We propose a method to probe the nature of phase transitions in lattice QCD
at finite temperature and density, which is based on the investigation of an
effective potential as a function of the average plaquette. We analyze data
obtained in a simulation of two-flavor QCD using p4-improved staggered quarks
with bare quark mass $m/T = 0.4$, and find that a first order phase transition
line appears in the high density regime for $\mu_q/T \sim 2.5$. The effective
potential as a function of the quark number density is also studied. We
calculate the chemical potential as a function of the density from the
canonical partition function and discuss the existence of the first order phase
transition line.Comment: 7 pages, 7 figures, talk presented at Lattice 200

Ejiri, Shinji

English

arXiv

We propose a method to probe the nature of phase transitions in lattice QCD at finite temperature and density, which is based on the investigation of an effective potential as a function of the average plaquette. We analyze data obtained in a simulation of two-flavor QCD using p4-improved staggered quarks with bare quark mass m/T = 0.4, and find that a first order phase transition line appears in the high density regime for {mu}{sub q}/T {approx}&gt; 2.5. The effective potential as a function of the quark number density is also studied. We calculate the chemical potential as a function of the density from the canonical partition function and discuss the existence of the first order phase transition line

EJIRI,S.

UNT Digital Library

__*,-4b BROQKHR6EN NATl  0 N A L L A B  0 R A T 0  R Y  BNL-79463-2007-CP Study of the critical point in lattice QCD at high temperature and density Shinji Ejiri' 'Department of Physics, Brookhaven National Laboratory, Upton, NY, USA 11973 Presented at the X X V  International Symposium on Lattice Field Theory Regensburg, Germany July 30 to August 4,2007 Physics Department Nuclear Theory Group Brookhaven National Laboratory P.O. Box 5000 Upton, NY 1 1973-5000 www.bnl.gov Notice: This manuscript has been authored by employees of Brookhaven Science Associates, L E  under Contract No. DE-AC02- 98CH10886 with the US. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. This preprint is intended for publication in a journal or proceedings. Since changes may be made before publication, it may not be cited or reproduced without the author's permission. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party's use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or Its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. Study of the critical point in lattice QCD at high temperature and density Shinji Ejiri" Physics Deparrmenl. Brookhaven National Laborutov, Upton. New York 11973, USA E-mail: e j iri@quark , phy . bnl . gov We propose a method to probe the nature of phase transitions in lattice QCD at finite temperature and density, which is based on the investigation of an effective potential as a fonction of the av- erage plaquette. We analyze data obtained in a simulation of two-flavor QCD using p4-improved staggered quarks with bare quark mass m / T  = 0.4, and find that a first order phase transition line appears in the high density regime for & f T 2 2.5. The effective potential as a function of the quark number density is also studied. We calculate the chemical potential as a function of the density from the canonical partition function and discuss the existence of the first order phase transition line. The XWlnternational$~mposium on Larrice Field Theory July 30 -August 4 2007 Regensburg, Germany 'Speaker. h'his work has been authored under contract DE-AC02-98CHI-886 with the U.S. Department of Energy. Study of the criticalpoint in lattice QCD at high temperature and density Shinji Ejiri 1. Effective potential as a function of the average plaquette The study of the QCD phase diagram at non-zero temperature ( T )  and chemical potential ( p q )  is one of the most important topics among studies of lattice QCD. In particular, the study of the endpoint of the first order phase transition line in the ( T , h )  plane is particularly interesting both from the experimental and theoretical point of view. The existence of such a critical point is suggested by phenomenological studies. The appearance of the critical endpoint in the ( T , p q )  plane is closely related to hadronic fluctuations in heavy ion collisions and may be experimentally examined by an event-by-event analysis of heavy ion collisions. Many trials have been made to prove the existence of the critical endpoint by first principle calculation in lattice QCD, however no definite conclusion on this issue is obtained so far. The purpose of this study is to clarify the existence of the endpoint of the first order phase transition line in the (T ,  &) plane. We propose a new method to investigate the nature of transition. We evaluate an effective potential as a function of the average plaquette (P)  I ,  and identify the type of transition from the shape of the potential. The partition function can be written as where Sg(P,p) is the gauge action, w(P) is the state density at & = 0 for each P, w(P')e-ss(p~fi) = w(P' ,p)  z 19U S ( P ' - P )  (detM)Nre6flN8rP, (1.2) and R(P, &) is the modification (reweighting) factor for finite h, which is defined by where (. . .)(p,&=o) means the expectation value at = 0, detM is the quark determinant, Nf is the number of flavors , and Nsite = N: x N, is the number of sites. We then define the effective potential as V ( P , p , h )  = - In(Rwe-Sg). If there is a first order phase transition point, where two different states coexist, the potential must have two minima at two different values of P. In this paper, we discuss whether the potential at = 0, i.e. In(w&g), which is quadratic function when the transition is a crossover, can change to a double-well potential by the reweighting factor for finite &, as illustrated in Fig. 1 (left). 2. Taylor expansion in terms of kqlT and Gaussian distribution However, the calculation of the quark determinant is quite expensive and is actually difficult except on small lattices. Moreover, the calculation of R(P, &) becomes increasingly more difficult 'For later discussions, we define the average plaquette P as P -Sg/(6/3Nsite). This is the average of the plaquette 2We restrict ourselves to discuss only the case when the quark matrix does not depend on fl explicitly, e.g. the 'N~fmnst be replaced in these equations to Nf/4 when we use a stlggered we quark action. over all elementary squares for the standard gauge action. standard Wilson and staggered quark actions, the +improved staggered quark action etc., for simplicity. 2 Study of the critical point in laftice QCD at high temperature and density Shinji Ejiri Crossover Reweighting factor double-well patentml t Figure 1: (left) Schematic figures of the effective potential and the reweighting factor. (right) The histogram ofthecomplexphaseforp,JT= l .Oand2.0atP=3.65.  for large & due to the sign problem, i.e. the statistical error becomes exponentially larger as clq increases. We avoid these problems by the following two ideas. One is that we perform a Taylor expansion of IndetM(&) in terms of & at j+  = 0 and calculate the expansion coefficients [I], The Taylor expansion coefficients are rather easy to calculate by using the stochastic noise method. Although we must cut off this expansion at an appropriate order in &, we can estimate the appli- cation range where the approximation is valid for each analysis [2,3]. While the application range of the Taylor expansion of In =OX& should be limited by the critical point because In =OX& is singular at the critical point, there is no such limit for the application range in the expansion of InR(P,&) because the weight factor should always be well-defined. The sign problem is avoided by the following idea. We consider a probability distribution func- tion as a function of the complex phase of the quark determinant 8, IF1 = I detM(&)/detM(0)lNf and P, i.e. G(P, IFI, 8). If we assume the distribution function in 8 is well-approximated by a Gaussian function, the sign problem in the calculation of InR(P,pq) is completely solved [4]. We define the complex phase by a Taylor expansion, 8 = N ~ N ~ N , ~ ~ = , I ~ ~ ~ ~ + , ( c ~ , / T ) ~ ~ + ~ .  Since the partition function is real even at non-zero density, the distribution function has the symmetry under the change from 0 to -8. Therefore, the distribution function is written by G(0) - exp[-(u# + a404 + a606 f . .  .)I.  Moreover, because Dn is a trace of a matrix which has space index [3], e.g. D I  = tr[M-'dM/d(&/T)],  the central limit theorem suggests that the distribution function is well-approximated by a Gaussian function, when the system size is suffi- ciently large in comparison to the correlation length between diagonal elements of the matrix. We plotted the distribution of the complex phase in Fig. 1 (right) and fitted by a Gaussian function (dashed line). It is found that this approximation is quite well, hence we consider the leading term of the expansion only, G(P, lFl, e) = d w  exp[-q(P, IFl)e*]. The coeRcient a2(P, IFI) 3 Study of the critical point in lattice QCD at high temperature and densiry Shinji Ejiri 8-3.52 3.60 3.65 3.80 4.W T/5-0.76 0.90 1.W 1.36 1.98 4 '0 .18 0.8 0.82 084 0.86 0.88 0.9 0.92 0.94 P __-.- '0.78 0.8 0.81 0.84 0.86 0.88 0.9 0.92 0.94 P Figure 2: (left) The plaquette histogram and the effective potential at InR(P,&,) for k / T  = 0.5 - 2.5. = 0. (right) The reweighting factor is given by 1/(2uz) = (0') for each Pand IFI. The numerator of Eq. (1.3) is then evaluated by Because l/u2 - O(Ns,te), the phase factor in R ( P , h )  decreases exponentially as a function of the volume. However, the operator in Eq. (2.2) is always real and positive for each configuration in this framework, hence the expectation value ofR(P,&) is always larger than its statistical error, namely the contribution InR(P,&) to the effective potential V ( P , p , p q )  is always well-defined. Therefore, the sign problem is completely avoided if we can assume the Gaussian distribution of 0. 3. Numerical results of the effective potential We calculate w(P, p)  and R(P, &) using data obtained by simulations in [3]. The Taylor expan- sion coefficients are computed up to O(pi). These are measured at sixteen simulation points from p = 3.52 to 4.00 for the bare quark mass mu = 0.1. The corresponding temperature normalized by the pseudo-critical temperature is in the range of T/T, = 0.76 to 1.98. The ratio of pseudo-scalar and vector meson masses is mps/mv E;: 0.7 at /3 = 3.65. The lattice size Nslte is 163 x 4. The number of configurations is 1000 - 4000 for each p. The probability distribution function w(P,P),  i.e. the histogram of P for each P, and the effective potential V(P,p) at & = 0 are given in Fig. 2(left). To obtain w(P,p), we grouped the configurations by the value of P into blocks and counted the number of configurations in these blocks, and the potential V(P,p) is normalized by the minimum value for each temperature. TheresultsforInR(P,&) areshownbysolidlinesinFig.2(right)for&/T=0.5,1.0,1.5,2.0 and 2.5. We find a rapid change in InR around P N 0.83, and the variation becomes larger as k / T  increases. The dashed lines in Fig. 2 (right) are the results that we obtained when the effect of the complex phase, i.e. exp[-1/(4q)], is omitted. These dashed lines correspond to the reweighting factor with non-zero isospin chemical potential p, and zero quark chemical potential & [5]. The variation of InR in terms of P becomes milder when the effect of the complex phase is omitted. 4 Study of the criticalpoint in lattice QCD at high temperature anddensity Shinji Ejiri 6000 4x10’ 2x10’ 4000 0 zoo0 -aid 0 0.18 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 “‘0s0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 P P Figure 3: The slope (lefi) and curvatllre (right) of InR(P,&). The dashed line is the curvature of - Inw. This explains the difference between the phase diagrams of QCD with non-zero quark chemical potential and non-zero isospin chemical potential [4]. We discuss the shape of the effective potential at non-zero pq.  The effective potential is ob- tained from V ( P , P , k )  = -lnw(P,p) -lnR(P,pq) substituting the data in Fig. 2. From Eq. (1.2), the slope of V(P,p,k) can be controlled by P ,  V(p,P,Pq) = V(PvP0, k )  - 6(P - P0)KircR (3.1) under a change of + p, however the curvature of the potential does not change by P .  We expect that; the curvature vanishes at the endpoint of the first order phase transition line by canceling &(lnw)/dP2 and &(lnR)/dP*. In order to analyze the sign of d2V/dP2(P,k) ,  we fitted the data of lnR by a quadratic function of P and calculate the first and second derivatives of InR(P, k )  at each P. The results of the slope and curvature are shown in Fig. 3 for each p q / T .  In the region around P - 0.83, d(lnR)/dP becomes larger as k / T  increases and InR(P,k) changes sharply in this region. The magnitude of the curvature of InR also becomes larger as k / T  increases. To evaluate &(lnw)/dP2(P), we assume w(P,p) in Fig. 2 (left) is a Gaussian function. In this case, the curvature at (P) is given by -&(lnw)/dPz = 6N,ie/xp, where XP 3 6Nsite((P- (P))’) is the plaquette susceptibility. The dashed line in Fig. 3 (right) is the result of -&(lnw)/dP’(P). It is found from Fig. 3 (right) that the maximum value of d * ( I n R ) / d ~ ( P , k )  at P = 0.80 becomes larger than -&(lnw)/dP’ for k / T  2 2.5. This means that the curvature of the effective potential vanishes at k / T  N 2.5 and becomes negative for large k / T .  namely the shape of the effective potential which is of quadratic type at k = 0 changes to a double-well type at large k / T .  For.the quantitative estimation of the endpoint of the first order phase transition, further investigation must be needed. However, this argument strongly suggests the existence of the first order phase transition line in the (7‘,&) plane. Further details of this analysis are given in [4]. . .  4. Canonical partition function Next, we want to abply the effective potential argument to the weight factor as a function of the quark number density p.  The physical meaning of this potential is clearer than that o f P  because 5 Study of the criticalpoint in lattice QCD at high temperature anddensiw Shinji Ejiri Figure 4: The chemical potential as a function of the quark number density. the weight factor for each quark number N corresponds to the canonical partition function Fc, F & ( T , k )  = eNklT%(T,N),  N = i N : ,  p/T3 = FN:. (4.1) N The canonical partition function can be given by an inverse Laplace transformation [6,7, 81, where is an appropriate real constant. Note that F&(T,pq +2ziT/3) = F x ( T , k ) .  Recently, this canonical partition function is calculated for Nf = 4 using the Glasgow method [9] .  However, with present day computer resources, the calculation by the Glasgow method is difficult except on small lattices. We consider approximations which is valid for large volume and low density in this approach, as discussed in the first part of this paper. We calculate the grand partition function by the Taylor expansion, Eq. (2.  l), where V = N,". We moreover use a saddle point approximation, which is valid for a large sys- tem. We find a saddle point zo in the complex p q / T  plane for each configuration, which satisfies [NfN, Cy=l nD,z"-' - dzzz0 = 0. The canonical partition function is given by for large V .  Here, D"(z) = ( d z / d l )  (NfNixr= ,Da9)  and D"(z0) = IDN(zo)le'". The chemical potential, i.e. the slope of the effective potential, is also evaluated by k = :  1 alnFc(T,fiV) T V  (20 exp [V (NfN C L l  DnG - @0)1 e-? G) ( T  -o E @ -  1. (4.5) - a i  ( e x p [ V ( N f N , ~ = I D , ~ - i z ~ ) ] e - ~ ~ )  (T,lb=O) 6 StU& of fhe criticalpoinf in lattice QCD at high temperahire anddensify Shinji Ejiri This equation is similar to the formula of the reweighting method for finite &. The operator in the denominator corresponds to a reweighting factor, and the chemical potential is an expectation value of the saddle point calculated with this modification factor. We analyze the data used in the previous section. The Taylor expansion coefficients up to O(p:) are used. The volume V = 163 would be sufficiently large, and we assume a Gaussian distribution function for the complex phase of the reweighting factor, again. We find a saddle point zo numerically for each configuration, assuming exists near the real axis in the low density region of the complex &/T plane. We use multi-p reweighting method [IO] combining all data obtained at 16 points of p. Configurations are generated with the provability of the Boltzmann weight in Monte-Carlo simulations, however the important configurations will change when the weight is changed by the reweighting method. For such a case, the multi-p reweighting is effective, since the important configurations are automatically selected among all configurations generated at multi-p, and also this method is useful for the interpolation between the simulation points. We plot the result of &/T in Fig. 4 as a function of p / T 3  for each temperature. The dot- dashed line is the value in the free gas limit. As seen in Fig. 2 (left), the configurations do not distribute uniformly in the range of P which is necessary in this analysis, and correct results cannot be obtained if the important Configurations are missing. At low temperature, the important value of P changes very much as p increases, therefore we plotted only the data when the expectation value of P is on the peaks of the histograms of P in Fig. 2 (left). The dashed lines are cubic spline interpolations of these data. It is found from this figure that a qualitative feature of ~ l p / T  changes around T / C  - 0.8, i.e. p q / T  increases monotonically as p increases above 0.8, whereas it shows an s-shape below 0.8. This means that there is more than one values of p / T 3  for one value of &/T below T/Tc - 0.8. This is a signature of a first order phase transition. The critical value of p q / T  is about 2.5, which is consistent with the result in the previous section. Although further studies including justifications of these approximations used in this analysis are necessary for more qualitative investigation, this result also suggests the existence of the first order phase transition line in the (T, pq)  plane. References [l]  C.R.Allton,elaL,Phys. Rev D66(2002)074507[hep-1at/0204010]. [Z] C.R. Allton, ef al,, Phys. Rev. D 68 (2003) 014507 [hep-lat/0305007].  [3] C.R. Allton, etaL,Phys. Rev. D71 (2005)054508 [hep-lat/0501030].  [4] S.Ejiri,arXiv:0706.3549 [5] J.B. Kogut and D.K. Sinclair, PoS (LAT2006) 1 4 7  [hep-lat/0609041];  arxiv:  0709.236. [6] D.E. Miller and K. Redlnh, Phys Rev. D 35 (1987) 2524. [7] A. Hasenfraa and D. Toussaint, Nucl. Phys B 371 (1992) 539. [SI A. Alexandm, M. Faber, I. Horvath and K.-F, Liu, Pbys. Rev D 72 (2005) 1145 13 [9] S.KratochvilaandP.deForcrand, PoS(LAT2005) 167  [hep-lat /0509143]  [hep-lat/0507020].  [ 101 A.M. Ferrenberg and R.H. Swendsen, Pbys. Rev. Left. 63 (1989) 1195. I 

STUDY OF THE CRITICAL POINT IN LATTICE QCD AT HIGH TEMPERATURE AND DENSITY.

The XXV Internaitonal Symposium on Lattice Field Theory July 30 - August 4 2007 Regensburug, GermanyWe propose a method to probe the nature of phase transitions in lattice QCD at finite temperature and density, which is based on the investigation of an effective potential as a function of the average plaquette. We analyze data obtained in a simulation of two-flavor QCD using p4-improved staggered quarks with bare quark mass m/T = 0.4, and find that a first order phase transition line appears in the high density regime for μq/T >_～2.5. The effective potential as a function of the quark number density is also studied. We calculate the chemical potential as a function of the density from the canonical partition function and discuss the existence of the first order phase transition line

Name not available

Study of the critical point in lattice QCD at high temperature and density

https://digital.library.unt.edu/ark:/67531/metadc898772/m2/1/high_res_d/920315.pdf

Study of the critical point in lattice QCD at high temperature and density

Abstract

Similar works

Full text

Available Versions

UNT Digital Library

Name not available