August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 International Journal of Modern Physics E Vol. 22, No. 8 (2013) 1350056 (17 pages) c World Scientific Publishing Company DOI: 10.1142/S0218301313500560 STRANGENESS PRODUCTION IN RELATIVISTIC HEAVY-ION COLLISIONS IN THE QUARK RECOMBINATION MODEL X. S. WEN and C. B. YANG∗ Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, P. R. China ∗[email protected] Received 6 January 2013 Accepted 30 June 2013 Published 13 August 2013 In this paper, yields of strange hadrons in relativistic heavy-ion collisions are investigated in the framework of the recombination model. Yield ratios of strange to nonstrange hadrons are studied. Strangeness enhancement is shown stronger for higher initial quark densities and for hadrons with more strange quarks. Keywords: Quark recombination model; strangeness production; heavy-ion collisions. PACS Number: 25.75.Gz 1. Introduction The study of particle production is an important subject in high-energy heavy-ion collisions. Physicists considered many mechanisms to describe the particle production processes, such as the string model1 and the independent parton fragmentation model.2 But the applicability of those models has some limitations. In the string model, the production of particles in hadronic collisions is treated as the breakup of strings between quark–antiquark pairs, and it can only describe the processes of particle production at low pT . The independent parton fragmentation model can only describe the processes of particle production at high pT . 3 Both models predict very small p/π ratio considering the much heavier mass of proton than that of pion. However, experimental results at the Relativistic Heavy-Ion Collider (RHIC) show that p/π can be as large as about 1 at pT ∼ 3 GeV/c. 4–6 In recent years, as a new approach to hadronization in relativistic heavy-ion collisions at RHIC, the quark recombination model7–12 has been successfully developed. The quark recombination model was first formulated over three decades ago by Hwa and Das as a model for hadronization in the fragmentation region in hadron–hadron collisions.13 Different from the two traditional models mentioned above, the quark recombination models ∗Corresponding author. 1350056-1 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 X. S. Wen & C. B. Yang can be applied in any pT region, and they can solve some RHIC puzzles, such as the unexpectedly high p/π ratio at pT about 3 GeV/c, the constituent quark number scaling of the elliptic flow, and the Cronin effect in d + Au collisions.15 It has been shown that the recombination of soft or thermalized quarks dominates at low pT hadron production in Au+ Au collisions at RHIC energies, while the fragmentation for hard partons dominates at very high pT . When the yields of final state hadrons are concerned, the recombination involving soft partons only is needed, because most of hadrons have quite low pT . Since the recombination model with instant hadronization violates the unitarity condition, a finite hadronization time is introduced in Refs. 16 and 17, with meson (baryon) production rate proportional to the square (cubic) of quark density. In that improved recombination model, yields of hadrons in the light quark sector are studied in Refs. 16 and 17. Similar considerations have been formulated in the ALCOR model.18–20 In particular, the pT spectra and ratios, such as π/p, have been calculated in Refs. 21 and 22. The study of strange particle production in elementary and nuclear collisions at relativistic energies has been an important topic for a few decades. In the initial state of the collision, no open strangeness exists and all strange particles in the final state are produced in the interactions. In relativistic heavy-ion collisions, the vacuum properties may be changed so that the confined quarks may be liberated, and strange quarks can be produced copiously and a chemical equilibrium state may be reached. Then the production of strange particles can be enhanced. Therefore, the enhancement of strange particles has been regarded as a possible signal for the formation of a new state of matter, quark–gluon plasma, in relativistic heavy-ion collisions,23 which can help us to understand the hadronization mechanism from the hot and dense quark matter created in the relativistic heavy-ion collisions. The strangeness production in heavy-ion collisions has been studied for a long time in the coalescence/recombination model.24–26 In this paper, we use the improved quark recombination model to investigate the production of various strange hadrons from quark–gluon plasma produced in ultrarelativistic heavy-ion collisions. This paper is organized as follows. In Sec. 2, basic formalisms for the problem are presented. In Sec. 3, results are shown on the yields and ratios for the three cases with different scenarios for the evolution of the system. Section 4 is a brief conclusion. 2. Basic Formalism In this section, we will introduce the basic framework of the quark recombination model with a nonzero hadronization time. In the improved recombination model, the rate for the yields of meson (baryon) from a given parton system are assumed proportional to square (cubic) of parton density, in the same essence as that in Ref. 16. For the production of mesons and baryons with momentum p, we have p dNM dpdt = Z dp1 p1 dp2 p2 F M (p1, p2, t)R M (p1, p2, p) (1) 1350056-2 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 Strangeness Production in RHIC in Quark Recombination Model and p dN B dpdt = Z dp1 p1 dp2 p2 dp3 p3 F B(p1, p2, p3, t)R B(p1, p2, p3, p), (2) where F M,B are the joint (2- and 3-quark) time dependent momentum distribution functions and RM,B are the recombination functions for the mesons and baryons. In this paper, we use p to represent the transverse momentum pT . In the above two equations, Q i dpi pi F M,B gives the average numbers of combinations for quark– antiquark pairs and three quarks. Therefore, there is no divergence at small pi . If one combines Eqs. (1) and (2) with additional input on how the parton distributions evolve during hadronization, one can predict the produced hadron spectra. Note that production of all hadron species contributes to the evolution of parton distributions. At present we do not have such input, so we cannot get spectra for all final state hadron species. When we are interested in the yields of final state hadrons, we only need to consider the contribution to produced hadrons from the recombination of dominant soft quarks with low pT , then the joint parton distributions can be written as F M(p1, p2, t) = V (t)f1(p1, t)f2(p2, t) = V (t)ρ1(t)ρ2(t)p1p2 exp[−(p1 + p2)/T ] , (3) F B(p1, p2, p3, t) = V (t)f1(p1, t)f2(p2, t)f3(p3, t) = V (t)ρ1(t)ρ2(t)ρ3(t)p1p2p3 exp[−(p1 + p2 + p3)/T ] , (4) with V (t) the spatial volume of the partonic system and ρ(t) the constituent parton density, T the inverse slope of parton distributions or apparent temperature enhanced by flow. Strictly speaking the low pT distributions of quarks are not an exponential in pT , but in the transverse kinetic energy ET . But from previous studies of the recombination models,9,10,12 one can learn that the quark mass effect on hadron spectra is not very important. When one calculates the total multiplicities, neglecting quark mass is a not too bad approximation, because the measure, pT dpT , suppresses low pT contribution from integrating over pT . With ET in place of pT , the following calculations can also be performed, but the expressions become much more complicated. So, we use, here, exponential form for the pT distributions. Extension to include finite quark mass is straightforward. Substituting the distribution functions into Eqs. (1) and (2) one can get the rates of yield for mesons and baryons as dNM dt = V (t)ρ1(t)ρ2(t) Z dp1 p dp2 p exp[−(p1 + p2)/T ]R M(p1, p2, p)pdp = AM V (t)ρ1(t)ρ2(t), (5) dN B dt = V (t)ρ1(t)ρ2(t)ρ3(t) Z dp1 p dp2 p dp3 p exp[−(p1 + p2 + p3)/T ] × R B(p1, p2, p3, p)p 2 dp = ABV (t)ρ1(t)ρ2(t)ρ3(t), (6) 1350056-3 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 X. S. Wen & C. B. Yang where AM = Z dp1 p dp2 p exp[−(p1 + p2)/T ]R M(p1, p2, p)pdp , (7) AB = Z dp1 p dp2 p dp2 p exp[−(p1 + p2 + p3)/T ]R B(p1, p2, p3, p)p 2 dp . (8) From last equations, one can see that parameters AM and AB encode the information of the distribution inverse slope T and the recombination functions which are determined by the valence quark structure of hadrons. Since we are working in u, d, s sector, we will consider yields of long life hadrons like pions, kaons, φ and proton etc. The feed-down effects from resonances can be attributed directly to the long life hadrons with the same flavor contents. So, we need the following recombination functions12: (1) for pion Rπ(p1, p2, p) = p1p2 p2 δ( p1+p2 p − 1), (2) for kaon RK(p1, p2, p) = [B(2, 3)]−1 p 2 1p 3 2 p5 δ( p1+p2 p − 1), with B(m, n) the Euler’s Beta function which is from the normalization of the wave function for the involved hadron,12 and (3) for φ meson Rφ(p1, p2, p) = [B(2, 2)]−1 p 2 1p 2 2 p4 δ( p1+p2 p − 1). In this paper, the symbol K represents both K+ and K0 , and K¯ for K− and K0. The symbol π represents π ± and π 0 . Similarly, for baryons we also need to specify their recombination functions. For proton, Σ, Ξ and Ω, their recombination functions are given by Rp(p1, p2, p3, p) = [B(3, 5)B(3, 2)]−1 p1p2 p 2 3 p3 p 2 δ p1 + p2 + p3 p − 1 , RΣ(p1, p2, p3, p) = [B(2, 5)B(2, 3)]−1 p1p2 p 2 2 p3 p 3 δ p1 + p2 + p3 p − 1 , RΞ(p1, p2, p3, p) = [B(2, 6)B(3, 3)]−1 p1 p 2 p2p3 p 2 3 δ p1 + p2 + p3 p − 1 , RΩ(p1, p2, p3, p) = [B(3, 6)B(3, 3)]−1 p1p2p3 p 3 3 δ p1 + p2 + p3 p − 1 . In the current investigation, we use Σ to represent the baryons with one strange quark and Ξ for baryons with two strange quarks. In other words, Σ is a collection of Λ, Σ +, Σ −, Σ 0 , and Ξ includes Ξ0 , Ξ −. Substituting these recombination functions into Eqs. (5) and (6) and carrying out the integrations one can obtain the values of AM,B for mesons and baryons mentioned above as Aπ = 1 6 T 2 , AK = Aφ = 1 5 T 2 = 6 5 Aπ , Ap = 1 20 T 3 , AΣ = 20 21 Ap, AΞ = Ap, AΩ = 12 11 Ap . (9) 1350056-4 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 Strangeness Production in RHIC in Quark Recombination Model Then, the rate equations for the yields of mesons and baryons can be written as dNπ dt = AπV ρqρq¯, dNp dt = ApV ρ3 q , dNp¯ dt = ApV ρ3 q¯ , dNK dt = 6 5 AπV ρqρs¯, dNΣ dt = 20 21 ApV ρ2 q ρs, dNΣ¯ dt = 20 21 ApV ρ2 q¯ρs¯ , dNK¯ dt = 6 5 AπV ρq¯ρs, dNΞ dt = ApV ρ2 sρq, dNΞ¯ dt = ApV ρ2 s¯ρq¯ , dNφ dt = 6 5 AπV ρsρs¯, dNΩ dt = 12 11 ApV ρs 3 , dNΩ¯ dt = 12 11 ApV ρs¯ 3 . (10) Since the hadronization process takes place at low temperature, no more quark and antiquark pairs can likely be produced, so that the numbers of the dressed quarks and antiquarks can be regarded as conserved. The conservation of total number for quarks (and antiquarks) in the hadronization process reads d(V ρq) dt = −3 dNp dt − 4 dNΣ dt dNΞ dt − 1.5 dNπ dt − dNK dt , d(V ρq¯) dt = −3 dNp¯ dt − 4 dNΣ¯ dt − dNΞ¯ dt − 1.5 dNπ dt − dNK¯ dt , d(V ρs) dt = −3 dNΩ dt − 4 dNΞ dt − 4 dNΣ dt − dNφ dt − 2 dNK¯ dt , d(V ρs¯) dt = −3 dNΩ¯ dt − 4 dNΞ¯ dt − 4 dNΣ¯ dt dNφ dt − 2 dNK dt . (11) Here, we have assumed symmetry between u and d quarks and q represents (u+d)/2. In the above expressions, the numbers in front of each term on the right-hand side need to be explained. As an example, let us study the term for pion yield. The number 1.5 in front of that term in the first and the second expressions comes from the facts that there are three kinds of pions (π ± and π 0 ) and that each pion has only one (anti)quark, thus in total three light quarks contribute 1.5 to the decrease of u and d quark numbers. All the other numbers in front of each term can be counted similarly. It should be noticed that what we need to know for our later calculations of hadron yields are ratios of AB/AM etc. The approximation caused by neglecting quark mass in calculating AM and AB can be approved for those ratios. The initial conditions are V (t = 0) = V0, ρq(t = 0) = ρ0, ρq¯(t = 0) = αρ0, ρs(t = 0) = ρs¯(t = 0) = βρ0, with V0, ρ0, α and β parameters for the problem. Equations (10) and (11) ensure the unitarity in hadronization process and form the fundamental formulas for the yields of mesons and baryons from the quark recombination model with unitarity constraint. Then, we can write down a set of equations for the evolution of parton densities as d(V ρq) dt = −ApV (3ρ 3 q + 80ρ 2 qρs/21 + ρqρ 2 s ) − AπV (3ρqρq¯/2 + 6ρqρs¯/5), 1350056-5 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 X. S. Wen & C. B. Yang d(V ρq¯) dt = −ApV (3ρ 3 q¯ + 80ρ 2 q¯ρs¯/21 + ρq¯ρ 2 s¯ ) − AπV (3ρqρq¯/2 + 6ρq¯ρs/5), d(V ρs) dt = −ApV (36ρ 3 s /11 + 4ρ 2 sρq + 80ρsρ 2 q /21) − AπV (6ρsρs¯/5 + 12ρsρq¯/5), d(V ρs¯) dt = −ApV (36ρ 3 s¯/11 + 4ρ 2 s¯ρq¯ + 80ρs¯ρ 2 q¯/21) − AπV (6ρsρs¯/5 + 12ρs¯ρq/5). (12) For simplicity and deriving results depending on least number of parameters, we define some variables nπ = Nπ/ρ0V0, np = Np/ρ0V0, np¯ = Np¯/ρ0V0, nK = NK/ρ0V0, nΣ = NΣ/ρ0V0, nΣ¯ = NΣ¯ /ρ0V0, nK¯ = NK¯ /ρ0V0, nΞ = NΞ/ρ0V0, nΞ¯ = NΞ¯ /ρ0V0, nφ = Nφ/ρ0V0, nΩ = NΩ/ρ0V0, nΩ¯ = NΩ¯ /ρ0V0. They are the average hadron multiplicities produced from per (light) constituent quark corresponding to the state just before hadronization. Equations (10) and (12) are not closed, because there are more unknown variables than the number of equations. To get any result, one needs to specify the behavior of evolution of the dense medium in the hadronization process. For this purpose, one can combine these equations with hydrodynamics. In Refs. 27 and 28, this problem has been considered when all those constituent quarks are used to form one species of mesons, with detailed analysis of density dependent cross-section and expansion dynamics. But for our present purpose, more than one kind of mesons and baryons are involved, and therefore, without involving detailed physics for the evolution, we choose arbitrary, in Sec. 3, three evolution scenarios for V for simplicity. 3. Results Now, we study hadron yields for three (nonrealistic) cases for the evolution of quark systems in the hadronization process. 3.1. For fixed volume of the system First, we consider the case with fixed volume: V (t) = V0. This case corresponds to a parton system neither with expansion nor shrinking. We can define some variables: ρ1 = ρq/ρ0, ρ2 = ρs/ρ0, r1 = ρq¯/ρq, r2 = ρs¯/ρs, u = Apρ0/Aπ, τ = Aπρ0t. Among these variables, u is proportional to the initial quark density and τ is proportional to the evolution time, r1 and r2 are the ratios of antiquark density over the one for the corresponding quarks, ρ1 and ρ2 are the nonstrange and strange quark density in unit of the initial light quark density. Then, Eq. (12) can be rewritten as dρ1 dτ = −u 3ρ 3 1 + 80 21 ρ 2 1ρ2 + ρ 2 2ρ1 − 3 2 ρ 2 1 r1 − 6 5 ρ1ρ2r2 , dρ2 dτ = −u 36 11 ρ 3 2 + 4ρ 2 2ρ1 + 80 21 ρ2ρ 2 1 − 6 5 ρ 2 2 r2 − 12 5 ρ1ρ2r1 , 1350056-6 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 Strangeness Production in RHIC in Quark Recombination Model dr1 dτ = −u 3ρ 2 1 r 3 1 + 80 21 ρ1ρ2r 2 1 r2 + ρ 2 2 r1r 2 2 − 3 2 ρ1r1 − 6 5 ρ2r1 − r1 ρ1 dρ1 dτ , dr2 dτ = −uρ2 36 11 ρ 2 2 r 3 2 + 4ρ1r1ρ2r 2 2 + 80 21 ρ 2 1 r2r 2 1 − 6 5 ρ2r2 − 12 5 ρ1r2 − r2 ρ2 dρ2 dτ and Eq. (10) reads dnπ dτ = ρ 2 1 r1, dnp dτ = uρ3 1 , dnp¯ dτ = uρ3 1 r 3 1 , dnK dτ = 6 5 ρ1ρ2r2, dnΣ dτ = 20 21 uρ2 1ρ2, dnΣ¯ dτ = 20 21 uρ2 1ρ2r 2 1 r2 , dnK¯ dτ = 6 5 ρ1ρ2r1, dnΞ dτ = uρ2 2ρ1, dnΞ¯ dτ = uρ2 2ρ1r 2 2 r1 , dnφ dτ = 6 5 ρ 2 2 r2, dnΩ dτ = 12 11 uρ3 2 , dnΩ¯ dτ = 12 11 uρ3 2 r 3 2 . Initial conditions for last equations are ρ1(t = 0) = 1, ρ2(t = 0) = β, r1(t = 0) = α, r2(t = 0) = 1. The obtained yields depend on values of parameters u, α and β. First, we input typical values for α and β as α = 0.9, β = 0.6 and investigate the yields as functions of u. A large value of u may be caused due to a high initial quark density or large effective inverse slope T for parton distributions. The obtained results for the yields are shown in Fig. 1 for (a) nonstrange hadrons, (b) hadrons with one strange quark, (c) hadrons with two strange quarks and (d) hadrons consisted of only strange quarks. 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 u (d) φ Ω 2Ω 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 u (d) φ Ω 2Ω 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 u (d) φ Ω 2Ω 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 u (d) φ Ω 2Ω 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 u (d) φ Ω 2Ω Fig. 1. Yields of hadrons as functions of u for the case with the fixed volume for given α = 0.9, β = 0.6 for (a) nonstrange hadrons, (b) hadrons with one strange quark, (c) hadrons with two strange quarks and (d) hadrons consisting of only strange quarks. 1350056-7 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 X. S. Wen & C. B. Yang 0.51 0.52 0.53 0.1 0.2 0.3 0.4 0.5 K/π R φ/π Σ/p Ξ/p Ω/p Fig. 2. Yield ratios of strange to nonstrange hadrons, φ/π, Σ/p, Ξ/p and Ω/p as functions of K/π for the same case as in Fig. 1. 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 β (d) φ Ω 2Ω Fig. 3. Yields of hadrons as functions of β for given u = 0.5, α = 0.9 for the same evolution scenario as in Fig. 1. From Eq. (6), one can see that baryons are produced in the earlier period of the hadronization process when the parton density is high. Therefore, as u increases (namely initial quark density is higher), more baryons can be produced relative to the mesons. Because the initial light antiquark (¯u, ¯d) density is α = 0.9 times lower than that for quarks (u, d), the yield of antiproton is about α 3 ≃ 0.73 times smaller 1350056-8 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 Strangeness Production in RHIC in Quark Recombination Model 0 0.2 0.4 0 0.05 0.1 0.15 0.2 0.25 K/π R φ/π p/π Σ/π p/π PHENIX p/π PHENIX p/π Fig. 4. Yield ratios of hadrons, φ/π, p/π, Σ/π and ¯p/π as functions of K/π for given u = 0.5, α = 0.9 for the case as in Fig. 3. Points with error bars are taken from PHENIX data.29 than that of proton in (a). Since ¯s quarks can recombine with u and d quarks to form K, less ¯s quarks can be used to form Σ, ¯ Ξ and ¯ Ω, so the yields of ¯ Σ, ¯ Ξ and ¯ Ω¯ are a little bit less than those of their antiparticles, although initially s and ¯s have the same density. For clarity, yields for ¯p, Σ, ¯ Ξ and ¯ Ω are enhanced by a factor ¯ 2. For a system with diluter strange quarks, the difference between yields of Σ, ¯ Ξ and ¯ Ω and their antibaryons is more obvious. We can also learn that, with t ¯ he increase of u, the proton yield can be larger than that for pion when u ∼ 2.7 and the antiproton yield can also be larger than that of pion, when u ∼ 3.3 which are not shown in the figure. As well-known, strangeness production and its enhancement in relativistic heavy-ion collisions are important topics on the formation signal of quark–gluon plasma (QGP) and chiral symmetry restoration. To see the enhancement, we define ratios R between yields of strange particles to nonstrange ones as: K/π = nK nπ , φ/π = nφ nπ , Σ/p = nΣ np , Ξ/p = nΞ np , Ω/p = nΩ np etc. The yield ratios as functions of u may reflect the evolution of strange productions relative to the nonstrange ones, when the initial parton density increases. For given α = 0.9, β = 0.6 used in Fig. 1, the ratio results are shown in Fig. 2 not as functions of u but as functions of K/π ratio. Numerically, as u increases from 0.2 to 0.8 with given α = 0.9 and β = 0.6 for the current evolution scenario, K/π decreases from 0.533 to 0.512, φ/π from 0.245 to 0.226, while Σ/p increases from 0.410 to 0.462, Ξ/p from 0.211 to 0.244, and Ω/p from 0.119 to 0.138. As shown in the figure, with the increase of K/π, φ/π increases slightly, while Σ/p, Ξ/p and ω/p decrease strongly. Thus, strange baryons are more sensitive to the initial conditions. 1350056-9 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 X. S. Wen & C. B. Yang Then, we consider the influence of β (the initial strange quark density relative to that of nonstrange ones) on hadron yields. For this purpose, we fix the initial density of light quark, thus fix u to an arbitrary value 0.5. As shown in Fig. 3, with the increase of strange quark density, the yields of strange hadrons increase while those for nonstrange hadrons decrease. The increase of the yields of strange hadrons with β is obvious from the basic formulas. The decrease of nonstrange hadrons with β results from the fact that, for larger β, a larger fraction of nonstrange quarks hadronize into K, Σ, Ξ, so that the number of available light quarks for nonstrange hadrons becomes less. In Fig. 4, we show the yield ratios of strange to nonstrange particles as functions of K/π, as in Fig. 2. With β increasing from 0.05 to 0.75 (by a factor of 15) at given u = 0.5 and α = 0.9, K/π increases from 0.042 to 0.681 by a factor 16.2, φ/π from 0.0016 to 0.399 by a factor of about 250, Σ/p increase from 0.0025 to 0.0367 by a factor of 14.7, while p/π decreases from 0.0908 to 0.0826, ¯p/π from 0.0374 to 0.0369. In addition, with the increase of strange quark number in a hadron, the enhancement becomes stronger. For a comparison with RHIC data, p/π and ¯p/π ratios from PHENIX29 are shown as functions of K/π in the figure. Our results for this case do not agree with the data. 3.2. For fixed light quark density For the second case, we assume that the light quark density is fixed in hadronization process. This assumption means that the partonic system shrinks during hadronization. For this case, we define new variables: µ = ln(V /V0), r1 = ρq¯/ρq, r2 = ρs/ρq, r3 = ρs¯/ρq, u = Apρ0/Aπ, τ = Aπρ0t. Then, the equations governing the evolution and hadron productions are from Eq. (12), dµ dτ = −u 3 + 80r2 21 + r 2 2 − 3 2 r1 − 6r3 5 , dr1 dτ = −ur1 3r 2 1 + 80 21 r1r3 + r 2 3 − 3 2 r1 − 6r1r2 5 − r1 dµ dτ , dr2 dτ = −ur2 36r 2 2 11 + 4r2 + 80 21 − 6r2r3 5 − 12r2r1 5 − r2 dµ dτ , dr3 dτ = −ur3 36r 2 3 11 + 4r3r1 + 80r 2 1 21 − 6r2r3 5 − 12r3 5 − r3 dµ dτ , dnπ dτ = r1 exp(µ), dnp dτ = u exp(µ), dnp¯ dτ = ur3 1 exp(µ), dnK dτ = 6 5 r3 exp(µ), dnK¯ dτ = 6 5 r1r2 exp(µ), dnΣ dτ = 20 21 ur2 exp(µ), 1350056-10 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 Strangeness Production in RHIC in Quark Recombination Model dnΣ¯ dτ = 20 21 ur2 1 r3 exp(µ), dnΞ dτ = ur2 2 exp(µ), dnΞ¯ dτ = ur2 3 r1 exp(µ), dnφ dτ = 6 5 r2r3 exp(µ), dnΩ dτ = 12 11 ur3 2 exp(µ), dnΩ¯ dτ = 12 11 ur3 3 exp(µ). The initial conditions are µ(t = 0) = 0, r1(t = 0) = α, r2(t = 0) = r3(t = 0) = β. Due to the assumption that the light quark density is a constant in the hadronization process, the parton system shrinks almost exponentially. To get some numerical results, we first fix α = 0.9 and β = 0.6 and investigate the yields of hadrons as functions of u. When u is large, the number of the quarks is abundant so the hadronization process terminates very quickly. In Fig. 5, we show hadron yields as functions of u for fixed parameters α = 0.9 and β = 0.6. From the figure, one can see that with the increase of u more baryons but less mesons can be produced, and that hadrons with more strange quarks have smaller yields, similar to the first case. Comparing to the first case, one can find from Fig. 5(a) that with the increase of u, the yields of baryons increase more rapidly, because the quark density is fixed at larger values, in favor of baryon production. The proton yield can be larger than that of pion, when u ∼ 0.96 and the antiproton yield can also be larger than that of pion when u ∼ 1.1. Similar to the results for the first case, the yields of Σ, ¯ Ξ and ¯ Ω are smaller than their antiparticles in (b), (c) and (d). ¯ In Fig. 6, ratios of hadron yields from the results in Fig. 5 are shown for fixed α = 0.9 and β = 0.6. The ratios are a little larger than those shown in Fig. 2. In the u range shown, K/π decreases from 0.582 to 0.555 (−4.9%) φ/π decreases from 0.274 to 0.259 (−6%), Σ/p from 0.379 to 0.320 (−18.7%), Ξ/p from 0.208 to 0.196 (−6%) and Ω/p from 0.227 to 0.213 (−6%). 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 u (d) φ Ω 2Ω Fig. 5. Yields of hadrons as functions of u for the case with fixed light quark density for given α = 0.9, β = 0.6. 1350056-11 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 X. S. Wen & C. B. Yang 0.55 0.56 0.57 0.58 0.59 0.6 0.1 0.2 0.3 0.4 0.5 K/π R φ/π Σ/p Ξ/p Ω/p Fig. 6. Yield ratios of strange- to nonstrange hadrons, φ/π, Σ/p, Ξ/p and Ω/p as functions of K/π for the same case as in Fig. 5. 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 β (d) φ Ω 2Ω Fig. 7. Yields of hadrons as functions of β for given u = 0.5, α = 0.9 for the case as in Fig. 5. Similarly, with fixed u but changing β, one can study the initial strange quark density dependence of hadron yields and their ratios. The results for yields and yield ratios are shown in Figs. 7 and 8. For fixed u but increasing β, more strange hadrons but less nonstrange hadrons can be produced. Comparing to the corresponding results for the first case, we can see that the yields of mesons become smaller but 1350056-12 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 Strangeness Production in RHIC in Quark Recombination Model 0 0.2 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 K/π R φ/π p/π Σ/π p/π PHENIX p/π PHENIX p/π Fig. 8. Hadron yield ratios φ/π, p/π, Σ/π and ¯p/π as functions of K/π for given u = 0.5, α = 0.9 for the case as in Fig. 7. Points with error bars are taken from PHENIX data.29 those of baryons become larger. This is because the quarks densities remain large in hadronization process for this case, more quarks are used to produce baryons and therefore less mesons can be produced. One can see, from Fig. 8, that the yield ratios increase with the increase of β. With the increase of β from 0.05 to 0.75, K/π increases from 0.0467 to 0.712 by a factor 15, φ/π increases from 0.0018 to 0.415 by a factor 230, Σ/p from 0.0158 to 0.243 by a factor 15, but p/π from 0.537 to 0.536 and ¯p/π from 0.434 to 0.435, thus they remain almost as a constant. As in Fig. 4, our ratio results for this case do not agree with experimental data. 3.3. For the case with hydrodynamic flow As the last case, we consider a more realistic evolution of the parton system. For the one-dimensional expansion case, more sophisticated treatment can be found in Ref. 19. Here, we assume that the partonic system expands according to Hubble’s rule from the hydrodynamics. Then, one can assume that the volume expansion rate is proportional to the volume, dV dτ = νV , (13) with ν a parameter for the rate for the exponential expansion of the volume. By defining the new variables r1 = ρq/ρ0, r2 = ρq¯/ρ0, r3 = ρs/ρ0, r4 = ρs¯/ρ0, one can get from Eqs. (5) and (6) dr1 dτ = − ur1 3r 2 1 + 80 21 r1r3 + r 2 3 − r1 3 2 r2 + 6r4 5 − r1ν , 1350056-13 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 X. S. Wen & C. B. Yang dr2 dτ = − ur2 3r 2 2 + 80 21 r2r4 + r 2 4 − r2 3 2 r1 + 6r3 5 − r2ν, dr3 dτ = −ur3 36r 2 3 11 + 80 21 r 2 1 + 4r3r1 − r3 6r4 5 + 12 5 r2 − r3ν , dr4 dτ = −ur4 36r 2 4 11 + 80r 2 2 21 + 4r4r2 − r4 6r3 5 + 12 5 r1 − r4ν , dnπ dτ = r1r2 exp(ντ), dnp dτ = ur3 1 exp(ντ), dnp¯ dτ = ur3 2 exp(ντ), dnK dτ = 6 5 r1r4 exp(ντ), dnΣ dτ = 20 21 ur2 1 r3 exp(ντ), dnΣ¯ dτ = 20 21 ur2 2 r4 exp(ντ), dnK¯ dτ = 6 5 r3r2 exp(ντ), dnΞ dτ = ur2 3 r1 exp(ντ), dnΞ¯ dτ = ur2 4 r2 exp(ντ), dnφ dτ = 6 5 r3r4 exp(ντ), dnΩ dτ = 12 11 ur3 3 exp(ντ), dnΩ¯ dτ = 12 11 ur3 4 exp(ντ), with r1(t = 0) = 1, r2(t = 0) = α, r3(t = 0) = r4(t = 0) = β. Here, we use ν = 0.1. In this case, the quark densities decrease a little faster than for the first case because of the expansion of the volume. Then, one can anticipate that baryon yields will be lower than those for the first case, while meson yields will be higher. To get some quantitative results, we first fix α = 0.9 and β = 0.6 and investigate the yields as functions of u. The results are shown in Fig. 9. Because of the ρ 3 dependence of baryon production rates, most baryons are produced, when the parton density is high, so the expansion rate ν has smaller influence on baryon production than 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 u (d) φ Ω 2Ω Fig. 9. Yields of hadrons as functions of u for given α = 0.9, β = 0.6 for the case with hydrodynamical expansion. 1350056-14 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 Strangeness Production in RHIC in Quark Recombination Model for mesons. One can see that with the increase of u the proton yield can be larger than that for pion when u ∼ 2.9, and the antiproton yield can also be larger than that of pion when u is further larger. Comparing Figs. 1 and 9 one can see that the meson’s yields are a little larger than those for the first case, and on the contrary, the yields of baryons and antibaryons are smaller, as anticipated. In Fig. 10, ratios of hadron yields are shown for fixed α = 0.9 and β = 0.6 from the results in Fig. 9. When K/π decreases from 0.602 to 0.592 (−2%) in the u range shown, φ/π decreases from 0.291 to 0.277 (−5%), Σ/p from 0.485 to 0.480 (−1%), Ξ/p from 0.265 to 0.260 (−2%) and Ω/p from 0.153 to 0.149 (−3%). For this case, one can consider the yields as functions of β when u is fixed. The results are shown in Fig. 11. The results in Fig. 11 show that the yields of strange hadrons increase and those of nonstrange hadrons decrease in the β range shown. Comparing to the first case, the yields of mesons are larger, and the yields of baryons and anti-baryons are smaller. Some yield ratios are shown in Fig. 12 as functions of K/π ratio, as in Figs. 4 and 8. One can see from the figure that the yield ratios of strange over nonstrange hadrons increase with the increase of β, while p/π and ¯p/π decrease a little with β. With the increase of β from 0.05 to 0.75, K/π increases from 0.0486 to 0.743 by a factor 15, φ/π increases from 0.0019 to 0.438 by a factor 230, Σ/p from 0.0110 to 0.155 by a factor 14, but p/π from 0.279 to 0.258 and ¯p/π from 0.208 to 0.196, thus both p/π and ¯p/π have extremely weak dependence on β. To get more detailed results, such as centrality dependence of the spectra, one needs other model results as input. A possible choice is the AZHYDRO calculation, see Ref. 30 for further reference. That will be discussed in a separate work. 0.59 0.595 0.6 0.605 0.1 0.2 0.3 0.4 0.5 K/π φ/π Σ/p Ξ/p Ω/p 0.59 0.595 0.6 0.605 0.1 0.2 0.3 0.4 0.5 K/π R φ/π Σ/p Ξ/p Ω/p Fig. 10. Yield ratios as functions of ratio K/π for given α = 0.9, β = 0.6 for the case as in Fig. 9. Points with error bars are taken from PHENIX data.29 1350056-15 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 X. S. Wen & C. B. Yang 0 0.2 0.4 0.6 n π p 2p K Σ 2Σ 0 0.2 0.4 0.6 0.8 0 0.1 0.2 φ Ξ 2Ξ 0.2 0.4 0.6 0.8 β φ Ω 2Ω 0 0.2 0.4 0.6 n π p 2p K Σ 2Σ 0 0.2 0.4 0.6 0.8 0 0.1 0.2 φ Ξ 2Ξ 0.2 0.4 0.6 0.8 β φ Ω 2Ω 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 β (d) φ Ω 2Ω 0 0.2 0.4 0.6 n (a) π p 2p (b) K Σ 2Σ 0 0.2 0.4 0.6 0.8 0 0.1 0.2 (c) φ Ξ 2Ξ 0.2 0.4 0.6 0.8 β (d) φ Ω 2Ω Fig. 11. Yields of hadrons as functions of β for given u = 0.5 and α = 0.9 for the case as in Fig. 9. 0 0.2 0.4 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 K/π R φ/π p/π Σ/π p/π PHENIX p/π PHENIX p/π Fig. 12. Yield ratios as functions of β for given u = 0.5, α = 0.9 for the case as in Fig. 11. Points with error bars are taken from PHENIX data.29 4. Conclusion The yields and yield ratios are studied in the framework of the quark recombination model for three artificial scenarios for the evolution of parton system. From the results one can see that the yields of baryons and mesons depend on the details of the evolution of the system, but strangeness enhancement can be observed as the initial strange quark density increases in all cases. In this paper, the initial condition of the density ratio between the light antiquark (the strange quark) is 1350056-16 Int. J. Mod. Phys. E 2013.22. Downloaded from www.worldscientific.com by HUAZHONG NORMAL UNIVERSITY on 09/03/13. For personal use only. August 29, 2013 11:5 WSPC/143-IJMPE S0218301313500560 Strangeness Production in RHIC in Quark Recombination Model constant for all u. This may be a reason why our calculated ratios for p/π and ¯p/π do not agree with the PHENIX data for the three cases considered. In real collisions, densities for the light and strange quarks may increase with colliding energy and centrality. Therefore, u, α, β must change their values in some specific correlated way. Such a correlation is not included in this paper. So, the formula used in this paper need being revised in some way in order to make quantitative comparison with experimental data. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 10775057 and 11075061, the Ministry of Education of China under Grant No. 306022 and the Programme of Introducing Talents of Discipline to Universities under Grant No. B08033. References 1. B. Andersson, G. Gustafson, G. Ingelman and T. Sjaostrand, Phys. Rep. 97 (1983) 31. 2. R. D. Field and R. P. Feynman, Nucl. Phys. B 136 (1978) 1. 3. R. C. Hwa and C. B. Yang, Phys. Rev. C 67 (2003) 034902. 4. PHENIX Collab. (K. Adcox et al.), Phys. Rev. Lett. 88 (2002) 242301. 5. PHENIX Collab. (S. S. Alder et al.), Phys. Rev. Lett. 91 (2003) 172301. 6. PHENIX Collab. (S. S. Alder et al.), Phys. Rev. C 69 (2004) 034909. 7. V. Greco, C. M. Ko and P. Levai, Phys. Rev. Lett. 90 (2003) 202302. 8. V. Greco, C. M. Ko and P. Levai, Phys. Rev. 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