Microscopic calculation of the neutrino mean free path inside hot neutron matter Isaac Vidaña CFisUC, University of Coimbra

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1 Microscopic calculation of the neutrino mean free path inside hot neutron matter Isaac Vidaña CFisUC, University of Coimbra Landau Fermi liquid theory in nuclear & many-body theory May 22 nd 26 th 2017, Trento (Italy)

2 Done a bit of time ago in collaboration with Jerôme Margueron (IPNL Lyon) Ignazio Bombaci (Università di Pisa) For details see: Phys. Rev. C 68, (2003)

3 The talk in few words ² Purpose: Study consistently the effect of temperature in the calculation of neutrino mean free path in pure neutron matter ² Method: Both the EoS and response function computed including shortand long-range correlations in a BHF+RPA framework. Particlehole residual interaction extracted from Brueckner theory in terms of Landau parameters. ² Results: Temperature increases the ph interaction but decreases the effect of RPA correlations in the dynamical structure functions. The lack of consistency between finite T EoS and the dynamical structure functions leads to a underestimation of the ν-mfp of ~ 30-50%.

4 SN, NS & Neutrinos Neutrinos play a crucial role in the physics of supernova, in the early evolution of neutron stars & binary merger of compact objects ² Large number of neutrinos produced by e-capture during the collapse of the pre-supernova core. Most of the initial gravitational binding energy is stored in neutrinos ² λ ν decreases as the radius of the neutron star shrinks from ~ 100 km to ~ 10 km becoming smaller than the NS radius neutrino trapping strong influence on the overall properties of hot & leptonrich newborn neutron star, substantially different from the cold & deleptonized one.

5 Proto-Neutron Stars: Composition Neutrino free µ ν = 0 Neutrino trapped µ ν 0 (Burgio & Schulze 2011) (Burgio & Schulze 2011) Neutrino trapped ê ê ü ü ü ü Large proton fraction Small number of muons Onset of Σ - (Λ) shifted to higher (lower) density Hyperon fraction lower in ν-trapped matter

6 Proto-Neutron Stars: EoS (Burgio & Schulze 2011) Nucleonic matter ² ν-trapping + temperature softer EoS Hyperonic matter ² ν-trapping + temperature stiffer EoS ² More hyperon softening in ν-untrapped matter (larger hyperon fraction)

7 Proto-Neutron Stars: Structure (Burgio & Schulze 2011) Nucleonic matter ν-trapping + T: reduction of M max (IV et al. 2003) N,Y,l,ν Hyperonic matter ν-trapping + T: increase of M max delayed formation of a low mass BH N,Y,l go to BH

8 Neutrino Interactions with Matter During their propagation in matter neutrinos can be: Z 0 W ± L NC = G F 2 l ν µ µ j Z L CC = G F cosθ c 2 l µ j W µ l ν µ = ψ ν γ µ ( 1 γ 5 )ψ ν, j µ Z = 1 2 ψ 4γ µ ( c v c A γ 5 )ψ 2 l µ = ψ l γ µ ( 1 γ 5 )ψ ν, j µ W = ψ 4 γ µ ( g v g A γ 5 )ψ 2 Scattered via weak coupling with baryon neutral currents Absorbed via weak coupling with baryon charged currents σ (E 1 ) V = 2 d 3 p 2 (2π ) 3 " 2 W fi = G F $ V + A # ( ) 2 (1 p 2 d 3 p 3 (2π ) 3 d 3 p 4 (2π ) f 2(E 3 2 )(1 f 3 (E 3 ))(1 f 4 (E 4 ))(2π ) 4 δ ( P 1 + P 2 P 3 P 4 )W fi cosθ 12 )(1 p 4 cosθ 34 )+ V " A E 2 E 4 ( ) 2 (1 p 2 ( ) m2 cosθ 23 )(1 p 4 cosθ 14 ) V 2 " A 2 E 2 E 4 % (1 cosθ 13 )' E 2 E 4 &

9 Neutrino Mean Free Path in Neutron Matter Here we consider only the ν-mfp in pure neutron matter only scattering processes contribute In the non-relativistic limit for neutrons the ν-mfp of non-degenerate neutrinos with initial energy E ν is given by: λ 1 (E ν,t ) = where: cosθ = ˆk i ˆk f, G F 2 32π 3 ( c) dk f " # c 2 V ( 1+ cosθ)s (0) q, q 0,T ( ) + c A 2 3 cosθ ( )S (1) q, q 0,T ( ) q = k i k f, q 0 = E νi E ν f transferred momentum & energy $ % S (0) ( q, q 0,T ) S (1) ( q, q 0,T )! # " # $ dynamical structure function describing the response of neutron matter to the excitations induced by neutrinos

10 BHF approach in a Nutshell BHF = lowest order of the BBG expansion ² Energy per particle E A (ρ, β) = 1 A τ n τ (k) 2 k k 2m τ 2A Free Fermi Gas ² Bethe-Goldstone Equation τ k n τ (k)re" U τ ( k # ) $ % Correlation Energy Infinite sumation of two-hole line diagrams Partial sumation of pp ladder diagrams ( ) τ1 τ 2 τ 3 τ 4 = V τ1 τ 2 τ 3 τ Ω V τ 1 τ 2 τ i τ j Q τ i τ j G ω τ i τ j [ ] ε τ (k) = 2 k 2 + Re U τ (k) 2m τ ω ε τ i ε τ j + iη G ω, τ = n, p ( ) τ i τ j τ 3 τ 4 U τ ( k ) = 1 Ω τ ' k' n τ ' (k ') kτ k 'τ ' G(ε τ ( k )+)ε τ ' ( k ') kτ k 'τ ' A ü Pauli blocking

11 Finite Temperature EoS ² The Bloch-De Dominicis expansion represents the natural extension to finite temperature of the BBG one, to which it leads in the zero temperature limit ² Baldo & Ferreira showed in fact that the dominant terms in the BD expansion were those that correspond to the T=0 of the BBG diagrams, where the temperature is introduced only through the Fermi-Dirac distribution Hartree & Fock terms in the BD expansion (Picture taken from Baldo & Ferreira, PRC 59, 682 (1999)) Corresponding T=0 diagrams in the BBG expansion

12 ² Therefore, temperature effects can be introduced at the BHF level in a very good approximation by just replacing " $ n τ (k) = # $ % 1, if k k Fτ 0, otherwise 1 f τ (k,t ) = 1+ exp ε τ (k,t ) µ τ (T ) {( ) / k B T} in the Pauli operator Q ττ =(1-f τ )(1-f τ ) & s.p. energies ε τ. ² Self-consistency implies that together with the BG equation and the U τ (k,t), µ τ (Τ) must be extracted at each step of the iterative process from the normalization condition k ρ τ = f τ (k,t )

13 Once self-consistency is achieved the total free energy per particle can be determined as where F A (ρ, β,t ) = E A (ρ, β,t ) T S (ρ, β,t ) A E A (ρ, β,t ) = 1 A τ f τ (k,t ) 2 k k 2m τ 2A τ k f τ (k,t )Re" U τ ( k # ) $ % and the entropy per particle is calculated in the quasi-particle approximation S A (ρ, β,t ) = 1 A τ k ( f τ (k,t )ln( f τ (k,t )) + ( 1 f τ (k,t ))ln( 1 f τ (k,t )))

14 BHF nucleon mean field T=0 MeV T=10 MeV ρ=0.1 fm -3 ρ=0.2 fm -3 Isospin splitting of mean field in ANM U n ~ U 0 + U sym β U p ~ U 0 U sym β Symmetry potential U sym = U n U p 2β Results for Av18+UIX

15 Parabolic approximation of the s.p. energy When calculating the response function we will approximate the k dependence of the s.p. energy by a quadratic function ε(k,t ) ~ 2 k 2 2m * (T ) +U(k = 0,T ), 1 m * (T ) = 1 m + 1 k U(k,T ) k k=k F (k F defined for T=0) Two short comments: ü U(k=0,T) increases when increasing T Gogny D1P T=10 T=20 T=40 T=80 ü m * (T) is nearly T-independent up to T=80 MeV BHF results for NSC97e

16 Finite Temperature EoS: Asymmetric Matter Results for Av18+UIX

17 Finite Temperature EoS: Neutron Matter Results for NSC97e

18 A couple of comments: Ø The parabolic assumption F A ( ρ, β,t ) = F A " ( ρ, 0,T ) + F A (ρ,1,t ) F % $ (ρ, 0,T )'β 2 # A & T=10 MeV is valid both at zero and finite temperature T=0 Results for Av18+UIX

19 Ø Symmetry energy at finite temperature can be well approximated by E sym (ρ,t ) F A (ρ,1,t ) F (ρ, 0,T ) A Results for Av18+UIX

20 Response Function at Finite T Ô Given a external prove the (linear) response of a system and the dynamical structure function at finite T are χ ( q, q 0, T ) = 1 Z n m e E n /k B T " n Ô m 2 $ $ q 0 E n + E m + iη # $ 2 % n Ô m ' q 0 + E n E m + iη ' & ' S( q, q 0, T ) = 1 Z e E n /k B T n Ô m 2 δ ( E n E m q 0 ), Z = e E m /k B T n m m Fluctuation-Dissipation Theorem S( q, q 0, T ) = 1 π Im χ q, q 0, T ( ) 1 e q 0 /k B T

21 BHF Response Function The BHF response function can be obtained from the polarization function Π 0 χ BHF ( q, q 0,T ) = ReΠ 0 ( q, q 0,T ) + isign(q 0 )Im Π 0 ( q, q 0,T ) Π 0 ( q, q 0,T ) = 2i d 3 p (2π ) 3 dω f (p,t ) 2π ω ε(p,t )+ iη p f (p + q,t ) ω + q 0 ε(p + q,t )+ iη p+q Integrating over ω in the complex plane χ BHF ( q, q 0,T ) = d 3 p (2π ) 3 ( ) f ( p + q,t ) ( ) + iη f p,t q 0 +ε(p,t ) ε p + q,t particle-hole propagator G BHF ( p, q, q 0,T )

22 BHF Dynamical Structure Function Using the parabolic approximation for ε(k) Im χ BHF & S BHF obtained in a closed form can be Im χ BHF ( q, q 0,T ) = 2 m * T 4πq( c) ln " 1+ /2)/kBT $ e(a+q e (A q 0 /2)/k B T # S BHF ( q, q 0,T ) = 1 π Im χ BHF ( q, q 0,T ) 1 e q 0 /k B T % & ', A = µ m* q 2 0 n 2q 2 q2 8m * ρ=0.25 fm -3, T=10 MeV q=10 MeV ü In-medium effects introduced through m * & µ n, which introduces, besides a density dependence, and additional T dependence on χ BHF & S BHF [MeV -1 fm -3 ] BHF (NSC97e) Gogny D1P

23 Particle-Hole Residual Interaction ² Quasi-particle energy & ph residual interaction E = E 0 + ε( k 1 )δn(k 1 )+ 1 V ph ( k 1, k 2 )δn(k 1 )δn(k 2 ) 2 k 1 k 1k2 ε( k 1 ) = δe δn(k 1 ) V ph ( k 1, k 2 ) = δ 2 E δn(k 1 )δn(k 2 ) ² Since k ν, q < k F & E ν, q 0 < E F Landau Fermi liquid theory ( ) V ph = f l + f ' l (τ 1 τ 2 )+ g l (σ 1 σ 2 )+ g ' l (σ 1 σ 2 )(τ 1 τ 2 ) P l ˆk1 ˆk 2 l PNM ( ) ² We consider monopole & dipole contributions to the ph in PNM 4 Landau parameters F l=0,1 = N 0 f l=0,1, G l=0,1 = N 0 g l=0,1 N 0 = m* k F π 2 2

24 Landau Parameters F 0, F 1, G 0 & G 1 ² At finite T the Landau parameters F 0 & G 0 can be related with the usual bulk properties of the nuclear medium as it is done for T=0 ( K = 9ρ 2 F /V ) = 9ρ 1+ F 0 ρ 2 N 0 ( ) χ 1 = 1 2 F /V = 1+ G 0, ρ µ 2 ρ 2 2 S = ρ ρ ρ s χ F ² F 1 is deduced from the effective mass m * m =1+ F 1 3 ² No simple relations for G 1 can be deduced from thermodynamical properties or general relations like e.g., forward scattering sum rule. So here we take G 1 =0

25 Density dependence of the Landau Parameters Results for: ü BHF (solid black (T=10 MeV) & dashed red (T=80 MeV)) with NSC97e ü T independent Gogny D1P (solid red) Ø F 0 & G 0 increase with ρ & T whereas F 1 decreases with ρ and is nearly constant in T Ø Main difference between BHF & Gogny in the spin-density fluctuation channel. G 0 falls to -1 in the Gogny case due to the existence of a ferromagnetic instability absent in the BHF calculation (see Artur s talk)

26 Temperature dependence of the Landau Parameters A comment: ² Finite T Landau parameters are non-standard essentially because k F cannot be properly defined. But, we think the prescription we have chosen does not change the general features of the present results ρ=0.25 fm -3 ² A better prescription will require to the use a momentum dependent residual interaction ² Here we want just to give an estimation of temperature effects when they are treated consistently As already said F 0 & G 0 increase with T whereas F 1 is rather constant

27 RPA Response Function The RPA response function can be obtained by solving the Bethe- Salpeter equation for the ph propagator (S) G RPA ( k 1, q, q 0,T ) = G BHF ( k 1, q, q 0,T ) + G BHF ( k 1, q, q 0,T ) d 3 k 2 (2π ) V (S,S') (S) 3 ph ( k 1, k 2 )G RPA S' ( k 2, q, q 0,T ) (S) χ RPA ( q, q 0,T ) = d 3 k 1 (2π ) 3 (S) G RPA ( k 1, q, q 0,T ) (S,S') (S) If V ph ( k 1, k 2 ) = δ SS' V ph ( q), the solution of the BS equation allows to write the response function in the simple form (S) χ RPA ( q, q 0,T ) = ( ) χ BHF q, q 0,T 1 χ BHF ( q, q 0,T )V ph (S) ( q) For details see: A. Pastore, D. Davesne & J. Navarro, Phys. Rep. 563, 1 (2015)

28 BHF+RPA Dynamical Structure Function (S) S RPA ( q, q 0,T ) = 1 (S) Im χ RPA π ( q, q 0,T ) 1 e q 0 /k B T ρ=0.25 fm -3, T=10 MeV q=10 MeV ü Main differences of the two interactions in the spin channel, as expected from the Landau parameters [MeV -1 fm -3 ] ü Spin zero sound weakly present for Gogny D1P while it seems very important for the BHF calculation [MeV -1 fm -3 ] BHF (NSC97e) Gogny D1P ü Pronounced spin zero sound at T=0 smeared by Landau damping when increasing T

29 Neutrino Mean Free Path from BHF response λ 1 (E ν,t ) = G 2 F dk 16π 2 f " # c 2 V ( 1+ cosθ)s (0) q, q 0,T ( ) + c A 2 3 cosθ ( )S (1) q, q 0,T ( ) $ % ² The MPF follows, approximately, the expected law T -2 ² Discrepancies between BHF & Gogny at low T due to density dependence of the effective mass. Moreover, Gogny is T-independent BHF (NSC97e) E ν =3T Gogny D1P ² BHF MPF is sensible to T essentially though the explicit T dependence of the response function. Small effect due to the finite T EoS ² Reduced medium effects at high T (E ν =3T) similar results of both interactions

30 Effect of the RPA correlations By comparing solid & dashed black lines (same ph interaction) one sees: ² Increase of T reduces effect of RPA correlations ² The lack of consistency between the finite T EoS and the Response Function leads to an underestimation of the ν-mfp of about 30-50% Temperature induces two opposite effects which tend to compensate & make R is rather constant in the range MeV: ² Increases the ph interaction (F 0 & G 0 increase with T) ² Decreases the effects of these correlations in the calculation of the dymanical structure factors BHF T=10 MeV BHF T=80 MeV Consistent BHF, LP (T=10 MeV) RF (T=80 MeV) Gogny D1P T=10 MeV Gogny D1P T=80 MeV E ν =3T ü R ~ 8 for BHF at high ρ. Also shown for T=0 EoS with TBF by Lombardo et al. ü R ~ 1 for Gogny due to the presence of spin instabilities at high ρ.

31 The Message of this Talk ² Study consistently the effect of temperature in the calculation of neutrino mean free path in pure neutron matter ² EoS & response function calculated including short- and longrange correlations in a BHF+RPA framework ² Interplay between T effects in the EoS (mean field & ph interaction deduced from it) & explicit T-dependence of the dynamical structure factors: the first one increases the role of RPA correlations wheres the second decreases it. Compensation of both effects R nearly constant in the range MeV ² Lack of temperature consistency leads to an underestimation of the ν-mpf of ~ 30-50%

32 You for your time & attention Arnau, Alessandro & Dany for the invitation ECT * for its support