The spectrum of 3d 3-states Potts model and universality Mario Gravina Univ. della Calabria & INFN collaborators: R. Falcone, R.Fiore, A. Papa SM & FT 2006, Bari

OUTLINE introduction 3d 3q Potts model numerical results conclusions 1) Svetitsky-Yaffe conjecture 2) Universal spectrum conjecture 3d 3q Potts model numerical results conclusions SM & FT 2006, Bari

1) SVETITSKY-YAFFE CONJECTURE Universality 1) SVETITSKY-YAFFE CONJECTURE SU(N) d+1 confinement-deconfinement Theories with different microscopic interactions but possessing the same underlying global symmetry have common long-distance behaviour Z(N) d order-disorder finite temperature what about 1st order phase transition? if transition is 2nd order SM & FT 2006, Bari

2) universal mass spectrum correlation function of local order parameter m1, m2, m3 … SM & FT 2006, Bari

universality conjecture Ising 3d theory 1 theory 2 lF4 3d SU(2) 4d theory 3 Agostini at al. 1997 Caselle at al. 1999 Fiore, Papa, Provero 2003 m1 m2 m3 m4 m1 m2 m3 m4 m4 m1 m4 m1 m4 m1 = m1 m2 m3 m4 = m3 m1 m3 m1 m3 m1 = = m2 m1 m2 m1 m2 m1 = = SM & FT 2006, Bari

We want to test these two aspects of universality 3d 3q POTTS MODEL h b bc hc L=48 1) 1st order transition L=70 2) 3d Ising point MONTE CARLO simulations CLUSTER ALGORITHM to reduce autocorrelation time SM & FT 2006, Bari

Potts model Z(3) breaking order-disorder PHASE TRANSITION SM & FT 2006, Bari

Phase diagram h=0 weak 1st order transition point Is the mass spectrum universal? 1st order critical lines 2nd order critical endpoint b Z(3) broken phase 2nd order critical ISING endpoint h=0 Does universality hold also for weak 1st order transition? comparison with SU(3) (work in progress) bc Falcone, Fiore, Gravina, Papa Z(3) symmetric phase h hc SM & FT 2006, Bari

h=0 – 1st order transition order parameter is the magnetization global spin SM & FT 2006, Bari

h=0 – 1st order transition at finite volume tunneling effects 0.5508 between symmetric and broken phase between degenerated broken minima 0.550565 complex M plane SM & FT 2006, Bari

h=0 – 1st order transition at finite volume To remove the tunneling between broken minima we apply a rotation only the real phase is present SM & FT 2006, Bari

Masses’ computation MASS CHANNELS by building suitable combinations of the local variable ZERO MOMENTUM PROJECTION by summing over the y and z slices VARIATIONAL METHOD to well separate masses contributions in the same channel (Kronfeld 1990) SM & FT 2006, Bari (Luscher, Wolff 1990)

0+ CHANNEL 2+ CHANNEL b=0.5508 h=0 m0+=0.1556(36) m2+=0.381(17) meff r SM & FT 2006, Bari

masses’ computation 1st order transition n=1/3 0+ channel 2+ channel 0.60 m0+(b1) m0+(b2) n=1/3 m0+ (b1)=0.1556 0.5508 b1=0.5508 0.550565 bt=0.550565 b 0+ channel 2+ channel in the scaling region b2 SM & FT 2006, Bari 0.550565 – 0.56 at least

mass ratio m2+ m0+ prediction of 4d SU(3) pure gauge theory at finite temperature screening mass ratio at finite temperature? b SM & FT 2006, Bari

2nd order Ising endpoint Karsch, Stickan (2000) h b bc hc z t Pc ISING pt h b bc hc temperature-like (bc,hc)= (0.54938(2),0.000775(10)) (tc,xc)= (0.37182(2),0.25733(2)) ordering field-like s@-r@-0.69 SM & FT 2006, Bari

2nd order endpoint 0.37233 0.37248 SM & FT 2006, Bari

local variable Correlation function order parameter SM & FT 2006, Bari

We separated contributions from two picks and calculated masses mass spectrum right-pick 0+ CHANNEL 2+ CHANNEL t=0.37248 x=0 We separated contributions from two picks and calculated masses m0+ m2+ 0.0749(63) 0.188(12) 3d ISING VALUE r SM & FT 2006, Bari

CONCLUSIONS We used 3d 3q Potts model as a theoretical laboratory to test some aspects of universality THANK YOU evidence found of universal spectrum 1) Ising point 2) weak 1st order tr. pt. prediction of SU(3) screening spectrum? left-pick? SM & FT 2006, Bari

SM & FT 2006, Bari

weak 1st order transition discontinous order parameter the jump is small Tt SM & FT 2006, Bari

Phase diagram h=0 weak 1st order transition point Mass spectrum is universal? 1st order critical lines 2nd order critical endpoint b Z(3) broken phase 2nd order critical ISING endpoint h=0 Universality also holds for weak 1st order transition? bc Z(3) symmetric phase h hc SM & FT 2006, Bari

Universality order parameter susceptibility correlation lenght Critical exponents Tc order parameter Tc susceptibility correlation lenght Tc SM & FT 2006, Bari