Deconﬁnement transition from “perturbation theory”

Deconfinement transition from “perturbationtheory”

M. Pelaez, U. Reinosa, J. Serreau, M. Tissier, A. Tresmontant,N. Wschebor

Ilha bela, September 2015

Deconfinement transition ...

Introduction

Yang-Mills theory, gauge-fixed in the Landau gauge with theFaddeev-Popov procedure, is described by a set of massless fields:Gluons (Aa

µ), ghosts (ca and ca) and a Lagrange multiplyer (ha)

L =1

4(F a

µν)2 + ∂µc

a(Dµc)a + ha∂µA

aµ

However, lattice simulations see unambiguously a gluon propagatorthat saturates at low momentum.


However...

Gluon propagator is massive! (Sternbeck et al ’07)!

0.001 0.01 0.1 1 10 100

q2[GeV

2]

0

5

10

15

20

D(q

2 )

L=64, 14 confsL=72, 20 confsL=80, 25 confs

Gluon propagatorL = 64, 72, 80, SA

The origin of the mass could:

result from solving Dyson-Schwinger equations;

be a consequence of a gluon condensate;

be related to Gribov ambiguity;

none of those...Deconfinement transition ...

Phenomenological model

The mass generation is a difficult issue. Once we are convinced itexists, how much physics can we understand?Introduce a mass for the gluon by hand in the (gauge-fixed)Lagrangian:

L =1

4(F a

µν)2 + ∂µc

a(Dµc)a + ha∂µA

aµ +

1

2m2

(

Aaµ

)2

(Here, we make the assumption that no extra field is needed)This is one particular representative of the Curci-Ferrari lagrangian.


Overview

Motivate the interest of this phenomenological model.

General arguments;Systematic comparison with Lattice correlation functions.

Applications to finite temperature physics

in the quenched approximation;with dynamic quarks, with and without chemical potential.


Nice properties of the model I

UV (p ≫ m) properties are unaffected by the gluon mass.

In particular, the theory is renormalizable to all orders (DeBoer et al). (gluon mass softly breaks the BRST symmetry)

the (running) gluon mass tends to zero in the ultraviolet(m(µ) ∝ gα(µ) with α > 0).

Feynman rules are identical to usual ones, except for themassive gluon propagator:

〈AµAν〉0(p) =(

δµν −pµpν

p2

)

1

p2 +m2

perturbation calculations are easy to perform.

Low momentum physics regularized by the gluon mass.


Infrared behavior

At very low momenta, gluons are frozen. Ghost loop dominates.

ΓAaµA

bν∼ δab(δµν)p

d−2

ΓAaµA

bνA

cρ∼ −f abc(ipµδνρ + · · · )pd−4

in d = 4, leads to log divergences, hard to see...in d = 3, gluon propag cte + |p|, 3-gluon vertex changes sign,consistent with lattice data.Interaction between ghosts is mediated by heavy gluons (seealso Weber). Effective interaction is suppressed by somepositive power of p at low momentum.

Par

amet

er e

xpan

sion

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

µ (GeV)


Ghost and gluon propagators

Need to compute 4 Feynman diagrams

Define 〈AµAν〉(p) =(

δµν − pµpνp2

)

G (p) 〈cc〉(p) = 1p2F (p).

Introduce 4 renormalization parameters and you get (s = p2/m2.):

G−1(p)/m2 = s + 1 +g2N

384π2s{

111s−1 − 2s−2 + (2− s2) log s

+ (4s−1 + 1)3/2(

s2 − 20s + 12)

log

(√4 + s −√

s√4 + s +

√s

)

+ 2(s−1 + 1)3(

s2 − 10s + 1)

log(1 + s)− (s → µ2/m2)}

,

F−1(p) = 1 +g2N

64π2

{

− s log s + (s + 1)3s−2 log(s + 1)− s−1 − (s → µ2/m2)}

,


Comparison with lattice data

For SU(2) (Cucchieri, Mendes ’08)

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

G(p

)

p (GeV)F

(p)

p (GeV)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Renormalization-group flow

From renormalization factors, deduce a set of coupled β functionsfor g and m:

In the UV (µ≫ m) βg ≃ − g3N16π2

113

In the IR (µ≪ m) βg ≃ + g3N16π2

16

For SU(3) (Bogolubsky ’09, Dudal ’10)

p G

(p)

2

p (GeV)

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8

F(p

)

p (GeV)

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8


Other correlation functions

By the same technique, we have computed (all tensorialcomponents) and compared with lattice data, when available:

3 gluon vertex and ghost-gluon vertex;quark propagator;quark-gluon vertex;

Agreement (Maximal error of 15-20%) in the quenchedapproximation.

In unquenched calculations (Skullerud et al), still ok, but lessprecise, because the quark-gluon vertex is larger (typically thedouble of the ghost-gluon vertex).

1-loop compares badly to lattice for the quark renormalizationfactor and for one of the structure tensors of the quark-gluonvertex (λ2).


Quark mass

The quark mass is enhanced in the infrared. But no chiralsymmetry breaking.

0 1 2 3 4 5

0.05

0.10

0.15

0.20

0.25

0.30

p HGeVL

MHpLHG

eVL


Quark-gluon vertex

0 1 2 3 4 5 6

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

p HGeVL

Λ1H

pL


The phase diagram of QCD I

In heavy ion collisions, and core of neutron stars, matterreaches extreme conditions, with temperatures of the order of∼ 1012 K, densities of ∼ 1018 kg/m3.

Typical values for strong interactions. In strong interactionsunits: T ∼ 1 GeV, ρ ≃ 1 GeV/fm3.

In the quenched approximation (no dynamic quarks), latticesimulations clearly show a phase transition at a temperature∼ 250 MeV, which is in the nonperturbative regime.

Extension to finite chemical potential is intricate because theaction is not real anymore.


Phase diagram of Yang-Mills II

To study the theory at finite temperature, compactify thetime direction, with periodicity β = 1/T .

There exist gauge transformations A → AU such that AU isperiodic, although U itself is not: U(β, x) = z U(0, x) with z

an element of the centre (for SU(N), z = e imπ/NI with

m ∈ {0, · · · ,N − 1}).Some quantities are invariant under the centre (such as F 2

µν).Other vary, such as the Polyakov loop

ℓ ∝ Tr Pe ig∫ β

0 A0(τ)dτ

〈ℓ〉 = exp(−βFq) where Fq is the free energy of a quark.

If the centre symmetry is realized, 〈ℓ〉 = 0, Fq = ∞, confinedphase.If the centre is spontaneously broken, 〈ℓ〉 6= 0, Fq finite,deconfined phase.


Phase diagram of Yang-Mills III

To encode the centre symmetry, convenient to

decompose the gluon field in a background A and afluctuating a field: A = A+ a (we choose A constant,temporal, in the Cartan subalgebra);

introduce the Landau-de Witt gauge ∂µaµ + gf abc Abµa

cµ = 0;

choose A such that 〈a〉 = 0.

This last condition is fulfilled by minimizing some potential V (A).Center transformations act on A!At leading order in g , the SU(2) potential reads (r3 = βgA3)

V = T 4

(

3

2Fmβ(r3) +

1

2F0(r3)− 1F0(r3)

)

+O(g2)

Fm(r) =

∫

q

log(1 + e−2√

m2+q2 − 2e−√

m2+q2 cos(r))

The Polyakov loop:

〈ℓ〉 = cos(r3/2) +O(g2)


Phase diagram of Yang-Mills IV

At high temperatures (red), V → +1F0(r3). (Weiss potential)At low temperatures (blue), V → −1

2F0(βgA).

−0.8

−0.4

0

0.4

0 π/2

VSU(2)(r)

πr

0.08

0.12

0.04

0

−0.040

VSU(2)(r)

π/2 π

r

r = π is invariant under center transformations and corresponds to〈ℓ〉 = 0.The leading order approximation captures the good physics!


Pressure I

0.1 0.2 0.3

0.01

0

0

0.001

0 0.2 0.4T 1lcT (GeV)

p (GeV−4)

T 2lc


Pressure II

0

0.1

0.2

0.3

0.1 0.2 0.3 0.4

p/T 4

T (GeV) T 1lc T 2l

c

π2

120


Phase diagram for SU(3) I

Two fields in the Cartan subalgebra: r3 and r8.

0

r3

2π−2π

0

−4π 4π

r8

4π√3

− 4π√3

Calculation of the potential and Polyakov loop generalizes easily.


Phase diagram for SU(3) II

Leads to first order transition

2.8 3 3.2 3.4 3.6 3.8

V

r3


Finite chemical potential

Add the quarks lagrangien density, with chemical potential:

∑

f

ψf ( /D +Mf + µγ0)ψf

Explicitely break centre symmetry.

Contributes to the potential as:

Vf (µ) = − T

π2

∫

∞

0dq q2

{[

log[1 + e−β(√

q2+M2f+µ)

]

+ (µ→ −µ)}

+O(g2)


Vanishing chemical potential, SU(3)

Consider 2 +1 quarks (only for large mass because we don’tcontrol chiral limit). Columbia plot:

1 0.9 0.8 0.8

0.9

1

Crossover

1st order

1− e−Msm

1− e−Mum


Imaginary chemical potential, SU(3)

The action and potential V are real, so lattice simulations canbe performed.

It is symmetric under a simultaneous transformation of thebackground field and of the chemical potential (Roberge,Weiss). µ/T = iπ/3 plays a particular role:

We retrieve the properties obtained in Lattice approaches (deForcrand, Philipsen), in particular at the tricritical point.


small quark mass

0.38

0.36

0.34

0.32

µi/Tπ/3 2π/30

T/m


intermediate quark mass

0.36

0.340

µi/Tπ/3 2π/3

T/m


large quark mass

0.36

0.3480 π/3 2π/3

µi/T

T/m


Real chemical potential

For real background fields, the potential is complex!

To be consistent, need to choose r3 real and r8 imaginary (Seealso Nishimura 2014). Then the potential is then real.

0.8

0.9

1

0.8 0.9 1

1− e−Ms

m

1− e−Mu

m

2

1

0 0.36 0.38

T/m

Fq/m

Fq/m


Conclusions

Curci-Ferrari seems to capture many “nonperturbative”properties of QCD within “perturbation theory”.

This would mean that the major nonperturbative ingredient isthe gluon mass.

We have a nice model to study low-energy properties of QCD.Tested in several situations.

Propagators in the Landau-de Witt gauge.Chiral symmetry breaking?Polyakov loop in other representations?Analytic structure of the correlation functions?Wilson loop?Two-loop calculations for the propagators?Transport coefficients?...

Can we generate the mass from first principles (relation withproblems with disorder in stat. phys.)?

Can we build a physical subspace?


Documents

Deconﬁnement transition from “perturbation theory”