Pauli blocking in the pion gas - a lesson for compact star physics 1 - PowerPoint PPT Presentation

Pauli blocking in the pion gas - a lesson for compact star physics 1 David Blaschke Institute of Theoretical Physics, University Wroc� law, Poland Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, Russia International Conference ”MESON 2018” Krak´ ow, 7 June 2018 1 Collab: N.-U. Bastian, A. Dubinin, A. Friesen, H. Grigorian, G. R¨ opke, L. Turko David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 1 / 26

QCD Phase Diagram with Clustering Aspects Strongly interacting matter LHC RHIC Energy Scan NICA / FAIR Lattice QCD e a v o n r e p u S nuclear neutron stars deconfinement saturation David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 2 / 26

Φ—Derivable Approach to the Cluster Virial Expansion   A A     � � − G − 1 � � � � � Ω = Ω l = Tr ln + Tr (Σ l G l ) + Φ[ G i , G j , G i + j ] c l , l l =1 l =1   i , j   i + j = l δ Φ G (0) − 1 − Σ A , Σ A (1 . . . A , 1 ′ . . . A ′ , z A ) = G − 1 = δ G A (1 . . . A , 1 ′ . . . A ′ , z A ) A A Stationarity of the thermodynamical potential is implied δ Ω δ G A (1 . . . A , 1 ′ . . . A ′ , z A ) = 0 . Cluster virial expansion follows for this Φ − functional ≡ Φ = Figure: The Φ functional for A − particle correlations with bipartitions A = i + j . David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 3 / 26

Green’s function and T-matrix: separable approximation The T A matrix fulfills the Bethe-Salpeter equation in ladder approximation T i + j (1 , 2 , . . . , A ; 1 ′ , 2 ′ , . . . A ′ ; z ) = V i + j + V i + j G (0) i + j T i + j , which in the separable approximation for the interaction potential, V i + j = Γ i + j (1 , 2 , . . . , i ; i + 1 , i + 2 , . . . , i + j )Γ i + j (1 ′ , 2 ′ , . . . , i ′ ; ( i + 1) ′ , ( i + 2) ′ , . . . , ( i + j ) ′ ) , leads to the closed expression for the T A matrix � − 1 , T i + j (1 , 2 , . . . , i + j ; 1 ′ , 2 ′ , . . . ( i + j ) ′ ; z ) = V i + j � 1 − Π i + j with the generalized polarization function � Γ i + j G (0) Γ i + j G (0) � Π i + j = Tr i j The one-frequency free i − particle Green’s function is defined by the ( i − 1)-fold Matsubara sum G (0) 1 1 1 (1 , 2 , . . . , i ; Ω i ) = � ω 2 − E (2) . . . ω 1 ...ω i − 1 i ω 1 − E (1) Ω i − ( ω 1 + ...ω i − 1 ) − E ( i ) (1 − f 1 )(1 − f 2 ) ... (1 − f i ) − ( − ) i f 1 f 2 ... f i = . Ω i − E (1) − E (2) − ... E ( i ) David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 4 / 26

Useful relationships for many-particle functions G (0) i + j = G (0) G (0) (1 , 2 , . . . , i ; Ω i ) G (0) � i + j (1 , 2 , . . . , i + j ; Ω i + j ) = ( i + 1 , i + 2 , . . . , i + j ; Ω j ) . i j Ω i Another set of useful relationships follows from the fact that in the ladder approximation both, the full two-cluster ( i + j particle) T matrix and the corresponding Greens’ function G i + j = G (0) � − 1 � 1 − Π i + j (1) i + j have similar analytic properties determined by the i + j cluster polarization loop integral and are related by the identity T i + j G (0) i + j = V i + j G i + j . (2) which is straightforwardly proven by multiplying Equation for the T i + j − matrix with G (0) i + j and using Equation (1). Since these two equivalent expressions in Equation (2) are at the same time equivalent to the two-cluster irreducible Φ functional these functional relations follow T i + j = δ Φ /δ G (0) i + j , V i + j = δ Φ /δ G i + j . Next we prove the relationship to the Generalized Beth-Uhlenbeck approach! David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 5 / 26

GBU EoS from the Φ − derivable approach Consider the partial density of the A − particle state defined as d 3 q ∂ Ω A ∂ d ω � � � � − G − 1 � � � n A ( T , µ ) = − = − d A ln + Tr (Σ A G A ) + Φ[ G i , G j , G i + j ] . (3) A (2 π ) 3 ∂µ ∂µ 2 π i , j Using spectral representation for F ( ω ) and Matsubara summation i + j = A � ∞ d ω Im F ( ω ) c A 1 � F ( iz n ) = , = f A ( ω ) = exp[( ω − µ ) / T ] − ( − 1) A 2 π ω − iz n ω − iz n −∞ zn with the relation ∂ f A ( ω ) /∂µ = − ∂ f A ( ω ) /∂ω we get for Equation (3) now d 3 q � d ω ∂ Φ[ Gi , Gj , GA ] � � − G − 1 � � 2 π f A ( ω ) ∂ n A ( T , µ ) = − d A � Im ln + Im (Σ A G A ) + � , (2 π )3 i , j ∂ω A ∂µ i + j = A where a partial integration over ω has been performed For two-loop diagrams of the sunset type holds a cancellation 2 which we generalize here for cluster states d 3 q d ω ∂ ∂ Φ[ G i , G j , G A ] � � � d A f A ( ω ) ( Re Σ A Im G A ) − = 0 . (2 π ) 3 2 π ∂ω ∂µ i , j i + j = A Using generalized optical theorems we can show that ( G A = | G A | exp( i δ A )) ∂ ∂ ∂δ A � � = − 2 sin 2 δ A � � − G − 1 � � G ∗ Im ln + Im Σ A Re G A = 2 Im G A Im Σ A A Im Σ A . A ∂ω ∂ω ∂ω The density in the form of a generalized Beth-Uhlenbeck EoS follows A A d 3 q d ω ∂δ i � � f i ( ω )2 sin 2 δ i � � n ( T , µ ) = n i ( T , µ ) = d i . (2 π ) 3 2 π ∂ω i =1 i =1 2B. Vanderheyden & G. Baym, J. Stat. Phys. (1998), J.-P. Blaizot et al., PRD (2001) David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 6 / 26

Example: Deuterons in Nuclear Matter The Φ − derivable thermodynamical potential for the nucleon-deuteron system reads Ω = − Tr { ln( − G 1 ) } − Tr { Σ 1 G 1 } + Tr { ln( − G 2 ) } + Tr { Σ 2 G 2 } + Φ[ G 1 , G 2 ] , where the full propagators obey the Dyson-Schwinger equations G − 1 (1 , z ) = z − E 1 ( p 1 ) − Σ 1 (1 , z ); G − 1 (12 , 1 ′ 2 ′ , z ) = z − E 1 ( p 1 ) − E 2 ( p 2 ) − Σ 2 (12 , 1 ′ 2 ′ , z ) , 1 2 with selfenergies and Φ functional δ Φ δ Φ Σ 1 (1 , 1 ′ ) Σ 2 (12 , 1 ′ 2 ′ , z ) = = δ G 1 (1 , 1 ′ ) ; δ G 2 (12 , 1 ′ 2 ′ , z ) , Φ = , fulfilling stationarity of the thermodynamic potential ∂ Ω /∂ G 1 = ∂ Ω /∂ G 2 = 0 . For the density we obtain the cluster virial expansion n = − 1 ∂ Ω ∂µ = n qu ( µ, T ) + 2 n corr ( µ, T ) , V with the correlation density in the generalized Beth-Uhlenbeck form � dE 2 π g ( E )2 sin 2 δ ( E ) d δ ( E ) n corr = . dE David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 7 / 26

Example: Deuterons in Nuclear Matter T = 5 MeV n = 0.001 fm -3 3 n = 0.003 fm -3 n = 0.01 fm -3 scattering phase shift n = 0.03 fm -3 n = 0.1 fm -3 2 1 0 0 10 20 30 40 50 60 70 energy E rel [MeV] Figure: Integrand of the intrinsic partition function as function of the c.m.s. energy in the deuteron channel. Mott dissociation and Levinson’s theorem! From G. R¨ opke, J. Phys. Conf. Ser. 569 (2014) 012014. David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 8 / 26

Cluster Virial Expansion for Quark-Hadron Matter within the Φ Derivable Approach � − G − 1 � � � � Ω = c i Tr ln + Tr (Σ i G i ) + Φ [ G Q , G M , G D , G B ] , i i = Q , M , D , B � d ω d 3 q � ∂δ i � � 1 − e − ω/ T �� 2 sin 2 δ i � = ω + 2 T ln d i ∂ω . (2 π ) 3 2 π i = Q , M , D , B Figure: Φ functional for the quark-meson-diquark-baryon system in 2-loop approx. Σ i = δ Φ [ G Q , G M , G D , G B ] . δ G i David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 9 / 26

The selfenergies ... Figure: Selfenergies for Greens functions of Q-M-D-B system in 2-loop approx. David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 10 / 26

Mott Dissociation of Pions in Quark Matter Figure: The Φ functional (left panel) for the case of mesons in quark matter, where the bosonic meson propagator is defined by the dashed line and the fermionic quark propagators are shown by the solid lines with arrows. The corresponding meson and quark selfenergies are shown in the middle and right panels, respectively. David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 11 / 26

Mott Dissociation of Pions in Quark Matter The meson polarization loop Π M ( q , z ) enters the definition of the meson T matrix − Π M ( q , ω + i η ) = | T M ( q , ω ) | − 1 e − i δ M ( q ,ω ) , T − 1 M ( q , ω + i η ) = G − 1 S which in the polar representation introduces a phase shift δ M ( q , ω ) = arctan( ℑ T M / ℜ T M ), that results in a generalized Beth-Uhlenbeck equation of state for the thermodynamics of the consistently coupled quark-meson system Ω = Ω MF + Ω M , where the selfconsistent quark meanfield contribution is Ω MF = σ 2 d 3 p � � � 1 + e − ( E p − Σ + − µ ) / T � � 1 + e − ( E p +Σ − + µ ) / T �� MF − 2 N c N f E p + T ln + T ln , 4 G S (2 π ) 3 The mesonic contribution to the thermodynamics is � d ω d 3 k 2 sin 2 δ M ( k , ω ) δ M ( k , ω ) � � � 1 − e − ω/ T �� Ω M = d M ω + 2 T ln , (2 π ) 3 2 π d ω David Blaschke (IFT, Wroc� law) Cluster Virial Expansion for Quark Matter 07.06.2018 12 / 26