QGP from the quantum ground-state of QCD? beautiful math or New - - PowerPoint PPT Presentation

SLIDE 1

QGP from the quantum ground-state of QCD?





beautiful math or “New Physics” of QCD?

Roman Pasechnik

• 1

SLIDE 2

• 2

Summary: stages of the “micro Big Bang”

0.2 0.4 0.6 0.8 1.0 1.2 J/λ4

• 2.0
• 1.5
• 1.0
• 0.5

0.5 1.0 Leff/λ4(⨯103) 0.2 0.4 0.6 0.8 1.0 1.2 J/λ4

• 2.0
• 1.5
• 1.0
• 0.5

0.5 1.0 Leff/λ4(⨯103) 0.2 0.4 0.6 0.8 1.0 1.2 J/λ4

• 2.0
• 1.5
• 1.0
• 0.5

0.5 1.0 Leff/λ4(⨯103)

• (almost) classical homogeneous 


CE gluon condensate evolution

• small inhomogeneities
• perturbative regime 


(short distances)

• energy “swap” from condensate


to the fluctuations (quasiclassical pic.)

• large inhomogeneities (plasma modes)
• parametric resonance effect 


(particle production mechanism)

• quantum ground state


formation (CE mostly 
 + initiation of CM)

• large distances/essentially


quantum dynamics

• domain-wall formation

Stage I Stage II Stage III

SLIDE 3

Short range strong interactions Asymptotic freedom Running QCD coupling Color charge anti-screening Confinement Color confinement!

QCD vacuum: short vs long distances

3

SLIDE 4

4

Stage I

SLIDE 5

5

Homogeneous gluon condensate: semi-classics

Aa

0 = 0

temporal (Hamilton) gauge

ea

i Aa k ≡ Aik

ea

i ea k = δik

ea

i eb i = δab

due to local SU(2) ~ SO(3) isomorphism

Aik(t, ⃗ x) = δikU(t) + Aik(t, ⃗ x)

F a

µν = ∂µAa ν − ∂νAa µ + gYM f abcAb µAc ν

Classical YM Lagrangian: Lcl = −1 4F a

µνF µν a

Corrections are small 
 for gYM<<1
 (short distances!)

U(t) ≡ 1 3δik⟨Aik(t, ⃗ x)⟩⃗

x ,

⟨Aik(t, ⃗ x)⟩⃗

x =

• Ω d3xAik(t, ⃗

x)

• Ω d3x

Basis for canonical (Hamiltonian) quantisation of “condensate+waves” system:

Zeroth-order in waves = “pre-quilibrium state”?

HYM ≃ HYMC = 3 2

• (∂0U)2 + g2U 4

, ∂0∂0U + 2g2 U 3 = 0

0.0 0.5 1.0 1.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 tTU UU0 Ut U0coskgU0t

t = − U

U0

dU

• g2U 4

0 − g2U 4 ,

U(0) = U0 , U ′(0) = 0

SLIDE 6

6

Stage II

SLIDE 7

7

Condensate+waves semi-classical system

Longitudinally polarised (plasma) mode becomes physical due to interactions with the homogeneous condensate!

• − δlk(∂0∂0U + 2g2U 3) + (−∂0∂0

Alk + ∂i∂i Alk − ∂i∂k Ali − gelmk∂i AmiU − 2gelip∂i ApkU − gelmi∂k AmiU + g2 AklU 2 − g2 AlkU 2 − 2g2δlk AiiU 2) + (−gelmp∂i Ami Apk − 2gelmp Ami∂i Apk − gelmp∂k Ami Api + g2 Ali AikU + g2 Ali AkiU + g2 Aik AilU

• −
• −
• − 2g2

Aii AlkU − g2δlk Api ApiU) + g2( Ali Apk Api − Api Api Alk) = 0 .

• Aik = ψik + eiklχl

tensor basis decomposition

χ⃗

p l = sσ l η⃗ p σ + nlλ⃗ p ,

ψ⃗

p ik = ψ⃗ p λQλ ik + ϕ⃗ p σ(nisσ k + nksσ i ) + (δik − nink)Φ⃗ p + ninkΛ⃗ p

Hwaves

YM

= 1 2

• ∂0ψλ ∂0ψ†

λ + ∂0φσ ∂0φ† σ + ∂0Φ ∂0Φ† + 1

2 ∂0Λ ∂0Λ† + ∂0ησ ∂0η†

σ

+ ∂0λ ∂0λ† + p2 ψλψ†

λ + p2

2 φσφ†

σ + p2 ΦΦ† + p2

2 ηση†

σ + p2 λλ†

− p2 2 eγσ(ησφ†

γ + φγη† σ) + igp U eσγηση† γ − igp U Qλγψλψ† γ

− igpUeσγφσφ†

γ − igp U (2Φλ† − 2λΦ† + Λλ† − λΛ†)

+ 2g2 U 2 ηση†

σ + 2g2 U 2 λλ† + g2 U 2 (4ΦΦ† + 2ΦΛ† + 2ΛΦ† + ΛΛ†)

• Full Hamiltonian

“condensate+waves” system evolution:

HYM = HYMC +

• ⃗

p

Hwaves

YM

SLIDE 8

8

Decay of the homogeneous condensate

HU = 3 2

• ∂0U∂0U + g2 U 4

, Hparticles = 1 2

• ⃗

p

• ∂0ψλ ∂0ψ†

λ + ∂0φσ ∂0φ† σ + ∂0Φ ∂0Φ† + 1

2 ∂0Λ ∂0Λ† + ∂0ησ ∂0η†

σ

+ ∂0λ ∂0λ† + p2 ψλψ†

λ + p2

2 φσφ†

σ + p2 ΦΦ† + p2

2 ηση†

σ + p2 λλ†

− p2 2 eγσ(ησφ†

γ + φγη† σ)

• ,

Hint = 1 2

• ⃗

p

• igp U eσγηση†

γ − igp U Qλγψλψ† γ

− igpUeσγφσφ†

γ − igp U (2Φλ† − 2λΦ† + Λλ† − λΛ†)

+ 2g2 U 2 ηση†

σ + 2g2 U 2 λλ† + g2 U 2 (4ΦΦ† + 2ΦΛ† + 2ΛΦ† + ΛΛ†)

• .

H H

1 2 3 4 5 1.0 0.5 0.0 0.5 1.0 tTU UU0

Ultra-relativistic gluon plasma production!

SLIDE 9

9

Stage III

SLIDE 10

10

QCD confinement as a dual Meissner effect

type-II superconductor N S

Superconductor Flux tube solenoid solenoid

usual Meissner effect Magnetic field cannot penetrate through a superconductor, except by burning out a narrow tube where the superconductivity is destroyed (the Abrikosov vortex)

Energy of the magnetic “monopole-antimonopole” pair is proportional to L (string potential) L

• the QCD vacuum as a condensate of chromo-magnetic monopoles


(c.f. condensation of BCS pairs in usual superconductors)

• quarks are sources of chromo-electric field
• inside the quark-antiquark tube the chromo-magnetic condensate is destroyed
• electric field is squeezed inside the tube (the Abrikosov-Nielsen-Olesen vortex)

The dual Meissner effect in QCD (analogous to that in dual superconductors):

SLIDE 11

11

Long distances: chromo-magnetic condensate

Quantum-topological (chromomagnetic) vacuum in QCD

✏vac∼10−2GeV4 Λcosm ∼ 10−47 GeV4

CM condensate: Ground-state at long distances:

We must be missing something very important!?

Vacuum in QCD has incredibly wrong energy scale… or

SLIDE 12

12

Effective YM action approach

chromoelectric (CE) condensate (Savvidy vacuum)

trace anomaly:

[111] G. K. Savvidy, Phys. Lett. 71B, 133 (1977). [37] H. Pagels and E. Tomboulis, Nucl. Phys. B 143, 485 (1978).

The energy-momentum tensor:

Lcl = −1 4F a

µνF µν a

Effective YM Lagrangian:

e Aa

µ ≡ gYMAa µ

d F a

µν ≡ gYMF a µν.

possesses well-kn ⇣

• d ln |¯

g2| d ln |J |/µ4 = β(¯ g2) 2

hich we r at J ⇤ > 0, he only p

NOTE: the RG equation

Leff= J 4¯ g2(J ), J =−Fa

µνFµν a ,

J ← →−J ,

|,¯ g2=¯ g2(|J |) T µ

µ =−(¯

g2) 2¯ g2 J

Equations of motion:

− → D ab

ν

Fµν

b

¯ g2 ✓ 1−(¯ g2) 2 ◆ =0, − → D ab

ν ≡

⇣ ab− → @ ν−f abcAc

ν

⌘ ,

T ν

µ = 1

¯ g2 h(¯ g2) 2 1 i⇣ Fa

µλFνλ a +1

4ν

µJ

⌘ ν

µ

(¯ g2) 8¯ g2 J .

appears to be invariant under

0.2 0.4 0.6 0.8 1.0 1.2 J/λ4

• 2.0
• 1.5
• 1.0
• 0.5

0.5 1.0 Leff/λ4(⨯103)

At least, for SU(2) gauge symmetry, the all-loop and one-loop effective Lagrangians are practically indistinguishable (by FRG approach)

[14] A. Eichhorn, H. Gies and J. M. Pawlowski, Phys. Rev. D 83 (2011) 045014 [Phys. Rev. D 83 (2011) 069903]. [15] P. Dona, A. Marciano, Y. Zhang and C. Antolini, Phys.

• Rev. D 93 (2016) no.4, 043012.

0.2 0.4 0.6 0.8 1.0 1.2 J/λ4

• 0.08
• 0.06
• 0.04
• 0.02

0.02 g-2

Effective Lagrangian: Inverse running coupling
 is a better expansion parameter!

SLIDE 13

13

Classical YM condensate Savvidy (CE) vacuum

Quantum corrections

“Radiation” medium

Unstable solution!

Asymptotic tracker solution!

Stable solution!

• YM ∝ 1/a4

QCD vacuum:
 a ferromagnetic undergoing spontaneous magnetisation (Pagels&Tomboulis)

CE condensate on non-stationary (FLRW) background

• In fact, both chromoelectric and chromomagnetic condensates 


are stable on non-stationary (FLRW) background of expanding Universe ”Time” CE instantons
 are formed first!

✏CE!+const t!1

Q ⌘ 32 11⇡2e(⇠ΛQCD)4T µ

µ [U]

= 6e h (U 0)2 1 4U 4i a4(⇠ΛQCD)4 Exact partial solution:

• n |Q| = 1,

e Q(U) = 1

−1

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14

“Mirror” symmetry of the ground state

αs(µ2) = αs(µ2

0)

1 + β0 αs(µ2

0) ln(µ2/µ2 0)

is invariant under

αs = ¯ g2 4π

For pure gluodynamics at one-loop: Choosing the ground state value of the condensate we observe that the mirror symmetry, indeed, holds provided i.e. in the ground state only!

µ2 ⌘ p |J | , µ , µ2

0 ⌘

p |J ⇤|

as the physical scale

e J ' J ⇤

⌘ p |J | ⌘ ↵s(µ2

0)

! ↵s(µ2

0)

Leff = J 4¯ g2

In a vicinity of the ground state, the effective Lagrangian

(1) = bN 48⇡2 ¯ g2

(1)

where b = 11

e J ' J ⇤ Z2: J ∗← →−J ∗, ¯ g2(J ∗)← →−¯ g2(J ∗), (¯ g2

∗)←

→−(¯ g2

∗),

SLIDE 15

15

Heterogenic quantum YM ground state: two-scale vacuum

The running coupling at one-loop with two energy scales

CE vacuum:

a classical s

• r (¯

g2

⇤) = 2.

• rresponds to

Mirror symmetry

CM vacuum: 2. On the oth

• (¯

g2

⇤) = 2, 2

e.o.m. is automatically satisfied! Reduces to the standard YM e.o.m. discussed in e.g. in instanton theory Trace anomaly: Trace anomaly:

D ⌘ ⇣

• A

⌘

• !

D ab

ν

Fµν

b

¯ g2

• =0,

¯ g2'¯ g2

∗

T µ

µ,CE= 1

¯ g2

∗

J ∗.

T µ

µ,CM=+ 1

¯ g2

∗

J ∗.

)= 96⇡2 bNln(|J |/4

±),

4

±⌘|J ∗|exp

h ⌥ 96⇡2 bN|¯ g2

1(J ∗)|

i .

L(1)

eff = bN

384⇡2 J ln ⇣|J | 4

±

⌘ ,

|J ∗|=2

+2 −

0.2 0.4 0.6 0.8 1.0 1.2 |J|/λ+4

• 2.0
• 1.5
• 1.0
• 0.5

0.5 1.0 Leff/λ+4(⨯103)

λ λ

not coincide, ar , |J ⇤|/4 =

1 e 3

hich we r at J ⇤ > 0, he only p

Cosmological CE attractor

2

+/2 −=e

+

¯ g2

1(J ) =

¯ g2

1(µ4 0)

1 +

bN 96π2 ¯

g2

1(µ4 0) ln(|J |/µ4 0)

One-loop:

2 4 6 8 |J|/λ-4

• 2.0
• 1.5
• 1.0
• 0.5

0.5 1.0 Leff/λ+4(⨯103)

J ∗<0

|J ∗|/4

−=e

Cosmological CM attractor

SLIDE 16

16

Post-confinement: stage IV

• Both CE and CM reach


their attractors

• CM/CE domains 


“crystallisation”

• CC is formed

SLIDE 17

17

✏vac ⌘ 1 4hT µ

µ ivac = ⌥Leff(J ⇤)

✏CE

vac

• J ∗>0 + ✏CM

vac

• J ∗<0 ⌘ 0

Exact compensation of CM and CE vacua as soon as the cosmological attractor is achieved!

Macroscopic evolution and vacua cancellation

1 2 3 4 5 6 0.00 0.01 0.02 0.03 0.04 QCDt UQCD

e Q(U) = −1 ”Time” CM instantons
 are formed latest! Asymptotic tracker solutions!

Confined phase De-confined phase

Standard Friedmann Cosmology with zeroth CC!

CE energy density CE EMT trace

System with very unusual dynamical properties! 17

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18

Take-home facts on QCD ground-state “crystal”:

• No ghost problem associated with negative coupling due to: 


(i) only gauge invariant quantities are used 
 (ii) local loss of Lorentz (e.g. rotational) invariance


• Nielsen-Olsen proof of instability of CE condensate on a rigid Minkowski 


in NOT in contradiction with our results: we consider YM evolution 


• n a dynamical (FLRW) spacetime while equilibrium is achieved only 

• asymptotically. 

• A possible decay of CE condensate into an anisotropic vacuum after 


a cosmological relaxation time would be exponentially suppressed and 
 is practically never realised


• Even starting from an initial non-zero energy-density, the evolution of localised 3-space 


“pockets” of the CE and CM condensates trigger a mutual screening, flowing towards
 a zero-energy density attractor and accompanying by a formation of the domain walls 
 corresponding to an asymptotic restoration of the Z2 (Mirror) symmetry and effectively 
 protecting the “false” CE vacua pockets from further decay (“time crystal” ground-state)


• The vacua cancellation mechanism seems to naturally marry the existing confinement


pictures related to a formation of a network of t’Hooft monopoles or chromovortices.
 In this approach, the scalar kink profile may correspond the J-invariant whose change
 may be related to the presence of monopole or vortex solutions localise inside
 the space-time domain walls. This implies the existence of space-time solitonic 


• bjects of a new type.