Thermodynamics of non-abelian gauge theories in the confined...

Thermodynamics of non-abelian gauge theories in the confined phase -SU(2) and SU(3)

Alessandro Nada

Universita degli Studi di Torino

Turin Lattice Meeting, 22/23-12-2014

Alessandro Nada (Universita di Torino) Thermodynamics of gauge theories close to Tc Turin Lattice Meeting, 22/23-12-2014 1 / 23

Introduction

One of the main features of SU(N) non-abelian gauge theories is the existence of adeconfinement phase transition, i.e. a temperature (or a density) above which the matter is“deconfined”, like the quark-gluon plasma (QGP) in Quantum Chromodynamics.

In this work we explore the thermodynamics of pure gauge theories in the confining phase whenapproaching the deconfinement transition, by means of Monte Carlo numerical simulations and astring-inspired model for the glueball spectrum.

The pure gauge sector retains many of the features of the full theory, without the problems thatthe regularization of fermions on the lattice induces.


Introduction

This kind of analysis was performed in 3+1 spacetime dimensions in 20091 for SU(3) and in 2+1dimensions in 20112 for SU(N) theories (with N = 2, 3, 4, 5, 6). A huge step forward in terms ofprecision of lattice data for SU(3) was made in 20122 on a large temperature range: thenumerical results in the confined phase will be the subject of our analysis.

Then the SU(2) pure gauge theory has been considered, in order to provide more insight on thethermodynamics of SU(N) theories and on the characteristics of their glueball spectra. Suchtheory is unique because only the C = +1 sector of the glueball spectrum exists, leading todifferent expectations for its behaviour in the confined phase.

1H. Meyer, High-Precision Thermodynamics and Hagedorn Density of States, 20092M. Caselle et al., Thermodynamics of SU(N) Yang-Mills theories in 2+1 dimensions I - The confining phase, 20113Sz. Borsanyi et al., Precision SU(3) lattice thermodynamics for a large temperature range, 2012


1 Lattice Thermodynamics

2 Glueball gas and closed string modelIdeal glueball gasClosed string modelSU(3) data analysis

3 SU(2) pure gauge theoryLattice setup and scale settingResults and theoretical descriptionSU(2) vs. SU(3)


Lattice Thermodynamics

Section 1




Lattice regularization

Only a truly non-perturbative approach such as lattice regularization can describe thedeconfinement transition and the confined phase of non-abelian gauge theories.

For SU(N) pure gauge theories on the lattice the dynamics are described by the standard Wilsonaction

SW = β∑

p=sp,tp

(1−1

NReTrUp)

where UP is the product of four Uµ SU(N) variables on the space-like or time-like plaquette P

and β = 2Ng2 .

The partition function is

Z =

∫ ∏x,µ

dUµ(x)e−SW

and the expectation value of an observable A

〈A〉 =1

Z

∫ ∏n,µ

dUµ(n) A(Uµ(n)) e−SW



Thermodynamic quantities

On a Nt × N3s lattice the volume is V = (aNs )3 (where a is the lattice spacing), while the

temperature is determined by the inverse of the temporal extent (with periodic boundaryconditions): T = (aNt )−1.

The thermodynamic quantities taken into account will be:

I the pressure p, that in the thermodynamic limit (i.e. for large and homogenous systems) canbe written as

p 'T

Vlog Z(T ,V )

I the trace of the energy-momentum tensor ∆, that in units of T 4 is

∆

T 4=ε− 3p

T 4= T

∂

∂T

( p

T 4

)Energy density ε = ∆ + 3p and entropy density s = ε+p

T= ∆+4p

Tcan be easily calculated.



Thermodynamics on the lattice

The pressure can be estimated by the means of the so-called “integral method”1:

p(T ) 'T

Vlog Z(T ,V ) =

1

a4

1

Nt N3s

∫ β(T )

0dβ′

∂ log Z

∂β′.

It can be written (relative to its T = 0 vacuum contribution) as

p(T )

T 4= −Nt

4∫ β

0dβ′ [3(Pσ + Pτ )− 6P0]

where Pσ and Pτ are the expectation values of spacelike and timelike plaquettes respectively andP0 is the expectation value at zero T .The trace of energy-momentum tensor is simply

∆(T )

T 4= T

∂

∂T

( p

T 4

)= −Nt

4T∂β

∂T[3(Pσ + Pτ )− 6P0] .

ε and s can be obtained indirectly as linear combinations.

1J. Engels et al., Nonperturbative thermodynamics of SU(N) gauge theories, 1990


Glueball gas and closed string model

Section 2

Glueball gas and closed string model


Glueball gas and closed string model Ideal glueball gas

Ideal glueball gas

The behaviour of the system is supposed to be dominated by a gas of non-interacting glueballs.The prediction of an ideal relativistic Bose gas can be used to describe the thermodynamics ofsuch gas. Its partition function for 3 spatial dimensions is

log Z = (2J + 1)2V

T

(m2

2π

)2 ∞∑k=1

(T

km

)2

K2

(k

m

T

)where m is the mass of the glueball, J is its spin and K2 is the modified Bessel function of thesecond kind of index 2.Observables such as ∆ and p thus can be easily derived:

p =T

Vlog Z = 2(2J + 1)

(m2

2π

)2 ∞∑k=1

(T

km

)2

K2

(k

m

T

)

∆ = ε− 3p = 2(2J + 1)

(m2

2π

)2 ∞∑k=1

(T

km

)K1

(k

m

T

)


Glueball gas and closed string model Ideal glueball gas

SU(3): trace of energy-momentum tensor

0

0.05

0.1

0.15

0.2

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

∆/T

4

T/Tc

lightest SU(3) glueballall SU(3) glueballs below the two-particle threshold

SU(3) - Borsanyi et al.

The contribution of all glueball states with mass m < 2m0++ is considered.


Glueball gas and closed string model Closed string model

Glueballs as rings of glue

A closed string model for the full glueball spectrum that follows the original work of Isgur andPaton12 can be introduced to account for the values of thermodynamic quantities near thetransition. In the closed-string approach glueballs are described in the limit of large masses as“rings of glue”, that is closed tubes of flux modelled by closed bosonic string states.

The mass spectrum of a closed strings gas in D spacetime dimensions is given by

m2 = 4πσ

(nL + nR −

D − 2

12

)where nL = nR = n are the total contribution of left- and right-moving phonons on the string.

Every glueball state corresponds to a given phonon configuration, but associated to each fixed nthere are multiple different states whose number is given by π(n), i.e. the partitions of n.

The density of states ρ(n) is expressed through the square of π(n):

ρ(n) = π(nL)π(nR ) = π(n)2 ' 12 (D − 2)D−1

2

(1

24n

) D+12

exp

(2π

√2(D − 2)n

3

)

in D spacetime dimensions.

1N. Isgur and J. Paton, A Flux Tube Model for Hadrons in QCD, 19852R. Johnson and M. Teper, String models of glueballs and the spectrum of SU(N) gauge theories in (2+1)-dimensions, 2002


Glueball gas and closed string model Closed string model

Spectral density

The Hagedorn temperature1 is defined as

TH =

√3σ

π(D − 2)

The spectral density as a function of the mass (i.e. ρ(m)dm = ρ(n)dn) can be expressed as

ρ(m) =(D − 2)D−1

m

(πTH

3m

)D−1

em/TH

where the characteristic Hagedorn-like exponential spectrum appears and can be used to describethe glueball spectrum above an arbitrary mass threshold.

The trace of the energy-stress tensor can be integrated on masses bigger than 2m0++ with thedegeneracy ρ(m) and summed to the contribution of the mass states computed on the lattice.

∆ =∑

m<2m0++

(2J + 1)∆(m,T ) +

∫ ∞2m0++

dm ρ(m) ∆(m,T )

1R. Hagedorn, Nuovo Cim. Suppl. 3, 147 (1965)


Glueball gas and closed string model SU(3) data analysis


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

∆/T

4

T/Tc


closed string modelSU(3) - Borsanyi et al.


Glueball gas and closed string model SU(3) data analysis

SU(3): pressure

0

0.005

0.01

0.015

0.02

0.025

0.03

0.7 0.75 0.8 0.85 0.9 0.95 1

p/T

4

T/Tc


closed string modelSU(3) - Borsanyi et al.


SU(2) pure gauge theory

Section 3

SU(2) pure gauge theory


SU(2) pure gauge theory Lattice setup and scale setting

Lattice setup and scale setting

All finite temperature simulations are performed on lattices with Nt = 5, 6, 8.The variation of temperature is obtained varying the β parameter through the renormalizationgroup equation

T

Tc= e

β−βc4Nb0

(β

βc

)− b12b0

2

Nt βc

4 2.2986(6)5 2.37136(54)6 2.4271(17)8 2.5090(6)

βc for different Nt1.

A few points are also determinedmodifying Nt while keeping β = βc . Inthis case

T ′ =Nt

N′tTc

where Nt is the temporal extent of thecorresponding βc .

A much more reliable scale setting is in the works, using Polyakov loop correlators to estimate thestring tension a

√σ.

1B. Lucini, M. Teper and U. Wenger, The high temperature phase transition in SU(N) gauge theories, 2003


SU(2) pure gauge theory Results and theoretical description

SU(2) results from simulations

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.7 0.75 0.8 0.85 0.9 0.95 1

∆/T

4

T/Tc

Nt = 5Nt = 6Nt = 8

variable Nt - βc = 2.3986 (Nt = 4)variable Nt - βc = 2.37136 (Nt = 5)variable Nt - βc = 2.4271 (Nt = 6)


SU(2) pure gauge theory Results and theoretical description


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.75 0.8 0.85 0.9 0.95 1

∆/T

4

T/Tc


closed string modelSU(2) - Nt = 6


SU(2) pure gauge theory SU(2) vs. SU(3)

SU(2) vs. SU(3)

Glueball spectrum

For N ≥ 3 the closed flux tube has two possible orientations that account for the C = +1/− 1sectors. Thus a further twofold degeneracy must be introduced.

Transition order and Hagedorn temperature

I for SU(2) in 3+1 dimensions the deconfining transition is second-order. This means thatTH = Tc .

I for SU(3) the deconfining transition is a first-order one, so TH 6= Tc . The prediction of theeffective Nambu-Goto action can be used: TH/

√σ = 0.691. This leads to

TH

Tc= 1.098



SU(2) vs. SU(3): results for trace of energy-momentum tensor

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.7 0.75 0.8 0.85 0.9 0.95 1

∆/T

4

T/Tc

all SU(2) glueballs below the two-particle thresholdclosed string model for SU(2)


SU(2) - Nt = 6SU(3) - continuum extrapolation



SU(2) vs. SU(3): results for trace of energy-momentum tensor

Logarithmic scale; SU(3) data at T = Tc much higher than SU(2) one

0.1

1

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

∆/T

4

T/Tc



SU(2) - Nt = 6SU(3) - continuum extrapolation



Conclusions

Predictions of the closed string model agree very well with SU(2) and SU(3) lattice data, stronglysupporting the hypotesis of a non-interacting glueball gas that controls the thermodynamics inthe confined phase.Moreover the lack of a gap between SU(2) and SU(3) results is well explained by the role of theHagedorn temperature in the closed string model.

Further developments

I Reliable scale setting through Polyakov loop correlators and interquark potential

I Continuum extrapolation

I Pressure (→ energy and entropy)

I SU(N > 3), G2


SU(2) lattice setup

N3s × Nt nβ β-range nconf

N4s

603 × 5 13 [2.25,2.37136] 1.5× 105

324

603 × 6 9 [2.3059,2.4271] 1.5× 105

324

723 × 6 21 [2.3059,2.4271] 1.5× 105

404

723 × 8 7 [2.4478,2.5090] 1× 105

404

Different lattice setups of SU(2) simulations in the 0.7 < T/Tc < 1 temperature range. In the first column is

shown the size of the lattice for finite-T (N3s × Nt ) and T = 0 (N4

s ) simulations. nβ denotes the number ofβ-values (i.e. the number of points) inside a certain β-range. nconf denotes the approximated size of thesample, that is T = 0 and finite-T statistics.


SU(3) glueball spectrum

JPC m/√σ m (MeV)

0++ 3.347(68) 1475(30)0++∗ 6.26(16) 2755(30)2++ 4.891(65) 2150(30)2++∗ 6.54(22) 2880(100)0−+ 5.11(14) 2250(60)2−+ 6.32(11) 2780(50)1+− 6.06(15) 2670(65)

SU(3) glueball masses 1

1H. Meyer, Glueball regge trajectories, 2004



taken from S. Borsanyi et al., Precision SU(3) lattice thermodynamics for a large temperaturerange, 2012



0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

∆/T

4

T/Tc

all SU(3) glueballs below the two-particle thresholdclosed string model

closed string model with TH / Tc = 1.024SU(3) - Borsanyi et al.

with a different estimate for the Hagedorn temperature1

1H. Meyer, High-Precision Thermodynamics and Hagedorn Density of States, 2009


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