相对论重离子碰撞中确定QCD相边界的若干问题

Several problems in determining the QCD phase boundary by relativistic heavy ion collisions

为了从相对论重离子碰撞实验确定量子色动力学(Quantum Chromo-dynamics,QCD)所预言的相变临界点和相边界,必须考虑实验数据中非临界涨落、有限系统尺度、有限演化时间的影响。本文综述了这三方面工作的主要内容、结果和意义。对于非临界涨落,主要讨论了由于有限事件数对观察量测量的影响,估计了在相对论重离子对撞机(Relativistic Heavy-Ion Collider,RHIC)能量扫描区,精确测量高阶守恒荷高阶矩所需要的事件数。提出用泊松分布描述有限末态粒子数所致的统计涨落,将统计涨落和实验结果比较,发现统计涨落贡献为主,必须扣除泊松主导的统计涨落。提出混合事件方法,定义动力学累积矩为原始样本的累积矩减去混合样本的累积矩,利用多相输运模型(A Multiphase Transport Model,AMPT)的default模型重建了一个与之相对应的混合事件样本,结果表明:动力学累积矩确实能很好地扣除泊松样的统计涨落,尤其是中心度bin宽度和探测器效率的影响。对于有限系统尺度的影响,利用三维三态Potts模型研究了各种有限系统尺度下,它的磁化强度的高阶感应率在一级相变、临界点,以及平滑过渡区域的行为。发现在固定外场,穿越相边界的时候,从二阶到六阶磁化率都会出现非单调行为,或符号的变化,而且在三个相变区域,非单调行为类似。因此,仅从非单调行为不能区分不同级数的相变。进一步研究了磁化率的有限尺度标度行为,它们的标度指数在不同级数相变中是不一样的,可以区分不同级数的相变。根据观测量的有限尺度标度性,给出了用固定点确定临界参数的定量方法,并将该方法应用到三维三态Potts模型模拟产生的数据分析,展示了方法的精确有效性。对于非平衡演化的影响,采用Metropolis算法模拟了三维Ising模型在临界点附近从非平衡到平衡的演化过程。发现其序参量在演变过程中以指数形式趋近其平衡值,这与动力学朗之万方程给出的结果相同。临界温度下的平均弛豫时间随系统尺度的z次幂发散,表明它能很好地表示动力学方程中的弛豫时间。非平衡演化过程中序参量的三阶矩和四阶矩会出现正、负值震荡,符号取决于观测时间,结果与平滑过渡区动力学模型一致。研究还发现,在平滑过渡区,非平衡演化持续时间非常短,非平衡对观察量的影响非常弱;但是在一级相变线上,非平衡弛豫的时间非常长,非平衡豫影响不可忽略。这些定性特征对实验确定QCD的临界点和相边界具有重要的指导意义。

The goal of relativistic heavy-ion collisions is to determine the phase boundary of quantum chromodynamics(QCD)phase transitions.Critically sensitive observables are suggested to be higher-order cumulants of conserved charges.The non-monotonous behavior of higher cumulants was observed at the relativistic heavy-ion collider(RHIC).However,it remains unclear whether these non-monotonous behaviors are critically related.We studied the influences of non-critical fluctuations,finite system size,and limited evolution time to determine if they cause non-monotonous behavior.First,we examined the minimum statistics required for measuring the fourth cumulant.The minimum statistic obtained using the centrality bin width correction(CBWC)method was 25 M.We suggest using a 0.1%centrality bin in the CBWC method instead of each Nch.With a 0.1 centrality bin width,1 M statistics are sufficient.We then pointed out the statistical fluctuations from the limited number of final particles.By assuming the independent emission of each positive(or negative)charged particle,the statistical fluctuations of positive(or negative)charged particles were presented by a Poisson distribution,and the statistical fluctuations of net-charged particles were their evolution.The obtained statistical fluctuations for net protons,net electronic charges,and net baryons were consistent with those from the Hadron Resonance Gas model.In addition,the measured cumulants at RHIC/STAR are dominated by these Poisson-like statistical fluctuations.At the end of this section,we suggest the pooling method of mixed events and demonstrate that the sample of mixed events accurately presents the contributions of the background.Dynamic cumulants were defined as the cumulant of the original sample minus that of the mixed sample.Dynamical cumulants were shown to simultaneously reduce the influence of the statistical fluctuations,centrality bin width effects,and detector efficiency.Second,because the system is finite,the correlation length at the critical point is not developed to infinity in contrast to the system at thermal limits.Using a Monte Carlo simulation of the three-dimensional three-state Potts model,we demonstrated the fluctuations of the second-and fourth-order generalized susceptibilities near the temperatures of the external fields of the first-,second-,and crossover regions.Both the second-and fourth-order susceptibilities showed similar peak-like and oscillationlike fluctuations in the three regions.Therefore,non-monotonic fluctuations are associated with the second-order phase transition and the first-order phase and crossover in a finite-size system.The exponent of finite-size scaling(FSS)characterizes the order of transitions or crossover.To determine the parameters of the phase transition using the FSS,we studied the behavior of a fixed point in the FSS.To quantify the behavior of the fixed point,we define the width of the scaled observables of different sizes at a given temperature and scaling exponent ratio.The minimum width reveals the position of the fixed point in the plane of the temperature and scaling exponent ratio.The value of this ratio indicates the nature of the fixed point,which can be a critical,first-order phase transition line point,or crossover region point.To demonstrate the effectiveness of this method,we applied it to three typical samples produced by a three-dimensional three-state Potts model.The results show that the method is more precise and effective than conventional methods.Possible applications of the proposed method are also discussed.Finally,because of the limited evolution time,some processes in relativistic heavy-ion collisions may not reach thermal equilibrium.To estimate the influence of the nonequilibrium evolution,we used the three-dimensional Ising model with the Metropolis algorithm to study the evolution from nonequilibrium to equilibrium on the phase boundary.The order parameter exponentially approaches its equilibrium value,as suggested by the Langevin equation.The average relaxation time is defined.The relaxation time is well represented by the average relaxation time,which diverges as the zth power of the system size at a critical temperature,similar to the relaxation time in dynamical equations.During nonequilibrium evolution,the third and fourth cumulants of the order parameter could be positive or negative depending on the observation time,which is consistent with the calculations of dynamical models at the crossover side.The nonequilibrium evolution at the crossover side lasts briefly,and its influence is weaker than that at the firstorder phase transition line.These qualitative features are instructive for experimentally determining the critical point and phase boundary in quantum chromodynamics.