Neutrino Response in Supernovae

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1 Neutrino Response in Supernovae SN1987A before and after C. J. Horowitz Indiana University Historic neutrino events from SN1987A. Filled points recognized with Noble Prize.

2 INT e Scattering Program 1997

3 Core Collapse Supernova Core of massive star collapses to form protoneutron star. νs form neutron star energizes shock that ejects outer 90% of star. Envelope ν July 5, 1054 Crab nebula Shock Hot bubble Crab Pulsar Proto-neutron star: hot, e rich Hubble ST

4 Nu Response in Supernovae Neutrino scattering related to electron scattering. Much experience from electron scattering labs such as Bates, Jlab very relevant for neutrinos. In general interested in response of low density neutron rich matter. What are the properties and modes of excitation of such systems? Radioactive beam experiments at RIA and elsewhere may provide important insight. Example: what are pygmy resonances and how might they be related to the excitation modes of neutron rich pasta?

5 Ground Rules for SN nu response Document corrections (>10%) to major nu interactions in relevant density regimes. Expect negative feedback! Nevertheless some 10% corrections matter: example weak magnetism is very important for composition of neutrino driven wind and nucleosynthesis. Long term goal: theoretical error bars for nu microphysics of supernova simulations (like standard solar model, but not as good).

6 Neutrino Response Density Regimes 1. Uniform nuclear matter at nuclear density and above: Relativistic RPA response including weak magnetism. 2. Nuclear pasta at just below normal density: Semiclassical molecular dynamics response. 3. Low density complex gas near neutrino sphere: Model independent Virial expansion equation of state and long wave length response. 4. Discussion: scaling and relation to universal unitary gas. D. Page

7 Consistent Neutrino Response Supernova simulations use large equation of state (EOS) table: pressure vs density, temperature and proton fraction. Neutrino opacities should be consistent with EOS. RPA (with same interaction) yields consistent linear response of mean field ground state. Example: Shen EOS based on TM1 relatativistic mean field interaction. Cross section Plus sign for neutrinos, minus for antineutrinos. Weak magnetism and third response function.

8 Mean Free Path Antineutrinos RPA Hartree RPA Hartree Neutrinos It can take a long time (>1 sec) for nu to diffuse from high densities. Shock will have succeeded or failed by then. High density nu opacity may not be so important for supernova dynamics.

9 Nuclear Pasta

10 Frustration and Nuclear Pasta Matter is correlated at short distances from attractive strong interactions and anticorrelated at large distances from coulomb repulsion. Normally these length scales are well separated so nucleons bind into nuclei segregated on a crystal lattice. Michelangelo

11 Nuclear Pasta At great densities, the attractive nuclear and repulsive coulomb interactions are comparable. Leads to a complex ground state. Can involve sphere (meat ball), rod (spaghetti), plate (lasagna), or other shapes. This nuclear pasta expected in neutron star crusts + supernovae. Pasta closely related to multifragmentation. Both driven by nuc attraction and coulomb repulsion. Pasta should have unusual properties and dynamics because of the frustration.

12 Simple Molecular Dynamics Model of Nuclear Pasta Heavy cluster thermal de Broglie wavelength is << inter cluster spacing semi-classical approx. good. Charge neutral system of n, p and e. [e provide screening length λ for Coulomb.] n, p interact via classical 2-body pot. v(r) = a Exp[-r 2 /Λ]+ b ij Exp[-r 2 /2Λ] + e i e j Exp[-r/λ]/r Parameters fit to binding E and saturation density of nuclear matter and symmetry energy. Use MDGRAPE hardware to run simulations with to nucleons.

13 Molecular Dynamics Simulation Small run with 4000 nucleons [800 p (red), 3200 n (clear)] Density ρ=0.05 fm -3 Visualization by FSU undergraduate J. Weiner. All pasta results for Y p =0.2 and T=1 MeV.

14 Simulation with 40,000 nucleons at T=1 MeV, ρ=0.01 fm -3, Y p =0.2 Isospin distillation: Y p ~0.3 in clusters (N/Z=7/3) fm -3 surface of the proton density Graphics by Brad Futch FSU

15 Simulation with nucleons at ρ=0.025 fm -3, (T=1 MeV, Y p =0.2) Graphics by Brad Futch FSU

16 Simulation with 100,000 nucleons at ρ=0.05 fm -3 showing pasta phase. Graphics by Brad Futch FSU

17 ν Neutron Scattering Roger Pynn

18 Neutrino Pasta Scattering Core collapse supernovae Incident beam radiate 99% of their energy in neutrinos. SN dynamics very sensitive to how neutrinos interact with the matter. Neutrino cross section per nucleon is dσ/dω = S(q) dσ/dω free Structure factor S is sum over all possible reflections. S(q) = i,j exp[iq.r ij ] Reflected beam

19 Static Structure Factor vs Density Coherent cluster scattering ρ=0.01 fm Ion screening 0.075

20 Ion Screening An impurity ion in a plasma is surrounded by a screening cloud of other ions. E+M charge of impurity, Z, is completely screened. This screening cloud also screens the weak charge of the ion, N. Screening greatly increases ν mean free path Screening cloud X Impurity Plasma (EM charge = Z, weak charge N) (EM charge Z, weak charge N) However if screening cloud made of mixture of ions with different ratio of weak to E+M charge, N/Z, then weak charge will not be completely screened. Opacity of mixtures can be larger then opacity of single species.

21 Liquid Vapor Phase Transition In a first order phase transition, low density vapor is in equilibrium with high density liquid. Large density fluctuations arise as liquid is converted to/ from vapor. Static structure factor at q=0 related to density fluctuations. P Expect very large S(0) in two-phase coexistence region of simple first order phase transition very short neutrino mean free paths in a supernova [J. Margueron, PRC70(2004)028801]. We find no large enhancement of S(0) system is not described by simple first order phase transition because ratio of liquid to vapor fixed by charge density of electrons. mixed phase ρ

22 Quark/ Hadron, Nucleon/Nucleus Duality When does nu scatter from nucleons or nuclei? Can describe system with different coordinates. Quark / hadron duality at J-Lab. SLAC e scattering data shows sum of nucleon resonances (solid) close to quark deep inelastic scattering (dashed). Describe low density clustered system in nucleon or nucleus coordinates

23 Nucleon Nucleus Duality Composition of MD simulation 2x10 13 g/cm 3 Cluster algorithm: A nucleon belongs to a cluster (nucleus) if it is within R c =3 fm of at least one other nucleon in the cluster. For supernova simulations need to calculate weak interactions of complex mixtures. Statistical models (such as Botvina et al) use detailed binding energies but approx. interactions between nuclei. Rare nuclei with big weak cross sections can be important.

24 Nuclear calculation, S ion (q) <N> F(q) 2 Full nucleon simulation Nuclear versus Nucleon response at 0.01 fm -3 (2x10 13 g/cm 3 )

25 Nuclear versus nucleon response at rho=0.025 fm -3 (5x10 13 g/cm 3 ) Nuclear calc. S ion (q) <N> F(q) 2 Full nucleon simulation Nuclear models of response (ground state only) appear to break down rapidly with increasing density at g/cm 3 and above.

26 Dynamical Response Function Can directly calculate S(q,w) from MD simulations Density in q space: ρ(q,t)= j e iq.r j(t). Average over ~10,000 configurations each with 100,000 nucleons. This is 10-20GB. 1-2 weeks on 4 accelerated MDGRAPE-2 boards.

27 Preliminary Response at ρ=0.05 fm -3, Y p =0.2 and T=1 MeV

28 S(q,w)=<N>F(q) 2 S ion (q,w) Preliminary Full nucleon response Response at ρ=0.01 fm -3, Y p =0.2 and T=1 MeV

29 Response of Ion Mixture Ion simulations q=0.113 fm -1 Rho=2e13 g/cm 3 Mixture with different N/Z ions has concentration fluctuations which diffuse slowly (without an E+M restoring force) and show up as a peak at low w.

30 Complex Low Density Vapor Phase in Neutrinosphere of Supernovae View sun with photons, see photosphere. View sun in neutrinos, (see angular resolution of ν-e scattering in Super-K). View SN in neutrinos, see surface of last scattering called the neutrinosphere. What is neutrinosphere like? Conditions at neutrinosphere: Temperature ~ 4 MeV from 20 SN1987a events. Mean free path λ=1/σρ R σ ~ G F2 E ν2 and E ν ~3T ρ ~ g/cm 3 [~10-4 fm -3 ] 90 deg.! What is the composition, equation of state, and neutrino response of the neutrinosphere? At low neutrinosphere density, Virial expansion gives model independent answers!

31 QCD Phase Diagram Quark Gluon Plasma Temperature Early Universe RHIC Hadrons Vapor Critical point Liquid Color Superconductor Supernova neutrinosphere Density Nuclei Neutron stars

32 Universal Behavior of Neutron Matter Consider a low density fermi system with very large scattering length a, and very small effective range r much less then inter-particle spacing. There are no length scales associated with interaction. Therefore system will exhibit universal behavior independent of details. To what extent does real neutron matter at low density approach this unitary limit? Real a=-19 fm, r=2.7 fm. Use Virial expansion to simply relate energy of neutron matter to nn scattering properties. A number of cold atom experiments to test universal behavior of fermions in this unitary limit.

33 Virial Expansion 101 Assume (1) system in gas phase and has not undergone a phase transition with increasing density or decreasing temp. (2) fugacity z=e µ/t with µ the chemical pot is small. Expand grand canon. partition function Q in powers of z: P=T lnq/v, n=z d/dz lnq/v P=2T/λ 3 [z+b 2 z 2 +b 3 z 3 + ], n=2/λ 3 [z+2b 2 z 2 +3b 3 z 3 + ] Here λ=thermal wavelength=(2π/mt) 1/2 2 nd virial coef. b 2 (T) calculated from 2 particle partition function: Q 2 = states Exp[-E 2 /T] E 2 is energy of 2 particle state. Thus b 2 depends on density of states.

34 Density of states Put system in big spherical box of radius R Relative mom. k from E 2 =k 2 /2m reduced. ψ(r 1 -r 2 =R infinity)=0=sin[kr+lπ/2+δ l (k)] or kr+lπ/2+δ l (k)=nπ. Distance between states k=π/(r+dδ/dk) so dn/de ~ 1/ k ~ R + dδ/dk with + for bose and for fermions. b 2 includes both bound states and scattering resonances on equal footing.

35 Neutron matter Equation of State Neutron matter virial nearly independent of temperature. b n (T)=0.303, 0.309, at T=2, 10, and 20 MeV. Error bars (dotted) from estimate of b 3 Crosses from microscopic FHNC calc. by Friedman + Pandharipande

36 Scaling of Neutron Matter EOS If b i (T) are independent of T the EOS will scale P/T 5/2 = f(n/t 3/2 ). Neutron matter P is only a function of n/t 3/2 instead of a function of n and T separately From P/T=g/λ 3 [z+b n z 2 + ] and n=g/λ 3 [z+2b n z 2 + ] with λ proportional tot -1/2. Unitary Limit: calculate b 2 with only s-wave scattering: a=-infinity, r=0. δ( 1 S 0 )=π/2 b 2 (T)=3/2 5/2 = independent of T. In unitary limit system clearly scales. Real neutron matter scales, to a very. good approx., but with a b n =0.3 that is 40% smaller then unitary limit. In scaling limit energy density ε=3/2p [Thomas et al have tested this for a universal system of cold 6 Li atoms.]

37 Nuclear Matter Is very different from neutron matter because of cluster formation. Deuterons appear as bound state in b 2. α particles will appear as bound state in b 4. Large α binding E α =28.3 MeV gives large e +E α/t contribution to b 4. Nucleon only virial expansion may be accurate only over a very reduced density range because of the abnormally large b 4. Solution: include α explicitly and work with system of p, n, and α s. Chemical equilibrium 2µ p +2µ n =µ α gives z α =z p2 z n2 e E α/t. Work to 2 nd order in z p, z n, z α. Can include heavier nuclei at even higher densities.

38 n, p, α system Need four virial coefficients: b n for neutron matter, b nuc for symmetric nuclear matter, b α for alpha system, b αn for interaction between an α and N. Virials from NN, Nα and αα elastic scattering phase shifts. α α Elastic Phase Shifts

39 Vapor phase is complex with large α Mass Fraction α particle mass fraction in nuclear matter vs density. Lattimer Swesty EOS is dashed while Sumi is an EOS based on a rel. mean field interaction (dotdashed). One can study this low density vapor in heavy ion collisions. T=4 MeV Texas A&M group observes large x α for particles emitted at intermed. vel.

40 Neutrino Response ν neutral current cross section dσ/dω = (G 2 E ν2 /16π 2 )[ (1+cosθ)S v + g a2 (3-cosθ)S a ] Vector response is static structure factor S v =S(q) as q 0 S(0)=T/(dP/dn) Axial or spin response from spin polarized matter. S a =(1/n) d/dz a (n + -n - ) n+ =n - Typical RPA calculations neglect alpha particles. Virial expansion provides model independent results for EOS, composition, and ν response of complex nuclear vapor. T=4 MeV Vector Total Virial Axial Burows + Sawyer RPA Y p =1/2

41 Scaling of ν Response S(0)=T/(dP/dn) = T (dn/dz) / (dp/dz) P/T = 2/λ 3 ( z + z 2 b 2 + ) n=2/λ 3 (z + 2z 2 b 2 + ) S(0) = (1+4zb 2 + ) / (1+2zb 2 + ) If virial coef. b 2, b 3, are indep. of T, S(0) scales: instead of separate function of n and T it is only function of z=e µ/t All virial coef b 2 =.53, b 3, b 4 are temp independent for unitary gas. For neutron matter: b n (T)=0.303, 0.309, (at T=2, 10, and 20 MeV) Nuclear matter does not scale because of cluster formation: alpha abundance depends strongly on T.

42 Scalings Scaling Naive system Mystery Why Scaling Scaling violations x Parton model Evolution Asym. freedom lnq 2 y Free F Gas F(y) not free FG??? 2p-2h? z Unitary gas b n not b unitary V has no scale NN effective range

43 Neutrino Response in Supernovae Pasta simulations: Jorge Piekarewicz (FSU), Angeles Perez-Garcia (Salamanca, Spain), Don Berry (IU High Performance Computing) Virial expansion: Achim Schwenk (U. Washington + TRIUMF) Graduate students: Liliana Caballero Helber Dussan Gang Shen Supported in part by DOE and state of Indiana.