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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
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
Gribov copiesGauge condition
equivalent configs
1
⇒ L = 14g2
FaµνFa
µν + ∂µca(Dµc)a + iha∂µAaµ Valid at best at high 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
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
Weyl chambers and center symmetry
Weyl chambers and center symmetry
Weyl chambers and center symmetry
Weyl chambers and center symmetry
Weyl chambers and center symmetry
One-loop result - SU(3)
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
One-loop result - SU(3)
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?
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