The spectrum of 3d 3-states Potts model and universality Mario Gravina Univ. della Calabria &...

The spectrum of 3d The spectrum of 3d 3-states3-states

Potts model and Potts model and universalityuniversality

Mario GravinaMario Gravina

Univ. della Calabria & Univ. della Calabria & INFNINFN

SM & FT 2006, BariSM & FT 2006, Bari

collaborators: R. Falcone, R.Fiore, A. Papacollaborators: R. Falcone, R.Fiore, A. Papa

OUTLINEOUTLINE

SM & FT 2006, BariSM & FT 2006, Bari

introductionintroduction

3d 3q Potts 3d 3q Potts modelmodelnumerical resultsnumerical results

conclusionsconclusions

1) Svetitsky-Yaffe conjecture1) Svetitsky-Yaffe conjecture

2) Universal spectrum 2) Universal spectrum conjectureconjecture

UniversalityUniversality

SM & FT 2006, BariSM & FT 2006, Bari

Theories with different microscopic Theories with different microscopic interactions but possessing the interactions but possessing the

same underlying global symmetry same underlying global symmetry have common long-distance have common long-distance

behaviourbehaviour

1) SVETITSKY-YAFFE 1) SVETITSKY-YAFFE CONJECTURECONJECTURE

SU(N) SU(N) d+1d+1confinement-confinement-

deconfinementdeconfinement

Z(N) dZ(N) dorder-disorderorder-disorder

if transition is 2nd orderif transition is 2nd order

finite temperaturefinite temperature

what about 1st order phase what about 1st order phase transition?transition?

2) universal mass 2) universal mass spectrumspectrum

rm1ji

1eapp)r(C

1m1

SM & FT 2006, BariSM & FT 2006, Bari

r ,rji ...eaeaeapp)r(C rm3

rm2

rm1ji

321

mm11, m, m22,, mm3 3

……

local order local order parameterparameter

ipcorrelation function correlation function ofof

universality conjectureuniversality conjecture

56.2m

m

0

2

SM & FT 2006, BariSM & FT 2006, Bari

mm

11

mm

22

mm

33

mm

44

mm

11

mm

22

mm

33

mm

44

mm

11

mm

22

mm

33

mm

44

theory 1theory 1 theory 2theory 2 theory 3theory 3

mm22

mm

11

mm22

mm

11

mm22

mm

11

mm44

mm

11mm33

mm

11

mm33

mm

11

mm44

mm

11 mm33

mm

11

mm44

mm

11

==

==

==

==

== ==

Ising 3dIsing 3d dd SU(2) 4dSU(2) 4dCaselle at al. 1999Caselle at al. 1999 Fiore, Papa, Provero Fiore, Papa, Provero

20032003

83.1m

m

0

0

Agostini at al. Agostini at al. 19971997

SM & FT 2006, BariSM & FT 2006, Bari

CLUSTER ALGORITHMCLUSTER ALGORITHMto reduce autocorrelation to reduce autocorrelation timetime

We want to test these two aspects of We want to test these two aspects of universalityuniversality

3d 3q POTTS MODEL3d 3q POTTS MODEL

1) 1st order 1) 1st order transitiontransition2) 3d Ising point2) 3d Ising point

?mm

0

2

MONTE CARLO simulationsMONTE CARLO simulations

L=4L=488L=7L=700 h

c

hc

Potts modelPotts model

SM & FT 2006, BariSM & FT 2006, Bari

iji

hijihH 2,1,0i

Z(3) breakingZ(3) breaking

order-disorderorder-disorder

PHASE TRANSITIONPHASE TRANSITION

Phase diagramPhase diagram

SM & FT 2006, BariSM & FT 2006, Bari

h

c

hc

1st order critical lines

2nd order critical endpoint

h=0h=0

weak 1st order transition weak 1st order transition pointpoint

2nd order critical ISING 2nd order critical ISING endpointendpoint

h=0h=0Z(3) symmetric phaseZ(3) symmetric phase

Z(3) broken phaseZ(3) broken phaseDoes universality hold Does universality hold also for weak 1st order also for weak 1st order

transition?transition?

Is the mass spectrum Is the mass spectrum universal?universal?

comparison with comparison with SU(3)SU(3)

(work in progress)(work in progress)Falcone, Fiore, Gravina, Falcone, Fiore, Gravina, PapaPapa

h=0 – 1st order transition h=0 – 1st order transition

.const.c.css32

Hji

ijji

ji

SM & FT 2006, BariSM & FT 2006, Bari

2,1,0i i3

2i

i es

order parameter is the order parameter is the magnetizationmagnetization

i

i3 sL1

M

global global spinspin

h=0 – 1st order transition h=0 – 1st order transition at finite volume at finite volume

t

SM & FT 2006, BariSM & FT 2006, Bari

tunneling effectstunneling effects

between symmetric between symmetric and broken phaseand broken phase

between degenerated between degenerated broken minimabroken minima

0.5505650.550565

0.55080.5508

complex M complex M planeplane

SM & FT 2006, BariSM & FT 2006, Bari

h=0 – 1st order transition h=0 – 1st order transition at finite volume at finite volume

To remove the tunneling To remove the tunneling between broken minima we between broken minima we apply a rotation apply a rotation

3

2i

i es

3

2i

i es

only the real only the real phase is presentphase is present

Masses’ computationMasses’ computation

r ,rji

)1r(C)r(C

ln)r(meff

SM & FT 2006, BariSM & FT 2006, Bari

...eaeaeass)r(C rm3

rm2

rm1jiij

321

VARIATIONAL METHODVARIATIONAL METHODto well separate masses contributions to well separate masses contributions

in the same channelin the same channel

by summing over the y and z slicesby summing over the y and z slicesZERO MOMENTUM PROJECTIONZERO MOMENTUM PROJECTION

MASS CHANNELSMASS CHANNELS

by building suitable combinations of by building suitable combinations of the local variablethe local variable

1eff m)r(m

(Kronfeld (Kronfeld 1990)1990)

)ss(ssziyii

)0(i

)ss(ssziyii

)2(i

(Luscher, Wolff (Luscher, Wolff 1990)1990)

SM & FT 2006, BariSM & FT 2006, Bari

0+ CHANNEL0+ CHANNEL

=0.5508 =0.5508 h=0h=0

2+ CHANNEL2+ CHANNEL

mm0+0+=0.1556(3=0.1556(36)6)mm2+2+=0.381(17)=0.381(17)

rr

mmeffeff

masses’ computationmasses’ computation

t

SM & FT 2006, BariSM & FT 2006, Bari

0.55080.5508

0.600.60

0+ channel0+ channel2+ channel2+ channel

1st order transition1st order transition

=1/3=1/3

)(m)(m

20

10

t2

t1

in the scaling in the scaling regionregion

t2

t1

mm000.1550.1556611=0.5508=0.5508

tt=0.550565=0.550565

mm00mm0+0+

((

0.550565 – 0.56 at0.550565 – 0.56 at leastleast

0.5505650.550565

mm0+0+mm2+2+

mass ratiomass ratio

SM & FT 2006, BariSM & FT 2006, Bari

)10(43.2mm

0

2

prediction of 4d SU(3) prediction of 4d SU(3) pure gauge theory at pure gauge theory at

finite temperature finite temperature screening mass ratio screening mass ratio at finite temperature?at finite temperature?

mm2+2+

mm0+0+

2nd order Ising endpoint2nd order Ising endpoint

hMEH

M~

E~

H iji

hijihH

SM & FT 2006, BariSM & FT 2006, Bari h

c

hc

rMEE~

sEMM

~

temperature-temperature-likelikeordering field-likeordering field-like

ISING ptISING pt

((cc,h,hcc)=)=

(0.54938(2),0.000775(10))(0.54938(2),0.000775(10))((cc,,cc)=)=

(0.37182(2),0.25733(2))(0.37182(2),0.25733(2))ssrr-0.69-0.69

Karsch, Stickan (2000)Karsch, Stickan (2000)

h

c

hc

Pc

2nd order endpoint2nd order endpoint

37182.0c

SM & FT 2006, BariSM & FT 2006, Bari

0.372330.37233 0.372480.37248

M~

M~

M~

local variablelocal variable

3

3i iihi 2

sm~

i ij

jihissEMM

~

i

i3 m~L1

M~

SM & FT 2006, BariSM & FT 2006, Bari

jiij m~m~)r(C

order order parameterparameter

Correlation Correlation functionfunction

mass spectrummass spectrum

)37(51.2mm

0

2

SM & FT 2006, BariSM & FT 2006, Bari

We separated contributions from We separated contributions from two picks and calculated massestwo picks and calculated masses

0+ CHANNEL0+ CHANNEL2+ CHANNEL2+ CHANNEL

right-pickright-pick

=0.37248 =0.37248

0.0749(63)0.0749(63)0.188(12)0.188(12)

56,2m

m

0

2

3d ISING VALUE3d ISING VALUE

mm2+2+mm0+0+

rr

CONCLUSIONSCONCLUSIONS

SM & FT 2006, BariSM & FT 2006, Bari

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

aspects of universalityaspects of universality

1) Ising point1) Ising point evidence found of evidence found of universal spectrumuniversal spectrum

2) weak 1st 2) weak 1st order tr. pt.order tr. pt.

prediction of SU(3) prediction of SU(3) screening screening spectrum?spectrum?

THANK YOUTHANK YOU

)10(43.2mm

0

2

left-pick?left-pick?

SM & FT 2006, BariSM & FT 2006, Bari

1st order transition1st order transition

SM & FT 2006, BariSM & FT 2006, Bari

Tt

discontinous order discontinous order parameterparameter

weakweak

the jump is the jump is smallsmall

Phase diagramPhase diagram

SM & FT 2006, BariSM & FT 2006, Bari

h

c

hc

1st order critical lines

2nd order critical endpoint

h=0h=0

weak 1st order transition weak 1st order transition pointpoint

2nd order critical ISING 2nd order critical ISING endpointendpoint

h=0h=0Z(3) symmetric phaseZ(3) symmetric phase

Z(3) broken phaseZ(3) broken phaseUniversality also holds Universality also holds

for weak 1st order for weak 1st order transition?transition?

Mass spectrum is Mass spectrum is universal?universal?

UniversalityUniversality

SM & FT 2006, BariSM & FT 2006, Bari

Critical Critical exponentsexponents

order parameterorder parameterTc

susceptibilitysusceptibilityTc

correlation lenghtcorrelation lenght

Tc