perturbative aspects of the qcd phase diagram€¦ · perturbative aspects of theqcd phase diagram...

Perturbative aspects of the QCD phase diagram

Urko Reinosa∗

(based on collaborations with J. Serreau, M. Tissier and N. Wschebor)

∗Centre de Physique Théorique, Ecole Polytechnique, CNRS.

GDR-QCD, November 10, 2016, IPNO.

Theoretical approaches to the QCD phase diagram

Non-exhaustive list:

- finite-temperature lattice QCD- Schwinger-Dyson equations- functional renormalization group

⎫⎪⎪⎪⎬⎪⎪⎪⎭non-perturbative

Perturbation theory?

Outline

I. A modified perturbation theory in the infrared?

(Vacuum, Landau gauge).

II. Application to the QCD phase structure.

(Finite T, Landau-DeWitt gauge).

A modified perturbation theoryin the infrared

Perturbation theory: common wisdom

The QCD running coupling decreases at high energies: asymptotic freedom.

As a counterpart, the (perturbative) running coupling increases when theenergy is decreased, and even diverges at a Landau pole known as ΛQCD:

LQCD

E

g2HEL

Yes but ... perturbation theory is based on the Faddeev-Popov procedure whichis at best valid at high energies and should be modified at low energies.

Perturbation theory: (not so) common wisdom

Perturbation theory defined only within a given gauge-fixing (ex: ∂µAaµ = 0).

The gauge-fixing is based on the Faddeev-Popov approach:

A

Equivalent configs AUµ

Field configs satisfyinggauge cond. (∂µAU

µ = 0)

1

⇒ L = 14g2

FaµνFa

µν + ∂µca(Dµc)a + iha∂µAaµ




A

Gribov copiesGauge condition

equivalent configs

1

⇒ L = 14g2

FaµνFa

µν + ∂µca(Dµc)a + iha∂µAaµ Valid at best at high energies!




A

Gribov copiesGauge condition

equivalent configs

1

⇒ L = 14g2

FaµνFa

µν + ∂µca(Dµc)a + iha∂µAaµ + LGribov ← not known so far

Could LGribov restore the applicability of perturbation theory in the IR?

How to constrain LGribov?

(Landau gauge) lattice simulations [Cucchieri and Mendes; Bogolubsky et al; Dudal et al; . . . ]are crucial: ⋆ free from the Gribov ambiguity;

⋆ provide valuable information to construct models for LGribov.

In particular, they predict:

⋆ a gluon propagator G(p) that behaves like a massive one at p = 0;

⋆ a ghost propagator F(p)/p2 that remains massless.

G(p

)

p (GeV)

0

1

2

3

4

5

6

7

8

9

0 0.5 1 1.5 2 2.5 3

F(p

)

p (GeV)

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8

Our model

As the dominant contribution to LGribov, we propose a gluon mass term:

L = 14g2

FaµνFa

µν + ∂µca(Dµc)a + iha∂µAaµ +

12

m2AaµAa

µ

Particular case of the Curci-Ferrari (CF) model. Most appealing features:

⋆ Just one additional parameter (simple modification of the Feynman rules).⋆ Perturbatively renormalizable and ∃ RG trajectories without Landau pole:

Landaupole

No Landau pole

60

40

20

0 0 2 4 6 8 10

g

m

⇒ perturbative calculations may be pushed down to the IR!

Vacuum correlators

Amazing agreement between LO perturbation theory in the CF model andLandau gauge lattice vacuum correlators [with m ≃ 500 MeV for SU(3)]:

G(p

)

p (GeV)

0

1

2

3

4

5

6

7

8

9

0 0.5 1 1.5 2 2.5 3

F(p

)

p (GeV)

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8

[Tissier, Wschebor, Phys.Rev. D84 (2011)]

What are the predictions of the model at finite temperature?

QCD phase structure

Back to the QCD phase diagram

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

chiral2nd orderZ(2)

deconfined2nd orderZ(2)

crossover

1st

d

tric

∞

∞

Pure

Order parameter(s): Polyakov loop(s)

` ≡ 13⟨trP eig ∫

1/T0 dτ A0⟩∝ e−βFquark

¯≡ 13⟨tr (P eig ∫

1/T0 dτ A0)

†⟩∝ e−βFantiquark

F = ¥

confined

F < ¥

deconfined

Tc

0

Temperature

Ord

erp

aram

eter

Back to the QCD phase diagram

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

chiral2nd orderZ(2)

deconfined2nd orderZ(2)

crossover

1st

d

tric

∞

∞

Pure

Deconfinement transition≡ SSB of center symmetry.

Problem: center symmetry is notmanifest in the Landau gauge.

Underlying symmetry: center symmetry

AUµ = UAµU† − iU∂µU†

U(τ + 1/T, x) = U(τ, x) ei 2π3 k (k = 0,1,2)

Under such a transformation: `→ ei2π3 k`

Broken

Symmetry

Unbroken

Symmetry

Tc

0

Temperature

Ord

erp

aram

eter

Changing to the Landau-DeWitt gauge

We move from the Landau gauge to the Landau-DeWitt gauge:

0 = ∂µAaµ → 0 = (Dµaµ)a { Dab

µ ≡ ∂µδab + f acbAcµ

aaµ ≡ Aa

µ − Aaµ

with A a given background field configuration.

Faddeev-Popov gauge-fixed action + phenomenological LGribov:

LA =1

4g2FaµνFa

µν + (Dµc)a(Dµc)a + iha(Dµaµ)a+12

m2aaµaa

µ

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶Landau gauge + ∂µ → Dµ and Aµ → aµ

Background and symmetries

In practice, one chooses A complying with the symmetries at finite T:

Aµ(τ, x) = Tδµ0 (r3λ3

2+ r8

λ8

2) ∈ Cartan subalgebra

It is also convenient to work with self-consistent backgrounds: 0 = ⟨aµ⟩A.⋆ shown to be the minima of a certain Gibbs potential Γ[A] ≡ ΓA[a = 0].⋆ this functional is center-symmetric:

Γ[AU] = Γ[A], ∀U(τ + 1/T) = ei 2π3 k U(τ)

As in the textbook situation (ex: Ising-model), SSB occurs when the ground stateof the Gibbs potential moves from a symmetric state to a symmetry breaking one.But what is the (center-)symmetric state in the present case?

Weyl chambers and center symmetry

One-loop result - SU(3)


T−4Γ[A] = 3F(r3, r8,m/T) − F(r3, r8,0) ∼ { −F(r3, r8) if T ≪ m2F(r3, r8) if T ≫ m


We obtain a first order transition in agreement with lattice predictions:

0 2 Π

3

4 Π

3

2 Π

0

[UR, J. Serreau, M. Tissier and N. Wschebor, Phys.Lett. B742 (2015) 61-68]

Summary of one-loop results

order lattice fRG our model at 1-loopSU(2) 2nd 2nd 2ndSU(3) 1st 1st 1stSU(4) 1st 1st 1stSp(2) 1st 1st 1st

Tc (MeV) lattice fRG(∗) our model at 1-loop(∗∗)

SU(2) 295 230 238SU(3) 270 275 185

(∗) L. Fister and J. M. Pawlowski, Phys.Rev. D88 (2013) 045010.(∗∗) UR, J. Serreau, M. Tissier and N. Wschebor, Phys.Lett. B742 (2015) 61-68.

Summary of two-loop results

order lattice fRG our model at 1-loop our model at 2-loopSU(2) 2nd 2nd 2nd 2ndSU(3) 1st 1st 1st 1stSU(4) 1st 1st 1st 1stSp(2) 1st 1st 1st 1st

Tc (MeV) lattice fRG(∗) our model at 1-loop our model at 2-loop(∗∗)

SU(2) 295 230 238 284SU(3) 270 275 185 254

(∗) L. Fister and J. M. Pawlowski, Phys.Rev. D88 (2013) 045010.(∗∗) UR, J. Serreau, M. Tissier and N. Wschebor, Phys.Rev. D93 (2016) 105002.

Adding (fundamental) quarks

We add Nf (heavy) quarks in the fundamental representation:

L = LLdW +LGribov +Nf

∑f=1

{ψf (∂/ − igAa/ ta +Mf + µγ0)ψf }

Center symmetry is explicitly broken by the boundary conditions:

ψ(1/T, x) = −ψ(0, x)

U(1/T, x) = e−i2π/3U(0, x)

⎫⎪⎪⎪⎬⎪⎪⎪⎭⇒ ψ′(1/T, x) = −e−i2π/3ψ′(0, x)

C-symmetry is also broken as soon as µ ≠ 0:r3

r8

µ = 0: Phase transition

µ = 0: Dependence on the quark masses

M > Mc ∶

2.8 3 3.2 3.4 3.6 3.8

V

r3

first order

M = Mc ∶

2.8 3 3.2 3.4 3.6 3.8

V

r3

second order

M < Mc ∶

2.8 3 3.2 3.4 3.6 3.8

V

r3

crossover

µ = 0: Columbia plot

1 0.9 0.8 0.8

0.9

1

Crossover

1st order

1 − e−Msm

1 − e−Mum

µ = 0: Columbia plot

Nf (Mc/Tc)our model (*) (Mc/Tc)lattice (**) (Mc/Tc)matrix (***) (Mc/Tc)SD (****)

1 6.74 7.22 8.04 1.422 7.59 7.91 8.85 1.833 8.07 8.32 9.33 2.04

(∗) UR, J. Serreau and M. Tissier, Phys.Rev. D92 (2015).(∗∗) M. Fromm, J. Langelage, S. Lottini and O. Philipsen, JHEP 1201 (2012) 042.(∗∗∗) K. Kashiwa, R. D. Pisarski and V. V. Skokov, Phys.Rev. D85 (2012) 114029.(∗∗∗∗) C. S. Fischer, J. Luecker and J. M. Pawlowski, Phys.Rev. D91 (2015) 1, 014024.

µ ∈ iR: Phase transitionLow T

µ ∈ iR: Phase transitionHigh T

µ ∈ iR: Roberge-Weiss transition

2

1

0

0.34

0.36 2

1 0

Arg ℓ

T/m

µi/T

µ ∈ iR: Dependence on the quark masses

M > Mc(0) ∶ 0.36

0.348

0 π/3 2π/3µi/T

T/m

M ∈ [Mc(iπ/3),Mc(0)] ∶ 0.36

0.34

0µi/T

π/3 2π/3

T/m

M = Mc(iπ/3) ∶

0.38

0.36

0.34

0.32

µi/Tπ/3 2π/30

T/m

Same structure as in the lattice study of [P. de Forcrand, O. Philipsen, Phys.Rev.Lett. 105 (2010)]

µ ∈ iR: Tricritical scaling

8

7

6 0 0.2 0.4 0.6 0.8 1 1.2

Mc/Tc

(π/3)2 − (µi/Tc)2

McTc= Mtric.

Ttric.+ K [(π

3 )2 + (µT )2]2/5

our model(∗) lattice(∗∗) SD(∗∗∗)

K 1.85 1.55 0.98Mtric.Ttric.

6.15 6.66 0.41

(∗) UR, J. Serreau and M. Tissier, Phys.Rev. D92 (2015).(∗∗) Fromm et.al., JHEP 1201 (2012) 042.(∗∗∗) Fischer et.al., Phys.Rev. D91 (2015) 1, 014024.

µ ∈ R: Complex backgrounds

We find that, for µ ∈ R, the background r8 should be chosen imaginary:

r3

r8

r3

r8

ir8

µ ∈ iR, r8 ∈ R µ ∈ R, r8 ∈ iR

µ ∈ R: Complex backgrounds

The choice r8 = ir8, r8 ∈ R is crucial for the interpretation of ` and ¯ in terms of Fq

and Fq to hold true [UR, J. Serreau and M. Tissier, Phys.Rev. D92 (2015)]:

`(µ) = e−βFq = er8√

3 + 2e−r8

2√

3 cos(r3/2)3

¯(µ) = e−βFq = e−r8√

3 + 2er8

2√

3 cos(r3/2)3

2

1

0

0.36 0.38

T/m

Fq/m

Fq/m

µ ∈ R: Columbia plot

The critical surface moves towards larger quark masses:

0.8

0.9

1

0.8 0.9 1

1 − e−Msm

1 − e−Mum

µ ∈ R: tricritical scaling

The tricritical scaling survives deep in the µ2 > 0 region:

18

14

10

0 40 80 120

Mc/Tc

(π/3)2 + (µ/Tc)2

Conclusions

Simple one-loop calculations in a model for a Gribov completion of theLandau-DeWitt gauge account for qualitative and quantitative featuresof the QCD phase diagram in the heavy quark limit:

⋆ Correct account of the order parameter in the quenched limit;⋆ Critical line of the Columbia plot at µ = 0;⋆ Roberge-Weiss phase diagram and its mass dependence for µ ∈ iR.

The case µ ∈ R requires the use of complex backgrounds.

Certain aspects require the inclusion of two-loop corrections.

TODO:

⋆ Chiral phase transition? Critical end-point in the (T, µ) diagram?⋆ Hadronic bound states? Glueballs?⋆ How to define the physical space?

perturbative aspects of the qcd phase diagram€¦ · perturbative aspects of theqcd phase diagram...

Documents