Pions in the quark matter phase diagram

Transcription

1 Pions in the quark matter phase diagram Daniel Zabłocki Instytut Fizyki Teoretycznej, Uniwersytet Wrocławski, Poland Institut für Physik, Universität Rostock, Germany Bogoliubov Laboratory of Theoretical Physics, Joined Institute for Nuclear Research Dubna, Russia HISS Dubna 28 - Dense Matter Dubna, 21 July 28 in collaboration with David Blaschke and Roberto Anglani 1

2 Aim of the talk investigation of the phase diagram of QCD beyond mean field level in NJL framework 2

3 1. introduction to the problem one challenging problem of quantum chromodynamics is the study of phase diagram T? µ 3

4 1.1. what we know about phase diagram QGP RHIC T c SPS E SPS low _ c c gg _ q q qq H M CSC Zhuang, P.F. et al [hep-ph] Shuryak and Zahed hep-ph/

5 2. the effective model how to give a reliable description in the region around the critical values of chemical potential? perturbation theory cannot be applied in this region we have to accept a good compromise. an effective model: the Nambu--Jona-Lasinio 5

6 2.1 Nambu--Jona-Lasinio The NJL model of QCD mimics the quark-quark interaction mediated by gluons with an effective point-like four fermion interaction cons absence of gluon in the Lagrangian; quarks are not confined; etc. pro a simple approach to the description of chiral symmetry breaking and phase transitions; analytical calculations possible 6

7 2.2 the starting point: the NJL Lagrangian For the description of hot, dense Fermi-systems, with strong short-range interactions we consider a Lagrangian with internal degrees of freedom (2-flavor, 3-color), with a current -current-type four-fermion interaction L = L + L qq + L q q L = q(i/ m + µγ )q L q q = G S [ ( qq) 2 + ( qiγ 5 τ q) 2] L qq = G D q = ψ 1 ψ 2 ψ 3 ψ 4 ( u d A=2,5,7 ) r g b [ qiγ5 Cτ 2 λ A q T ] [ q T icγ 5 τ 2 λ A q ] m,u = m,d = m µ u = µ d = µ τ = (τ 1, τ 2, τ 3 ) C = iγ 2 γ G D Scalar diquark coupling strength Scalar and pseudoscalar coupling strength 7 G S

8 the partition function [ ] L eff = σ2 + π 2 4G S Z = A A 4G D 2.3 the partition function [ ] Z = [ β [dq] [d q] exp dτ d 3 x L Hubbard-Stratonovich auxiliary fields [ ] [ β [dq] [d q] [d A ] [d A ] [dσ] [dπ] exp dτ ] d 3 x L ] [ ] + q(i/ m + µγ )q q(σ + iγ 5 τ π)q + i A 2 qt icγ 5 τ 2 λ A q i A 2 qiγ 5Cτ 2 λ A q T here the sum over repeated indices is implied. In ( the) Nambu-Gorkov formalism, it is possible to get a more com ( ) [ Nambu-Gorkov formalism Ψ 1 ( ) ( q Ψ 2 q c 1 ) (] ) q [ q c q c (x) C q T (x). 2 [ ] [ β Z = [d A ] [d A ] [dσ] [dπ] exp dτ d 3 x ( σ2 + π 2 A ) ] A [dψ] [ d 4G S 4G Ψ ] [ β ] exp dτ d 3 x Ψ S 1 Ψ ( D with the inverse propagator defined as ( ( ) ) S 1 i / + µγ m σ iγ 5 τ π A γ 5 τ 2 λ A A γ 5τ 2 λ A i/ µγ m σ iγ 5 τ t π Ω = T ln Z Z = [ β [d A ] [d A {exp ] [dσ] [dπ] dτ d 3 x ( σ2 + π 2 4G S A ) ] A exp [ Tr ( ln S 1)]} 4G D 8

9 3. the mean field approximation ( ) Z = [ β [d A ] [d A {exp ] [dσ] [dπ] dτ how to calculate this? d 3 x ( σ2 + π 2 4G S A ) ] A exp [ Tr ( ln S 1)]} 4G D the mean field approximation (MFA) decompose bosonic collective fields into a ] homogeneous MF part fluctuation part + order parameter: characterization of phase structure MF + δ Ω m = Ω = m m = 8G S m = 8G D σ σ MF + σ Z MF = exp d 3 p 1 (2π) 3 d 3 p (2π) 3 E p [ βv ( σ2 MF + π2 MF 4G S { [1 2nF (E p ) ] ξ p [ 1 2nF (E p ) E p E p + 1 2n F (E + p ) E + p ( ) MF MF 4G D + [ 1 2n F (E + p ) ] ξ+ p ] correlations )] exp [ Tr ( ln S 1 )] MF E p + [ } ( ) + n F ( ξ p + ) n F (ξp ) ] 9 E ± p = (ξ ± p ) with ξ ± p = E p ± µ, E p = m 2 + p 2 associated to the red-green quark (E ) and antiquarks (E

10 3.1 results of MFA 4 m " D = 1. 4 m " D = 1. m,! [MeV] 3 2! m,! [MeV] 3 2! µ = 2 MeV µ = 31 MeV 1 T = MeV T = 1 MeV T = 15 MeV [ [ ] µ [MeV] T [MeV] Λ = MeV, m = MeV. C phase ( ). T G S Λ 2 = Buballa M. // Phys. Rept. 25 V.47.p.25. H. Grigorian, Phys. Part. Nucl. Lett. 4, 223 (27) [arxiv:hep-ph/62238]. Kapusta J., 1, Cambridge University Press, (1

11 the phase diagram MF 2 T [MeV] 15 1 "SB 5 BEC =.75 =1. =1.3 Crossover BCS 2SC ! [MeV]

12 ( ) Z = 4 what about fluctuations? [ β [d A ] [d A {exp ] [dσ] [dπ] dτ d 3 x ( σ2 + π 2 4G S A ) ] A exp [ Tr ( ln S 1)]} 4G D ln-expansion around MF values S 1 = S 1 MF + Σ, Σ ( σ iγ5 τ π δ A γ 5 τ 2 λ A δa γ 5τ 2 λ A σ iγ 5 τ t π ) Tr[ln(S 1 )] = Tr[ln(S 1 MF + Σ)] = Tr{ln[S 1 MF (1 + S MF Σ)]} = Tr ln S 1 MF + Tr ln[1 + S MF Σ] = Tr ln S 1 MF + Tr[S MF Σ 1 2 S MF ΣS MF Σ +... ] Π ππ π Π σσ Π σδ2 Π σδ 2 σ Tr (S MF ΣS MF Σ) = (π, σ, δ2, δ 2, δ5, δ7) Π δ 2 σ Π δ 2 δ 2 Π δ 2 δ2 δ 2 Π δ2 σ Π δ2 δ 2 Π δ2 δ2 δ 2 Π δ 5 δ 5 δ 5 Π δ 7 δ 7 δ 7 12

13 4.1 meson polarization functions and masses Π ππ (q, q) = 2 d 3 p (2π) 3 + t p,t k t p t k s p,s k T + (s p, s k ) E s p p E s k p+q { nf (s p ξ s p p ) n F (s k ξ s k p+q) q s k ξ s k p+q + s p ξ s p p n F (t p E s p p ) n F (t k E s k p+q) q t k E s k p+q + t p E s p p n F (s p ξ s p p ) n F (s k ξ s k p+q) q + s k ξ s k p+q s p ξ s p p ( tp t k E s p p E s k p+q + s p s k ξ s p p ξ s k p+q 2) Π σσ (q, q) = 2 d 3 p (2π) 3 + t p,t k t p t k { T nf (s p ξ s p p ) n F (s k ξ s k p+q (s p, s k ) ) q s p,s s k ξ s k k p+q + s p ξ s p p E s p p E s k p+q n F (t p E s p p ) n F (t k E s k p+q ) q t k E s k p+q + t pe s p p + n F (s p ξ s p p ) n F (s k ξ s k p+q ) q + s k ξ s k p+q s p ξ s p p ( t p t k E s p p E s k p+q + s ps k ξ s p p ξ s k p+q 2) } Similar equations can be derived for the other matrix elements Sun et al. Phys. Rev. D (27) in the 2-color limit Ebert et al. Phys. Rev. C (25) in the T= limit 13 T ± (s p, s k ) = 1 ± s p s k p k m 2 E p E k

14 4.2 the pion mass = 1. T=1 MeV 6 m " [MeV] [MeV] $ # m # 2m m m=! % Γ π = ( ReΠππ m 2 π ) 1 ImΠ ππ m π ! [MeV] 2 1 2G S Π(q, q = ) = T [MeV] 15 1 "SB 14 5 BEC =.75 =1. =1.3 Crossover BCS 2SC ! [MeV]

15 the phase diagram MF 2 T [MeV] 15 1 "SB 5 BEC =.75 =1. =1.3 Crossover BCS 2SC ! [MeV]

16 the phase diagram revisited 25 η D = 1. 2 T 4fm/c T 4 fm T [MeV] 15 1 χsr T 5!SB 2SC Γ π = ( ) 1! [MeV] ReΠππ ImΠ ππ m 2 π m π T (τ = Γ 1 π ) 16

17 RHIC phenomenology T [MeV] A #SB! " =! " >2 MeV 5 $ D = 1. 2SC ! [MeV] B QGP probed in RHIC is far to be a perfect liquid; an explanation: strong correlations in the plasma Shuryak and Zahed (PRL, 23) 17 Region dominated by strong correlated states with a lifetime > 1 fm/c

18 summary and outlook fluctuations are included in Gaussian approximation beyond MF; systematical treatment in the non-perturbative regime possible some properties of mesons are studied diquark calculations almost finished new insight for phase diagram; important for HIC and CSs investigate σ δ -mixing constraints of color and electrical neutrality and beta-equilibrium to be implemented (HIC and CSs) investigation of BEC-BCS crossover (strong coupling); see lectures of P.F. Zhuang the same formalism can be applied to Nuclear MF theory under investigation together with G. Röpke and D. Blaschke 18

19 acknowledgments Thanks for your attention 19