Heavy Flavor Enhancement as a Signal of Color Deconfinement

We argue that the color deconfinement in heavy ion collisions may lead to enhanced production of hadrons with open heavy flavor (charm or bottom). We estimate the upper bound of this enhancement.

where N AB coll (b) is the average number of primary nucleon collisions, which is determined by the geometry of the colliding nuclei, σ N N →HF +X is the total cross section of the HF hadron pair production in N + N collisions and σ inel N N is the total inelastic cross section of N + N interaction. (Note that in high energy collisions the HF production is dominated by the creation of hadrons with open HF. The HF quarkonia correspond to a tiny fraction of the total HF yield and can be safely neglected in our consideration.) There are however some indirect indications that an essential deviation from the standard formula (1) may exist. Recent analysis of the dimuon spectrum measured in central Pb+Pb collisions at 158 A GeV by NA50 Collaboration [1] reveals a significant enhancement of the dilepton production in the intermediate mass region (1.5÷2.5 GeV) over the standard sources. The primary 2 interpretation attributes this observation to the enhanced production of open charm [1]: about 3 times above the direct extrapolation (1) from N+N data. Similar result has been recently obtained in the framework of the statistical coalescence model [3][4][5]. This model connects the multiplicities of hadrons with open and hidden charm. It was found [4,5] that an enhancement of the open charm by the factor of about 2 ÷ 4 over the direct extrapolation is needed to explain the data on the J/ψ multiplicity. It was suggested in Ref. [5] that this enhancement may appear due to the broadening of the phase space available for the open charm because of the presence of strongly interacting medium.
In the present letter we demonstrate that a deconfined medium (quark-gluon plasma (QGP) or its precursor) can make an essential influence on the hadronization of HF (anti-)quarks. This leads to an enhancement of the HF hadron production in A+B collisions in comparison to the direct extrapolation (1) from the N+N data. We restrict ourselves to a rough estimation of the upper bound of possible HF enhancement due to the color deconfined medium.
The process of production of a HF hadron pair can be subdivided into two stages: the hard production of a HF quark-antiquark pair (QQ) and its subsequent hadronization into observed particles. Therefore, there is an essential difference between HF hadron production and, e.g., hard dilepton production (the Drell-Yan process): created Q and Q can and even have to interact with the surrounding quarks and gluons to be transformed into observed HF hadrons.
To get an intuitive picture of possible medium effects let us start from the open HF production in e + e − annihilation. The HF QQ pair created at the first stage, hadronizes into observed particles. The hadronization has a nonperturbative nature. Its dynamics can be qualitatively understood in the framework of the string picture. When the distance between Q and Q reaches the range of the confinement forces, a string connected these colored objects is formed. If the e + e − center-of-mass (c.m.) energy √ s (equal to the invariant mass of QQ pair M QQ ) lies well above the corresponding HF meson threshold 2m M (equal to 2m D or 2m B for cc and bb quarks, respectively), Q and Q break the string into two (or more) peaces, so that the final state contains a HF hadron pair (and possibly a number of light hadrons). However, when the e + e − c.m. energy exceeds the heavy quark threshold ( √ s > 2m Q ) but lies below the corresponding HF meson threshold 2m M ( √ s < 2m M ), the string cannot be broken and the open HF hadron pair can not be formed.
Let us imagine now the e + e − annihilation inside a deconfined medium. Due to the Debye screening, no string is formed between colored objects in this case. If the heavy Q and Q are created, they can fly apart within the medium as if they were free particles. It does not matter whether their initial invariant mass M QQ exceeds the corresponding hadron threshold or not. The created QQ pair will be able to form a HF hadron pair at the stage of QGP hadronization. This means that the e + e − annihilation inside the QGP would produce HF hadrons even if the collision energy is not sufficient for producing these hadrons in the vacuum.
In N+N or A+B collisions the HF QQ pairs are produced due to hard parton interactions. The calculations in the leading order of the perturbative quantum chromodynamics (pQCD) show that a great fraction of QQ pairs are created with invariant masses M QQ below the corresponding meson threshold 2m M even at the largest RHIC energy. If this QQ pair creation takes place in the deconfined medium, which is expected to be formed in high energy A+B collisions, the presence of such a medium makes possible a hadronization of these pairs. This should lead to an enhancement of the HF hadron production in A+B collisions in comparison to the standard result (1) obtained within the direct extrapolation of the N+N data.
There are of course essential differences between the open HF hadron production in the e + e − annihilation and in N+N or A+B collisions. Even in N+N collisions, when no deconfined medium is expected, the created QQ pair can interact with the spectator partons and has therefore a chance to form HF hadron pair even if its primary invariant mass was insufficient for this process. Moreover, in contrast to the e + e − annihilation, the most of QQ pairs are created in the color octet state and therefore they have to interact with the spectators to form a color neutral final state. Instead of breaking the string, the Q and Q can form hadron states by means of coalescence with light spectator (anti-)quarks 3 .
As no theoretical descriptions of this complicated process exist, we restrict ourselves to a rough estimation of the upper bound of possible HF hadron enhancement due to the color deconfined medium. We assume that • In the case of N+N collisions, no subthreshold QQ pairs contribute to the HF hadron production 4 .
confined medium is formed, all QQ pairs hadronize into particles with the open HF 5 .
The first assumption looks reasonable at low collision energies, whereas to justify the second one high energies are evidently preferable. This means that assuming validity of the both statements we overestimate the expected HF enhancement effect and, therefore, the above assumptions give its upper bound.
We make now the numerical estimates which follow from the above assumptions. The total cross section of heavy QQ pair production by colliding nucleons is given by the formula (see e.g. Ref. [6]) where s is the squared c.m. energy of the colliding nucleons, x 1 (x 2 ) is the fraction of the momentum of the first (second) nucleon carried by the parton 1 (2), f 1 and f 2 are the fractional-momentum distribution functions or structure functions, µ F is the factorization scale, σ 12→QQ (ŝ) is the cross section of heavy quark-antiquark pair production by interacting partons at squared centerof-mass energyŝ. For ultrarelativistic nucleons,ŝ is given by the formulaŝ = x 1 x 2 s. The sum in the right hand side of Eq.(2) runs over all the pairs of parton types, that give nonzero contribution to the production cross section. We restrict ourselves to the leading order of pQCD. In this case, two basic processes of heavy flavor creation have to be taken into account: the gluon fusion gg → QQ and the light quark-antiquark annihilation qq → QQ. So the sum in Eq.(2) includes (1, 2) = (g, g), (q, q), (q, q), where q in its turn runs over the light flavors q = u, d, s. The corresponding parton cross sections are given by the formulas [6]: where χ = 1 − 4m 2 Q /ŝ , µ R is the renormalization scale and m Q is the mass of the heavy quark. The masses of light quarks are neglected.
Eq.(2) can be rewritten in the form where the differential cross section with respect to the squared invariant massŝ = M 2 QQ of the QQ pair is given by the formula The probability distributions of QQ pairs with respect toŝ are shown in Fig. 1 and Fig. 2 for charm and bottom, respectively. The computation were done using the CERN library of parton distribution functions PDFLIB [7]. The default set of structure functions MRS (G) [8] was chosen. The HF quark masses are fixed as m c = 1.25 GeV for charm and m b = 4.2 GeV for bottom, the c.m. energy of the colliding parton pair was used as the renormalization and factorization scales: µ F = µ R = √ŝ . We estimate now the upper bound of the HF enhancement in A+B collisions. We assume that in N+N collisions the HF QQ pairs cannot hadronize, unless its c.m. energy exceeds the corresponding HF hadron threshold. Therefore, to calculate the total HF hadron production cross section we cut the integral in Eq.(5) at its lower bound by the corresponding meson threshold 6 : where m M is the mass of the lightest meson containing corresponding HF quark (D-meson for the charm and B-meson for the bottom), m N is the nucleon mass.
In contrast, when two nucleons interact in the deconfined medium (as in high energy A+B collision), our assumption states that all QQ pairs survive and form HF hadrons at the stage of the QGP hadronization. Therefore, the cross section σ N N →HF +X in the formula (1) should be replaced by the cross section σ N N →QQ+X . Hence for the upper bound of the enhancement factor we use the formula The behavior of E max (s) for charm and bottom is shown in Fig.3. It is seen that the largest effect is expected at low energies. Therefore an experimental study of the effect should be done at the minimum energy, where the deconfinement medium is expected to be formed and the inclusive cross-section of HF production is large enough to make its measurement feasible.
The upper bound of open charm enhancement at SPS energy is by the factor of about 5 ÷ 6. This means that the enhanced production of open charm hadrons by the factor 2 ÷ 4 found in Ref. [1] and Refs. [4,5] can be explained by the influence of the deconfined medium.
We conclude that the deconfined medium, which is expected to be formed in nucleus-nucleus collisions can influence the process of hadronization of heavy quarks, this leads to the enhanced production of hadrons with open heavy flavors (charm and bottom). The rough estimation of the upper bound of the effect at SPS energies is found to be large enough to explain the indirect experimental data [1] and the phenomenological evaluations [4,5]. We consider the enhancement of the heavy flavor yield as a possible signal of the color deconfinement.