Correlation functions of QC 2 D at finite density Ouraman Hajizadeh, - - PowerPoint PPT Presentation

Correlation functions of QC2D at finite density

Ouraman Hajizadeh, Tamer Boz, Axel Maas, Jon-Ivar Skullerud September, 2017 Zalakaros

Overview

1

Motivation

2

Similarities to SU(3)

3

SU(2) at finite density

4

Correlation functions

5

Finite temperature vs finite density

6

conclusion and outlook

Motivation

Why QC2D?

the simplest non-abelian gauge theory with fermions accessible at finite density on the lattice Insightful to the case of real QCD, based on the similarities. Informative about the behavior of non-abelian gauge theories in the different thermodynamic regimes. benchmark for other non-perturbative approaches to study QCD in different thermodynamic regimes.

Similarities to SU(3)

Phase Diagram

200 400 600 800

µq (MeV)

50 100 150 200 250

T (MeV)

0.2 0.4 0.6 0.8 µa 24 16 12 8 Nτ BEC?

BCS?

Quarkyonic

QGP Hadronic

<qq> <L>

(Tamer Boz, Seamus Cotter, Leonard Fister, Dhagash Mehta ,Jon-Ivar Skullerud, EPJ 2013 )

hadronic phase at low density and temperature QGP at high temperature and/or density quarkyonic phase at medium density and low temperature

Similarities to SU(3)

Order parameters and different phases

Three main distinct phases Hadronic: < qq >= 0, < L >∼ 0 low T and µ confined, chiral symmetry broken Quarkyonic: < qq >≠ 0, < L >∼ 0 medium µ, low T: quarks are bulk degrees of freedom (superfluid) but confined Quark-Gluon Plasma: at high T < qq >= 0, < L >≠ 0 ? a deconfined strongly interacting quark matter at high density and low temperature < qq >≠ 0, < L >≠ 0

Similarities to SU(3)

Polyakov loop: an order parameter in unquenched QCD? Polyakov loop: order parameter of deconfinement transition of YM theory with static quarks. < L >≠ 0: deconfined phase: finite free energy for static quarks. in unquenched QCD: quarks are dynamical so we have string

• breaking. the free energy is always finite.

what is the role of < L > in unquenched QCD, if it is always nonzero?

SU(2) at finite density

Polyakov loop vs quark density

0.2 0.4 0.6 0.8

µa

1 2 3 4

<L>

Nτ=24 Nτ=16 Nτ=12 Nτ= 8

0.2 0.4 0.6 0.8 1 0.05 0.1

0.2 0.4 0.6 0.8 1

µa

0.5 1 1.5 2 2.5

nq/nSB

lat Nτ=24 Nτ=12 Nτ=8

(S. Cotter, P. Giudice, S. Hands, J-I. Skullerud, Phys. Rev. D87 034507 (2013))

a dramatic increase in < L > resembling the ”deconfinement” transition in YM theory. for large T and/or µ at T = 0 for µ > µd < L >≠ 0 for large µ and/or T: deconfined phase, static quark free energy is finite. µda = 0.7 for the lowest T. nq enters a strongly interacting region for µ > µd,L(µd,T) = L(0,Td) not compatible with the deconfinement picture at high temperatures and zero density: perturbative regime.

Correlation functions

Phase diagram vs correlation functions phase diagram in terms of fundamental degrees freedom correlation functions may contain information on thermodynamic features, e.g. phase transitions phase transitions occur in medium at finite T and µ

Correlation functions

finite temperature-density formalism Euclidean coordinate: no time evolution: equilibrium temperature: compactified ”time” direction density: boundary condition heat bath is at rest: its four velocity only is nonzero along time direction. finite T or µ effects: different dressing functions (gL,gT) for correlation functions projected along or transversal to the time direction. for p ≫ T or p ≫ µ the difference of gT and gL is getting negligible: restoration of manifest Lorentz symmetry.

Correlation functions

Gluon propagator at finite density

ap (p)

L

D

• 2

a 1 2 3 4 5 6 7

Soft mode =0 µ T=0, =0 µ T=1/8, =0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0,

SU(2) transverse gluon propagator

ap 0.5 1 1.5 2 2.5 3 3.5 (p)

L

D

• 2

a 1 2 3 4 5 6 7 8 9

Soft mode =0 µ T= =0 µ T=1/8, =0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0,

SU(2) longitudinal gluon propagator

IR enhancement for medium chemical potentials (µ0 < µ < µd) compared to the vacuum, within this region: almost µ independence IR screening of large µ (µ > µd)

Correlation functions

Gluon propagator vs quark density

0.2 0.4 0.6 0.8 1

µa

0.5 1 1.5 2 2.5

nq/nSB

lat Nτ=24 Nτ=12 Nτ=8

ap (p)

L

D

• 2

a 1 2 3 4 5 6 7

Soft mode =0 µ T=0, =0 µ T=1/8, =0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0,

SU(2) transverse gluon propagator

(Simon Hands, Seamus Cotter, Pietro Giudice and Jon-Ivar Skullerud, XQCD 2012)

the screening region of chemical potential for gluon propagator corresponds to the region of strongly interacting quark matter for µ > µd ( nq

nSB > 1)

the region µd > µ > µ0 with almost constant value for the gluon propagator covers the ”weakly” interacting region nq ∼ nSB

Finite temperature vs finite density

Gluon Propagator at finite temperature YM

p [GeV] 0.5 1 1.5 2 2.5 3 ]

• 2

(p) [GeV

L

D 5 10 15 20 25 Temperature

c

0.361T

c

0.440T

c

0.451T

c

0.549T

c

0.603T

c

0.733T

c

0.903T

c

0.968T

c

0.986T

c

T

c

SU(2) longitudinal gluon propagator below T

p [GeV] 0.5 1 1.5 2 2.5 3 ]

• 2

(p) [GeV

L

D 2 4 6 8 10 12 14 16 18 20 22 Temperature

c

T

c

1.02T

c

1.04T

c

1.10T

c

1.81T

c

2.2T

c

SU(2) longitudinal gluon propagator above T

(Christian S. Fischer, Axel Maas, and Jens A. Mueller, EPJ 2010)

IR limit of longitudinal propagator responds strongly to the phase transition: the drop above Tc

Finite temperature vs finite density

p [GeV] 0.5 1 1.5 2 2.5 3 ]

• 2

(p) [GeV

T

D 1 2 3 4 5 6 Temperature

c

0.361T

c

0.440T

c

0.451T

c

0.549T

c

0.603T

c

0.733T

c

0.903T

c

0.968T

c

0.986T

c

T c

SU(2) transverse gluon propagator below T

p [GeV] 0.5 1 1.5 2 2.5 3 ]

• 2

(p) [GeV

T

D 0.5 1 1.5 2 2.5 3 3.5 4 Temperature

c

T

c

1.02T

c

1.04T

c

1.10T

c

1.81T

c

2.2T c

SU(2) transverse gluon propagator above T

(Christian S. Fischer, Axel Maas, and Jens A. Mueller, EPJ 2010)

transverse propagator is less sensitive to the phase transition below and above Tc almost temperature independent

Finite temperature vs finite density

Finite density propagator

ap (p)

L

D

• 2

a 1 2 3 4 5 6 7

Soft mode =0 µ T=0, =0 µ T=1/8, =0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0,

SU(2) transverse gluon propagator

ap 0.5 1 1.5 2 2.5 3 3.5 (p)

L

D

• 2

a 1 2 3 4 5 6 7 8 9

Soft mode =0 µ T= =0 µ T=1/8, =0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0,

SU(2) longitudinal gluon propagator

no considerable difference between longitudinal and transverse propagator around µd. different from finite T case.

Finite temperature vs finite density

Screening mass at finite temperature

Screening mass: Ms =

1 √ D(0), D(0) is the propagator at zero momentum.

t

• 0.05

0.05 0.1 [GeV]

s E

M 0.1 0.2 0.3 0.4 0.5 0.6

3

4x46

3

6x48 Phase transition vicinity in four dimensions t

• 0.1
• 0.05

0.05 0.1 [GeV]

E s

M 0.1 0.2 0.3 0.4 0.5 0.6 Phase transition vicinity

SU(2) left, SU(3) right

(Axel Maas, Jan M. Pawlowski, Lorenz von Smekal, Daniel Spielmann, Phys. Rev. D 85, 034037 (2012))

the 2nd order phase transition is indicated by the continuous increase

• f the screening mass of SU(2) gluon propagator as well as the 1st
• rder transition by a jump in the screening mass of SU(3) gluon

propagator around Tc

Finite temperature vs finite density

Screening mass at finite temperature

t

• 0.05

0.05 0.1 [GeV]

s E

M 0.5 0.55 0.6 0.65 0.7

Magnetic screening mass

(Axel Maas, Jan M. Pawlowski, Lorenz von Smekal, Daniel Spielmann, Phys. Rev. D 85, 034037 (2012))

magnetic screening mass does not indicate effects of the phase transition

Finite temperature vs finite density

Screening mass at finite density

µ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 [GeV]

• 1/2

(0)

L

D 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Electric screening mass for SU(2)

µ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 [GeV]

• 1/2

(0)

L

D 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Magnetic screening mass for SU(2)

no significant difference between electric (longitudinal) and magnetic (transverse) screening mass the effect of transition is not observed at finite density in the electric screening mass compared to finite T. the response of the magnetic screening mass to the phase transition is more observable than the electric part, in contrast to the finite T case.

Finite temperature vs finite density

Ghost Dressing function no difference between finite T and µ caes. IR enhancement no obvious temperature dependence.

ap 0.5 1 1.5 2 2.5 3 3.5 4 G(p) 1 1.5 2 2.5 3 3.5 4 4.5

Matsubara frequency 1 =0 µ T=0, =0 µ T=1/8, =0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0,

SU(2) ghost dressing function

Finite temperature vs finite density

ghost-gluon vertex at finite temperature no temperature dependence around the phase transition.

p [GeV] 0.5 1 1.5 2 2.5 3 3.5 4 /3) π (p,p,

A c c

G 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Ghost-gluon vertex, all momenta equal Ghost-gluon vertex, all momenta equal

p [GeV] 0.5 1 1.5 2 2.5 /2) π (p,p,

A c c

G 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Ghost-gluon vertex, orthogonal momenta with two equal c

T/T 1.02 1.04 1.06 1.08 1.1 1.55 2.

Ghost-gluon vertex, orthogonal momenta with two equal p [GeV] 0.5 1 1.5 2 2.5 /2) π (p,0,

A c c

G 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Ghost-gluon vertex, one momentum vanishing Ghost-gluon vertex, one momentum vanishing p [GeV] (Ghost) 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 k [ G e V ] ( G l u

• n

) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 /2) π (p,k,

A c c

G 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Ghost-gluon vertex, orthogonal momenta Ghost-gluon vertex, orthogonal momenta

(Leonard Fister, Axel Maas, Phys. Rev. D 90, 056008 (2014))

Finite temperature vs finite density

ghost-gluon vertex at finite density

ap 0.5 1 1.5 2 2.5 3 /3) π (p,p,

A c c

G 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ghost-gluon vertex, all momenta equal

ap 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /2) π (p,p,

A c c

G 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ghost-gluon vertex, orthogonal momenta with two equal ap 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /2) π (p,0,

A c c

G 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Soft mode =0 µ T=0, =0 µ T=1/8, =0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0, Ghost-gluon vertex, one momentum vanishing ap (Ghost) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 ak (Gluon) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 /2) π (p,k,

A c c

G 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ghost-gluon vertex, orthogonal momenta

no significant difference between finite T and finite µ IR constant in all cases

Finite temperature vs finite density

Running coupling running coupling derived from ghost-gluon vertex

ap 0.5 1 1.5 2 2.5 3 3.5 4 (p) α 0.5 1 1.5 2 2.5 3 3.5

=0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0,

Longitudinal coupling constant

ap 0.5 1 1.5 2 2.5 3 3.5 4 (p) α 0.5 1 1.5 2 2.5 3 3.5

=0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0,

Transverse coupling constant

similar to the coupling in the vacuum no signature of quarkyonic phase (nq ∼ nSB)

Finite temperature vs finite density

three gluon vertex of SU(2) YM in the vacuum, for different lattices

p [GeV] 1 2 3 4 5

/2) π (p,p,

3

A

G

1 2 Three-gluon vertex, orthogonal momenta with two equal p [GeV] 1 2 3 4 5

/2) π (0,p,

3

A

G

1 2

Three-gluon vertex, one momentum vanishing

p [GeV] 1 2 3 4 5 6 7

/3) π (p,p,

3

A

G

1 2

Three-gluon vertex, all momenta equal

(Attilio Cucchieri, Axel Maas, Tereza Mendes,Phys.Rev.D77:094510,2008)

IR suppression of tree level element of the three gluon vertex for three different kinematics.

Finite temperature vs finite density

three gluon vertex of unquenched SU(2) in the vacuum

ap 0.5 1 1.5 2 2.5 3 /3) π (p,p,

A

G 10 − 8 − 6 − 4 − 2 − 2 4 6 8 10

Three-gluon vertex, all momenta equal

ap 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /2) π (p,p,

A

G 10 − 8 − 6 − 4 − 2 − 2 4 6 8 10 Three-gluon vertex, orthogonal momenta with two equal ap 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /2) π (0,p,

A

G 10 − 8 − 6 − 4 − 2 − 2 4 6 8 10

=1.6 β =1.7 β =1.9 β

Three-gluon vertex, one momentum vanishing 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 10 − 5 − 5 10 Three-gluon vertex, orthogonal momenta

behavior compatible with the YM case

Finite temperature vs finite density

magnetic three gluon vertex at finite temperature

p [GeV] 0.5 1 1.5 2 2.5 3 3.5 /3) π (p,p,

A

G

• 3
• 2
• 1

1 2 3

Three-gluon vertex, all momenta equal Three-gluon vertex, all momenta equal

p [GeV] 0.5 1 1.5 2 2.5 /2) π (p,p,

A

G

• 3
• 2
• 1

1 2 3 Three-gluon vertex, orthogonal momenta with two equal

c

T/T 1.02 1.04 1.06 1.08 1.1 1.55 2.

Three-gluon vertex, orthogonal momenta with two equal p [GeV] 0.5 1 1.5 2 2.5 /2) π (0,p,

A

G

• 3
• 2
• 1

1 2 3 Three-gluon vertex, one momentum vanishing Three-gluon vertex, one momentum vanishing q [GeV] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 p [GeV] 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 /2) π (p,q,

A

G

• 3
• 2
• 1

1 2 3

Three-gluon vertex, orthogonal momenta Three-gluon vertex, orthogonal momenta

(Leonard Fister, Axel Maas, Phys. Rev. D 90, 056008 (2014))

pronounced temperature dependence IR enhancement close to Tc, in contrast to T = 0 case. sensitivity to the transition like electric propagator: surprising for the magnetic vertex

Finite temperature vs finite density

three gluon vertex at finite density

p [GeV] 0.5 1 1.5 2 2.5 3 /3) π (p,p,

A

G

• 1
• 0.5

0.5 1 1.5 2

Three-gluon vertex, all momenta equal

p [GeV] 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /2) π (p,p,

A

G

• 1
• 0.5

0.5 1 1.5 2 Three-gluon vertex, orthogonal momenta with two equal p [GeV] 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /2) π (0,p,

A

G

• 1
• 0.5

0.5 1 1.5 2 Soft mode =0 µ T=0, =0 µ T=1/8, =0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0, Three-gluon vertex, one momentum vanishing 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

• 1
• 0.5

0.5 1 1.5 2

Three-gluon vertex, orthogonal momenta

within the statistics no special trend is observed for the whole set of data

Finite temperature vs finite density

three gluon vertex at finite density, data with less than 50% error

p [GeV] 0.5 1 1.5 2 2.5 3 /3) π (p,p,

A

G

• 1
• 0.5

0.5 1 1.5 2

Three-gluon vertex, all momenta equal

p [GeV] 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /2) π (p,p,

A

G

• 1
• 0.5

0.5 1 1.5 2 Three-gluon vertex, orthogonal momenta with two equal p [GeV] 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 /2) π (0,p,

A

G

• 1
• 0.5

0.5 1 1.5 2 Soft mode =0 µ T=0, =0 µ T=1/8, =0.3 µ T=0, =0.4 µ T=0, =0.5 µ T=0, =0.6 µ T=0, =0.7 µ T=0, =0.75 µ T=0, =0.8 µ T=0, =0.9 µ T=0, Three-gluon vertex, one momentum vanishing 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

• 1
• 0.5

0.5 1 1.5 2

Three-gluon vertex, orthogonal momenta

IR suppression no obvious trend, different from T = 0 and µ = 0 no dependence on the chemical potential in contrast to the effect of temperature.

conclusion and outlook

conclusion significant difference between finite temperature and finite density behavior of gluonic sector close to the phase transition ghost sector is insensitive to medium: no significant temperature or density dependence for ghost-gluon vertex and ghost propagator no signal of free or weakly interacting region from the ggv remarkable simplification for functional methods due to the decoupling of ghosts and gluons from matter

conclusion and outlook

• utlook

more extensive finite temperature study in the unquenched regime to be compared to the quenched data in order to determine the effect of the dynamical quarks in the medium on the gauge sector searching for the underlying mechanism responsible for qualitatively different critical behavior of gluonic sector at finite density and finite temperature studying the matter sector and the related couplings in order to find traces of weakly interacting region of quark matter