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

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

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

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

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universality conjecture SM & FT 2006, Bari m1m1m1m1 m2m2m2m2 m3m3m3m3 m4m4m4m4 m1m1m1m1 m2m2m2m2 m3m3m3m3 m4m4m4m4 m1m1m1m1 m2m2m2m2 m3m3m3m3 m4m4m4m4 theory 1 theory 2 theory 3 m2m2m2m2 m1m1m1m1 m2m2m2m2 m1m1m1m1 m2m2m2m2 m1m1m1m1 m4m4m4m4 m1m1m1m1 m3m3m3m3 m1m1m1m1 m3m3m3m3 m1m1m1m1 m4m4m4m4 m1m1m1m1 m3m3m3m3 m1m1m1m1 m4m4m4m4 m1m1m1m1 = = = = == Ising 3d d d SU(2) 4d Caselle at al Fiore, Papa, Provero 2003 Fiore, Papa, Provero 2003 Agostini at al. 1997

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

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

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

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

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

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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

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

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SM & FT 2006, Bari 0+ CHANNEL = h=0 = h=0 2+ CHANNEL m 0+ =0.1556(36) m 2+ =0.381(17) r m eff

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masses computation SM & FT 2006, Bari channel 2+ channel 1st order transition =1/3 =1/3 in the scaling region m = = t = t = m 0 m 0 m 0+ ( m 0+ ( – 0.56 at least m 0+ m 2+

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

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2nd order Ising endpoint SM & FT 2006, Bari h c hchc temperature-like ordering field-like ISING pt ( c,h c )= ( (2), (10)) ( c, c )= ( (2), (2)) s r Karsch, Stickan (2000) h c hchc PcPc

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2nd order endpoint SM & FT 2006, Bari

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

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

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

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SM & FT 2006, Bari

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

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

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

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