Collaborators Based on work done with Awaneesh Singh, Sanjay Puri - PowerPoint PPT Presentation

K INETICS OF C HIRAL TRANSITION IN HOT AND DENSE QUARK MATTER Hiranmaya Mishra Theoretical Physics Division, Physical Research Laboratory, Ahmedabad, India IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 1

Collaborators Based on work done with Awaneesh Singh, Sanjay Puri School of Physical Sciences Jawaharlal Nehru University, New Delhi ( Nucl. Phys. A864 (2011)176-201, Phys. Atom. Nucl. 75 (2012) 689, Nucl. Phys. A908 (2013)12-28; Eur. Phys. letts 2013 (to appear) ) IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 2

O UTLINE • Introduction • QCD phase digram • Vacuum with quark condensates in NJL model and phase diagram • Ginzburg-Landau expansion and TDGL equation • Quench through second order transition and domain growth • Quench through first order transition (bubble nucleation and spinodal decomposition) • Summary and Outlook IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 3

QCD UNDER EXTEREME CONDITION • Extreme conditions exist in the universe. (Compact astrophysical objects, Cosmology) • Exploring QCD phase diagram is important to understand the phase we live in • Fundamental properties of QCD IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 4

QCD PHASE DIAGRAM (S CHEMATIC ) IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 5

I NTRO . CONTD . . . Mostly attention has been focussed on Critical dynamics (time dependent behaviour in the vicinity of critical point) far from equllibrium dynamics (dynamics subsequent to a quench from the disordered phase with vanishing quark condensate to the ordered phase) We shall discuss the far from equllibrium dynamics and focus on the late stage of the phase separation kinetics of quark matter and the scaling properties of the correlation functions. IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 6

CSB AND VAC . STRUCTURE IN NJL MODEL (HM and S.P . Misra, Phys Rev. D48,(1993)5376) ψψ ) 2 + ( ¯ L NJL = i ¯ / ψ + G [( ¯ ψiγ 5 τψ ) 2 ] ψ∂ Two flavor, massless. � q 0 ( k ) † σ · ˆ q 0 ( − k ) d k − h.c. ) | 0 � | vac � = exp( k h ( k )˜ q 0 | 0 � = 0 Determine the condensate function h( k ) by minimising energy (T=0, µ =0),/free energy ( T � = 0 , µ = 0 )/, thermodynamic potential ( T � = 0 , µ � = 0 ). | k | = − 2 g � ¯ tan 2 h ( k ) = M ψψ � | k | 1 g = G (1 + 4 N c ) IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 7

NJL MODEL C ONTD . · · · Thermodynamic potential 12 � � k 2 + M 2 − | k | ) d k Ω = − ( (2 π ) 3 12 � − [log(1 + exp( − βω − ) + log(1 + exp( − βω + )] d k (2 π ) 3 M 2 + (1) 4 g √ k 2 + M 2 ∓ ν , ν = µ − Gρ v /N c . ω ∓ = Mass gap equation M = 2 g 2 N c N f � M √ k 2 + M 2 [1 − n − ( k , β, µ ) − n + ( k , β, µ )] d k (2 π ) 3 IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 8

P HASE DIAGRAM ; NJL MODEL 400 400 300 300 M (MeV) M (MeV) 200 200 100 100 (a) (b) 0 0 0 100 200 300 400 0 50 100 150 200 µ (MeV) T (MeV) a b Mass ∼ G � ¯ ψψ � as a function of µ for T=0 (Fig a) and as a function of T for µ = 0 (Fig b) IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 9

P HASE DIAGRAM ; NJL MODEL CONTD . · · · 125 (a) µ = 311.00 MeV µ = 321.75 MeV II µ = 328.00 MeV 100 µ = 335.00 MeV Massless quarks tcp T (MeV) 75 (M = 0) 50 Massive quarks I (M ≠ 0) 25 S 1 S 2 0 240 260 280 300 320 µ (MeV) Phase diagram of the Nambu-Jona-Lasinio model in the ( µ, T )-plane for zero current quark mass. A line of first-order transitions (I) meets a line of second-order transitions (II) at the tricritical point (tcp). ( µ tcp , T tcp ) ≃ (282 . 58 , 78) MeV. The dot-dashed lines S 1 and S 2 denote the spinodals or metastability limits for the first-order transitions. IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 10

P HASE DIAGRAM ; NJL MODEL CONTD . · · · 2nd order tricritical pt. spinodal spinodal 1st order (triple line) T 1 > T > T c , M>0 is a metastable state (superheated liq.) T 2 < T < T c , M=0 is metastable state (supercooled gas) IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 11

G INZBURG L ANDAU EXPANSION OF FREE ENERGY In the mean field approx. close to the phase boundary, the thermodynamic potential may be expanded in power series of the order parameter M upto logarithmic corrections: Sasaki,Friman,Redlich,PRD77, 034024 (2008); Iwasaki,PRD 70, 114031(2004) · · · Ω (0) + a 2 M 2 + b 4 M 4 + d 6 M 6 + · · · ≡ f ( M ) . Ω ( M ) = ˜ ˜ a, b, d —functions of ( µ, T ) Gap equation: f ′ ( M ) = aM + bM 3 + dM 5 = 0 . Soln.s  M 0 = 0 , √  b 2 − 4 ad ± = − b ± M 2 .  2 d IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 12

G L FREE ENERGY – CONTD · · · For b > 0 transition is second order. Stationary pt.s are M = 0 (for a > 0 ) OR M=0, ± M + (for a < 0 ) For b < 0 phase transition is first order with the soln.s of gap eq.s a > b 2 / 4 d, M = 0 , b 2 / 4 d > a > 0 , M = 0 , ± M + , ± M − , (2) M = 0 , ± M + , a < 0 . Condn. of degeneracy of two minima ( Ω( M = 0) = Ω( M = M + ) or a c = 3 b 2 / (16 d ) ) determines T c . T 1 ( T 2 ) is determined by a = b 2 / 4 d ( a = 0 ). IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 13

G INZBURG L ANDAU PHASE DIAGRAM in the (b,a) space 0.04 100 2 /16d a c (I) = 3|b| II M = 0 tcp a c (II) = 0 80 a s1 = 0 0.03 T (MeV) µ = 311.00 MeV 2 /4d 60 a s2 = |b| µ = 321.75 MeV µ = 328.00 MeV 40 µ = 335.00 MeV I S 2 20 M ≠ 0 S 1 0.02 0 a/(d Λ 4 ) 240 260 280 300 320 340 µ (MeV) S 2 0.01 M = 0 I II 0 S 1 tcp M = / 0 -0.01 -0.3 -0.15 0 0.15 2 ) b/(d Λ Phase diagram in ( b , a )-space for the GL free energy. A line of first-order transitions (I) meets a line of second-order transitions (II) at the tricritical point (tcp), which is located at the origin. The equation for I is a c = 3 | b | 2 / (16 d ) , and that for II is a c = 0 . The dashed lines denote the spinodals S 1 and S 2 IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 14

D YNAMICAL EQUATIONS (TDGL EQNS ) Consider a system which is rendered thermodynamically unstable by a rapid quench from the disordered (symmetric) phase to the ordered (broken-symmetric) phase. The unstable homogeneous state (with M ≃ 0 ) evolves via the emergence and growth of domains rich in the preferred phase (with M � = 0 ). Such far-from-equilibrium evolution, is termed phase ordering dynamics or domain growth or coarsening . Most problems in this area historically arise from condensed matter systems. Equally fascinating is the kinetics of chiral transition! IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 15

TDGL CONTD . · · · Since coarsening system is inhomogeneous one includes a gradient term in the GL free energy � � F ( M ) + K � 2 � � � Ω [ M ] = d� r ∇ M 2 The evolution of the system is described by the Langevin equation with an inertial term: ∂ 2 γ ∂ r, t ) = − δ Ω [ M ] ∂t 2 M ( � r, t ) + ¯ ∂t M ( � + θ ( � r, t ) δM which models the relaxational dynamics of M ( � r, t ) to the minimum of Ω [ M ] (dissipative which damps the system towards the equillibrium configuration). γ : damping coefficient. θ ( � r, t ) represents the Langevin noise force assumed to be Gaussian and white satisfying the fluctuation-dissipation relation � θ ( r , t ) � = 0 and γTδ ( r ′ − r ′′ ) δ ( t ′ − t ′′ ) � θ ( r ′ , t ′ ) θ ( r ′′ , t ′′ ) � = 2¯ IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 16

TDGL CONTD · · · Rescaling M 0 M ′ , � M = M 0 = | a | / | b | , ξ� � r ′ , r � = ξ = K/ | a | , t 0 t ′ , � t = t 0 = 1 / | a | , | a | M 0 θ ′ . θ = (3) Dropping primes, we obtain the dimensionless TDGL equation: ∂ 2 r, t ) + γ ∂ r, t ) = − sgn (a) M − sgn (b) M 3 − λM 5 + ∇ 2 M + θ ( � ∂t 2 M ( � ∂t M ( � r, t ) , where λ = | a | d/ | b | 2 > 0 . IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 17

Early time behavior r ) = ¯ Consider the deterministic equation ( θ = 0 ) around an extremum pt. ( M ( � M + φ ( � r ) ) in the Fourier space ∂ 2 k, t ) + γ ∂ ∂t 2 φ ( � ∂t φ ( � k, t ) + ( − α + k 2 ) φ ( � k, t ) = 0 , α = − f ′′ ( M ) , ( α > 0 ¯ M - local Max; α < 0 – local Min.) General soln. k, t ) = A 1 e Λ + ( � k ) t + A 2 e Λ − ( � φ ( � k ) t γ 2 + 4( α − k 2 ) � k ) = − γ ± Λ ± ( � . 2 For α > 0 - instability for long wavelength ( k < √ α )(exponential growth of fluctuations) For α < 0 , no instability: fluctuations are exponentially damped. The damping is relaxational for k 2 < ( γ 2 − 4 | α | ) / 4 and oscillatory for k 2 > ( γ 2 − 4 | α | ) / 4 IOP HEP SEMINAR, Bhubaneswar June 5, 2013 – p. 18