Phase structure of finite density Phase structure of finite density - PowerPoint PPT Presentation

Phase structure of finite density Phase structure of finite density lattice QCD by a histogram method Q y g Shinji Ejiri Shinji Ejiri Niigata University WHOT-QCD collaboration S. Ejiri 1 , S. Aoki 2 , T. Hatsuda 3,4 , K. Kanaya 2 , Y. Nakagawa 1 , H. Ohno 2,5 , H. Saito 2 , and T. Umeda 6 1 Niigata Univ., 2 Univ. of Tsukuba, 3 Univ. of Tokyo, 4 RIKEN, 1 Niigata Univ 2 Univ of Tsukuba 3 Univ of Tokyo 4 RIKEN 5 Bielefeld Univ., 6 Hiroshima Univ. YIPQS HPCI i YIPQS-HPCI international molecule-type workshop on New-type of i l l l k h N f Fermions on the Lattice (YITP, Kyoto, Feb.9-24, 2012)

Phase structure of QCD at high temperature and density Lattice QCD Simulations quark-gluon plasma phase T • Phase transition lines RHI • Equation of state LH SP PS C C RHIC low-E FAIR • Direct simulation: • Direct simulation: A AGS Impossible at  0. deconfinement? chiral SB? quarkyonic? hadron phase phase color super color super conductor? nuclear matter  q

Probability distribution function  Distribution function (Histogram) X : order parameters, total quark number, average plaquette etc.         Z m , T , dX W X , m , T , histogram  In the Matsubara formalism,             S N Z m , T , DU det M m , e g f                          S S N N W X , m , T , DU X- X det M m , e g f  where det M : quark determinant, S g : gauge action. h d M k d i S i  Useful to identify the nature of phase transitions  Useful to identify the nature of phase transitions e.g. At a first order transition, two peaks are expected in W ( X ). 

 -dependence of the effective potential         ,        Z T , dX W X , T , ( ) ln ( ) V X W X eff X : order parameters, total quark number, average plaquette, quark determinant etc. Crossover      V X , T , Critical point Correlation length: short eff V ( X ): Quadratic function Correlation length: long Correlation length: long Curvature: Zero T T QGP 1 st order phase transition 1 order phase transition hadron Two phases coexist CSC? CSC? D Double well potential bl ll i l 

Quark mass dependence of the critical point Quenched N f =2    2 nd order 2 nd order 1 st order 1 st order Physical point y p m s Crossover Crossover 1 st order   0 0 0  m ud • Where is the physical point? • Extrapolation to finite density – investigating the quark mass dependence near  =0 • Critical point at finite density?

Equation of State • Integral method    3 p 1 ln Z – Interaction measure e ac o easu e    ,  4 3 ln T VT a P computed by plaquette (1x1 Wilson loop) and the derivative of det M . – Pressure at  =0 p 1 4  ln Z 3 T T VT VT • Integral    p p 3 p     a       d ln a   4 4 4 4 4 4 T T T T T T a 0 a a 0 a 0 : start point p =0 3           p   p   1 1 Z Z ( ( ) ) N N det d t M M ( ( ) ) • Pressure at  0,         t 0 ln ln     4 4 3     T T VT Z ( 0 ) N det M ( 0 )   s 0 1        X dX X W X , m , T , X P or det M ( ) det M ( 0 ) • with    m , T , Z

Plan of this talk • Test in the heavy quark region – H. Saito et al. (WHOT-QCD Collab.), Phys.Rev.D84, 054502(2011) – WHOT-QCD Collaboration, in preparation • Application to the light quark region at finite density Application to the light quark region at finite density – S.E., Phys.Rev.D77, 014508(2008)) – WHOT-QCD Collaboration, in preparation WHOT QCD Collaboration, in preparation (Lattice 2011 proc.: Y. Nakagawa et al., arXiv:1111.2116)

Distribution function in the heavy quark region WHOT QCD C ll b Ph WHOT-QCD Collab., Phys.Rev.D84, 054502(2011) R D84 054502(2011) • We study the critical • We study the critical surface in the ( m ud , m s ,  ) space in the heavy quark space in the heavy quark region. • Performing quenched • Performing quenched simulations + Reweighting. Reweighting. – plaquette gauge action + Wilson quark action

(  , m,  )-dependence of the Distribution function • Distributions of plaquette P (1x1 Wilson loop for the standard action )                              N 6 N P W W P P , , , , m m , , DU DU P- P P P det det M M m m , , e e f site       (Reweight factor         R P , , m , m , W P , , m , W P , , m , 0 0 0 0 0   N    f   det M m ,     P- P           det , 0 M m N       f     0 det det M M m m ,                              ( ( , 0 0 ) ) 6 6 N N P P ' ' 6 6 N N P P ' ' R P e e 0 site 0 site 0           P- P det M m , 0    0 ( , 0 ) 0 P ' Effective potential:     V                                     V P P , , , , m m , , ln ln W W P P , , m m , , V V P P , , , , m m , , 0 0 ln ln R R P P , , , , m m , , m m , , eff ff eff ff 0 0 0 0 0 0 0 0     N     f         det M m , ,            l ln R R P P 6 6 N N P P ln l     site 0   det M m , 0 0 P

Distribution function in quenched simulations Effective potential in a wide range of P : required Effective potential in a wide range of P : required. Plaquette histogram at K =1/ m q =0. Derivative of V eff at  =5.69   5  points, quenched. 24 3 N 4 , site dV     dV                    dV eff / dP is adjusted to  =5.69, using  eff eff 6 N j , g 2 2 1 1 site site 2 2 1 1 dP dP dP dP These data are combined by taking the average.

Effective potential near the quenched limit WHOT-QCD, Phys.Rev.D84, 054502(2011) WHOT QCD Phys Rev D84 054502(2011) dV Quenched Simulation first order eff ( m q =  , K=0) dP dP K ~1/ m q for large m q crossover Quark mass smaller 24 3  5  points, N f =2 4 lattice, • detM: Hopping parameter expansion • detM: Hopping parameter expansion,   N f =2: K cp =0.0658(3)(8)     det M K        4 N 3 N  N ln   N 288 N K P 12 2 N K t t      f f site s R    det 0 M T T c  real part of Polyakov loop l t f P l k l 0 0 . 02 02 m • First order transition at K = 0 changes to crossover at K > 0. 

Endpoint of 1 st order transition in 2+1 flavor QCD N f =2: K cp =0.0658(3)(8)           det d M M K K        4 N 3 N  2 ln 2 288 N K P 12 2 N K   t t   site s R   det M 0 N f =2+1         2 det M K det M K   ud s ln     3   det M 0                4 4 4 4 N N 3 3 N N N N  288 N 2 K K P 12 2 N 2 K K t t t site ud s s ud s R The critical line is described by t    N N N 2 2 K K K K 2 2 K K t t t t t  ud s cp( N f 2)

Finite density QCD in the heavy quark region              a a † † U x e q U x , U x e q U x in det M 4 4 4 4          T T * * e q , e q Polyakov loop                   det det M M K K ,             4 N 3 N / T / T *  N ln N 288 N K P 6 2 N K e e   t t   f f site s   det M 0 , 0                               N N N N 4 4 3 3  N N 288 288 N N K K P P 12 12 2 2 N N K K cosh h T T i i sinh i h T T t t t t f site s R I phase Polyakov loop        i  i  distribution R I • We can extend this discussion to finite density QCD. to finite density QCD.