Hadron formation from interaction among quarks
aa r X i v : . [ h e p - ph ] O c t Hadron formation from interaction among quarks
Z. G. Tan Department of Electronic and Communication Engineering,Changsha University,Changsha,410003,P.R.China Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central ChinaNormal University, Wuhan 430079, China
C. B. Yang Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central ChinaNormal University, Wuhan 430079, China
Abstract T his paper deals with the hadronization process of quark system. A phenomenological potential is intro-duced to describe the interaction between a quark pair. The potential depends on the color charge ofthose quarks and their relative distances. Those quarks move according to classical equations of mo-tion. Due to the color interaction, coloring quarks are separated to form color neutral clusters which aresupposed to be the hadrons. The study of particle production is an important subject in high-energy heavy-ion collisions. Physicistsconsidered many mechanisms to describe the particle production processes,such as the string model [1]and the independent parton fragmentation model [2]. Both models can not explain the novel phenomenaexperimental observed at BNL/RHIC, such as an unexpectedly large p/π ratio of about 1 at p T about3 GeV/ c [3], since they both predicted a very small p/π ratio of about 0.2. In recent years, as a newapproach to hadronization, the quark recombination model has been proposed [4], which can be appliedin any p T region, and can solve the puzzles from RHIC, such as the unexpectedly high p/π ratio and theconstituent quark number scaling of the elliptic flow.In the implementations of the quark recombination model as in [4], it is assumed that the hadronizationtakes zero time, thus the quark distributions does not change in the process. Then some analyticalexpressions for the spectra of the final state particles can be derived. However, since the yield of meson(baryon) is proportional to the square (cubic) of the quark density, it will be four (eight) time largerif the quark density is doubled. So the naive quark recombination model violates the unitarity in thehadronization. To shun this difficulty, a finite hadronization time is introduced in Refs. [5]. It is assumedthere that the production rate for a species of meson is proportional to the product of densities ofcorresponding quark and antiquark. Similar for baryon production rate. Thus, the essence of the originalquark recombination model is kept in the revised model. Because of a finite time for hadroinization,production of particles will reduce the number of quarks and thus influence later particle production.Therefore, all particle productions are correlated. In [5], only the yields of particles have been studied,with the production correlation fully considered. Frequently, one would like to learn the transverse spectraof produced particles. Such a job cannot be done analytically, and some Monte Carlo method must beused.Another problem in the quark recombination model is about the interactions among quarks in quark-gluon-plasma (QGP) produced in ultrarelativistic heavy ion collisions. The QGP created at BNL/RHIC E-mail:[email protected]
The interaction among quarks in hadronization should, in principle, be described by the basic theory,quantum choromodynamics (QCD). When hadronization is concerned, as in the case discussed in this pa-per, the color interactions among quarks can not be calculated perturbatively, and many non-perturbativeeffects play role since hadronization process is a low momentum transfer process. Without first-principleguidance, one can use a phenomenological potential to describe the quark interaction, as done for boundstate problems. Such a potential should depend on the colors carried by the interacting quarks. It hasbeen shown that the potential corresponding to a single-gluon exchange is inversely proportional to theseparation r between quarks when r is small. When r is large, a string may be formed between twoquarks and the corresponding potential is ∝ r . Thus the potential is assumed as U ij = c ij a (cid:18) r + br (cid:19) , (1)where c ij = c ji is the color factor related quark i and quark j , while a and b are two parameters of themodel. If one chooses a >
0, then some properties of c ij can be claimed. Since a quark and an anti-quarkwith the same color can combine to form a meson, they must attract one another, thus one may expectthat for such a quark pair c ij >
0. If a quark and an anti-quark do not carry the same color, c ij < c ij > c ij <
0. Now one can consider theinteraction between a quark and a hadron. Since hadrons are color neutral, a quark would have no netforce acting on a hadron if the hadron were a point particle. This condition can be satisfied if one chooses c r ¯ r = − c rr = − c rb = − c rg . Similarly for other color combinations. Thus one can put all c ij as a matrix C = Q − − − (2)with Q = − Q = 1. The diagonal elements are forinteraction between quarks with the same color, and the off-diagonal ones for those interactions withdifferent colors. Then when two or three quarks move close enough to make a colorless cluster, the totalforce that act on an other quark is nearly zero, as shown in FIG.1.The next step is to fix values of parameters a and b in Eq.(1). While parameter a determines thestrength of the total interaction between a pair of (anti) quarks, the parameter b tells the relative strengthof the two interactions. Since the interactions among quarks are not considered in most event generators,one may impose that the sum of the interaction potentials is equal to zero, in order to conserve the totalenergy of the system. This additional requirement can be used to determine the value of parameter b . Itcan be understood that the value of parameter a will be responsible for the spatial separation betweenquarks in a hadron. In this paper, we are only interested in whether color neutral clusters can be formedwhen color interactions are taken into account. Thus the value of a is not too relevant.2igure 1: According to Eq.2, the total force that a baryon acts on a quark (lower) or a anti-quark (upper)is about zero. In order to see the feasibility of our model, we deal with the hadronization of a quark system with 30quark and antiquark pairs. It is assumed that the quarks have the same flavor. The generalization toinclude more flavors is straightforward. The total energy is assumed to be E=60 GeV. We assume thatthey are taken from a large thermal equilibrium distribution E = X i gVπ ¯ h Z f i ( T, p ) p | ~p | dpN = X i gVπ ¯ h Z f i ( T, p ) | ~p | dp, (3)where the summation runs over all flavors considered in the problem and g = N c N s = 6 is the degeneracynumber for a quark. f i ( T, p ) is the distribution function for quarks of a specific flavor f ( T, p ) = 11 + e p /T , (4)with p = p | ~p | + m i . In the calculations, only u, d and s quark flavors are considered and the corre-sponding masses are chosen as 0.3GeV for u and d quarks, and 0.5GeV for s quarks, very close to theone used in the constituent quark model. In fact, the main results obtained below have little dependenceon the number of flavors used.For given E and N , a initial volume V and the temperature T can be obtained from Eq.(3). We put N = 60 and E = 60GeV in this paper. Then the value for V is about 8.0 fm and that for T is 0.28GeV.We assume that the quarks are uniformly distributed in a spherical space with volume V . Direct samplingaccording to Eq. (3) for momentum distribution is not easy. But one can do it in another way. One canassign to each quark a momentum with constant magnitude but a random direction to keep the totalmomentum zero. Then let the quarks undergo a certain number (for example 1000) elastic collisions inside3he spatial volume V with periodic boundary conditions used, as described in [6]. One can check that themomentum distribution obtained is like the one in Eq.(3). Because of Eq. 4, the momentum distributionis spherically symmetric. Now we add the color property to each quark randomly. For convenience we use1,2,3 to represent the colors, say, 1 for red, 2 for green,and 3 for blue, and -1,-2,-3 for the correspondinglyanti-colors. To make the whole system color neutral, one can simultaneously assign the opposite colorsto quark and antiquark for each pair. Potential of the whole system can calculate according to U = N − X i =1 N X j = i +1 U ij , (5)where U ij is calculated according to Eq. (1) with the fixed parameter a = 0 .
85 GeV/fm − , which isdouble that suggested in [7]. It should be mentioned that the value of parameter a has no influence to themain results in this paper. The choice of this specific value for a is just as a try. b could be determinedby requiring U = 0 at the the moment we assign colors to quarks. At any time t , we can calculate theforce acted on quark i from quark j ~f ij = − ▽ U ij = c ij a ~r ij r ij − b~r ij r ij ! . (6)The total force acted on quark i from all other quarks is ~F i = X j = i ~f ij . (7)After the system moves dt forward, particle i will move according to the classical dynamics, and itsmomentum and position are ~p i ( t + dt ) = ~p i ( t ) + ~F i dt ,~r i ( t + dt ) = ~r i ( t ) + ~v i dt , (8)where ~v i = ~p i /E i and E i = p | ~p i | + m i . The evolution of the system runs over a time period longenough to enable quark clusters well separated from one another. Since interaction among quarks depends strongly on positions of quarks, analytical calculation for allquarks’ trajectories is impossible. Thus numerical calculation has to be done with a finite time step∆ t >
0. As a result of the finiteness of time step ∆ t , numerical errors must occur, and the total energy ofthe system is, generally, not conserved exactly. In principle, only the ∆ t → dt → t = 0 . c both the potential and the kinetic energy of the system do not change anymore. Thatmeans that the system has been split into some clusters of color neutral and the interactions among thoseclusters is very weak. Thus one can say that hadrons or pre-hadrons has been formed then. In FIG.3,we show two instant pictures for the initial and final state distributions of the quarks in the space. Fromthe final state picture many clusters with different size could be found clearly.Numerically, one can judge a cluster as a meson or a baryon or multi-quark state easily by countingthe quark numbers in the cluster. The species of the meson or baryon could be obtained by studying thecomposition of each cluster. Comparing the distribution of particles in configuration space before andafter hadronization as shown in FIG.4, we found that each cluster is exactly a hadron.4 E ( G e V ) E k E p E t Figure 2: Following the evolution of the system, the energy is transformed between the potential andkinetic. 5 z (f m ) x (fm)y (fm) −100 0−1000100−100−50050100 z (f m ) x (fm)y (fm) Figure 3: Quark distributions for the initial and final state in the evolution of the system. Left part isfor the initial distribution of quarks and right one for that after a time of 100fermi/ c .6
100 0 100−1000100−100−50050100 z (f m ) x (fm)y (fm) −100 0−1000100−100−50050100 z (f m ) x (fm)y (fm) Figure 4: Hadron spatial distribution obtained by replacing each cluster (in the left penal) by a hadron(in the right penal). Where a solid ball represents a particles, while a hollow one for an anti-particle.The size of the ball is used to distinguish meson and baryon. A bigger symbol is for a baryon.7
Conclusion
The hadronization of a quark system is considered as a dynamical process. An interaction potential isintroduced which depends on the colors and spatial separation of quarks. Quarks move according toclassical equations of motion. It is found that various hadrons could be formed naturally by gatheringcoloring quarks to color neutral clusters. In the process, the total energy is conserved within a highaccuracy. Thus this paper shows that such a method may be useful for studying more sophisticatedhadronization processes in heavy ion collisions.This work was supported in part by the National Natural Science Foundation of China under GrantNos. 11075061 and 11221504, by the Ministry of Education of China under Grant No. 306022 , and bythe Programme of Introducing Talents of Discipline to Universities under Grant No. B08033,also by theOpen innovation fund of the Ministry of Education of China under Grant No. QLPL2014P01.
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] K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. , 242301 (2002); S.S. Adler etal.[PHENIX Collaboration], Phys. Rev. Lett. , 172301 (2003).[4] V. Greco, C.M. Ko and P. Levai, Phys. Rev. Lett. , 202302 (2003); R.J.Fries, B. Mueller, C.Nonaka and S.A. Bass, Phys. Rev. Lett. , 202303 (2003); R.C. Hwa and C.B. Yang, Phys. Rev. C 67 , 034902 (2003).[5] C.B. Yang, J. Phys.
G 32 (2006) L11; C.B. Yang and H. Zheng, J. Phys.
G 34 (2007) 2063.[6] Z.G. Tan, A. Bonasera, C.B. Yang, D.M. Zhou and S. Terranova, Int. J. Mod. Phys.
E 16 , 2269(2007).[7] D.H. Perkins,