Neutrino Interactions in Neutron Star Matter Omar Benhar INFN and Department of Physics Sapienza Università di Roma I-00185 Roma, Italy Based on work done in collaboration with Andrea Cipollone, Alessandro Lovato, and Cristina Losa PAFT14, Vietri sul mare, April 14, 2014
Outline Motivation Definition of neutrino mean free path Models of nuclear dynamics phenomenological hamiltonianas and effective interactions mean field approximation: nucleon effective mass long range correlations and collective excitations Matter response and neutrino mean free path Summary and prospects Omar Benhar (INFN, Roma) PAFT14 2 / 23
Motivation The role of weak interactions driving the opacity of neutron star matter to neutrinos has long been recognized (see, e.g. Burrows & Sawyer, 1998) Neutrino opacity, conveniently parametrized in terms of neutrino mean free path, is a critical input required for simulations of neutrino transport, which are in turn needed to determine gravitational wave emission from Overview proto-neutron stars (flow chart courtesy of Morgane Fortin) Equation of state P = P (n, T, Y ) Diffusion coefficient D = D (n, T, Y ) Evolution equations : Transport equations Structure equations Profiles in P, n, T and Y Oscillations and GW emission Frequency and damping time of the quasinormal modes Omar Benhar (INFN, Roma) PAFT14 3 / 23
Neutrino cross section and mean free path The mean free path λ is obtained from 1 d 3 λ = ρ q W(q, ω), (2π) 3 where W(q, ω) is the scattering rate at momentum transfer q and energy transfer ω. Consider neutral current interactions, as an example with W(q, ω) = G2 F 4π 2 1 EE L µνw µν, L µν = k µ k ν + k ν k µ g µν (kk ) + iɛ µανβ k α k β, k (k 0, k) and k (k 0, k ) being the four momenta of the incoming and outgoing neutrino, respectively Omar Benhar (INFN, Roma) PAFT14 4 / 23
The matter response tensor The calculation of the tensor W µν = 0 J µ n n J ν 0 δ (4) (p 0 + k 0 p n k 0 ) n requires a consistent description of the target internal dynamics, determining the initial and final states, 0 and n, as well as of the nuclear weak current J µ J µ = [ j µ V (i) j µ A (i)] In the non relativistic limit i j µ V = ψ nγ µ ψ n ψ nψ n δ µ o, j µ A = ψ nγ µ γ 5 ψ n ψ nσ i ψ n δ µ i, Omar Benhar (INFN, Roma) PAFT14 5 / 23
In the non relativistic limit, the scattering rate reduces to W(q, ω) = G2 F (1 4π 2 + cos θ)s ρ (q, ω) + C2 A 3 (3 cos θ)sσ (q, ω), the density and spin responses being given by S ρ (q, q 0 ) = 1 n O ρ N q 0 2 δ(q 0 +E 0 E n ), S σ (q, q 0 ) = Sαα(q, σ q 0 ), with n S σ αβ (q, q 0) = 1 N O ρ q (i) = n n O σ α q 0 0 O σ β q n δ(q 0 + E 0 E n ), e iq r i, O σ q (i) = e iq r i σ i. i i α Omar Benhar (INFN, Roma) PAFT14 6 / 23
0 and n are eigenstates of the nuclear hamiltonian p 2 i H = T + V = 2m + v ij +... i j>i Nucleon-nucleon interaction in the 1 S 0 channel, dominant in the neutron-neutron sector Omar Benhar (INFN, Roma) PAFT14 7 / 23
Correlated basis function formalism The eigenstates of the nuclear hamiltonian are approximated by the set of correlated states, obtained from independent particle model states [e.g. Fermi Gas (FG) states for nuclear matter] n = F n FG n FG F F n FG 1/2 = 1 F n FG, F = S Nn the structure of the two-nucleon correlation operator reflects the complexity of nuclear dynamics f ij = [f TS (r ij ) + δ S1 f tt (r ij )S ij ]P ST S,T=0,1 P ST spin isospin projector operator, S ij = σ α i σβ j j>i r α ij rβ ij shapes of f TS (r ij ) and f tt (r ij ) determined form minimization of the ground-state energy r 2 ij f ij δ αβ Omar Benhar (INFN, Roma) PAFT14 8 / 23
Nucleon-nucleon potential and correlation functions Omar Benhar (INFN, Roma) PAFT14 9 / 23
The CBF formalism can be exploited to obtain a well behaved effective interaction defined through 0 H 0 0 0 = 0 FG T + V eff 0 FG with and V eff = v eff (ij) j>i v eff (ij) = v n eff (r ij)o n ij, n O n ij = [11, (τ i τ j )] [11, (σ i σ j ), S ij ]. The resulting v eff is well behaved, and suitable for perturbation theory in the Fermi gas basis. Omar Benhar (INFN, Roma) PAFT14 10 / 23
CBF effective interaction v(r) [MeV] 1000 800 600 400 200 (a) v eff S=0,T =1 (r) v bare S=0,T =1 (r) v(r) [MeV] 1000 800 600 400 200 (b) v eff S=1,T =1 (r) v bare S=1,T =1 (r) 0 0-200 0 0.5 1 1.5 2 2.5 r [fm 1 ] -200 0 0.5 1 1.5 2 2.5 r [fm 1 ] v(r) [MeV] 80 60 40 20 0 (c) v eff t,t =1 (r) v bare t,t =1 (r) -20-40 0 0.5 1 1.5 2 2.5 r [fm 1 ] Omar Benhar (INFN, Roma) PAFT14 11 / 23
The mean field approximation Consider a non interacting Fermi gas: energy spectrum of single-particle states e 0 (k) = k2 2m = 1 m = 1 de 0 k dk In the mean-field (Hartree-Fock) approximation e(k) = k2 2m + U(k), U(k) = kk v eff kk a n(k ) k {F} where n(k ) is the Fermi distribution and [ kk a = kk k k ]/ 2. The effective mass is defined through 1 m = 1 de k dk = 1 m + du dk As a first approximation, interaction effects can be described within the Fermi gas model, replacing m m Omar Benhar (INFN, Roma) PAFT14 12 / 23
Consider, for simplicity, a system of neutrons interacting through a spin-independent potential d 3 k [ U(k) = 2 (2π) 3 n(k ) d 3 x v eff (x) 1 1 ] e ik x e ik x 2 [ = ρ d 3 x v eff (x) 1 1 ] e ik x l(k F x), 2 where ρ = kf 3 /3π2 is the density and l(x) = 1 d 3 k eik x ρ (2π) 3 n(k), the Fermi distribution at temperature T = 1/β and chemical potential µ being defined as n(k) = { 1 + exp β[e(k) µ] } 1 At T = 0, n(k) = θ(µ e(k)), µ = e(k F ), l(x) = 3 sin x x cos x x 3. Omar Benhar (INFN, Roma) PAFT14 13 / 23
Pure neutron matter at T = 0 and ρ = ρ 0 = 0.16 fm 3 Comparison between Hartree-Fock and kinetic energy spectra Momentum dependence of the effective mass Omar Benhar (INFN, Roma) PAFT14 14 / 23
Pure neutron matter at T = 0 Density-dependence of m (k F )/m, obtained using two slightly different effective interactions Omar Benhar (INFN, Roma) PAFT14 15 / 23
β-stable matter at T = 0 Consider charge neutral npe matter. At fixed baryon density, the proton fraction x p is determined from Proton and neutron spectra at ρ = ρ 0 µ n = µ p + µ e Omar Benhar (INFN, Roma) PAFT14 16 / 23
β-stable matter at T = 0 Density dependence of the proton and neutron effective masses at the Fermi surface Omar Benhar (INFN, Roma) PAFT14 17 / 23
T-dependence of neutron and proton effective mass Omar Benhar (INFN, Roma) PAFT14 18 / 23
Neutrino cross sections in matter at T = 0 Recall 1 λ = 1 σρ, σ = d 3 k d3 σ d 3 k Figura 4.6. Sezione Differential cross sections at ρ = 0.2 fm 3 d urto di erenziale relativistica in funzione dell energia trasferita ottenuta usando le masse e ettive dell interazione CBF per la materia stabile (lin continua) confrontata con il risultato ottenuto usando le masse nude (linea tratteggiat 4.2 Sezione d urto La 63 densitá barionica é nb =0.2 fm 3, l energia del neutrino iniziale é k0 =5Mev e momento trasferito é q =2.5 Mev. Figura 4.6. Sezione d urto di erenziale relativistica in funzione dell energia trasferita q0 ottenuta usando le masse e ettive dell interazione CBF per la materia stabile (linea continua) confrontata con il risultato ottenuto usando le masse nude (linea tratteggiata). La densitá barionica é nb =0.2 fm 3 Figura 4.7. Singoli contributi dei neutroni (linea tratteggiata e con punti), protoni (lin, l energia del neutrino iniziale é k0 =5Mev eil tratteggiata) ed elettroni (linea continua) alla sezione d urto di erenziale relativistica momento trasferito é q =2.5 Mev. Omar Benhar (INFN, Roma) funzione dell energia trasferita q0 a densità barionica nb =0.2 PAFT14 fm 3. 19 / 23
Beyond the mean field approximation Within the mean field approximation neutrino interactions lead to the excitation of one particle-one hole (1p1h), implying q 0 = e(p) e(h) At low momentum transfer, such that q 1 d, where d is the average interparticle distance, this scheme is no longer applicable. The occurrence of collective excitations must be taken into account Propagation of the particle-hole pair, giving rise to the collective mode, can be described replacing ph n = N C i p i h i The energy of the state n and the coefficients C i are obtained diagonalizing the N N hamiltonian matrix i=1 H ij = (E 0 + e(p i ) e(h i ))δ ij + p i h i V eff p j h j Omar Benhar (INFN, Roma) PAFT14 20 / 23
Neutrino responses at T = 0 and ρ = ρ 0 Recall 1 λ = G2 F 4 ρ d 3 q [ (1 + cos θ)s ρ (2π) 3 (q, q 0 ) + CA 2 (3 cos θ)sσ (q, q 0 ) ] S ρ (q, ω) [MeV 1 ] Density and spin responses at k 0 = 0.1fm 1 0.012 0.01 0.008 0.006 0.004 0.002 Landau CTD CHF S σ (q, ω) [MeV 1 ] 0.15 0.125 0.1 0.075 0.05 Landau CTD transverse CTD longitudinal CHF transverse CHF longitudinal 0 0 2 4 6 8 10 ω [MeV] 0.025 0 0 2 4 6 8 10 ω [MeV] Omar Benhar (INFN, Roma) PAFT14 21 / 23
Neutrino mean free path at T = 0 and ρ = ρ 0 Recall 1 λ = G2 F 4 ρ 2.6 2.4 d 3 q [ (1 + cos θ)s ρ (2π) 3 (q, q 0 ) + CA 2 (3 cos θ)sσ (q, q 0 ) ] CTD full expression CTD simplified expression CTD without collective mode λ/λf G 2.2 2 1.8 1.6 5 10 15 20 25 30 35 40 E ν [MeV] Omar Benhar (INFN, Roma) PAFT14 22 / 23
Summary & Outlook The understanding of neutrino transport properties requires a realistic description of nuclear dynamics. Within the mean field approximation, dynamical effects can be included replacing the bare masses of the nucleons with effective masses, computed using effective interactions obtained from phenomenological nuclear hamiltonians At low momentum transfer, the excitation of collective modes plays an important role, leading to a large increase of the response in the spin channel. A unified description of the response of cold nuclear matter to neutrino interactions, applicable in a broad kinematical range, is available. The extension to non zero temperatures, needed for applications to proto neutron stars, appears to be feasible. Omar Benhar (INFN, Roma) PAFT14 23 / 23
Background slides Omar Benhar (INFN, Roma) PAFT14 24 / 23
Alternative approach: Landau theory Landau theory of normal Fermi liquids can also be employed to obtain the density and spin responses of pure neutron matter the value of the Landau parameters can be obtained from the quasiparticle interaction, which can be in turn expressed in terms of matrix elements of the effective interaction f σσ pp = f pp + g pp (σ σ ) + f pp S 12 (p p ) = pσ p σ V eff pσ p σ pσ p σ V eff p σ pσ this formalism allows for a consistent treatment of single quasi particle excitations and collective modes, and can be easily extended at non zero temperatures T << T F Omar Benhar (INFN, Roma) PAFT14 25 / 23
Neutrino mean free path in neutron matter Mean free path of non degenerate neutrinos interacting through neutral current at zero temperature 1 λ = G2 F d 3 4 ρ q [ (1 + cos θ)s(q, ω) + C 2 (2π) 3 A (3 cos θ)s(q, ω) ] where S and S are the density (Fermi) and spin (Gamow Teller) response, respectively the collective mode is only excited in the spin channel Omar Benhar (INFN, Roma) PAFT14 26 / 23
Mean free path of a non degenerate neutrino in neutron matter. Left: density-dependence at k 0 = 1 MeV and T = 0 ; Right: energy dependence at ρ = 0.16 fm 3 and T = 0, 2 MeV Omar Benhar (INFN, Roma) PAFT14 27 / 23
Density and temperature dependence of the mean free path of a non degenerate neutrino at k 0 = 1 MeV and ρ = 0.16 fm 3 Omar Benhar (INFN, Roma) PAFT14 28 / 23