We present results of a simulation of 2 flavour QCD on a $16^3\times4$
lattice using p4-improved staggered fermions with bare quark mass $m/T=0.4$.
Derivatives of the thermodynamic grand canonical partition function
$Z(V,T,\mu_u,\mu_d)$ with respect to chemical potentials $\mu_{u,d}$ for
different quark flavours are calculated up to sixth order, enabling estimates
of the pressure and the quark number density as well as the chiral condensate
and various susceptibilities as functions of $\mu_{u,d}$ via Taylor series
expansion. Results are compared to high temperature perturbation theory as well
as a hadron resonance gas model. We also analyze baryon as well as isospin
fluctuations and discuss the relation to the chiral critical point in the QCD
phase diagram. We moreover discuss the dependence of the heavy quark free
energy on the chemical potential.Comment: 4 pages, 7 figures, talk presented at Quark Matter 2005, Budapes

Allton, C. R.

Döring, M.

Ejiri, S.

Hands, S. J.

Kaczmarek, O.

Karsch, F.

Laermann, E.

Redlich, K.

English

arXiv

We present results of a simulation of 2 flavour QCD on a 16{sup 3} x 4 lattice using p4-improved staggered fermions with bare quark mass m/T = 0.4. Derivatives of the thermodynamic grand canonical partition function Z(V,T,{mu}{sub u},{mu}{sub d}) with respect to chemical potentials {mu}{sub u,d} for different quark flavours are calculated up to sixth order, enabling estimates of the pressure and the quark number density as well as the chiral condensate and various susceptibilities as functions of {mu}{sub u,d} via Taylor series expansion. Results are compared to high temperature perturbation theory as well as a hadron resonance gas model. We also analyze baryon as well as isospin fluctuations and discuss the relation to the chiral critical point in the QCD phase diagram. We moreover discuss the dependence of the heavy quark free energy on the chemical potential

Doring, M.

ALLTON, C. R.

UNT Digital Library

BROOKH EN N A T I  0 N A L LAB 0 R A T 0  RY BNL-75084-2005-CP The QCD Equation of State for Two Flavours at Non-Zero Chemical Potential S. Ejiri, C.R. Allton, M. Doring, S. J. Hands, 0. Kaczmarek, F. Karsch, E. Laermann, and K. Redlich Presented at the Quark Matter 2005 I 8th lnternatiorzal Conference on Nucleus-Nucleus Collisions Budapest, Hungary August 4-9,2005 October 2005 Physics Department Brookhaven National Laboratory P.O. Box 5000 Upton, NY 11 973-5000 www. bnl.gov Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. 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. The QCD equation of state for two flavours at non-zero chemical potential S. Ejiri"*, C.R. Alltonb, M. Doring'd, S.J. Handsb, 0. Kaczmarek', F. I<arschCd, E. Laermann' a.nd K. Redlichef. "Depa.rtment of Physics, The University of Tokyo, Tokyo 113-0033, Japan bDepartment of Physics, University of Wales Swansea, Swansea SA2 8PP, UK 'Fa.ku1tat fur Physik, Universitat Bielefeld, D-33615 Bielefeld, Germmy dPhysics Department, Brookhaven National Laboratory, Upton, NY 11973, USA "Institute of Theoretical Physics, University of Wroclaw, PL-50204 Wroclaw, Poland fPhysics Department, Theory Division, CERN, CH-1211 Geneva 23, Switzerland We present results of a simulation of 2 flavour QCD on a 163 x 4 lattice using p4- improved staggered fermions with bare quark mass m/T = 0.4. Derivatives of the ther- modynamic grand canonical partition function Z( V, T ,  ,uu, P d )  with respect to chemical potentials p u , d  for different quark flavours are calculated up to sixth order, enabling es- timates of the pressure and the quark number density as well as the c h i d  condensate and various susceptibilities as functions of p!,,d via Taylor series expansion. Results are compared to high temperature perturbattion theory as well as a hadron resonance gas model. We also a*nalyze baryon as well a,s isospin fluctuations and discuss the relation to the chiral critica.1 point in the QCD phase diagram. We moreover discuss the dependence of the heavy quark free energy on the chemical potential. 1. Introduction It is important to study QCD at high temperature and high density by numerical simulations of lattice QCD. In particular, studies of the equation of state (EoS) ca.n provide basic input for the analysis of the experimental signatures for QGP formation, e.g. the EoS will control the properties of any hydrodynamic expansion. We extend studies of the EoS to non-zero baryon number density. Simulation a.t non-zero ba.ryon density is known to be difficult; however recent studies have found that a. Taylor expansion with respect to chemical potential ,u-L,,d is an efficient technique to investigate the low density regime, interesting for heavy-ion collisions [ 1, 21. In the calculation of the Taylor expansion coefficients, i.e. calculation of the derivatives at ,uu,d = 0, the technical difficulty for non-zero ,uu,d does not arise and a quantitative *Spea.ker. SE is supported by Grants-in-Aid of the Japanese MEXT No. 15540254. KR is supported by KBN under grant 2P03 (06925). FK is partly supported by a contract DE-AC02-98CH1-886 with the US. DOE. RiID is supported by DFG grant GRK-881. 2 S. Ejiri et  al. 1 0.5 0 1 1.2 1.4 -0.5 0.R T J C  0.4 0.2 0 I I I , I , ,  0.8 1 1.2 1.4 0.8 1 1.2 I .4 -0.2 T/r, T/r, Figure 1. The ratios of Taylor expansion coefficients for pressure (quark number suscep- tibility), isospin susceptibility and chiral condensate. study becomes possible within the error by truncation of higher order terms. Since ther- modynamic quantities can be defined by derivatives of the partition function, e.g. chiral condensate ($$) = (T/V)  (d In Z/dm) ,  quark number density nq = (T/V) (a In Z/dpq) ,quark number susceptibility xq = (T/V)(d2 In 2/8p; ) ,  and isospin susceptibility XI = (T/V)(a2 lnZ/6'$>, where pq = (puu+pd)/2 and p~ = (p, - -pd) /2 ,  the calculation of first derivatives yields basic QCD thermodynamics observables, and the calculation of higher derivatives is a natural extension. In this study, we evaluate the Taylor expansion coefficients of thermodynamic quantities and compare these results to the predictions from the perturbation theory in the high temperat,ure limit and from the hadron resonance gas model in the low temperature hadron phase. Also fluctuations of quark number, isospin number and electric charge are discussed near the transition temperature T,. They are estimated by xq ,  X I  and charge susceptibility xc. = xq/36 + x1/4, and are related to the critical endpoint expected at non-zero pq. Moreover we study the Taylor expansion coefficients of chiral condensate and heavy quark free energy. The details of simulations are given in [ 21. 2. Quark gluon gas and hadron resonance gas We define the expansion coefficients by p/T4 = (In Z) / (VT3)  E,"==, c ~ ( , u ~ / T ) ~ ,  We expect the equation of state to approach that of a free quark-gluon gas (Stefan- Boltzmann (SB) gas) in the high temperature limit. The coefficients in the SB limit for 2 flavour QCD with PI = 0 are well known as c2 = e; = 1, e4 = ci = 1/(2n2) a,nd c, = e: = 0 for n 2 6. Moreover the sign of c6 is negative starting at 0 ( g 3 )  in pertur- bation theory. On the other hand, in the low temperature phase QCD is well described by a hadron resonance gas model in which the pressure is obtained by summing over the contributions from all resonance states of hadrons. In this model, the contribution to p/T4 from each individual hadron which has baryon number Bi is in proportion to exp(3Bipq/T), hence the pressure can be written as p/T4 = G(T)  + F ( T )  cosh(3pq/T), where G(T)  and F ( T )  are the mesonic and baryonic components of p / T 4  at pq = 0. Similarly, we obtain xq/T2 = 9F(T)  cosh(3pq/T), x1/T2 = GI(T)  + F I ( T )  cosh(3pq/T) xq[xr]/T2 = E,"==, n(n - 1)Cn[C;I(Pq/T>n-2 mid ($q!J)/T3 fE,"=, c$qPq/T)n for PI  = 0. The QCD equation of state for two flavours at non-zero chemical potential 3 4 3 2 1 0.8 1 1.2 1.4 1.6 1.8 2 3 .  2 1 0.8 1 1.2 1.4 1.6 1.8 2 Figure 2. The quark number susceptibility xq/T2 (left) and isospin susceptibility x1/T2 (right) for various p, /T.  To is T, at p, = 0. and (4$>/T3 = Gdg(T) + Fqg(T) cosh(3pq/T). Here, the mesonic component for x, is zero because mesons have baryon number zero. Therefore, c4/c2 = 3/4, c6/c4 = 3/10, = 3/10 and et'/& = 3/4 in the region where the hadron resona,nce gas provides a good a.pproximation. We investigate these coefficients. The results for cn+2/c,, ~ i + ~ / c i  and c$f2/c$' are shown in Fig. 1. We find that these results are consistent with the prediction from the hadron resonance gas model for T/Tc 5 0.96 and approach the SB values, i.e. ~ 4 / ~ 2  = ci/cL = 1/27r2, c6/c4 = c ~ / c ~  = 0, in the high temperature limit. Also c6 is negative at high temperature as expected in the perturbation theory. These results suggest that the models of free quark-gluon gas and hadron gas seem to explain the behaviour of thermodynamic quantities well except in the narrow regime near T,. 3. Susceptibilities at non-zero p, Next, we calculate quark number and isospin susceptibilities in a range of 0 5 p,/T 5 1. The data connected by solid lines in Fig. 2 are obtained by xq/T2 = 2c2 + 1 2 ~ 4 ( p , / T ) ~  + 3 0 ~ s ( p , / T ) ~  and the corresponding equation for X I .  Dashed lines are the results from o(pi) expansion. Since the statistical error of c6 is still large near T,, the location and height of the peak are less accurately determined, and also the error from the truncation of higher order terms of p,/T seems to be visible for large pq/T. However, as seen in Fig. 1, c6 changes its sign at T,. This means the peak position of x, moves left, which is corresponding to the change of T, as a function of p,. T,(p,/T = l)/Tc(pa/T = 0) in [ 31 is about 0.93. Moreover, xq increases with higher orders of the expansion for T 5 T, which confirms the existence of a peak. This suggests the presence of a critical endpoint in the (T, p,) plane. On the other hand, XI in Fig. 2 does not show any singular behaviour. This is consistent with the sigma model prediction that only isosinglet degrees of freedom become massless at the critical endpoint. 4 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 S. Ejiri et  al. 1 1.5 3 3.5 4 1 1.5 2 .3 3.5 4 Figure 3. (a,) The ratios mi/rnA and mp/rnA a.s functions of T/Tc I ,  a.nd (b) the singlet screening mass at non-zero pq. 4. Screening mass at non-zero pq Finally we want to discuss the free energy of a static quark anti-quark pair at finite temperature and density. We extract the singlet free energy FiQ and colour averaged free energy Fro by the Polyakov loop correlation functions. For T > T,, the free energy is ex- pected to be exponentially screened at large distances, A F a l ( r ,  T ,  pq) = Frdl(00, T ,  pq)- bye screening mass mavJ in terms of p q / T  at pq = 0,  maVJ = ~ ~ = o m ~ J ( p q / T ) n .  We plot the data of rn,i/mh and my/rnA in Fig. 3(a). We find mi = (1/2)my. This is ex- pected from perturbation theory, which suggests that the leading order contribution to the colour singlet free energy is given by one gluon exchange while the colour averaged free energy is domina.ted by two gluon exchange. Moreover the perturbative Debye screen- ing mass ~ Z D  is given by mD(T, &) = rn~,o(T),/l+ 3Nf/[(2Nc + N~)T~](,U,/?')~, with m~,o(T)  = g ( T ) T , / w .  Here N, and N f  are the number of colour and flavour respectively. The solid line in Fig. 3(a) is the perturbative predict,ion for mi/m& The ratio m?j/mA is found to be consistent with perturbation theory for T 2 2T,. In Fig. 3(b) we show the pq-dependence of the singlet screening mass ml(pq, T) /T  for a sma.11 values of p q / T ,  where only contributions from mi and mi are included. Further details of this study are given in [ 41. F"?1 QQ (r,  T ,  pq) N e-mav'l(T~~~)r..  We calculate the Taylor expansion coefficients of the De- REFERENCES 1. C.R. Allton, S. Ejiri, S.J. Ha,nds, 0. Ihczmarek, F. Karsch, E. Laermann and C. Schmidt, Phys. Rev. D68 (2003) 014507. 2. C.R. Allton, M. Doring, S. Ejiri, S.J. Hands, 0. Kaczmarek, F. Karsch, E. Laermann and K. Redlich, Phys. Rev. D71 (2005) 054508. 3. C.R. Allton, S. Ejiri, S. J. Ha.nds, 0. Kaczmarek, I?. Karsch, E. Laermann, Ch. Schmidt a,nd L. Scorzato, Phys. Rev. D66 (2002) 074507. 4. M. Doring, S. Ejiri, 0. Kaczmarek, F. Karsch and E. Laermann, hep-lat/05nQnni 

QCD EQUATION OF STATE FOR TWO FLAVOURS AT NON-ZERO CHEMICAL POTENTIAL.

Ejiri S, Allton CR, Doring M, et al. The QCD equation of state for two flavors at non-zero chemical potential. Nucl.Phys. A. 2006;774:837-840.We present results of a simulation of 2 flavour QCD on a $16^3\times4$ lattice using p4-improved staggered fermions with bare quark mass $m/T=0.4$. Derivatives of the thermodynamic grand canonical partition function $Z(V,T,\mu_u,\mu_d)$ with respect to chemical potentials $\mu_{u,d}$ for different quark flavours are calculated up to sixth order, enabling estimates of the pressure and the quark number density as well as the chiral condensate and various susceptibilities as functions of $\mu_{u,d}$ via Taylor series expansion. Results are compared to high temperature perturbation theory as well as a hadron resonance gas model. We also analyze baryon as well as isospin fluctuations and discuss the relation to the chiral critical point in the QCD phase diagram. We moreover discuss the dependence of the heavy quark free energy on the chemical potential

Doring, Matthias

Hands, S. J

Kaczmarek, Olaf 

Karsch, Frithjof

Laermann, Edwin

Publications at Bielefeld University

The QCD equation of state for two flavors at non-zero chemical potential

S.  Ejiri

C.R. Allton

M. Döring

S.J. Hands

O. Kaczmarek

F. Karsch

E. Laermann

K. Redlich

Crossref

The QCD equation of state for two flavours at non-zero chemical potential

We present results of a simulation of 2 flavour QCD on a $16^3\times4$ lattice using p4-improved staggered fermions with bare quark mass $m/T=0.4$. Derivatives of the thermodynamic grand canonical partition function $Z(V,T,\mu_u,\mu_d)$ with respect to chemical potentials $\mu_{u,d}$ for different quark flavours are calculated up to sixth order, enabling estimates of the pressure and the quark number density as well as the chiral condensate and various susceptibilities as functions of $\mu_{u,d}$ via Taylor series expansion. Results are compared to high temperature perturbation theory as well as a hadron resonance gas model. We also analyze baryon as well as isospin fluctuations and discuss the relation to the chiral critical point in the QCD phase diagram. We moreover discuss the dependence of the heavy quark free energy on the chemical potential.We present results of a simulation of 2 flavour QCD on a $16^3\times4$ lattice using p4-improved staggered fermions with bare quark mass $m/T=0.4$. Derivatives of the thermodynamic grand canonical partition function $Z(V,T,\mu_u,\mu_d)$ with respect to chemical potentials $\mu_{u,d}$ for different quark flavours are calculated up to sixth order, enabling estimates of the pressure and the quark number density as well as the chiral condensate and various susceptibilities as functions of $\mu_{u,d}$ via Taylor series expansion. Results are compared to high temperature perturbation theory as well as a hadron resonance gas model. We also analyze baryon as well as isospin fluctuations and discuss the relation to the chiral critical point in the QCD phase diagram. We moreover discuss the dependence of the heavy quark free energy on the chemical potential.We present results of a simulation of 2 flavour QCD on a $16^3\times4$ lattice using p4-improved staggered fermions with bare quark mass $m/T=0.4$. Derivatives of the thermodynamic grand canonical partition function $Z(V,T,\mu_u,\mu_d)$ with respect to chemical potentials $\mu_{u,d}$ for different quark flavours are calculated up to sixth order, enabling estimates of the pressure and the quark number density as well as the chiral condensate and various susceptibilities as functions of $\mu_{u,d}$ via Taylor series expansion. Results are compared to high temperature perturbation theory as well as a hadron resonance gas model. We also analyze baryon as well as isospin fluctuations and discuss the relation to the chiral critical point in the QCD phase diagram. We moreover discuss the dependence of the heavy quark free energy on the chemical potential.We present results of a simulation of 2 flavour QCD on a $16^3\times4$ lattice using p4-improved staggered fermions with bare quark mass $m/T=0.4$. Derivatives of the thermodynamic grand canonical partition function $Z(V,T,\mu_u,\mu_d)$ with respect to chemical potentials $\mu_{u,d}$ for different quark flavours are calculated up to sixth order, enabling estimates of the pressure and the quark number density as well as the chiral condensate and various susceptibilities as functions of $\mu_{u,d}$ via Taylor series expansion. Results are compared to high temperature perturbation theory as well as a hadron resonance gas model. We also analyze baryon as well as isospin fluctuations and discuss the relation to the chiral critical point in the QCD phase diagram. We moreover discuss the dependence of the heavy quark free energy on the chemical potential.We present results of a simulation of 2 flavour QCD on a $16^3\times4$ lattice using p4-improved staggered fermions with bare quark mass $m/T=0.4$. Derivatives of the thermodynamic grand canonical partition function $Z(V,T,\mu_u,\mu_d)$ with respect to chemical potentials $\mu_{u,d}$ for different quark flavours are calculated up to sixth order, enabling estimates of the pressure and the quark number density as well as the chiral condensate and various susceptibilities as functions of $\mu_{u,d}$ via Taylor series expansion. Results are compared to high temperature perturbation theory as well as a hadron resonance gas model. We also analyze baryon as well as isospin fluctuations and discuss the relation to the chiral critical point in the QCD phase diagram. We moreover discuss the dependence of the heavy quark free energy on the chemical potential.We present results of a simulation of 2 flavour QCD on a 															16								3														×							4						 lattice using p4-improved staggered fermions with bare quark mass 							m							/							T							=							0.4						 . Derivatives of the thermodynamic grand canonical partition function 							Z							(							V							,							T							,															μ								u														,															μ								d														)						 with respect to chemical potentials 															μ																	u									,									d																					 for different quark flavours are calculated up to sixth order, enabling estimates of the pressure and the quark number density as well as the chiral condensate and various susceptibilities as functions of 															μ																	u									,									d																					 via Taylor series expansion. Results are compared to high temperature perturbation theory as well as a hadron resonance gas model. We also analyze baryon as well as isospin fluctuations and discuss the relation to the chiral critical point in the QCD phase diagram. We moreover discuss the dependence of the heavy quark free energy on the chemical potential

Allton, C.R.

Hands, S.J.

CERN Document Server

We present results of a simulation of 2 flavour QCD on a $16^3\times4$ lattice using p4-improved staggered fermions with bare quark mass $m/T=0.4$. Derivatives of the thermodynamic grand canonical partition function $Z(V,T,\mu_u,\mu_d)$ with respect to chemical potentials $\mu_{u,d}$ for different quark flavours are calculated up to sixth order, enabling estimates of the pressure and the quark number density as well as the chiral condensate and various susceptibilities as functions of $\mu_{u,d}$ via Taylor series expansion. Results are compared to high temperature perturbation theory as well as a hadron resonance gas model. We also analyze baryon as well as isospin fluctuations and discuss the relation to the chiral critical point in the QCD phase diagram. We moreover discuss the dependence of the heavy quark free energy on the chemical potential

Chris Allton

Cronfa at Swansea University

http://digital.library.unt.edu/ark:/67531/metadc785894/

The QCD equation of state for two flavours at non-zero chemical
  potential

The QCD equation of state for two flavours at non-zero chemical potential

Abstract

Similar works

Full text

Available Versions

UNT Digital Library

Publications at Bielefeld University

Crossref

CERN Document Server

Cronfa at Swansea University