![Page 1: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/1.jpg)
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.
![Page 2: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/2.jpg)
Theoretical approaches to the QCD phase diagram
Non-exhaustive list:
- finite-temperature lattice QCD- Schwinger-Dyson equations- functional renormalization group
⎫⎪⎪⎪⎬⎪⎪⎪⎭non-perturbative
Perturbation theory?
![Page 3: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/3.jpg)
Outline
I. A modified perturbation theory in the infrared?
(Vacuum, Landau gauge).
II. Application to the QCD phase structure.
(Finite T, Landau-DeWitt gauge).
![Page 4: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/4.jpg)
A modified perturbation theoryin the infrared
![Page 5: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/5.jpg)
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.
![Page 6: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/6.jpg)
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
![Page 7: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/7.jpg)
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!
![Page 8: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/8.jpg)
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?
![Page 9: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/9.jpg)
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
![Page 10: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/10.jpg)
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!
![Page 11: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/11.jpg)
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?
![Page 12: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/12.jpg)
QCD phase structure
![Page 13: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/13.jpg)
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
![Page 14: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/14.jpg)
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
![Page 15: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/15.jpg)
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
![Page 16: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/16.jpg)
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?
![Page 17: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/17.jpg)
Weyl chambers and center symmetry
![Page 18: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/18.jpg)
Weyl chambers and center symmetry
![Page 19: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/19.jpg)
Weyl chambers and center symmetry
![Page 20: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/20.jpg)
Weyl chambers and center symmetry
![Page 21: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/21.jpg)
Weyl chambers and center symmetry
![Page 22: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/22.jpg)
Weyl chambers and center symmetry
![Page 23: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/23.jpg)
One-loop result - SU(3)
![Page 24: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/24.jpg)
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
![Page 25: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/25.jpg)
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]
![Page 26: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/26.jpg)
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.
![Page 27: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/27.jpg)
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.
![Page 28: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/28.jpg)
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
![Page 29: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/29.jpg)
µ = 0: Phase transition
![Page 30: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/30.jpg)
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
![Page 31: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/31.jpg)
µ = 0: Columbia plot
1 0.9 0.8 0.8
0.9
1
Crossover
1st order
1 − e−Msm
1 − e−Mum
![Page 32: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/32.jpg)
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.
![Page 33: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/33.jpg)
µ ∈ iR: Phase transitionLow T
![Page 34: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/34.jpg)
µ ∈ iR: Phase transitionHigh T
![Page 35: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/35.jpg)
µ ∈ iR: Roberge-Weiss transition
2
1
0
0.34
0.36 2
1 0
Arg ℓ
T/m
µi/T
![Page 36: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/36.jpg)
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)]
![Page 37: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/37.jpg)
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.
![Page 38: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/38.jpg)
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
![Page 39: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/39.jpg)
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
![Page 40: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/40.jpg)
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
![Page 41: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/41.jpg)
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
![Page 42: Perturbative aspects of the QCD phase diagram€¦ · Perturbative aspects of theQCD phase diagram Urko Reinosa∗ (based on collaborations with J. Serreau, M. Tissier and N. Wschebor)](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f7528ddde34aa7b685bf335/html5/thumbnails/42.jpg)
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?