Cluster Production in pBUU - Past and Future Pawel Danielewicz - PowerPoint PPT Presentation

Introduction Real-Time Theory Applications Future Conclusions Cluster Production in pBUU - Past and Future Pawel Danielewicz National Superconducting Cyclotron Laboratory Michigan State University Transport 2017: International Workshop on Transport Simulations for Heavy Ion Collisions under Controlled Conditions FRIB-MSU, East Lansing, Michigan, March 27 - 30, 2017 Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Boltzmann Equation Model (BEM/pBUU) Degrees of freedom ( X ): nucleons, deuterons, tritons, helions ( A ≤ 3), ∆ , N ∗ , pions Fundamentals: Relativistic Landau theory (Chin/Baym) Energy functional ( ǫ ) Real-time Green’s function theory Production/absorption rates ( K < , K > ) ∂ f ∂ t + ∂ǫ ∂ f r − ∂ǫ ∂ f p = K < ( 1 ∓ f ) − K > f p r r p ∂ p p ∂ r ∂ r r ∂ p production absorption rate Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Single-Particle Energies & Functional ∂ f ∂ t + ∂ǫ ∂ f r − ∂ǫ ∂ f p = K < ( 1 ∓ f ) − K > f p r r p ∂ p p ∂ r ∂ r r ∂ p The single-particle energies ǫ are given in terms of the net energy functional E { f } by, δ E ǫ ( p ) = δ f ( p ) In the local cm, the mean potential is U opt = ǫ − ǫ kin p 2 + m 2 � and ǫ kin = Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Energy Functional The functional: E = E vol + E gr + E iso + E Coul E gr = a gr � d r ( ∇ ρ ) 2 where ρ 0 For covariant volume term, ptcle velocities parameterized in local frame: v ∗ ( p , ρ ) = p � �� � 2 p 2 + m 2 1 + c ρ 1 ρ 0 ( 1 + λ p 2 / m 2 ) 2 precluding a supraluminal behavior, with ρ - baryon density. The 1-ptcle energies are then � p dp ′ v ∗ + ∆ ǫ ( ρ ) ǫ ( p , ρ ) = m + 0 Parameters in the velocity varied to yield different optical potentials characterized by values of effective mass, m ∗ = p F / v F . Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Structure Interface Potential from p-scattering (Hama et al. PRC41(90)2737) & parameterizations Ground-state densities from electron scattering and from functional minimization. From E ( f ) = min : � ρ � � � p F ( ρ ) − 2 a gr ∇ 2 0 = ǫ − µ ρ 0 ⇒ Thomas-Fermi eq. Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Many-Body Theory Transport eq. for nucleons follows from the eq. of motion for the 1-ptcle Green’s function (KB eq.). Transport eq. for deuterons ( A = 2) from the eq. for 2-ptcle Green’s function?? Wigner function in second quantization � d r e − i pr � ˆ ψ † H ( R − r / 2 , T ) ˆ f ( p ; R , T ) = ψ H ( R + r / 2 , T ) � where �·� ≡ � Ψ | · | Ψ � and | Ψ � describes the initial state. Evolution driven by a Hamiltonian. Interaction Hamiltonian: H 1 = 1 � ˆ d x d y ˆ ψ † ( x ) ˆ ψ † ( y ) v ( x − y ) ˆ ψ ( y ) ˆ ψ ( x ) , 2 Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Evolution Contour � t 0 � � �� dt ′ ˆ � ˆ ˆ � T a H 1 I ( t ′ ) O H ( t 1 ) � = exp − i O I ( t 1 ) t 1 � t 1 � � �� dt ′ ˆ × T c H 1 I ( t ′ ) exp − i � t 0 � t 0 � � � � dt ′ ˆ ˆ H 1 I ( t ′ ) = � T exp − i − O I ( t 1 ) � , t 0 Expectation value expanded perturbatively in terms of V and noninteracting 1-ptcle Green’s functions on the contour � � ψ I ( x , t ) ˆ ˆ ψ † iG 0 ( x , t , x ′ , t ′ ) = � T I ( x ′ , t ′ ) � Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Single-Particle Evolution Wigner function corresponds to a particular case of the Green’s function on contour: � d r e − i pr ( ∓ i ) G < ( R + r / 2 , T , R − r / 2 , T ) f ( p ; R , T ) = If we find an equation for G , this will also be an equation for f . Dyson eq. from perturbation expansion: G = G 0 + G 0 Σ G Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Outcome of Evolution Formal solution of the Dyson eq: � ∓ iG < ( x , t ; x ′ , t ′ ) 1 G + ( x , t ; x 1 , t 1 ) d x 1 d t 1 d x ′ 1 d t ′ = × ( ∓ i )Σ < ( x 1 , t 1 ; x ′ 1 , t ′ 1 ) G − ( x , t ; x 1 , t 1 ) and ∓ i Σ < ( x , t ; x ′ , t ′ ) = � ˆ j † ( x ′ , t ′ )ˆ j ( x , t ) � irred where the source j is � H 1 � ˆ ψ ( x , t ) , ˆ ˆ j ( x , t ) = Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Quasiparticle Limit Under slow spatial and temporal changes in the system, the Green’s function expressible in terms of the Wigner function f and 1-ptcle energy ǫ p p ; x + x ′ , t + t ′ � ∓ iG < ( x , t ; x ′ , t ′ ) ≈ e i ( p ( x − x ′ ) − ǫ p ( t − t ′ )) � � d p f 2 2 Then also Boltzmann eq: ∂ t + ∂ǫ p ∂ r − ∂ǫ p ∂ f ∂ f ∂ f ∂ p = − i Σ < ( 1 − f ) − i Σ > f ∂ p ∂ r ∓ i Σ < : Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions 2-Particle Green’s Function Transport eq. for deuterons ( A = 2) from the eq. for 2-ptcle Green’s function?? � ˆ 1 t ′ ) ˆ 2 t ′ ) ˆ ψ ( x 2 t ) ˆ iG < ψ † ( x ′ ψ † ( x ′ = ψ ( x 1 t ) � 2 For the contour function: G 2 = G + G v G 2 where G – irreducible part of G 2 (w/o two 1-ptcle lines connected by the potential v ; anything else OK) In terms of retarded Green’s function G < 2 : iG < 1 + v G + i G < � 1 + v G − � � � 2 = 2 2 Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Deuteron Quasiparticle Limit In the limit of slow spatial and temporal changes, deuteron contribution to the 2-ptcle Green’s function: � ˆ 1 t ′ ) ˆ 2 t ′ ) ˆ ψ ( x 2 t ) ˆ iG < ψ † ( x ′ ψ † ( x ′ = ψ ( x 1 t ) � 2 � x 1 + x 2 x ′ 1 + x ′ � � 2 d ( r ′ ) φ d ( r ) e i p − e − i ǫ d ( t − t ′ ) d p f d ( p R T ) φ ∗ ≃ 2 2 + · · · , where R = 1 4 ( x 1 + x 2 + x ′ 1 + x ′ 2 ) , r = x 1 − x 2 φ d and f d – internal wave function and cm Wigner function · · · ≡ continuum Transport eq from integral quantum eq of motion: ∂ f d ∂ T + ∂ǫ d ∂ f d ∂ R − ∂ǫ d ∂ f d ∂ p = K < ( 1 + f d ) − K > f d ∂ p ∂ R Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Wave Equation From Green’s function eq, the equation for wavefunction: ( ǫ d ( P ) − ǫ N ( P / 2 + p ) − ǫ N ( P / 2 − p )) φ d ( p ) � d p ′ v ( p − p ′ ) φ d ( p ′ ) = 0 − ( 1 − f N ( P / 2 + p ) − f N ( P / 2 − p )) In zero-temperature matter, discrete states lacking over a vast range of momenta phenomenological cut-of f Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Cluster Production & Absorption > > < = φ ∗ v i G < v φ ?? Production & absorption rates: i K Leading contribution � K < = d r d r ′ φ ∗ d v � ˆ 1 t ′ ) ˆ ψ ( x 1 t ) � � ˆ 2 t ′ ) ˆ ψ † ( x ′ ψ † ( x ′ ψ ( x 2 t ) � v φ d Leading-order in the quasiparticle expansion: neutron & proton come together and make a deuteron. If system approximately uniform and stationary, the process not allowed by energy-momentum conservation. Process possible in a mean field varying in space, but, in nuclear case, the high-energy production rate low – tested in Glauber model. Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions 3-Nucleon Collisions First correction to the pure 1-ptcle state, from a coupling to p-h excitations, yields a contribution to the d-production due to 3-nucleon collisions. Still more nucleons involved in production of heavier clusters. Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Deuteron Production Detailed balance: |M npN → Nd | 2 = |M Nd → Nnp | 2 ∝ d σ Nd → Nnp Thus, production can be described in terms of breakup. Problem: Breakup cross section only known over limited range of final states - Interpolation/extrapolation needed Impulse approximation works at high incident energy Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Renormalized Impulse Approximation Renormalization factor for squared matrix element to get breakup cross section right as a function of energy d σ Nd → Nnp ∝ F σ NN | φ d ( p ) | 2 Clusters in pBUU Danielewicz

Introduction Real-Time Theory Applications Future Conclusions Single-Particle Spectra proton & deuteron inclusive spectra histograms: calculations using |M npN → Nd | 2 = |M Nd → npN | 2 ∝ d σ Nd → npN and � f � < 0 . 2 cut-off for deuterons Clusters in pBUU Danielewicz