Width of the QCD transition in a Polyakov-loop DSE model

Width of the QCD transition in a Polyakov-loop DSE model D. Horvati´ ca , D. Blaschkeb , D. Klabuˇcara , O. Kaczmarekc a Theoretical

Physics Department, University of Zagreb, Croatia for Theoretical Physics, University of Wroclaw, Poland c Department of Physics, Bielefeld University, Germany

b Institute

Three Days on Quarkyonic Island, Wroclaw, Poland May 20, 2011

Dyson-Schwinger approach to quark-hadron physics ◮

◮

the bound state approach which is nopertubative, covariant and chirally well behaved (e.g., GMOR Gell-Mann-Oakes-Renner 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

◮

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

◮

Homogeneous Bethe-Salpeter (BS) equation for a meson 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 D eff (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 .

Separable model

◮

◮

for separable interaction allowed form of the solution for the pseudoscalar BS amplitude is   P P S (P 2 ) f0 (q 2 ) ΓP S (q; P ) = γ5 iEP S (P 2 ) +F

this becomes a 2 × 2 matrix eigenvalue problem K(P 2 )f = λ(P 2 )f where the eigenvector is f = (EP S , FP S ), and the kernel is Z  d4 q 2 2  ˆ 4D0 tr f0 (q ) ti Sf (q+ ) tj Sf ′ (q− ) Kij (P 2 ) = − 4 3 (2π) t = (iγ5 , γ5 Pˆ ), tˆ = (iγ5 , −γ5 Pˆ /2P 2 ), q± = q ± P/2

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 )

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 d3 p 8D1 X 2 T f1 (p2n ) p ~ 2 [1 + af (T )f1 (p2n )] d−1 f (pn , T ) 9 (2π)3 n Z d3 p 8D1 X 2 2 T f (p2 ) ωn [1 + cf (T )f1 (p2n )] d−1 f (pn , T ) 3 1 n 3 (2π) n Z 16D0 X d3 p 2 e f + bf (T )f0 (p2n )] d−1 T f (p2 ) [m f (pn , T ) 3 0 n 3 (2π) n

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 )

Chiral restoration temperature ◮

Chiral restoration critical temperature TCh is TCh = 128 MeV

◮

M. Blank and A. Krassnigg, Phys. Rev. D 82, 034006 (2010) ω [GeV] Model Tc Model Tc Model Tc Model Tc

◮

(145 MeV in rank − 1)

N/A MN 169

0.3

0.4

0.5

AWW1 82 MT1 82 MR1 120

AWW2 94 MT2 94 MR2 133

AWW3 101 MT3 96 MR3 144

C. S. Fischer, J. Luecker, J. A. Mueller, [arXiv: 1104.1564], backreaction of the quarks onto the Yang-Mills sector

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) T Trp~,n,α,f,D



Sf−1 (pα n, T )

=

ln{Sf−1 (pα n , T )}

1 α − Σf (pα n , T ) · Sf (pn , T ) 2

−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 = φ 3 λ3 + φ 8 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 h i ¯ U (Φ, Φ) 1 = − a(T )Φ∗ Φ+b(T ) ln 1 − 6 + 4(Φ∗ 3 + Φ3 ) − 3(Φ∗ Φ)2 4 T 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.

T0 = 270 MeV ◮

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

Polyakov loop parametrization

◮

we can also consider polynomial ansatz for the Polyakov-loop potential

b2 (T ) ¯ b3 U ¯ 3 + b4 ΦΦ ¯ 2 Φ3 + Φ =− ΦΦ − 4 T 2 6 4    3  2 T0 T0 T0 b2 (T ) = a0 + a1 + a3 . + a2 T T T a0 = 6.75 , a1 = −1.95 , a2 = 2.625 , a3 = −7.44 b3 = 0.75 ,

b4 = 7.5

DSE + Polyakov loop ◮

Thermodynamic potential with Polyakov loop Ω(T )

=

−

¯ + U (Φ, Φ)

au (T )2

4T

X

+ C1u Z 3 X d p

2T

X

X

(2π)3

n α=0,±

−

n α=0,±

◮

Z

d3 p (2π)3

C2u h

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 =

◮

cu (T )2

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

◮

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

◮

Bernd-Jochen Schaefer, Jan M. Pawlowski, Jochen Wambach, Phys.Rev.D76:074023,(2007) Nf T0 [MeV]

0 270

1 240

2 208

2+1 187

3 178

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 .

◮

For T0 = 270 MeV, the critical temperatures are Tc = 195 MeV for Nf = 2 + 1 and Tc = 198 MeV for Nf = 2

◮

For T0 = 187 MeV → Tc = 164 MeV for Nf = 2 + 1, but ...

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

0.8 0.9

1 1.1 1.2 1.3 0.8 0.9 T/Tc

1 1.1 1.2 1.3 T/Tc

Results 200

Tc [MeV]

190 180 170 st

1 order transition crossover crossover

160 150 180

200

220 240 T0 [MeV]

260

280

Results

m(T)/m(0) , Φ(T)

1 0.8 0.6

T0=187 MeV T0=210 MeV T0=270 MeV

0.4 0.2 0 0.1

0.15 T [GeV]

0.2

Results qs

M [GeV]

1 0.8 0.6 0.4 0.2

2q

σ K 2π π

0

M [GeV]

1 0.8

σ

0.6

K

0.4

π

0.2 0 0

qs

2π

0.2

2q 0.4

0.6 0.8 T / Tc

1

1.2

Outlook

◮

Investigate effects of PL potential parametrization

◮

Investigate effects of changing form factors

◮

Include hadronic fluctuations beyond the mean field

◮

A.E. Radzhabov, D. Blaschke, M. Buballa, M.K. Volkov, [arXiv:1012.0664]