Determining Characteristic Parameters Of Phase Boundary By Fixed Points Of Finite-size System

The big bang theory holds that matter in the early universe was basically composed of QGP.Quantum Chromodynamics is the gauge theory of strong interactions between quark and gluon in hadrons and nucleons.It predicts that the quarks confined in the hadrons will be deconfined at high energies and formed quark gluon plasma.Due to the short duration of QGP,it cannot easily be observed directly.In order to ascertain and research the QGP,relativistic heavy-ion collision experiments create the extreme condition(high temperature and high density environment)where hadron matter has phase transition from hadron state to quark gluon plasma.Research results show that deconfined quark-gluon matter has been produced in the collision experiments.But for the QCD phase structure,the research proceedings have been ongoing.In especial,the location of the QCD critical point is still blurred.In order to determine the critical point of QCD and the phase boundary experimental-ly,the relativistic Heavy Ion Collider(RHIC)at Brookhaven National Laboratory(BNL)have run the beam energy scanning program.Through the STAR detector,the related fluctuation signals of QCD phase transition have been observed.Considering the critical related observables,the high-order cumulants of conserved charges are more sensitive to the correlation length and the fluctuation signals are preserved to the final state with the evolution process in the collisions.So in recent years they have been suggested to locate the QCD critical point as observables.Up till now,It has been measured at RHIC/BES I that the the fourth order cumulant of net-proton shows a clear non-monotonic energy de-pendence.And from lattice QCD results,the baryon number susceptibilities computed at vanishing chemical potentials have non-monotonic behavior as temperature varies.Fur-thermore,the non-monotonic results of high-order cumulants in the chiral potential model can be generalized to the case of non-zero chemical potential.So the non-monotonic be-havior mentioned above is not unique characteristic to critical fluctuations.In fact,the physical realizable system has finite size and limited evolution time.Due to the limited evolution time,critical slowing down cause that the system may not have reached thermalized equilibrium.Finite size effect impact on the critical fluctuation sig-nals,such as the peak location of susceptibility is pseudo-critical point which shifts off the critical point in thermodynamic limit.Therefore,the influence of these two factors on the observed results cannot be ignored.For finite system,given the observables,the variable is scaled reduced temperature in the conventional method of finite size scaling.Selection of the best scaling is done through the observed overlap of the size-scaled observables' curves.Then we can estimate the ac-curacy of the pre-assumed values of characteristic parameters of phase transition,which are phase transition temperature and scaling exponent ratio of phase transition.This tradi-tional approach has limitations.Selection of the best scaling is through visual observation.And because of the size-scaled factor,the overlap region of the size scaled observables'curves will enlarge at reduced temperature of phase transition region.Therefore,through the method,the measurement for characteristic parameters of phase transition often lack in precision.Then we choose the temperature as variable,instead of scaled reduced temperature.Any deviation from the phase transition temperature,the scaled curves for different sizes separate from each other.When approach to the phase transition temperature,the curves gradually converge.And only at the phase transition temperature,they intersect at a fixed point,where the corresponding temperature and scaling exponent ratio are the phase tran-sitional ones.In the renormalization group theory,the critical point is unstable fixed point.And in the first order phase transition line,fixed point also exists,which is discontinuous.If the finite size scaling can be generalized to the crossover region,the scaling exponent ratio is zero,just because of observable independent of system size in the transition region Although the scaled curves overlap in the transition region,there exist trival fixed points.So if the fixed point exists in the phase plane,the corresponding phase transition temperature and scaling exponent ratio of phase transtion can be determined near the phase boundary.The characteristics of scaling exponent ratios are different in various phase transition cases,where the ratios are respectively a fraction for critical point,an integer related to the system dimension in first order phase transition line and zero at crossover.Therefore,the value of these ratios can tell the nature of fixed point and the order of phase transition.How to analyze the behavior of fixed points near the phase boundary?In this work,We will give a quantitative method for locating fixed points.Firstly,we quantify the dis-tribution width between the scaled observables' curves for different sizes.Given the tem-perature and scaling exponent ratio,the sample standard deviation of scaled observables'points is on behalf of distribution width between these points.And we define a dimension-less quantity D(T,?),which is the function of temperature T and scaling exponent ratio a.The value of D(T,?)describes the relative distance of all points to their mean position,ie the width.When T and a are both the transitional values,all points will overlap.D(T,a)reaches its minimum Dmin(T,?),around unity.And we consider the fixed point exists here.Then,the corresponding temperature and scaling exponent ratio for the minimum of D(T,?)are the characteristic parameters for phase transition.Deviated from the charac-teristic parameters for phase transition,the points of curves move away from each other,and the value of D(T,?)increases.To verify the feasibility for the quantitative description of behavior of fixed point in the phase plane,we introduce the three-dimensional three-state potts model,which is of pure gauge and effective potential.Firstly,the mean magnetization which is order param-eter in potts model is taken as the observable.And we give three sample(critical point,first order phase transition line and crossover region)in external field.Then as varied with the temperature and scaling exponent ratio,the width D(T,?)for the different size-scaled curves of the mean magnetization is presented in the contour plots.Thereinto,the contour plots of the cases in both critical point and point on first order phase transition line have a minimum Dmin(T,?),around unity.Then,we consider the scaled curves in two cases intersect at fixed point.Combined with the projections of the width on the temperature(s-caling exponent ratio)axis,we can get the accurate the characteristic parameters of phase transition,namely phase transition temperature and scaling exponent ratio of phase transi-tion.And the corresponding scaling exponent ratios are the fraction and the integer zero.In the transition region of crossover,D(T,?)is independent of T.Its value is determined by a only.D(T,?)is featured with band at a=0 and the minimums Dmin(T,?)correspond to trival fixed points.It indicates that trival fixed points are aligned in a line parallel to T with ?=0 in this case and the scaled curves independent with system size.Visibly,the value of the width D is sensitive to the trend of the behavior of fixed point in the phase boundary nearby.The quantitative method can well quantifies the behaviour of fixed point,determining characteristic parameters of phase transition and distinguish the order of phase transition.Then if the quantitative method is applied into the RHIC BES,its relevant issues such as the choice of appropriate observable and the correctly estimation of the system size should be carefully discussed in heavy-ion collisions.Finally,We hope it will be helpful in locating the fixed point of QCD Phase transition from experimental side.