Neutrino Emission from Magnetized Proto-Neutron Stars in Relativistic Mean Field Theory

0.4 MB·English

by  Tomoyuki Maruyama

Neutrino Emission from Magnetized Proto-Neutron Stars in
Relativistic Mean Field Theory
Tomoyuki Maruyama,1,2,3 Nobutoshi Yasutake,4 Myung-Ki Cheoun,5,3
Jun Hidaka,3 Toshitaka Kajino,3,6 Grant J. Mathews,7 and Chung-Yeol Ryu5
2 1College of Bioresource Sciences, Nihon University, Fujisawa 252-8510, Japan
1
0 2Advanced Science Research Center,
2
Japan Atomic Energy Research Institute, Tokai 319-1195, Japan
n
a
J 3National Astronomical Observatory of Japan,

7
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
1

] 4Research Institute for the Early Universe, University of Tokyo,
E
H Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
.
h 5Department of Physics, Soongsil University, Seoul, 156-743, Korea
p
- 6Department of Astronomy, Graduate School of Science,
o
r
t University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
s
a
[ 7Center of Astrophysics, Department of Physics,

1 University of Notre Dame, Notre Dame, IN 46556, USA
v
5
(Dated: January 18, 2012)
5
4
3 Abstract
.
1
0 We make a perturbative calculation of neutrino scattering and absorption in hot and dense hy-
2
1 peronicneutron-star matter in the presenceof a strongmagnetic field. We findthat theabsorption
:
v
cross-sections show a remarkable angular dependence in that the neutrino absorption strength is
i
X
r reduced in a direction parallel to the magnetic field and enhanced in the opposite direction. This
a
asymmetry in the neutrino absorbtion can be as much as 2.2 % of the entire neutrino momentum
for an interior magnetic field of 2 1017G. We estimate the pulsar kick velocities associated
∼ ×
with this asymmetry in a fully relativistic mean-field theory formulation. We show that the kick
velocities calculated here are comparable to observed pulsar velocities.
PACS numbers: 25.30.Pt,21.65.Cd,24.10.Jv,95.85.Sz,97.60.Jd,
1
I. INTRODUCTION
Hot and dense hadronic matter is a topic of considerable current interest in nuclear
and particle physics as well as astrophysics because of its associated exotic phenomena. In
particular, many studies have addressed the possible exotic phases of high density matter.
Neutron stars are thought to be among the most realistic possible sites for the presence of
such high density matter. For example, the possible existence of an anti-kaon condensation
in neutron stars has been suggested [1], and the possible implications for astrophysical
phenomena have been widely discussed [2–5].
In this context, a number of works have discussed effects of a softening of the equation of
state (EOS) on both the static and dynamic properties of neutron stars [6–13]. Other work
has considered the thermal evolution of neutron stars by neutrino emission [14–19]. Among
them, Reddy et al. [20] have studied neutrino propagation in proto-neutron stars (PNSs) as
a means to examine the hyperon phase in the high density region.
On the other hand, magnetic fields are thought to play an important role in many astro-
physical phenomena such as asymmetry in supernova (SN) remnants, pulsar kicks [21], and
the existence of magnetars [22, 23]. Indeed, strong magnetic fields seem to be a crucial part
of the still unknown mechanism for non-spherical SN explosions and the origin of the high
velocity [24] that proto-neutron stars (PNS) receive at birth.
Although several post-collapse instabilities have been studied as a possible source of
non-spherical explosions and pulsar kicks, the unknown origin of the initial asymmetric per-
turbations and the uncertainties in the numerical simulations make this possibility difficult
to unambiguously verify [25, 26]. Another viable candidate is the possibility of asymmetric
neutrino emission either as a result of parity violation in the weak interaction [27, 28] or as a
result of an asymmetric magnetic field [29] in strongly magnetized PNSs. Recent theoretical
calculations [30, 31] have suggested that even a 1% asymmetry in neutrino emission out of
∼
the total neutrino luminosity of 1053 ergs could be enough to explain the observed pulsar
∼
kick velocities.
Reddy et al. [20] studied neutrino propagation in PNS matter including hyperons. How-
ever, these models did not include the effects of a strong magnetic field. Although Lai et
al. [30, 31] studied the effects of a magnetic field on asymmetric neutrino emission, the
neutrino-nucleon scattering processes were calculated in a non-relativistic framework [30].
2
In Ref. [32] we reported for the first time our calculated results of the absorption cross-
sections in hot dense magnetized neutron-star matter including hyperons in fully relativistic
mean field (RMF) theory [33, 34].
Inthis paper, we give more detailed informationof theneutrino scattering andabsorption
cross-sections in magnetized neutron-star matter studied in the RMF theory. In the present
RMF framework we take into account the Fermi motion of the baryons and electrons, their
recoil effects, the distortion effects of Fermi spheres made by the magnetic field, and the
effects of the energy difference of the mean field between the initial and final baryons.
We then solve the Boltzmann equation for neutrino transport in a 1D model and discuss
implications of the present results for the pulsar kicks.
In Sec. II we introduce the EOS of nuclear matter based on the RMF theory. In Sec.
III we explain the neutrino scattering and absorption cross-sections in baryonic matter with
the strong magnetic fields. Then, we show numerical results for neutrino reactions and
propagation in matter at finite temperature and discuss the possible associated pulsar kicks
of magnetized PNSs in Sec. IV. In Sec. V we summarize our work in this paper.
3
II. NEUTRON-STAR MATTER IN THE RELATIVISTIC MEAN-FIELD AP-
PROACH
First consider neutron-star matter including nucleons, Lambdas, electrons and electro-
neutrinos (ν ) without a magnetic field in the RMF approach. The Lagrangian density for
e
the matter field is written as
= + , (1)
Matter RMF Lep
L L L
where the first and second terms are the RMF and lepton parts, respectively. The lepton
part includes only the kinetic energy of the electrons and the electron neutrino ν . The RMF
e
part is given by
= ψ¯ (iγ ∂µ M )ψ +g ψ¯ ψ σ +g ψ¯ γ ψ ωµ
RMF N µ N N σ N N ω N µ N
L −
+ψ¯ (iγ ∂µ M )ψ +gΛψ¯ ψ σ +gΛψ¯ γ ψ ωµ
Λ µ − Λ Λ σ Λ Λ ω Λ µ Λ
1 C
U[σ]+ m2ω ωµ IV ψ¯ γ τ ψ ψ¯ γµτ ψ , (2)
− 2 ω µ − 2M2 N µ a N N a N
N
where ψ , ψ , σ, and ω areethe nucleon, Lambda, sigma-meson and omega-meson fields,
N Λ
respectively, M , M , and m are the masses of corresponding particles, γ is the Dirac
N Λ ω µ
matrix, and τ is the Pauli matrix in the isospin spaces. U[σ] is the self-energy potential of
a
the scalar mean-field given in Refs. [7, 41]. The last term describes the vector-type iso-vector
e
interaction between two nucleons and is equivalent to ρ-meson exchange [33]. [Note that
here and in the following we adopt natural units, i.e. ~ = c = 1]
From the Euler-Lagrange equations of the above Lagrangian, the Dirac spinor of the
baryon u (p,s) is obtained as a solution to the following equation:
b
[p/ M∗ U (b)γ ]u (p,s) = 0, (3)
− b − 0 0 b
where M∗ is the baryon effective mass, and U (b) is the time component of the mean-field
b 0
vector potential. We hereafter introduce the Feynman dagger p/ γ pµ for convenience. In
µ
≡
the RMF approach the nucleon and Lambda effective masses are given by
∗
M = M U (N),
N N − s
∗
M = M U (Λ), (4)
Λ Λ − s
with the scalar mean-field potentials:
U (N) = g σ , U (Λ) = gΛ σ . (5)
s σh i s σh i
4
The scalar mean-field σ is given by
h i
∂
U˜[ σ ] = g [ρ (p)+ρ (n)]+gΛρ (Λ), (6)
∂ σ h i σ s s σ s
h i
with the scalar densities, ρ (b) (b = p,n,Λ), defined by
s
2 M∗
ρ (b) d3p n(+)[e(+)(p)]+n(−)[e(−)(p)] b , (7)
s ≡ (2π)3 b b b b E∗(p)
Z h i b
where e(±) are the single particle (+) and antiparticle ( ) energies, E∗(p) = p2 +M∗2,
b − b b
(±) (±)
and the Fermi distributions, n (e ), are defined as p
b b
1
(±) (±)
n (e ) = (8)
b b (±)
1+exp[(e ε )/T]
b ± b
withthetemperatureT andthechemical potential ε . Inaddition, thebaryonsingle-particle
b
energy is written as e(±)(p) = E∗(p) U (b), and the U (b) are given by
b b ± 0 0
g C
U (p) = ω g (ρ +ρ )+gΛρ + IV (ρ ρ ), (9)
0 m2 ω p n ω Λ M2 p − n
ω N
g (cid:8) (cid:9) C
U (n) = ω g (ρ +ρ )+gΛρ IV (ρ ρ ), (10)
0 m2 ω p n ω Λ − M2 p − n
ω N
gΛ (cid:8) (cid:9)
U (Λ) = ω g (ρ +ρ )+gΛρ (11)
0 m2 ω p n ω Λ
ω
(cid:8) (cid:9)
in terms of the proton, neutron and Lambda number densities, ρ , ρ and ρ .
p n Λ
In this work, neutron-star matter at finite temperature will include protons, neutrons,
Lambdas, electrons and neutrinos with the conditions of charge neutrality and beta equi-
librium being satisfied. Then, the proton number density is equal to the electron number
density, ρ = ρ , and the chemical potentials obey the following condition
p e
ε = ε = ε +ε . (12)
n Λ p e
In addition, we assume that the lepton fraction is fixed to be Y = (ρ +ρ )/ρ , where ρ
L e ν B ν
is the neutrino number density, and ρ = ρ +ρ +ρ .
B p n Λ
IntheactualcalculationsweusetheparametersetsofPM1-L1[35],whichgivethebinding
energyperbaryonBE = 16MeV,theeffective massM∗/M = 0.7andtheincompressibility
N N
K = 200MeV at ρ = 0.17 fm−3 in nuclear matter, and the sigma- and omega-Lambda
0
couplings are 2/3 of those for the nucleon, gΛ = 2/3g . A similar relation is found
σ,ω σ,ω
naturally within the quark meson coupling (QMC) model [40].
5
600
(a) T = 20 MeV (c) T = 40 MeV
V) p, n
Me 400 p, n, Λ
A (
E / T 200
0
0.8
(b) xp (d) xp
0.6 xν xΛ
xν
ν
Λ,
p, 0.4
x
0.2
0.0
0 2 4 6 0 2 4 6
ρ / ρ ρ / ρ
B 0 B 0
FIG.1: (Coloronline)Upperpanels(a)and(c)showthedensitydependenceofthetotalenergyper
baryon E /A in neutron-star matter for T = 20 MeV (a) and 40 MeV (c). Solid and long-dashed
T
lines represent the calculated results in systems with and without Lambda particles. The lower
panels (b) and (d) show the number fractions of protons x , Lambda particles x , and neutrinos
p Λ
x for T = 20 MeV (b) and 40 MeV (d). Solid, dot-long-dashed, and short-dashed lines represent
ν
the calculated proton, Lambda, and neutrino number fractions, respectively, in a system including
p,n and Λ. Long-dashed and dotted lines represent the calculated proton and neutrino number
fractions in a system without Lambdas. In the present calculations we use the parameter-set
PM1-L1 [35] for the RMF and the lepton fraction is set to Y = 0.4.
L
InFig.1 we show the energy per nucleon intheupper panels (a andc) andtheprotonand
Lambda fractions in the lower panels (b and d) at T = 20MeV (a and b) and T = 40MeV
(c and d). In these calculations the lepton fraction is taken to be Y = 0.4. The solid and
L
dashed lines represent the results for matter with and without Lambdas, respectively. In
addition the dot-dashed lines in the lower panels indicate the Lambda fraction. We see that
the Lambda fraction appears when ρ & 2ρ and affects remarkably the EOS for ρ & 3ρ .
B 0 B 0
6
III. CROSS-SECTIONS FOR NEUTRINO REACTIONS IN MAGNETIZED
PROTO-NEUTRON STAR MATTER
We separate the Lagrangian density into the following parts
= + + + , (13)
RMF Lep Mag W
L L L L L
where the first, second, third and fourth terms are the RMF, lepton, magnetic, and weak
interaction parts, respectively. Detailed expressions for these terms are explained in the
following subsections.
A. Dirac Wave Function in a Magnetic Field
We assume that there is a uniform magnetic field along the z-direction B = Bzˆ and
for B<1018 G, the effect of the magnetic field on baryons is small enough to be treated
∼
perturbatively. The magnetic part of the Lagrangian density is written as
= + , (14)
Mag BM eM
L L L
where the first andsecond terms describe themagnetic interactions of baryons andelectrons,
respectively.
Ignoring the contribution from the convection current and considering only the spin-
interaction term, the baryon magnetic-interaction Lagrangian density can be written as
= µ ψ¯ σ ψ Fµν = µ ψ¯ σ ψ B (15)
BM b b µν b b b z b
L
b b
X X
with the electromagnetic tensor given by Fµν = ∂µAν ∂νAµ, where Aµ is the electro-
−
magnetic vector potential, σ = [γ ,γ ]/2i, σ = diag(1, 1,1, 1) and µ is the baryon
µν µ ν z b
− −
magnetic moment. We then obtain the baryon wave functions by solving the following Dirac
equation:
∗
[p/ M U (b)γ µ Bγ σ ]u (p,s) = 0 . (16)
− b − 0 0 − b 0 z b
The single particle energies e (p,s) and the Dirac spinors in the limit of a weak magnetic
b
field are given as follows:
2
e (p,s) = p2 + p2 +M∗2 +µ Bs +U (b)
b z T b b 0
s
(cid:18)q (cid:19)
E∗(p)+U (b)+∆E∗(p)s , (17)
≈ b 0 b
7
with
p2 +M∗2
∆E∗(p) = T µ B , (18)
b E∗(p) b
p b
and
1
u (p,s)u¯ (p,s) = [E∗(p)γ p γ +M∗](1+sγ a/(p))
b b 4E∗(p) b 0 − · 5
b
p µ B
+ z b (σ p M∗γ γ )
4E∗3(p) · − b 5 0
b
sµ B
+ b E∗(p)γ0 +M +p γ3 p γ1 p γ2 ,(19)
8E∗(p) p2 +M2 − b z − x − y
b T
(cid:0) (cid:1)
with p
1
a(p) (a ,a ,a ) = (p ,0,0,E∗(p)) . (20)
≡ 0 T z p2 +M∗2 z b
T b
Detailed derivations of these expressions are presented in the appendix A. The second and
p
third terms of Eq. (19) do not appear in the non-relativistic framework, but their contribu-
tions are negligibly small and omitted in the present work.
When we calculate the electron contribution in Eq. (14), we have to use another treat-
ment. This is because electron mass is very small, and its current is almost a Dirac current:
= eψ γ ψ Aµ, (21)
LeM − e µ e
where ψ is the electron field. Also, the effect of a strong magnetic field on electrons may
e
not be a small perturbation. The electron energy in the presence of a strong magnetic field
is generally given by
e (n,k ;s) = k2 +m2 +eB(2n+1+s), (22)
e z z e
p
where n enumerates the Landau levels of the electrons.
In the limit of B 0, the electron wave function becomes a plane wave. Therefore, we
→
can use the same expression as Eq. (19) for electrons, except for the spin vector. The upper
component of the electron Dirac spinor is an eigenvector of the matrix σ . The spin vector
z
in the rest frame of the electron is then (0;0,0,1). In the matter frame the boosted spin
vector can be written as
k k k k2
a(k) = a (k) z , z T ,1+ z , (23)
e
≡ m m (E (k)+m ) m (E (k)+m )
(cid:18) e e e e e e e (cid:19)
where k and k are the components along the z-direction and perpendicular to the z-
z T
direction, respectively.
8
When √2eB ε , the summation over n can be approximated as an integration over
e
≪
energy, i.e.
1 1
dx , (x = 2eB(n+ )) . (24)
T T
→ 2eB 2
n Z
X
Note that the variable x corresponds to p2 in the limit of B 0. Then, the expectation
T T →
ˆ
value of a operator is given by
O
2eB
< ˆ >= dk n (e (n,k ,s)) (n,k ,s)
z e e z z
O (2π)2 O
s n Z
XX
1
dx dk n (e (x ,k ,s)) (x ,k ,s)
T z e e T z T z
≈ (2π)2 O
s Z Z
X
1
d3kn (e (k,s)) (k,s), (25)
e e
≈ (2π)3 O
s Z
X
where the electron energy is approximately e k2 +m2 eBs.
e ≈ e −
Actual calculations are performed in the limit opf m 0, so that the electron energy and
e
→
the spin vector can be approximated by
eBs m
e k2 +m2 k + eµ Bs , (26)
e ≈ e − 2 k2 +m2 ≈ | | k e
q e | |
1 k k k2
a (k) k , zpT, z , (27)
e z
≈ m k k
e (cid:18) | | | |(cid:19)
where µ = e/2m .
e e
−
In this paper the temperature is taken to be rather low relative to the baryon rest mass,
T 40 MeV. In this case the anti-lepton and anti-baryon contributions are negligibly small
≤
in the neutrino reactions. Therefore, we ignore the contributions from antiparticles, and
omit the superscript ’+’ in the single particle energies e(±)(p) and the Fermi distribution
b
n(±)(p,s).
b
B. Neutrino Reaction Cross-Sections
We consider neutrino reactions in neutron-star matter consisting of electrons and baryons
(i.e. protons, neutrons and Lambdas). We assume individual collisions between the initial
neutrino and the constituent particles, and calculate the neutrino scattering (ν ν )
e e
→
and absorption (ν e−) cross-sections in the impulse approximation. Furthermore we
e
→
only consider rather low temperatures, T ε as discussed above, and then ignore the
b
≪
9
contribution from antiparticles. In addition, we treat this system as partially spin-polarized
owing to the magnetic field. The cross-section can then be described in terms of the initial
and final lepton momenta as k and k , as
i f
d3σ G2 d3p
= F V [1 n (e (k ,s ))] i W (k ,k ,p ,p )
dk3 16π2 − l l f l (2π)3 BL i f i f
f Xi,f slX,si,sf Z
n (e (p ,s ))[1 n (e (p ,s ))]
× i i i i − f f f f
(2π)δ3(k +p k p )δ( k +e e e ), (28)
× i i − f − f | i| i − l − f
where V is the volume of the system, and the index l denotes the final lepton species. The
indices i and f denote the initial and final particles for both baryons and electrons which
comprise the neutron-star matter. The function W in Eq. (28) is defined as
BL
1
W = LµνN (29)
BL 4 k k E∗(p )E∗(p ) µν
| i|| f| i i f f
with
1
Lµν = Tr (k/ +m )(1+γ a/ s )γµ(1 γ )k/ γν(1 γ ) , (30)
f l 5 l l 5 i 5
4 { − − }
and
1
∗
N = Tr (p/ +M )(1+γ a/ s )γ (c c γ )
µν 4 f f 5 f f µ V − A 5
(cid:8) (p/ +M∗)(1+γ a/ s )γ (c c γ ) , (31)
× i i 5 i i ν V − A 5 }
where m is the mass of the final lepton.
l
Since we assume that the magnetic field is weak, the Fermi distribution and the delta
function in the above equations can be expanded in terms of the magnetic field B. Then the
cross-section can be presented as a sum of two contributions with and without the magnetic
field
d3σ d3σ0 d3∆σ
S,A S,A S,A
= + , (32)
dk3 dk3 dk3
f f f
where the indices S and A indicate the cross-sections for scattering and absorption, respec-
tively. Note that σ0 is independent of B, and ∆σ is proportional to B. By considering
S,A S,A
the spin-dependence, we can express the W as follows
BL
W = W +W s +W s +W s +W s s +W s s +W s s . (33)
BL 0 i i f f e l 2 i f 3 l i 4 l f
Note that W , W and W only appear when the final lepton is an electron.
e 3 4
10