1. Effect of curvature on confinement-deconfinement phase transition By Ashok Goyal & Deepak Chandra * Department of Physics & Astrophysics University of Delhi, Delhi-110 007 * S.G.T.B. Khalsa College, Univ. of Delhi, Delhi- 110 007.

2. INTRODUCTION We study the following topics: 1.Dynamics of the 1 st order phase transition bubble nucleation through bubble nucleation in an expanding QGP. interactions 2.Effects of interactions in the QGP and the hadron gas. curvature term 3.Effects of the curvature term in the free energy.

3. 4.Comparison of the QGP phase transition with the early Universe early Universe phase transition. 5. Conclusions.

4. Bubble Nucleation W= – 4πr 3 (P h – P q ) /3 + 4πr 2 σ – 8π(γ q – γ h )r σ = Surface Tension, γ = γ q – γ h is the Curvature Coefficient. E c = gr/(3π)∫ 0 ∞ dk k{1 + exp(k – μ B )/T} -1 where γ ≈ γ q ≈ T 2 /16 assuming γ h « γ q and μ B «T. g = statistical wt. The surface tension is mainly due to the finite strange quark mass.

5. The pressure in the QGP has been calculated by using the thermodynamic potential given by Kapusta (Nucl.Phys. B148, 461 (1979). For the hadron phase we use the masses of the low lying 33 baryons and 45 mesons whose masses & degeneracy factors were taken from the Particle Data Group. We account for the short range repulsive forces by considering the finite volume of the baryons and antibaryons equal to that of a proton given as V p =m p /(4B) where B is the bag pressure. The hadronic pressure and baryon densities corrected for finite volume effects are given as:

6. P h = ( ∑ b P b pt )/(1 + ∑ b n b pt V p ) + ( ∑ b ′ P b ′ pt )/(1 + ∑ b ′ n b ′ pt V p ) + ∑ m P m pt n B = ∑ b n b pt /(1 + ∑ b n b pt V p ) + ∑ b ′ n b ′ pt /(1 + ∑ b ′ n b ′ pt V p ) where b, b ′ and m stand for baryons, antibaryons and mesons respectively. The pressure equilibrium between the two phases sets the critical temperature T c which depends on the parameter B and interactions. (B ¼ = 300 MeV gives T c ≈ 188 MeV)

7. Critical Bubble Radii and Free Energies Below the Critical Temperature T c, the (maxima ) bubbles have radius and free energy: r c+ = (σ/ΔP)( 1 + (1 – β) ½ ) W c+ = 4πσ 3 /(3ΔP 2 )[ 2 + 2(1 – β) ³⁄ ² – 3β ] Whereas the (minima) bubbles are given by: r c- = (σ/ΔP)( 1 – (1 – β) ½ ) W c- = 4πσ 3 /(3ΔP 2 )[ 2 – 2(1 – β) ³⁄ ² – 3β ] Where β = 2ΔPγ/σ 2 ≤ 1 is positive definite.

8. Above the Critical Temperature T c, the (minima ) bubbles have radius and free energy : r c+> = (σ/ΔP)( –1 + (1 + β) ½ ) W c+> = 4πσ 3 /(3ΔP 2 )[ 2 – 2(1 + β) ³⁄ ² + 3β ] Where restriction β = 2ΔPγ/σ 2 ≤ 1 is not present. This ensures that the phase transition begins as soon as the the plasma is formed by the nucleation of these equilibrium sized bubbles of hadron phase. If β becomes >1 below T c, then nucleation of both types of critical bubbles disappear. The transition now proceeds by the expansion of the already nucleated bubbles r c+, r c- and r c+>.

9. The phase transition now gets rolling as the value of the critical radius and free energy needed for creation of expanding bubbles becomes zero.

10. Nucleation rate & The dynamical equations The nucleation rate is given by: I = I 0 Exp( – W c /T) Where the prefactor I 0 in early Universe studies is given by ( W c /2πT) ³⁄ ² T 4. Csernai & Kapusta have given I 0 in QCD by: I 0 =(16/3π)(σ/3T) ³⁄ ² (ση q r c )/{ξ q 4 (Δω) 2 }

11. Where η q =14.4T 3 is the shear viscosity in the plasma phase & ξ q is a correlation length ≈0.7 fm. Δω is the difference in enthalpy densities. hadron fractionCsernai & Kapusta have given a kinetic equation to calculate the hadron fraction as: h(t) = ∫ t c t dt′ I + (T(t′)){1 – h(t′)}V(t′,t) + ∫ t c t dt′ I – (T(t′)){1 – h(t′)} 4π r c– 3 (T(t′))/3 + ∫ t 0 t c dt′ I +> (T(t′)){1 – h(t′)} 4π r c+> 3 (T(t′))/3…..(1) Where V(t′,t) is the volume of an expanding bubble at time t which was nucleated at the earlier time t′ taking bubble growth into account. The other two terms are for equilibrium sized bubbles.

12. In our case we have contributions to h(t) from I +, I – below T c and I +> above T c. V(t′,t) = 4π/3{r c+ (T(t′)) + ∫ t′ t dt′′υ(T(t′′))} 3 The velocity of the bubble walls υ(T) = υ 0 (1 – T/ T c ) ³⁄ ² The other rate equation is given by: de/dt = – ω/t …….(2) energyenthalpy where e(T) and ω(T) are the energy & enthalpy densities given as: e(T)=h(t)e h (t) + (1-h(t))e q (t) We have solved the two coupled equations (1) and (2) for different values of the bag pressure B and the surface tension σ.

13. Free energy W in units of MeV as a function of bubble radius r in fm. In Fig 1(a) solid (with γ) and dashed (without γ) curves are for B ¼ = 235 MeV and σ = 50 MeV fm –2. Long dashed (with γ) and dotted (without γ) curves are for B ¼ = 235 MeV and σ = 25 MeV fm –2.In Fig. 1(b) solid (with γ) and dashed (without γ) curves are for B ¼ = 235 MeV and σ = 7 MeV fm –2. Long dashed (with γ) and dotted (without γ) curves are for B ¼ = 300 MeV and σ = 7 MeV fm –2. Both curves 1(a) and 1(b) are at a temperature of 1.01 T c.

14. 1© and 1(d) curves are labeled as in Figs. 1(a) and 1(b) respectively except that both are at a temperature of 0.99 T c.

15. Temperature T/ T c as a function of time t in fm. In Figs. 2(a) and 2(b) solid, dashed, dotted and long dashed curves are labeled as in Figs 1(a) and 1(b) respectively.

16. Log of the nucleation time τ in units of fm/c as a function of temperature. Figs. 3(a) and 3(b) are labeled as in Figs. 1(a) and 1(b) respectively.

17. Shows critical bubble radius r c in fm as a function of T/ T c. They are the minimum free energy equilibrium critical bubbles. The solid and long dashed curves are both with γ correction. These bubbles do not exist without the curvature correction. Figures 4(a) and 4(b) are labeled as in Figs. 1(a) and 1(b) respectively.

18. Figures 4(c) and 4(d) show critical bubble radius r c in fm as a function of T/ T c. They are the maximum free energy expanding critical bubbles. They are labeled as in Figs. 1(a) and 1(b) respectively.

19. The hadron fraction h as a function of time t in fm. Curves in Figs. 5(a) and 5(b) are labeled as in Figs. 1(a) and 1(b) respectively.

20. RESULTS We Find that the inclusion of interactions in both the phases lowers the critical temperature, so higher value of the bag pressure are favored for reasonable values of T c. As was shown by Mardor and Svetitsky, equilibrium sized bubbles are created well over T c if the curvature term is included and this continues even below T c where they are joined by maxima bubbles which grow. For B ¼ = 300 MeV, σ = 7 MeV fm –2 (σ 1/3 = 65 MeV) and T=0.99T c the free energy has no extremum, only a maximum value at 0. The critical radius is 0 and all bubbles can expand to complete the phase transition. In such cases β>1 soon after T c is crossed and no fresh bubbles appear.

21. supercooling duration The degree of supercooling depends on the parameters. Important feature here is that the prefactor I 0 is the deciding factor for supercooling and the duration of the transition rather than the exponent. This is in contrast to the early Universe studies where the exponent decides the behavior. There is a delay in the completion of the phase transition if the γ term is included compared to no γ case. This is because there is a possibility of β>1 just below T c which completely inhibits any further nucleation causing the delay. However γ has a mixed behavior depending on B. For low B it delays the phase transition but large B actually speeds up the hadronization if the γ term is included, in contrast to the no γ term. Decreasing the surface tension increases the supercooling of the plasma by a significant amount because it causes a suppression of nucleation. This is reverse of the early Universe case where the exponent drives the nucleation and not I 0.

22. Finally, for the ideal Maxwell construction, inclusion of interactions reduces the range of the transition but the presence of the curvature term introduces a delay compared to the ideal case.