Toward a unified description of equilibrium and dynamicsof neutron star matter
Omar Benhar
INFN and Department of Physics“Sapienza” Universita di Roma
I-00185 Roma, Italy
Based on work done in collaboration withA. Carbone, A. Cipollone, G. De Rosi, C. Losa, and A. Lovato
INFN, Laboratori Nazionali del Gran SassoDecember 3rd, 2014
Motivation & Outline
? The validity of the models of neutron star matter must be gauged beyondtheir ability to predict acceptable values of M & R
? Need a unified approach providing a consistent description of
. EOS
. transport properties
. neutrino interactions
. supefulid gap
. . . .
? The paradigm of nuclear many-body theory
. ab initio approach
. effective interaction approach
. bridging the gap: effective interactions from the ab inito approach
? Summary & Outlook
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 2 / 31
The paradigm of nuclear many-body theory
? Nuclear matter is described as a collection of pointlike protons andneutrons interacting through the hamiltonian
H =∑
i
p2i
2m+
∑j>i
vij +∑k>j>i
Vijk
? It has long been realized∗ that the independent particle – or mean field –approximation, which amounts to replacing∑
j>i
vij +∑k>j>i
Vijk →∑
i
Ui ,
fails to take into account the effects of nucleon-nucleon correlations,which are known to play an important role in determining nuclearstructure and dynamics.
∗“The limitation of any independent particle model lies in its inability to encompass thecorrelation between the positions and spins of the various particles in the system” [Blatt &Weiskopf (AD 1952)].
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 3 / 31
The ab initio many-body approach
? The potentials are determined by a fit to the properties of the exactlysolvable two- and three-nucleon systems
. vij strongly constrained by deuteron properties and nucleon-nucleon (NN)scattering data: the ANL v18 model, as an example
vij =∑
p=1,18
vp(rij)Opij
Opij = [11, (σi · σj), Sij,L · S,L2,L2(σi · σj), (L · S)2] ⊗ [11, (τi · τj)] ,
[1, (σi · σj), Sij] ⊗ Tij , (τzi + τzj)
. The three-nucleon potential is determined fitting the properties of thethree-nucleon system
Vijk = V2πijk + VR
ijk
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 4 / 31
Results of the ab initio approach
? Proton-neutron differentialx-section at Ecm = 100 MeV ? Energy level of light nuclei from
Green’s Function Monte CarloAb-initio few-nucleon calculation
-100
-90
-80
-70
-60
-50
-40
-30
-20
Ener
gy (M
eV)
AV18AV18+IL7 Expt.
0+
4He0+2+
6He 1+3+2+1+
6Li3/2−1/2−7/2−5/2−5/2−7/2−
7Li
0+2+
8He 2+2+
2+1+
0+
3+1+
4+
8Li
1+
0+2+
4+2+1+3+4+
0+
8Be
3/2−1/2−5/2−
9Li
3/2−1/2+5/2−1/2−5/2+3/2+
7/2−3/2−
7/2−5/2+7/2+
9Be
1+
0+2+2+0+3,2+
10Be 3+1+
2+
4+
1+
3+2+
3+
10B
3+
1+
2+
4+
1+
3+2+
0+
0+
12C
Argonne v18with Illinois-7
GFMC Calculations24 November 2012
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 5 / 31
Nuclear matter EOS (no adjustable parameters involved)
? Binding energy per particle ofisospin-symmetric nuclear matter(SNM)
0
20
40
60
80
100
0 0.1 0.2 0.3 0.4 0.5
Ener
gy p
er n
ucle
on (
MeV
)
ρ (fm-3)
FHNC: v8’+TM’1FHNC: v8’+TM’2FHNC: v8’+TM’3
? Binding energy per particle ofpure neutron matter (PNM)
0
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5
Ener
gy p
er n
ucle
on (
MeV
)
ρ (fm-3)
FHNC: v8’+TM2
AFDMC: v8’+TM2
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 6 / 31
The efective interaction approach
? The bare potential is replaced with an efective potential, suitable for usewithin the framework of perturbation theory in the Fermi gas basis. TheSkyrme potential, as an example
veffij = δ(ri − rj)t(k,k′)
k =i2
(−→∇1 −
−→∇2) , k′ =
i2
(←−∇1 −
←−∇2)
? The above definition can be generalized to include spin-dependence. Theparameters involved are adjusted in such a way as to reproduce selectednuclear properties, as well as the equilibrium properties of isospinsymmetric nuclear matter.
? The ground state expectation value of the hamiltonian can be written inthe form of a energy-density functional
〈H〉 = 〈∑
i
p2i
2m+
∑j>i
veffij 〉 = E(ρp, ρn)
.Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 7 / 31
Results of the effective interaction approach
? Equation of state (EOS) of SNM and PNM computed using differentSkyrme- and Gogny-type effective interactions, compared to thevariational results obtained from the Argonne-Urbana hamiltonians.
? The effective interactions, while being capable to provide a reasonabledescription of the EOS, are limited by the approximations involved intheir definition, lacking a direct connection to the underlying nuclearinteractions.
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 8 / 31
Ab initio effective interaction
? In the ab initio approach the uncertainty associated with the dynamicalmodel is decoupled from the approximations involved in many-bodycalculations
? Once the nuclear hamiltonian is determined, in principle its eigenstatescan be obtained from the solution of the Schrodinger equation
H |n〉 = En |n〉
? Calculation of nuclear observables do not involve any additionalparameters
? The Schrodinger equation can only be solved for nuclei with massnumber A ≤ 12. Approximations are required for larger A, as well as foruniform nuclear matter in the A→ ∞ limit.
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 9 / 31
Correlated Basis Function (CBF) formalism
? The eigenstates of the nuclear hamiltonian are approximated by the set ofcorrelated states, obtained from the eigenstates of the Fermi Gas (FG)model
|n〉 =F|nFG〉
〈nFG|F†F|nFG〉1/2 =
1√Nn
F |nFG〉 , F = S∏j>i
fij
? the structure of the two-nucleon correlation operator reflects thecomplexity of nuclear dynamics
fij =∑
p
fp(rij)Opij
? the operators Onij are the same as those entering the definition of the NN
potential
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 10 / 31
Cluster expansion and FHNC equations
? The ground state expectation value of the hamiltonian is written as a sumof contributions associated with subsystems (clusters) consisisting of anincreasing number of particles
〈H〉 =〈0|H|0〉〈0|0〉
= EFG +∑n≥2
(∆E)n
? The relevant terms of the cluster expansion can be summed up at allorders solving a set of integral equations known as Fermi Hyper-NettedChain (FHNC) equations
? the shapes of the fp(rij) are determined form the minimization of theground-state expectation value of the hamiltonian
E0 ≥ minF
〈0FG|F†HF|0FG〉
〈0FG|0FG〉
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 11 / 31
Alternative approach: the CBF effective interaction
? Within CBF, the effective interaction is defined through
〈H〉 =〈0|F†(T + V)F|0〉〈0|F†F|0〉
= 〈0FG|T + Veff |0FG〉
? At two-body cluster level
Veff =∑j>i
veff(ij)
veff(ij) = f †ij
[−
1m
(∇2fij) −2m
(∇fij) · ∇ + vijfij
]
? Three-nucleon interactions can be taken into account extending thedefinition to include three-body cluster contributions
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 12 / 31
CBF effective interaction at SNM equilibrium density
-200
0
200
400
600
800
1000
0 0.5 1 1.5 2 2.5
v(r
)[M
eV]
r [fm−1]
(a) veffS=0,T=1(r)
vbareS=0,T=1(r)
-200
0
200
400
600
800
1000
0 0.5 1 1.5 2 2.5
v(r
)[M
eV]
r [fm−1]
(b) veffS=1,T=1(r)
vbareS=1,T=1(r)
-40
-20
0
20
40
60
80
0 0.5 1 1.5 2 2.5
v(r
)[M
eV]
r [fm−1]
(c) vefft,T=1(r)
vbaret,T=1 (r)
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 13 / 31
EOS of PNM obtained using the CBF effective interaction
0
10
20
30
40
50
60
0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32
E/A
[MeV
]
ρ [fm−3]
FHNC/SOC
veff12
|||||||||3b
AFDMC
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 14 / 31
Transport properties
? Landau-Abrikosov-Khalaktnikov formalism: Boltzman equation
∂n∂t
+∂n∂r
∂ε
∂p−∂n∂p
∂ε
∂r= I(n)
n = n0 + δn , n0 = 1 + exp[β(ε − µ)]−1
? The collision integral I(n) depends on the probability of the in mediumNN scattering process
W =16π2
m?2
(dσdΩ
)
? The description of transport properties require dynamical modelsproviding an accurate description of NN scattering in the nuclearmedium, constrained by the available data in the zero-density limit
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 15 / 31
Shear Viscosity of pure neutron matter
? Abrikosov-Khalatnikov (AK) estimate of the shear viscosity in thelow-temperature limit
ηAK =15ρm?v2
Fτ2
π2(1 − λη)
? Quasiparticle lifetime
τT2 =8π4
m∗31〈W〉
,
? Angle-averaged collision probability
〈W〉 =
∫dΩ
2πW(θ, φ)
cos (θ/2), λη =
〈W[1 − 3 sin4 (θ/2) sin2 φ]〉〈W〉
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 16 / 31
In medium neutron-neutron cross section
? From Fermi’s golden rule
W(p,p′) = 2π∣∣∣veff(p − p′)
∣∣∣2 ρ(p′)
dσdΩp′
=m?2
16π2
∣∣∣veff(p − p′)∣∣∣2
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 17 / 31
Single particle spectrum and effective mass
? Calculations carried out within the Hartree-Fock approximation usingthe CBF effective interaction
e(k) =k2
2m+
∑k′〈kk′|veff |kk′〉a ,
1m?
=1k
de(k)dk
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 18 / 31
? Density dependence of ηT2 n pure neutron matter
? Note: the SLya effective interaction, adjusted to reproduce the themicroscopic EOS, predicts ηT2 ∼ 6 × 1013 g cm−1 s−1 MeV2 at nuclearmatter equilibrium density, to be compared with the result obtained fromthe CBF effective interaction ηT2 ∼ 1.4 × 1015 g cm−1 s−1 MeV2 .
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 19 / 31
Thermal conductivity of pure neutron matter
? The transport coefficients computed using the CBF effective interactionis remarkably close to the result obtained within the G-matrix approachusing the same bare NN potential. Note: three-body interactions are nottaken into account.
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 20 / 31
Neutrino interactions in nuclear matter
? Neutral current interactions in neutron matter
JµZ =∑
i
jµi , jµi = ψniγµ(1 − CAγ
5)ψni
? In the non relativistic limit
J0Z → Oρ
q =∑
i
eiq·ri , JZ → Oσq =∑
i
eiq·riσi
? Neutrino scattering rate and response functions
W(q, ω) =G2
F
4π2
(1 + cos θ)Sρ(q, ω) +C2
A
3(3 − cos θ)Sσ(q, ω)
,Sρ(q, ω) =
1N
∑n
|〈n|Oρq|0〉|
2δ(ω + E0 − En) , Sσ(q, ω) =∑α
Sσαα(q, ω)
Sσαβ(q, ω) =1N
∑n
〈n|Oσαq |0〉〈0|O
σβq |n〉δ(ω + E0 − En)
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 21 / 31
Density and spin responses of pure neutron matter
? The target response tensor
WµνA =
∑n
〈0|JµZ†|n〉〈n|JνZ |0〉δ(ω + E0 − En)
must be computed using correlated initial and final states, which amountsto compute the transition matrix element of the effective operator
JµA =1
√N0Nn
F†JµAF
between FG states |n)
? In the one particle-one hole sector
|n〉 =1√Nph
F|ph) , 〈n|JνA|0〉 → (ph|JµA|0)
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 22 / 31
Including long range correlations
? Allow for propagation of the particle-hole pair, giving rise to theexcitation of collective modes. Replace
|n〉 →N∑
i=1
Ci |pihi)
? The energy of the state |n〉 and the coefficients Ci are obtaineddiagonalizing the N × N hamiltonian matrix
Hij = (E0 + epi − ehi)δij + (hipi|veff |hjpj)
with the CBF effective interaction and the Hartree-Fock spectrum
ek =k2
2m+
∑k′〈kk′|veff |kk′〉a
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 23 / 31
Alternative approach: Landau theory
? Landau theory of normal Fermi liquids can also be employed to obtainthe density and spin responses of pure neutron matter
? the value of the Landau parameters can be obtained from thequasiparticle interaction, which can be in turn expressed in terms ofmatrix elements of the effective interaction
fσσ′pp′ = fpp′ + gpp′(σ · σ′) + fpp′S12(p − p′)= 〈pσ p′σ′|veff |pσ p′σ′〉 − 〈pσ p′σ′|veff |p′σ′ pσ〉
? this formalism can be easily extended to non zero temperatures, in therange T << TF
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 24 / 31
Charged current interactions at low-momentum transfer
? Fermi (density, left) and Gamow-Teller (spin, right) contributions to theresponse of pure neutron matter at nuclear matter equilibrium density(ρ0 = 0.16 fm−3) and momentum transfer |q| = 0.1 fm−1
0
0.002
0.004
0.006
0.008
0.01
0.012
0 2 4 6 8 10
Sρ(q
,ω)
[MeV
−1 ]
ω [MeV]
LandauCTDCHF
0
0.025
0.05
0.075
0.1
0.125
0.15
0 2 4 6 8 10
Sσ(q
,ω)
[MeV
−1 ]
ω [MeV]
LandauCTD transverse
CTD longitudinalCHF transverse
CHF longitudinal
? the collective mode is only excited in the spin channel
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 25 / 31
Neutrino mean free path in neutron matter at ρ = ρ0
1.6
1.8
2
2.2
2.4
2.6
5 10 15 20 25 30 35 40
λ/λ
FG
Eν [MeV]
CTD full expressionCTD simplified expression
CTD without collective mode
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 26 / 31
Responses and Neutrino mean free path from Landau theory
? Dependence on momentum transfer at ρ = ρ0
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 27 / 31
? Mean free path of a non degenerate neutrino in neutron matter
. Left: density-dependence at k0 = 1 MeV and T = 0
. Right: energy dependence at ρ = 0.16 fm−3 and T = 0, 2 MeV
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 28 / 31
? Density and temperature dependence of the mean free path of a nondegenerate neutrino at k0 = 1 MeV and ρ = 0.16 fm−3
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 29 / 31
Neutron pairing in the 1S0 channel
? Gap equation
∆(k) = −1π
∫k′2dk′
v(k, k′)∆(k′)[(e(k′) − µ)2 + ∆2(k′)
]1/2
v(k, k′) =
∫r2drj0(kr)veff(r)j0(k′r)
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 30 / 31
Summary & Outlook
? Resolving the degeneracy associated with models of the EOS providingsimilar values of neutron stars’ mass and radius will require the study ofdifferent properties
? This analysis will in turn require the development of novel approaches,allowing for a consistent description based on a unified dynamical model
? Effective interactions obtained from realistic nuclear hamiltoniansprovide a powerful tool to carry out calculations of a number of differentquantities, ranging from the EOS to single particle properties and inmedium scattering probabilities
? The model dependence associated with the many-body approachemployed to obtain the effective interaction apperas to be remarkablyweak
Omar Benhar (INFN, Roma) INFN, LNGS December 3rd, 2014 31 / 31