Study of the influence of external effects on the properties of QCD - - PowerPoint PPT Presentation

Study of the influence of external effects on the properties of QCD by means of lattice simulations

• A. Yu. Kotov

(based on the PhD thesis)

JINR 20 April 2016

Motivation

Temperature Baryonic density Chiral density Magnetic field

Lattice simulations in QCD Allow to study strongly coupled systems Based on the first principles of QFT Acknowledged approach to QCD Very powerful method due to the development of computer systems

Discussed problems

Viscosity of Quark-Gluon Plasma Two-Color QCD with nonzero baryon density QCD with nonzero chiral density Superconductivity of QCD vacuum in superstrong magnetic fields

Viscosity of Quark-Gluon Plasma

Viscosity Fx = −η · du

dy · S, η–viscosity

Viscosity is connected with Txy

Hydrodynamical description

One heavy ion collision produces a huge number of final particles Large number of particles ⇒ hydrodynamical description can be used In hydrodynamics transport coefficients control flow of energy, momentum, electrical charge and

• ther quantities

Elliptic flow at STAR (Nucl. Phys. A 757, 102 (2005))

dN dφ ∼ (1 + 2v1cos(φ) + 2v2cos2(φ)), φ-scattering angle

QGP is close to ideal liquid (η

s = (1 − 3) 1 4π)

• M. Luzum and P. Romatschke, Phys. Rev. C 78, 034915 (2008)

Comparison of different liquids, arXiv:nucl-ex/0609025 QGP is the most superfluid liquid The aim: first principle calculation of transport coefficients

Previous lattice calculations (SU(3) gluodynamics) Karsch, F. et al. Phys.Rev. D35 (1987)

• A. Nakamura, S. Sakai Phys. Rev. Lett. 94, 072305 (2005)
• H. B. Meyer, Phys.Rev. D76 (2007) 101701
• H. B. Meyer, Phys.Rev. D76 (2007) 101701

η s = 0.134 ± 0.033 (T/Tc = 1.65) η s = 0.102 ± 0.056 (T/Tc = 1.24)

Green-Kubo formula T12T12E(τ) =

∞

• ρ(ω)coshω( 1

2T − τ)

sinh ω

2T

dω η = π lim

ω→0

ρ(ω) ω

Lattice calculation of transport coefficients Lattice measurement of the correlator C(t) = T12(t)T12(0) Calculation of the spectral function ρ(ω) from C(t) = T 5 ∞ dωρ(ω)

ch

• ω

2T −ωt

• sh
• ω

2T

• Hydrodynamical approximation ρ(ω)|ω→0 ∼ η

πω

Viscosity η = π limω→0

ρ(ω) ω

SU(2) gluodynamics, T/Tc ≃ 1.2

Correlation function

Calculation of the spectral function

C(t) = T 5 ∞ dωρ(ω)

ch

• ω

2T −ωt

• sh
• ω

2T

• Properties:

ρ(ω) ≥ 0, ρ(−ω) = −ρ(ω) Asymptotic freedom: ρ(ω)|NLO

ω→∞ = 1 10 dA (4π)2 ω4

• 1 − 5Ncαs

9π

• 7/8 of the whole correlator at t =

1 2T

Hydrodynamics: ρ(ω)|ω→0 = η

πω

Ansatz for the spectal function ρ(ω) = η

πωθ(ω0 − ω) + θ(ω − ω0)Aρasym(ω)

Ansatz for the spectal function ρ(ω) = η

πωθ(ω0 − ω) + θ(ω − ω0)Aρasym(ω)

χ2/dof ∼ 1, A = 0.723 ± 0.004, ω0 = 2.7GeV

η s = 0.18 ± 0.04

Other variants of the spectral function ρ(ω) = η

πω + th2 ω ω0 Aρasym(ω)

χ2/dof ∼ 1, A = 0.723 ± 0.003, ω0 = 2.0GeV

η s = 0.09 ± 0.03

Other variants of the spectral function

η s < 0.18 ± 0.04 η s ∈ (0.09, 0.18) η s = 0.134 ± 0.057

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 η/s KSS bound, Exp min Exp max

SU(3) gluodynamics, 16 × 323

η/s vs T

0.05 0.1 0.15 0.2 0.25 0.3 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 η/s T/Tc Exp max KSS bound, Exp min Fit 1 Fit 2

Results (T/Tc = 1.2)

η s = 0.134 ± 0.057 SU(2) η s = 0.178 ± 0.06 SU(3) η s = 1 4π ≃ 0.08 N=4 SYM λ = ∞ (Phys. Rev. Lett. 87 (2001) 081601) η s = (1 − 3) 1 4π ≃ 0.08 − 0.24 Experiment (Phys. Rev. C 78, 034915 (2008) η s ∼ 2 Perturbative result (JHEP 11 (2000) 001) η s = 0.102 ± 0.056 (SU(3), Phys.Rev. D76 (2007) 101701)

0.05 0.1 0.15 0.2 0.25 0.3 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 η/s T/Tc Exp max KSS bound, Exp min Fit 1 Fit 2

Conclusion Model-independent calculation of viscosity in SU(2) and SU(3) gluodynamics (no free parameters) Results are in agreement with experimental estimations QGP is strongly coupled system close to SYM and far from weakly interacting plasma

Two-Color QCD with nonzero baryonic density Phase diagram of QCD

a

Lattice QCD

Z =

• DψD ¯

ψDAµe−Sg− ¯

ψDψ =

• DAµe−Sg det D

Lattice QCD + µB − → Sign problem! det D / ∈ R

Lattice Monte-Carlo Calculations

AA Gauge group: SU(2) − → No sign problem! det D†(µ∗

B) = det

• (τ2Cγ5)D(µB)(τ2Cγ5)
• с C = γ2γ4 - charge conjugation matrix

det D(µB) ∈ R, > 0.

Phase diagram of SU(2) QCD

• J. B. Kogut, D. Toublan, D.K. Sinclair, Nucl.Phys. B642 (2002)

181-209

µB, confinement and chiral symmetry breaking

Chiral condensate:

0.01 0.1 1.6 1.8 2 2.2 2.4 2.6

<ψ –ψ> β

163x6 lattices, ma=0.01 µa=0.0 µa=0.2 µa=0.5

µB, confinement and chiral symmetry breaking

Polyakov loop:

0.05 0.1 0.15 0.2 0.25 0.3 0.35 1.6 1.8 2 2.2 2.4 2.6

<L> β

163x6 lattices, ma=0.01 µqa=0.0 µqa=0.2 µqa=0.5

Conclusion

First results about SU(2) QCD phase diagram µB disfavours chiral symmetry breaking Critical temperature decreases

QCD with nonzero chiral density

Topological fluctuations in QCD

(a) arXiv:1111.6733, P.V. Buividovich,

• T. Kalaydzhyan, M.I. Polikarpov

Anomaly: ∂µj(5)

µ

= CF (a)

µν ˜

F µν

(a) −

→ Nonzero chiral density ρ5

CME

Possible manifestation: Chiral Magnetic Effect (CME) ρ5 & B →

• 
• B

A A A

•  = Nc

2π2 µ5

B A A A

• K. Fukushima, D. Kharzeev, H. J. Warringa,

PRD 78, arXiv: 0808.3382 (hep-ph) Phase is important!

Phase diagram of QCD with nonzero µ5

Effective models (NJL, PNJL, PLSMq etc) arXiv: 1102.0188, 1110.4904, 1305.1100, 1310.4434 Dyson-Schwinger equations arXiv:1505.00316 Large Nc Universality arXiv:1111.3391 Lattice QCD (no sign problem)

Results

SU(2), Nf = 4 fermionic flavours Lattice size 6 × 203, mq = 12 MeV

0.02 0.04 0.06 0.08 0.1 0.12 160 180 200 220 240 260 280 L T, MeV

µ5 = 0 MeV µ5 = 150 MeV µ5 = 300 MeV µ5 = 475 MeV µ5 = 950 MeV

5 10 15 20 25 30 160 180 200 220 240 260 280 < − ψψ>/T3 T, MeV

µ5 = 0 MeV µ5 = 150 MeV µ5 = 300 MeV µ5 = 475 MeV µ5 = 950 MeV

Polyakov loop chiral condensate

• Results. Susceptibilities

0.3 0.4 0.5 0.6 160 180 200 220 240 260 280 χL T, MeV

µ5 = 0 MeV µ5 = 475 MeV µ5 = 950 MeV

0.4 0.8 1.2 1.6 160 180 200 220 240 260 280 χ< −

ψψ>

T, MeV

µ5 = 0 MeV µ5 = 475 MeV µ5 = 950 MeV

Polyakov loop susceptibility chiral susceptibility

Critical temperature vs µ5

200 210 220 230 240 200 400 600 800 1000 Tc, MeV µ5, MeV

Chiral sus Polyakov loop sus

Results for SU(3) gauge group and Nf = 2 Wilson fermions

0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Tc(µ5)/Tc(0) µ5a

Conclusions

200 210 220 230 240 200 400 600 800 1000 Tc, MeV µ5, MeV

Chiral sus Polyakov loop sus

A A Phase diagram with nonzero µ5 was studied in two theories: SU(2), Nf = 4 и SU(3), Nf = 2 Tc ↑ when µ5 ↑ The same behaviour in the chiral limit The transition seems to become sharper No splitting of χS and confinement transitions

Superconductivity of QCD vacuum in strong magnetic fields

In the background of strong magnetic field QCD vacuum turns into a superconductor (due to condensation of charged ρ-mesons)

• M. Chernodub, arXiv:1008.1055, arXiv:1101.0117

Superconductivity of QCD vacuum in strong magnetic fields

1 Emerges spontaneously at magnetic fields larger then critical

Bc ≈ 1016T eBc ≈ m2

ρ ≈ 0.6 GeV2

2 No Meissner effect (though vortices are formed) 3 Zero resistance along magnetic field 4 Isolator in other (perpendicular) directions

Approaches to problem

1 General ideas 2 Effective models (M. Chernodub, arXiv:1008.1055,

arXiv:1101.0117);

3 Gauge-gravity duality (N. Callebaut, D. Dudal, H. Verschelde,

arXiv:1105.2217; M. Ammon, J. Erdmenger, P. Kerner, M. Strydom , arXiv:1106.4551; ...)

4 Numerical calculations

Naive approach

Energy of ρ-meson in magnetic fiels E 2 = m2

ρ + eB(2n + 1 − 2Sz) + p2 z

Zero Landau level: n = 0 pz = 0 Spin along the field: Sz = 1      E 2 = m2

ρ − eB

If eB > eBc = mρ2 ≈ 0.6GeV 2 ⇒ E 2 < 0 ⇒ Condensation

Structure of the condensate

Effective bosonic model B = 1.01Bc

(b) Charged ρ-mesons

• M. Chernodub, J. Doorsselaere, H. Verschelde, arXiv:1111.4401.

Similar results in holography Y.-Y. Bu, J. Erdmenger, J. P. Shock,

• M. Strydom, arXiv:1210.6669

Numerical calculations in quenched QCD

• Two quark flavours:

u, d

• ρ-meson operator:

ρµ = ¯ uγµd

• Spin ±1 along magnetic field:

ρ± = 1

2(ρ1 ± iρ2)

• Correlator:

G±(z) = ρ†

±(0)ρ±(z)

• Condensate:

lim|z|→∞ G+(z) = |ρ|2

Superconducting condensate

Quenched QCD

• eB = 0

eB = 1.07 GeV2 eB = 1.28 GeV2 eB = 2.14 GeV2

1.7 1.8 1.9 2.0 2.1 2.2 0.001 0.002 0.005 0.010 0.020 L, fm

Η, GeV3

Mass (arXiv:hep-lat/9803003) mρ ∼ 1.1GeV

Superconducting vortices

Observables:

• Field:

ρµ = ¯ uγµd

• Correlator:

ρ+(x) → φ(x) = ρ†

+(0)ρ+(x)f

• Energy density:

E(x) = |Dµφ(x)|2

|φ(x)|2 , Dµ = ∂µ − ieAµ

• Electric current:

jµ(x) = φ∗Dµφ−φ(Dµφ)∗

2i|φ(x)|2

• Vortex density:

v(x) = sing arg φ(x) = ǫab

2π ∂ ∂xa ∂ ∂xb arg φ(x)

Superconducting vortices. Energy density. eB = 0.36GeV 2

(c) ⊥ B (d) B

arXiv:1301.6590

Superconducting vortices. Energy density. eB = 1.07GeV 2

(e) ⊥ B (f) B

Superconducting vortices. Energy density. eB = 2.14GeV 2

(g) ⊥ B (h) B

Electric current around the vortices

Correlations between positions of vortices

Conclusions

In a sufficiently strong magnetic field ρ-meson condensate is formed simultaneously New type of topological defects, "ρ-vortices emerge Liquid of ρ-vortices is observed in quenched lattice calculations(cf. theory: trigonal lattice)

Total conclusion

Viscosity in SU(2) and SU(3) gluodynamics was measured (in particular, with respect to the temperature) Influence of the baryonic chemical potential on the temperature of the confinement-deconfinement transition was studied The phase diagram of QCD with nonzero chiral density was investigated The hypothesis about superconductivity of QCD vacuum in strong magnetic fields was studied