The entropy puzzle and the quark combination model Jun Song, 1 Zuo-tang Liang, 1 Yu-xin Liu, 2 Feng-lan Shao, 3 and Qun Wang 4 $^{1}$School of Physics, Shandong University, Jinan, Shandong 250100, People’s Repulic of China $^{2}$Department of Physics, Peking University, Beijing 100871, People’s Repulic of China $^{3}$Department of Physics, Qufu Normal University, Qufu, 273165, People’s Repulic of China $^{4}$Interdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China We use two available methods, the Duhem-Gibbs relation and the entropy formula in terms of particle phase space distributions, to calculate the entropy in a quark combination model. The entropy of the system extracted from the Duhem-Gibbs relation is found to increase in hadronization if the average temperature of the hadronic phase is lower than that of the quark phase. The increase of the entropy can also be confirmed from the entropy formula if the volume of the hadronic phase is larger than 2.5-3.0 times that of the quark phase. So whether the entropy increases or decreases during combination depends on the temperature before and after combination and on how much expansion the system undergoes during combination. The current study provides an example to shed light on the entropy issue in the quark combination model. The quark combination model [1, 2] (QCM) or its variants such as the quark recombination or coalescence model are used to describe multi-hadron processes in high energy electron-positron, hadron-hadron and nucleus- nucleus collisions. Recently they are successful in ex- plaining the data at Relativistic Heavy Ion Collider (RHIC), which showed that the combination is more preferable than the fragmentation for the hadronization mechanism in intermediate transverse momentum region in heavy ion collisions at high energies [3–5]. The ba- sic idea of the QCM is that hadrons are combined from quarks and anti-quarks, i.e. two quarks merge into a me- son (a 2-to-1 process) and three quarks into a baryon (a 3-to-1 process). One of the most frequently asked ques- tions about the validity of the QCM is: does the entropy decrease in such 2-to-1 and 3-to-1 processes? The ques- tion arises naturally from the common sense that the en- tropy is normally reduced in a particle number decreas- ing process. The entropy problem is essential to nucleus- nucleus collisions since the issue of the entropy produc- tion is closely related to thermalization and has been the concern of the researchers for many years. There have been previous attempts or arguments about the entropy in the QCM-like models in literature [6–9], but no sys- tematic investigation has yet been made. In this paper we will address the entropy problem in the course of the quark combination in heavy ion collisions. As an exam- ple we will use a specific version of the QCM which is developed by the Shandong group [10–13]. The advan- tage of the Shandong QCM is its exclusive nature and simplicity, i.e. it can give all hadrons in an event. Con- sidering that the entropy can be regarded as a measure of degrees of freedom, the exclusive nature of Shandong QCM is suitable and applicable to the entropy problem. The building blocks for the QCM are constituent quarks and anti-quarks of the light flavors u , d and s with the momentum distributions f$_{i}$ ( p ) ≡ f$_{i}$ ( y, p $_{T}$) [14–16]. All hadrons come out with fixed momenta from combi- nation of their constituent quarks. Note that our QCM (or any of QCM-like models) does not deal with how the Suppose we have N$_{i}$ quarks or anti-quarks with the fla- vor i = u,d,s,u,d,s , the total number of quarks and anti- quarks are then N$_{qq}$ = N$_{q}$ + N$_{q}$ with N$_{q}$ = N$_{u}$ + N$_{d}$ + N$_{s}$ and N$_{q}$ = N$_{u}$ + N$_{d}$ + N$_{s}$ . In mid-rapidity and high energies almost all quarks and anti-quarks are excited from vacuum in pairs, we have N$_{q}$ ≈ N$_{q}$ . In lower en- ergies there are remenants of nucleons participating in the collisions which are left in the mid-rapidity region and leads to N$_{q}$ > N$_{q}$ . Since the strange quarks are heavier they are less produced and we can use a fac- tor λ$_{s}$ = N$_{s}$/N$_{u}$ ≈ N$_{s}$/N$_{d}$ < 1 to characterize the suppression of strange quarks relative to up and down quarks. The procedure to randomly combine all quarks and anti-quarks into hadrons is quite straightforward: we line up all these quarks and anti-quarks randomly into a 1-dimensional queue and let them combine into the groud state hadrons following the combination rule [10– 13]. The momentum distribution of a specific hadron is the convolution of those of its constituent quarks. Here the groud state hadrons include (3 / 2) + decuplet and (1 / 2) + octet baryons and 0 − pseudoscalar and 1 − vector mesons. The short-life resonances are allowed to decay into the long-life or final state hadrons which are recorded in the detectors. The experimental data are used to fix the inputs of the model. The charged multiplicity can fix N$_{q}$ and N$_{q}$ . The ratio of multiplicity of kaons to pions can fix λ$_{s}$ . Note that we focus on the unit rapidity re- gion at the central rapidity | y | < 0 . 5 . One of the most important element of the model is the transverse momen- constituent quarks acquire masses which is closely related to the physics of chiral phase transition, although the QCM-like models can be regarded as a phenomenologi- cal and microscopic description of the phase transition. There are no gluons in the model because the role of a gluon can be replaced by a pair of quarks. Actually hadronization is a dynamic process which takes place lo- cally at a freeze-out temperature, while our QCM (or any of QCM-like models) lacks the real time feature but treats hadronization in a global or an effective way. In this sense the QCM is similar to the statistical models. arXiv:1001.2801v2 [nucl-th] 23 Apr 2010