Lecture III Supernova Neutrinos A Neutron Star is Born 1500 km Shock wave ~1051ergs 3X107 km B. E. ~2-3 1053 ergs 10 km proto-neutron star: t~1-2 s 100 km Epochs in the evolution of a neutron star. Prakash et al., Phys. Rept. (1990) Traversing the Phase Diagram T (MeV) T 170 QGP 50 Novel High Density Phases: 10 Hyperons, Kaons Quark Matter .. Nuclear Matter Nuclei Neutron Stars 930 ~1200 ? ? µΒ (MeV) Traversing the Phase Diagram T (MeV) T 170 QGP 50 Novel High Density Phases: 10 Hyperons, Kaons Quark Matter .. Nuclear Collapse Matter Nuclei Neutron Stars 930 ~1200 ? ? µΒ (MeV) Traversing the Phase Diagram T (MeV) T 170 QGP PNS Evolution 50 Novel High Density Phases: 10 Hyperons, Kaons Quark Matter .. Nuclear Collapse Matter Nuclei Neutron Stars 930 ~1200 ? ? µΒ (MeV) Supernova Neutrinos Future: 3 ×1053 ergs = 1058 × 20 MeV Neutrinos Can detect ~10,000 neutrinos from galactic supernova Past: SN 1987a: ~ 20 neutrinos ..in support of supernova theory 28 np~ 6.7 × 10 =number of protons per ton Mtons= detector mass in tons -44 2 σref ~ 2 × 10 cm = weak cross-section D ~ 10 kpc = distance to supernova Supernova Neutrinos Future: 3 ×1053 ergs = 1058 × 20 MeV Neutrinos Can detect ~10,000 neutrinos from galactic supernova Past: SN 1987a: ~ 20 neutrinos ..in support of supernova theory 28 np~ 6.7 × 10 =number of protons per ton Mtons= detector mass in tons -44 2 σref ~ 2 × 10 cm = weak cross-section D ~ 10 kpc = distance to supernova Dimensional Analysis with G, M, and R R 2GM Compactness: s = 0.3 0.4 R Rc2 − GM m 1 R Binding Energy: n = s m = 140 200 MeV R 2 R n − R3 R R R 3/2 τFree Fall = = 80 ms − GM c R 1000 km s R R R τHydro = τFree Fall = c − c R s s Weak Interaction Time-Scales 2 2 2 2 4GF me E 44 E 2 σweak = 2 =1.7 10− 2 cm π me × me 2 1 5 me λν (T )= =3 10 2 cm nσweak(T ) × T 2 2 2 R 4 R T τ = = 10− s Diffusion cλ 10 km m2 ν e Protoneutron Star Evolution Initial state of the PNS. Pons et al. (1999) Snap shots of the evolution Neutrino diffusion deleptonizes and cools the PNS. Typical time-scales: Pons et al. (1999) Snap shots of the evolution Neutrino diffusion deleptonizes and cools the PNS. Typical time-scales: ⎛ t ⎞ T(t) ≈ T(t = 0) ⎜1 − ⎟ ⎝ τC ⎠ R2 τC ≈ CV c λν 3 µ Y (t) ≈ ν ν 6π 2 ⎛ t ⎞ ≈ Y(t = 0) exp⎜ − ⎟ ⎝ τD ⎠ 3 ∂Y R2 τ = L D π 2 ∂Y c λ ν νe Pons et al. (1999) € Snap shots of the evolution Neutrino diffusion deleptonizes and cools the PNS. Typical time-scales: ⎛ t ⎞ T(t) ≈ T(t = 0) ⎜1 − ⎟ ⎝ τC ⎠ R2 τC ≈ CV c λν 3 µ Y (t) ≈ ν ν 6π 2 ⎛ t ⎞ ≈ Y(t = 0) exp⎜ − ⎟ ⎝ τD ⎠ 3 ∂Y R2 τ = L D π 2 ∂Y c λ ν νe Pons et al. (1999) € Response of Interacting System ν ν emission ω,q ω =q ω scattering ∆R q 2π 1 q = > λ ∆R 2π 2π ω = > τ = τ τcollision collision Collision Time 2π 2π ω = < τ τcollision Response of Interacting System ν ν emission ω,q T ω =q ω scattering ∆R q 2π 1 q = > λ ∆R 2π 2π ω = > τ = τ τcollision collision Collision Time 2π 2π ω = < τ τcollision Response of Interacting System ν ν emission ω,q T ω =q ω scattering ∆R q 2π 1 q = > λ ∆R 2π 1 q = < λ ∆R ∆R 2π 2π ω = > τ = τ τcollision collision Collision Time 2π 2π ω = < τ τcollision Response of Interacting System ν ν emission ω,q T ω =q ω scattering ∆R q 2π 1 q = > λ ∆R 2π 1 q = < λ ∆R ∆R 2π 2π ω = > τ = τ τcollision collision Collision Time 2π 2π ω = < τ τcollision Linear Response Theory Consider a weak perturbation of the system by an external field The external field can couple to any property of the system. For illustration lets assume it couples to density, then This will produce a density fluctuation Eigenstate of Htot Eigenstate of Hsystem If Hext is weak it is possible to show that Density-density correlation function ΠR is the Fourier transform of DR and it is called the polarization function or the generalized susceptibility. Neutrino Rates ν ν emission ω,q ω =q ω scattering q d 2σ E ≈ G2 Im L (k,k + q) Πµν (q) V dcosθ dE ʹ F E ʹ [ µν ] Lµν (k,k + q) = Tr [ lµ (k) lν (k + q) ] r d 4 p µν ( ,q ) Tr [ j µ (p) jν (p q) ] Π ω = ∫ 4 + (2π) Dynamic structure factor: spectrum of density, spin and current fluctuations. € Neutrino Scattering µ µ Neutrinos couple to j (x) = ψ(x) γ (cV − cAγ5) ψ(x) density and spin NR + µ0 + i µi → cV ψ ψ δ − cA ψ σ ψ δ € Burrows,Reddy, Thomspon (2006) Sum Rules: Thermodynamic (compressibility) sum Rule: dω (1− eβhω ) ⎛ ∂ρ⎞ lim ∫ Sρ (q,ω) = ρ ⎜ ⎟ q →0 2π hω ∂p ⎝ ⎠T This relates response to the equation of state - provides a good consistency check. € F-Sum Rule: dω (1− eβhω ) 1 S (q, ) [[H, (q)], (q)] ∫ ω ρ ω = 2 ρ ρ 2π 2h 2h € S(q,ω) of a classical dense liquid The density-density correlation for N particles is Ensemble average Positions at t Positions at t=0 Need to specify equations of motion ie rj(t). Classical limit: One simple method is Molecular Dynamics. Response in Strongly Coupled Systems Charged ions in a screened plasma. Z2 e2 Γ= =8 akT Correlations are ωplasmon =0.3MeV q =3MeV important. (Fortunately) nuclear interactions are relatively weaker. Carlson & Reddy (2004) RPA - vertex corrections Self-consistent approximation to a mean-field ground state Mean Free Paths in the RPA r r p + q Γµ Γν = r p € € € r r G (p + q ) Γµ µν Γν € r G (p) € νµ € € € + r r r r G (p +€q ) µσ p + q Γµ Γν r r V p Horowitz & Wehrberger, (1991) G (p) στ Burrows & Sawyer, (1999) € σµ Reddy, Pons, Prakash, Lattimer, (1999) € € € € € € First-Order Transitions Glendenning (1992) (nuclear -> quark) Alford, Rajagopal, Reddy & Wilczek (2001) (nuclear ->CFL) exotic a A Mixed phase µ e = −µQ − − + b nuclear + € B µ B Mixed Phase: Heterogeneous phase with Nuclear Exotic € structures of size 5-10 fm. Neutrino Mean Free Path in a Mixed Phase ν’ ν Scattering from quark droplets in a quark-hadron transition. 2 dσ G = N F S Q2 E2 (1 + cos(θ)) dcos(θ) D 16π q W ν Reddy, Bertsch & Prakash, (2000) € Neutrino Propagation in Superconducting phases r 4 Πµν(q, qo) = ∫ d p Tr [ G(p) ΓµG(p + q)Γν] = p+q Γµ Γν € p + p+q Γµ Γν p Gap modifies excitation spectrum. No excitations below the gap ? Carter & Reddy (2000) Neutrino Propagation in Superconducting phases r 4 Πµν(q, qo) = ∫ d p Tr [ G(p) ΓµG(p + q)Γν] = p+q Γµ Γν € p + p+q Γµ Γν p Gap modifies excitation spectrum. No excitations below the gap ? Carter & Reddy (2000) Collective(Goldstone) modes p+q = q + γ Γµ Γµ µ λ(q) p Generalized Ward Identity : NG ˜ −1 ˜ −1 NG ∑qµΓµ = τ3 G (p) − G (p + q)τ3 µ if lt λ(q) is finite q→0 r Γ(q = (q ,qo) is singular when r Bogoliubov, Nuovo Cimento, 7, 6 (1958) vF Anderson, Phys. Rev. 112, 1900 (1958) qo = |q | 3 Nambu, Phys. Rev. 117, 648 (1960) € Collective(Goldstone) modes p+q = q + γ Γµ Γµ µ λ(q) p Generalized Ward Identity : Low energy response NG ˜ −1 ˜ −1 NG ∑qµΓµ = τ3 G (p) − G (p + q)τ3 dominated by the µ Goldstone (a.k.a. if lt λ(q) is finite q→0 Bogoliubov-Anderson ) r Γ(q = (q ,qo) is singular when mode r Bogoliubov, Nuovo Cimento, 7, 6 (1958) vF Anderson, Phys. Rev. 112, 1900 (1958) qo = |q | 3 Nambu, Phys. Rev. 117, 648 (1960) € Spectrum of Density Fluctuations in Superfluids Goldstone Mode Pair Breaking 2Δ Color-Flavor Locked Phase BCS pairing of all 9 quarks: Δ ≈ 100 MeV ! Egluon ≈ gµ Equark ≈ 2Δ € E € GB: SUC(3)⊗SUL (3)⊗SUR (3) Energy Δ ≈ mlight ms µ € E (1) = 0 GB: UB Excitation Spectrum SU(3)color ⊗ SU(3)L ⊗ SU(3)R ⊗ U(1)B ⇓ € SU(3)color+L+R ⊗ Z2 € Alford, Rajagopal & Wilczek, (1999) Goldstone Modes in CFL dispersion relations : 1 E = v p (v ≈ ) H 3 E = m2 + v2 p2 Schafer Phys. Rev .D 65, 074006 (2002) π π Manuel & Tytgat, Phys. Lett. B 479, 190 (2000) 2 ms 2 2 2 Son & Stephanov, Phys. Rev. D 61, 074012 (2000) EK m = ± + mk + v p 2µ € Coupling to Neutrinos φo ν φ- e- Z W- ν ν e- φ ν φ Z W- ν ν φ φ dispersion relations : 1 E = v p (v ≈ ) H 3 2 2 2 Eπ = mπ + v p 2 ms 2 2 2 EK m = ± + mk + v p 2µ Reddy, Sadszikowski & Tachibana (2003) € Neutrino Emit Cherenkov Radiation + φo φ ν νe GFfH G f ν’ F π e- Dominant Process CFL matter is opaque. Reddy, Sadszikowski & Tachibana (2003) CFL matter is opaque. Reddy, Sadszikowski & Tachibana (2003) Neutron Star Tomography - The Dream !.