The spectrum of 3 d 3 states Potts

OUTLINE introduction 1) Svetitsky-Yaffe conjecture 2) Universal spectrum conjecture 3 d 3 q Potts

1) SVETITSKY-YAFFE Universality CONJECTURE Theories with different microscopic SU(N) d+1 Z(N) d interactions but

2) universal mass spectrum correlation function of local order parameter m 1, m 2,

universality conjecture theory Ising 3 d 1 theory l. F 4 3 d 2

We want to test these two aspects of universality 3 d 3 q POTTS

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

Phase diagram b h=0 Is the mass spectrum universal ? lines point 1 st

h=0 – 1 st order transition order parameter is the magnetization global spin SM

h=0 – 1 st order transition at finite volume tunneling effects 0. 5508 0.

h=0 – 1 st order transition at finite volume To remove the tunneling between

Masses’ computation MASS CHANNELS by building suitable combinations of the local variable ZERO MOMENTUM

2+ 0+ CHANNEL b=0. 5508 h=0 meff . 1556 (36) m 2+ (17) 0+=0.

masses’ computation m 2+ 0+ 1 st order transition 0. 60 m 0+(b 1)

mass ratio m 2+ m 0+ prediction of 4 d SU(3) pure gauge theory

2 nd order Ising endpoint Karsch, Stickan (2000) b bc b z Pc ISING

2 nd order endpoint 0. 37233 SM & FT 2006, Bari 0. 37248

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

right-pick mass spectrum t=0. 37248 x=0 2+ CHANNEL 0+ We separated contributions from m

CONCLUSIONS We used 3 d 3 q Potts model as a theoretical laboratory to

weak 1 st order transition discontinous order parameter the jump is small Tt SM

Phase diagram b h=0 Mass spectrum is universal ? lines point weak 1 st

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

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

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

2) universal mass spectrum correlation function of local order parameter m 1, m 2, m 3 … SM & FT 2006, Bari

universality conjecture theory Ising 3 d 1 theory l. F 4 3 d 2 Agostini at al. 1997 Caselle at al. 1999 m 4 m 1 m 3 m 1 m 2 m 1 SM & FT 2006, Bari m 4 = m 4 m 1 m 2 = m 1 m 3 m 1 = m 2 m 1 m 3 SU(2) theory 4 d 3 Fiore, Papa, Provero 2003 m=4 m 3 m 2 m=1 = m 4 m 1 m 4 m 3 m 1 m 3 m 2 m 1

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

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

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

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

h=0 – 1 st order transition at finite volume tunneling effects 0. 5508 0. 550565 between symmetric and broken phase between degenerated broken minima complex M plane SM & FT 2006, Bari

h=0 – 1 st 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)

2+ 0+ CHANNEL b=0. 5508 h=0 meff . 1556 (36) m 2+ (17) 0+=0. 381 r SM & FT 2006, Bari

masses’ computation m 2+ 0+ 1 st order transition 0. 60 m 0+(b 1) m 0+(b 2) n=1/3 0. 5508 0. 550565 SM & FT 2006, Bari m 0+ (b 1)=0. 1556 b 1=0. 5508 bt=0 b. 550565 0+scaling 2+ channelregion in the 0. 550565 – 0. 56 at least b 2

mass ratio m 2+ m 0+ prediction of 4 d SU(3) pure gauge theory at finite temperature screening mass ratio at finite temperature? b SM & FT 2006, Bari

2 nd order Ising endpoint Karsch, Stickan (2000) b bc b z Pc ISING pt t hc temperature-like ordering field-like SM & FT 2006, Bari h b c (b , h )= (0. 54938(2), 0. 000775(10)) (bc , hc )= (0. 54938(2), 0. 000775(10)) (tc, xc)= (0. 37182(2), 0. 25733(2)) h s@-r@-0. 69 c h

2 nd order endpoint 0. 37233 SM & FT 2006, Bari 0. 37248

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

right-pick mass spectrum t=0. 37248 x=0 2+ CHANNEL 0+ We separated contributions from m 0+ 2+ two picks and calculated masses 0. 0749(63) 0. 188(12) 3 d ISING VALUE SM & FT 2006, Bari r

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

SM & FT 2006, Bari

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

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

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