effective polyakov loop models for qcd-like theories at

Effective Polyakov loop models

for QCD-like theories at finite chemical potential

Philipp Scior

with Lorenz von Smekal

Theoriezentrum, Institut für Kernphysik, TU Darmstadt

Lattice 2015, Kobe

18 July 2015 | Philipp Scior, Lorenz von Smekal | 1

Motivation

I Exploration of the QCD phase diagram at

finite µ

I Use effective Polyakov loop theory to

circumvent the sign problem

I Here: Use effective theory for QCD-like

theories

→ qualitative and quantitative comparison

with full theory at finite chemical potential

0 200 400 600 800

µq (MeV)

0

50

100

150

200

250

T (

MeV

)

0 0.2 0.4 0.6 0.8µa

24

16

12

8N

τ

BEC?

BCS?

Quarkyonic

QGP

Hadronic

<qq>

<L>

Boz, Cotter, Fister, Mehta, Skullerud

EPJ A 49, 11 (2013)


Effective Polyakov Loop Theory

I 3d SU(N) spin model sharing universal behavior at deconfinement transition

with underlying gauge theory

I Less computational cost, especially with dynamical fermions

I Finite density −→ Complex Langevin

I Can be derived from combined strong coupling and hopping expansion by

integrating out spacial links

Fromm, Langelage, Lottini, Neumann, Philipsen, PRL 110 (2013) 12

Langelage, Neumann, Philipsen, JHEP 1409 (2014) 131


Effective Action

Use strong coupling and hopping expansion: Contributions to the effective action

are graphs winding around the lattice in time direction

Gauge Action:

LiLj

⇒ Sgeff = λ

∑〈ij〉

LiLj + ...

Low temperature, strong coupling: λ ≤ 10−16 −→ fermionic partition function


Heavy Fermions and Hopping Expansion

det[D] = det[1− κH] = exp(−∞∑i=1

1iκi Tr[H i ])

Leading order:

−Seff = Nf

∑~x

log(1 + hTrW~x + h2)2



Order κ2:

−Seff = Nf

∑~x

log(1 + hTrW~x + h2)2 − 2Nf h2

∑~x ,i

TrhW~x

1 + hW~xTr

hW~x+i

1 + hW~x+i



Order κ4:



−Seff = Nf

∑~x


∑~x ,i

TrhW~x

1 + hW~xTr

hW~x+i

1 + hW~x+i

+ 2N2fκ4N2

τ

N2c

∑~x ,i

TrhW~x

(1 + hW~x )2 TrhW~x+i

(1 + hW~x+i )2

+ Nfκ4N2

τ

N2c

∑~x ,i ,j

TrhW~x

(1 + hW~x )2 TrhW~x−i

1 + hW~x−iTr

hW~x−j

1 + hW~x−j

+ 2Nfκ4N2

τ

N2c

∑~x ,i ,j

TrhW~x


1 + hW~x−iTr

hW~x+j

1 + hW~x+j

+ Nfκ4N2

τ

N2c

∑~x ,i ,j

TrhW~x


1 + hW~x+iTr

hW~x+j

1 + hW~x+j



−Seff = Nf

∑~x


∑~x ,i

TrhW~x

1 + hW~xTr

hW~x+i

1 + hW~x+i

+ 2N2fκ4N2

τ

N2c

∑~x ,i

TrhW~x


(1 + hW~x+i )2

+ Nfκ4N2

τ

N2c

∑~x ,i ,j

TrhW~x


1 + hW~x−iTr

hW~x−j

1 + hW~x−j

+ 2Nfκ4N2

τ

N2c

∑~x ,i ,j

TrhW~x


1 + hW~x−iTr

hW~x+j

1 + hW~x+j

+ Nfκ4N2

τ

N2c

∑~x ,i ,j

TrhW~x


1 + hW~x+iTr

hW~x+j

1 + hW~x+j

+ Nf (2Nf − 1)κ4N2τ

∑x ,i

h4

(1 + hTrW~x + h2)(1 + hTrW~x+i + h2).


Cold and Dense QC2D with Heavy Quarks

I Very heavy quarks: mq = 10.0014 GeV

I Diquark mass is fixed to md = 20 GeV→ small binding energy

I Deconfinement transition with unphysical lattice saturation

0

0.5

1

1.5

2

2.5

3

3.5

4

9.9 9.92 9.94 9.96 9.98 10 10.02 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

a3n

<|L

|>

µ in GeV

a3n<|L|>

1x10-14

1x10-12

1x10-10

1x10-8

1x10-6

0.0001

0.01

1

9.9 9.92 9.94 9.96 9.98 10 10.02 0.01

0.1

1

a3n

<|L

|>

µ in GeV

a3n<|L|>



I Behaviour of a3n is well described

by a LO mean field model

a3n = 4Nf1 + Le

mq−µ

T

1 + 2Lemq−µ

T + e2mq−µ

T

I No Fit! All values taken from

Simulations or are input

I 〈L〉 ≈ 5 · 10−5 < 〈|L|〉 = 0.006

〈L〉 determined by histograms

1x10-18

1x10-16

1x10-14

1x10-12

1x10-10

1x10-8

1x10-6

0.0001

0.01

1

100

9.88 9.9 9.92 9.94 9.96 9.98 10 10.02a3n

µ in GeV

4 MeV6 MeV9 MeV



I Most precise description of the

second exponential by

a3n = 4Nf exp(

2µ−md

T

)

I Bound State?

I Model includes Confinement

1x10

1x10

1x10

1x10

1

9.88 9.9 9.92 9.94 9.96 9.98 10 10.02

an

µ in GeV

a n

4 Nf L̃ exp

(

µ−mq

T

)

4Nf exp(

2µ−md

T

)


Phase Diagram

I Diquark onset line is not a

phase-boundry!

I Line terminates at µ = md2 according

to Silverblaze property

I Deconfinement transiton terminates

at µ > md2

I Hints for BEC at lower

Temperatures?

0

2

4

6

8

10

9.92 9.94 9.96 9.98 10 10.02

T in M

eV

µ in GeV

linear extrapolationonset points

0

2

4

6

8

10

9.985 9.99 9.995 10

md/2 mq

T in M

eV

µ in GeV

deconfinementpseudo-critical points

diquark-onset

0

2

4

6

8

10

9.985 9.99 9.995 10

md/2 mq

T in M

eV

µ in GeV


Extrapolation to T = 0

I T = 0 endpoint is in perfect

agreement with µ = md2

I Endpoint is independent of lattice

spacing

9.99

9.992

9.994

9.996

9.998

10

10.002

10.004

10.006

10.008

10.01

0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115

µc

in G

eV

a in fm

linear extrapolation


G2 QCD

I Smallest exceptional Lie Group

I All representation are real (β = 4)

I No sign problem: real and positive for Nf = 1

I YM theory has 1st order phase transition

I Spectrum: bosonic and fermionic baryons


Phase diagram

Density Polyakov loop

2.4 2.45 2.5 2.55 2.6 2.65 2.7

aµ

1/22

1/16

1/12

1/10

1/8

1/Nt

2 4 6 8

10 12 14

<a3n>

2.4 2.45 2.5 2.55 2.6 2.65 2.7

aµ

1/22

1/16

1/12

1/10

1/8

1/Nt

0.5

1

1.5

2

2.5

<|L

|>

I Cold and dens region of the G2 effective theory phase diagram

I β/Nc = 1.4, κ = 0.0357, Nt = 8, ...24


Cold and Dens Region

I Deconfinement and unphysical

lattice saturation

I three regions with different

exponential growth

I diquark onset

I fermionic baryon onset

1x10-10

1x10-8

1x10-6

0.0001

0.01

1

100

2.2 2.3 2.4 2.5 2.6 2.7 2.8 0.01

0.1

1

a3n

<|L

|>

aµ

compare with Wellegehausen, Maas,

Wipf, von Smekal PRD 89 (2014) 4

with a smaller quark mass κ = 0.147


Summary Conclusion

I Effective Polyalov loop theory for two-color QCD up to order κ4

I Thermal two-quark excitation onset at T > 0

I Extrapolation T → 0 gives md/2 according to Silverblaze Property

I However no signal for BEC

I Effective Polyalov loop theory for G2

I Two- and three-quark excitations in agreement with possible spectrum


Analytic relations for Cold and Dense regime

h = exp[

Nτ

(aµ + ln 2κ + 6κ2 u − uNτ

1− u

)],

amd = −2 ln(2κ)− 6κ2 − 24κ2 u1− u

+ 6κ4 +O(κ4u2,κ2u5) .


effective polyakov loop models for qcd-like theories at

Documents