωccc (solid line) and J/ψ (dashed line) at middle rapidity in central Pb+Pb collisions at √ sNN = 2.76 TeV.The calculation for J/ψ is scaled by a factor of 10 −3 .