The QCD transition temperature in a Polyakov-loop DSE model*

The QCD transition temperature in a Polyakov-loop DSE model∗ D. Horvati´ ca , D. Blaschkeb,c,d , D. Klabuˇcara , A. E. Radzhabovc ∗ Karl-Franzens-Universit¨ at a Theoretical

Graz

Physics Department, University of Zagreb, Croatia b University of Rostock, Germany c JINR Dubna, Russia d University of Wroclaw, Poland

October 8, 2008

Outline

◮

Dyson-Schwinger approach to quark-hadron physics

◮

Separable model at T = 0 and T 6= 0

◮

Difficulties and how to solve them

◮

Polyakov loop + DSE

◮

Results

◮

Summary and outlook

Dyson-Schwinger approach to quark-hadron physics ◮

= the bound state approach which is nopertubative, covariant and chirally well behaved (e.g., GMOR relation: 2 e q = −h¯ q qi/fπ2 ) limm e q →0 Mq q¯/2m ◮ ◮

direct contact with QCD through ab initio calculations phenomenological modeling of hadrons as quark bound states (e.g., here)

◮

coupled system of integral equations for Green functions of QCD

◮

... but ... equation for n-point function calls (n+1)-point function ... → cannot solve in full the growing tower of DS equations

◮

→ various degrees of truncations, approximations and modeling is unavoidable (more so in phenomenological modeling of hadrons, as here)

Dyson-Schwinger approach to quark-hadron physics ◮

Gap equation for propagator Sq of dressed quark q

+

=

λa µ γ 2

◮

Γaν (l, p)

Sq (l)

Homogeneous Bethe-Salpeter (BS) equation for a M eson q q¯ bound state vertex Γqq¯

M Γqq¯

=

K

M Γqq¯ Sq

Gap and BS equations in RLA truncation

Sf (p)

−1

→ Sf (p) =

4 = iγ · p + m ef + 3

Z

d4 q 2 eff g Dµν (p − q)γµ Sf (q)γν (2π)4

−ip Af (p2 ) + Bf (p2 ) −ip  + mf (p2 ) 1 = 2 = 2 ip Af (p2 ) + Bf (p2 ) p Af (p2 )2 + Bf (p2 )2 p + mf (p2 )2

λ(P 2 )Γff¯′ (p, P ) = −

4 3

Z

d4 q Deff (p − q)γµ Sf (q + (2π)4 µν

P 2

)Γff¯′ (q, P )Sf (q −

P4

◮

Euclidean space: {γµ ,γν} = 2δµν , 㵆 = γµ , a·b =

◮

P is the total momentum

◮

meson mass is identified from λ(P 2 = −M 2 ) = 1

◮

eff (k) an “effective gluon propagator” - modeled ! Dµν

P )γν 2

i=1 ai bi

Separable model eff : ◮ To simplify calculations, take the separable form for Dµν

eff Dµν (p − q) → δµν D(p2 , q 2 , p · q)

D(p2 , q 2 , p · q) = D0 f0 (p2 )f0 (q 2 ) + D1 f1 (p2 )(p · q)f1 (q 2 ) ◮ two strength parameters D0 , D1 , and corresponding form factors fi (p2 ). In the separable model, gap equation yields



Bf (p2 )

=

 Af (p2 ) − 1 p2

=

m ef + 8 3

Z

16 3

Z

Bf (q 2 ) d4 q D(p2 , q 2 , p · q) 2 2 2 4 (2π) q Af (q ) + Bf2 (q 2 )

(p · q)Af (q 2 ) d4 q D(p2 , q 2 , p · q) 2 2 2 . (2π)4 q Af (q ) + Bf2 (q 2 )

◮ This gives Bf (p2 ) = m e f + bf f0 (p2 ) and Af (p2 ) = 1 + af f1 (p2 ), reducing to nonlinear equations for constants bf and af .

A simple choice for ‘interaction form factors’ of the separable model:

◮

Interaction form factors: f0 (p2 ) = exp(−p2 /Λ20 ) f1 (p2 ) =

◮

◮

1 + exp(−p20 /Λ21 ) 1 + exp((p2 − p20 ))/Λ21

gives good description of pseudoscalar properties if the interaction is strong enough for realistic DChSB when mu,d (p2 ∼ small) ∼ the typical constituent quark mass scale ∼ mρ /2 ∼ mN /3.

Extension to T 6= 0 ◮

◮

At T 6= 0, the quark 4-momentum p −→ pn = (ωn , p~), where ωn = (2n + 1)πT are the discrete ( n = 0, ±1, ±2, ±3, ... ) Matsubara frequencies, so that p2n = ωn2 + p~ 2 . Dressed quark propagator Sf (pn , T ) = [i~γ · p~ Af (p2n , T ) + iγ4 ωn Cf (p2n , T ) + Bf (p2n , T )]−1

= ◮

−i~γ · p~ Af (p2n , T ) − iγ4 ωn Cf (p2n , T ) + Bf (p2n , T ) . p~ 2 A2f (p2n , T ) + ωn2 Cf2 (p2n , T ) + Bf2 (p2n , T )

There are now three amplitudes due to the loss of O(4) symmetry, and at sufficiently high T ≥ Td denominator can vanish. −→ For T ≥ Td quarks can be deconfined!

Extension to T 6= 0 ◮

◮

The solutions have the form Bf = m e f + bf (T )f0 (p2n ), 2 Af = 1 + af (T )f1 (pn ), and Cf = 1 + cf (T )f1 (p2n ) af (T )

=

cf (T )

=

bf (T )

=

Z 8D1 X T 9 n Z 8D1 X T 3 n Z 16D0 X T 3 n

d3 p 2 f1 (p2n ) p ~ 2 [1 + af (T )f1 (p2n )] d−1 f (pn , T ) (2π)3 d3 p 2 2 f1 (p2n ) ωn [1 + cf (T )f1 (p2n )] d−1 f (pn , T ) (2π)3 d3 p 2 e f + bf (T )f0 (p2n )] d−1 f0 (p2n ) [m f (pn , T ) (2π)3

where df (p2n , T ) is given by 2 2 2 df (p2n , T ) = p ~ 2 A2f (p2n , T ) + ωn Cf (pn , T ) + Bf2 (p2n , T )

Deconfinement and chiral restoration temperature

◮

Chiral restoration critical temperature TCh is TCh = 128 MeV

◮

(145 MeV in rank − 1)

Deconfinement temperature Td m e q [MeV]

Td [MeV]

0

97

5.5

107 (145 in rank-1)

115

194

Deconfinement and chiral restoration temperature

◮

Chiral restoration critical temperature TCh in other models

◮

results from A. Krassnigg and M. Joergler

Model MN1 Gaussian

1 2

I.S. 0.2809 0.76835 0.57625 0.461

[Munczek, Nemirovsky 1983] [Alkofer, Watson, Weigel 2002]

TCh [MeV] 169 1 83 95 97

ω 0.3 0.4 0.5

D 0.5618 1.5367 1.1525 2 0.922 2

Chiral symmetry restoration at T = TCh mq H0L@GeVD

0.6

TCh =128MeV

ms H0L

0.5 0.4 0.3

mu H0L m0 H0L

0.2 0.1 T@GeVD 0.05

0.1

0.15

0.2

0.25

0.3

DSE + Polyakov loop ◮

Consider DSE model for quark dynamics and generalize it by coupling to the Polyakov loop potential

◮

Use rank-2 separable form of the effective gluon propagator with the same form factors fixed by phenomenology Central quantity for the analysis of the thermodynamical behavior is the thermodynamical potential

◮

Ω(T )

¯ − = U(Φ, Φ)

  1 −1 α α α − T Trp~,n,α,f,D ln{Sf (pn , T )} − Σf (pn , T ) · Sf (pn , T ) 2

Sf−1 (pα n, T )

−1 α α = Sf,0 (pn , T ) − Σ−1 f (pn , T )

DSE + Polyakov loop

◮

Polyakov loop in a case of a constant background field: Φ=

Rβ 1 1 a Trc Tτ ei 0 dτ λa A4 (~x,τ ) = Trc eiφ/T Nc Nc

φ = φ3 λ3 + φ8 λ8 ◮

◮

Here φ is related to Euclidean background field and diagonal in color space ¯ to be real which sets φ8 = 0 We require Φ = Φ     φ φ 1  1 φ3 −i T3 i T3 ¯ Φ=Φ= +e = . 1+e 1 + 2 cos Nc Nc T

DSE + Polyakov loop ◮

Polyakov-loop potential i h ¯ 1 U(Φ, Φ) 3 ∗ 2 ∗ ∗3 + Φ ) − 3(Φ Φ) = − a(T )Φ Φ+b(T ) ln 1 − 6 + 4(Φ T4 2

with a(t) = a0 + a1



T0 T



+ a2



T0 T

2

,

b(T ) = b3



T0 T

3

.

Corresponding parameters are a0 = 3.51, ◮

a1 = −2.47,

a2 = 15.22,

b3 = −1.75.

Due to the coupling to the Polyakov loop the fermionic Matsubara frequencies ωn are shifted (pαn )2 = (ωnα )2 + p~ 2 ,

ωnα = ωn + αφ3 ,

α = −1, 0, +1

DSE + Polyakov loop ◮

Thermodynamic potential with Polyakov loop Ω(T )

=

−

¯ + U (Φ, Φ)

au (T )2

4T

X

X

C1u Z

2T

X

X

Z

n α=0,±

−

n α=0,±

◮

cu (T )2 C2u h

3

d p (2π)3

(2π)3

d3 p

2

+

bu (T )2 C3u

+

as (T )2 C1s

α 2

2

α 2

+

cs (T )2

2

C2s α 2

+

bs (T )2 C3s 2

α 2

ln p ~ Au ((pn ) , T ) + (ωn ) Cu ((pn ) , T ) + Bu ((pn ) , T )

u

s

C1 =

2D1 27 4D1 27

i

h i 2 2 α 2 α 2 2 α 2 2 α 2 ln p ~ As ((pn ) , T ) + (ωn ) Cs ((pn ) , T ) + Bs ((pn ) , T )

Where Cfi (f = u, d, s and i = 1, 2, 3) are: C1 =

◮

+

u

,

C2 =

,

C2 =

s

2D1 9 4D1 9

u

,

C3 =

,

C3 =

s

4D0 9 8D0 9

Thermodynamic potential still suffers from the divergency due to the zero-point energy. We employ a substraction scheme.

DSE + Polyakov loop Ωreg (T ) = Ω(T ) − Ωfree (T ) + Ωreg free (T ) + Ω(0) ◮

where Ωfree (T ) = −4T

X X Z n α=0,±

−2T

X X Z n α=0,±

◮

  2 d3 p 2 α 2 , ) + m e ln p ~ + (ω s n (2π)3

and Ωreg free (T ) = −4T

X

X

f =u,d,s α=0,± ◮

 2  d3 p α 2 2 ln p ~ + (ω ) + m e − n u (2π)3

with Eq =

q p~ 2 + m e 2q

   Ef − αφ3 d3 p ln 1 + exp − (2π)3 T

DSE + Polyakov loop

◮

◮

Obtain gap equations by minimizing the thermodynamic potential with respect to Aq , Bq , Cq and φ3 ¯ one obtains simple solution for by minimizing only U(Φ, Φ) gluon sector p ¯ a(T ) + 2 a(T )2 + 9b(T )a(T ) dU(Φ, Φ) =0 ⇒ Φ= dφ3 3a(T )

◮

and from that deconfinement temperature of 270 MeV

◮

For Nf flavors we have 3Nf + 1 coupled gap equations

◮

Now results for quark and gluon sectors coupled

Results

◮

The critical temperatures for chiral symmetry restoration (TCh ) and deconfinement (Td ) which differ by a factor of two when the quark and gluon sectors are considered separately get synchronized and become coincident when the coupling is switched on Tc = TCh = Td .

◮

The critical temperatures Tc = 194.8 MeV for Nf = 2 + 1 and Tc = 198.2 MeV for Nf = 2 are in agreement with recent lattice QCD simulations.

◮

Varying the current quark mass of the model results in a dependence of Tc on the light pseudoscalar meson mass: d in fair agreement with lattice QCD. Plus ... Tc = a + b Mps

Results

susceptibilities

40

dΦ(Τ)/dT dmu,d(T)/dT

Nf=2+1

Nf=2

dms(T)/dT

30 20 10 0.18

0.19

0.20

temperature T[GeV]

0.19

0.2

temperature T[GeV]

Results

w Polyakov loop

w/o Polyakov loop

40

dΦ(Τ)/dT dmu,d(T)/dT

susceptibilities

Nf=2+1

dms(T)/dT

30 20

Nf=2+1

10 0.18

0.19

0.20

temperature T[GeV]

0.1

0.15

0.2

temperature T[GeV]

Results Data taken form P. Petreczky, [arXiv:0705.2175]

Results 215

Tc [MeV]

210

205

200 Nf=2+1 Nf=2 fit 1 fit 2

195

190

0

100

200

300

Mps [MeV]

400

500

600

Results from BSE w/o PL MP @GeVD Td,u TCh

1.2

Td,s

1 0.8 Ms s 0.6

MΣ

0.4

MK

0.2

MΠ T@GeVD 0.05

0.1

0.15

0.2

Results from BSE with PL 0.8

Tc ,Td =195 MeV

Ms s

MP @GeVD

0.6

MΣ

MK

0.4

0.2

MΠ

0 0.18

0.185

0.19

0.195 T@GeVD

0.2

0.205

0.21

Summary and outlook

◮

Sketched Dyson-Schwinger approach to quark-hadron physics & a convenient concrete model

◮

Coupled quark and gluon sector via Polyakov loop potential

◮

Few remarkable results

◮

Calculate full thermodynamics (mesonic contribution to thermodynamic potential ...)

◮

How to apply it to other DSE models ...