Isayev kipt khnu17

Constraining central magnetic
field strength in strange quark
stars
Alexander Isayev
Kharkov Institute of Physics
Technology

XIII Conference "Problems of Contemporary
Nuclear Energetics"
2October 18, 2017
The main focus: nuclear matter in the deconfined state. If nuclear
matter is hot (high temperature) or dense (high baryon density) enough,
quarks will be released from the hadronic bags – the deconfinement
phase transition. The conditions of high temperature are realized in high
energy heavy-ion collisions at RHIC and LHC, and the conditions of high
density of about several times nuclear saturation density are met in
heavy neutron stars.
Strange quark matter: matter consisting of deconfined u, d, s quarks.
At zero temperature and pressure, the energy per baryon in
strange quark matter (SQM) for a certain range of the model QCD-related
parameters can be less than that for the most stable 56Fe nucleus, and,
hence, SQM can be the true ground state of matter.
A. Bodmer, Phys. Rev. D 4, 1601 (1971).
E. Witten, Phys. Rev. D 30, 272 (1984).
E. Farhi and R. L. Jaffe, Phys. Rev. D 30, 2379 (1984).
Introduction

Nuclear Energetics"
3October 18, 2017
Astrophysical implications. SQM can be encountered in:
1. Strange quark stars, self-bound by strong interactions
(in contrast to neutron stars bound by gravitational forces).
2. Hybrid stars: SQM is metastable at zero pressure, but it can appear in
the high-density core of a neutron star as a result of the deconfinement
phase transition. In this case, the stability of SQM is provided by the
gravitational pressure from the outer hadronic layers.
3. Small nuggets, called strangelets (can be found in cosmic rays).
Introduction
When a critical size droplet of SQM is formed in the interior of a neutron
star, the conversion of a neutron star to strange quark star or hybrid star
proceeds via the laminar combustion, accompanied by quite a strong
neutrino signal ~ 1051-1052 erg/s. As a result, the neutrino luminosity
has the quasi-plateau, corresponding to the period of laminar
combustion, which can represent a firm signature for the existence of
quark matter inside compact stars.

Nuclear Energetics"
4October 18, 2017
Another important aspect: compact stars are endowed with the magnetic
field.
Neutron stars observed in nature are magnetized objects with the
magnetic field strength at the surface in the range 109-1013 G.
R.C. Duncan, and C. Thompson (1992): even more strongly magnetized
objects, called magnetars, can exist in the universe with the field strength of
about 1014-1015 G at the surface. Golden candidates for magnetars: soft
gamma-ray repeaters and anomalous X-ray pulsars. Recently it was
suggested that a magnetized hybrid star, or a magnetized strange quark star
can be a real source of the SGRs or AXPs.
In the core of a magnetar the magnetic field strength may be even larger,
H ~ 1018-1020 G (from a scalar virial theorem).
Introduction

Nuclear Energetics"
5October 18, 2017
Introduction
Possible mechanisms to generate such strong magnetic fields:
• turbulent dynamo amplification mechanism in a compact star with fast
rotating core;
• spontaneous ordering of nucleon or quark spins in the dense core of a
compact star
To note: In such ultrastrong magnetic fields, because of the breaking of the
rotational O(3) symmetry by the magnetic field, the total pressure becomes
anisotropic having different values along and perpendicular to the field
direction. The pressure anisotropy should be accounted for in strong
magnetic fields characteristic to the cores of magnetars.

Nuclear Energetics"
6October 18, 2017
Introduction
Main objectives:
• to determine the domain in the model parameter space for which
magnetized SQM is absolutely stable, i.e., its energy per baryon is less
than that for the most stable 56Fe nucleus ;
• to study the impact of a strong magnetic field on the thermodynamic
properties of strange quark matter under the conditions relevant to the
the interior of strange quark stars with account of the effects of the
pressure anisotropy;
Related articles:
A.A. Isayev, Phys. Rev. C 91, 015208 (2015)
A.A. Isayev, J. Phys. Conf. Ser. 607, 012013 (2015)
A.A. Isayev, Int. J. Mod. Phys. A 29, 1450173 (2014)

Nuclear Energetics"
7October 18, 2017
General Formalism: Magnetized Strange Quark Matter
A theoretical framework to study SQM: the MIT bag model.
In the simplified version of the MIT bag model, quarks are considered as free
fermions moving inside a finite region of space called a ”bag”. The effects of
the confinement are accomplished by endowing the finite region with a
constant energy per unit volume, the bag constant B. The bag constant B can
be also interpreted as the inward pressure - the ”bag pressure”, needed to
confine quarks inside the bag.
The Lagrangian density for a relativistic system of noninteracting quarks (u,d,s)
and leptons (e-) in an external magnetic field reads
Dirac equation for the quark and electron spinors
1 2
1
16
( =0,1,2,3)
, , ,
[ ( ) ] ,
,
i i i i
u d s e
L i q A m F F
F A A A Hx
μ μν
μ μ μν
μν μ ν ν μ μ μ
ψ γ ψ
π
δ μ
= ∂ − − −
= ∂ − ∂ =
∑
0[ ( ) ] , , , ,i i ii q A m i u d s eμ
μ μγ ψ∂ − − = =

Nuclear Energetics"
8October 18, 2017
The energy spectrum of free relativistic fermions in an external magnetic field has the
form
2 2 1
2 sgn
2 2
| | , ( )i
z i i i
s
k m q H n qνε ν ν= + + = + −
ν=0,1,2,… enumerates the Landau levels, n is principal quantum number, s=+1
corresponds to a fermion with spin up, s = - 1 to a fermion with spin down.

Nuclear Energetics"
9October 18, 2017
The grand thermodynamic potential density for nonmagnetized fermions of ith
species at finite temperature
{ } ( )
3
3
1 1
2
( )/ ( )/
ln lni i i iT T
i i
d k
g T e eε μ ε μ
π
− − − +
⎡ ⎤ ⎡ ⎤Ω = − + + +⎣ ⎦ ⎣ ⎦∫
For magnetized fermions of ith species:
( )
23
3 2 2 2
1 1 0
1
2
2 2 2 2 22
| |
... ... | | ... ...
( ) ( )
z z i
i z
s n s n
dk dk q Hd kd k
q H dkπ
π π π π ππ
∞ ∞ ∞
⊥
=± =±−∞ −∞
→ → =∑∑ ∑∑∫ ∫ ∫ ∫ ∫
{ }2
1 0
1 1
2
, ,( )/ ( )/| |
ln ln
i i
n s i n s iT Ti i
i z
s n
q g H
T dk e e
ε μ ε μ
π
∞
− − − +
=±
⎡ ⎤ ⎡ ⎤Ω = − + + +
⎣ ⎦ ⎣ ⎦∑∑∫
Hence
or
{ }02
0 0
2 1 1
2
( )/ ( )/
,
| |
( ) ln ln
i i
i iT Ti i
i z
q g H
T dk e eν νε μ ε μ
ν
ν
δ
π
∞∞
− − − +
=
⎡ ⎤ ⎡ ⎤Ω = − − + + +
⎣ ⎦ ⎣ ⎦∑ ∫

Nuclear Energetics"
10October 18, 2017
2
02
0
2 2 2
2 2
2
4
2
means the integer part of the argument
2
3 for quarks (# o
max
,
, , ,
,
, , ,
max
| |
( ) ln ,
| | , ,
, [...]
| |
,
i i
F iii i
i i F i
i
i
i i i F i i
i i i
i
i
kq g H
k m
m
m m q H k m
m
I I
q H
g
ν
ν
ν ν ν
ν ν
ν ν ν
μ
δ μ
π
ν μ
μ
ν
=
⎧ ⎫+⎪ ⎪
Ω = − − −⎨ ⎬
⎪ ⎪⎩ ⎭
= + = −
⎡ ⎤−
= ⎢ ⎥
⎣ ⎦
=
∑
f colors)
1 for electrons,
⎧
⎨
⎩
The grand thermodynamic potential density for fermions of ith species in the
external magnetic field at zero temperature reads
The number density of fermions of ith species i
i
i T
ρ
μ
⎛ ⎞∂Ω
= −⎜ ⎟
∂⎝ ⎠
max
,0 ,2
0
| |
(2 )
2
i
ii i
i F
q g H
k
ν
ν ν
ν
ρ δ
π =
= −∑
2
02
0
2 2 2
2 2
2
4
2
means the integer part of the argument
2
3 for quarks (# o
max
,
, , ,
,
, , ,
max
| |
( ) ln ,
| | , ,
, [...]
| |
,
i i
F iii i
i i F i
i
i
i i i F i i
i i i
i
i
kq g H
k m
m
m m q H k m
m
I I
q H
g
ν
ν
ν ν ν
ν ν
ν ν ν
μ
δ μ
π
ν μ
μ
ν
=
⎧ ⎫+⎪ ⎪
Ω = − − −⎨ ⎬
⎪ ⎪⎩ ⎭
= + = −
⎡ ⎤−
= ⎢ ⎥
⎣ ⎦
=
∑
f colors)
1 for electrons,
⎧
⎨
⎩
i
i
i T
ρ
μ
⎛ ⎞∂Ω
= −⎜ ⎟
∂⎝ ⎠

Nuclear Energetics"
11October 18, 2017
The energy density for fermions of ith species at zero temperature
2
2 2
,
8
,
8 8
i
i
l i t i
i i
H
E E B
H H
p B p HM B
π
π π
= + +
= − Ω − − = − Ω − + −
∑
∑ ∑
In the MIT bag model, the total energy density, longitudinal pl and transverse pt
pressures in quark matter are given by [Phys. Rev. C 82, 065802 (2010)]
M=Σi Mi is the total magnetization, Mi is the partial magnetization
i i i iE μ ρ= Ω +
2
02
0
2
4
max
,
, , ,
,
| |
( ) ln
i i
F iii i
i i F i
i
kq g H
E k m
m
ν
ν
ν ν ν
ν ν
μ
δ μ
π =
⎧ ⎫+⎪ ⎪
= − +⎨ ⎬
⎪ ⎪⎩ ⎭
∑
i
i
iM
H μ
∂Ω⎛ ⎞
= −⎜ ⎟
∂⎝ ⎠
The total pressure in magnetized SQM is anisotropic. There are two sources
of the pressure anisotropy:
• the matter part ~ M
• the field part ~ H2/8π (the Maxwell term)
In strong magnetic fields, for nonferromagnetic medium the field part is dominating
over the matter part and represents the major source of the pressure anisotropy.

Nuclear Energetics"
12October 18, 2017
Absolute stability window for magnetized SQM
The absolute stability window: at zero external pressure and temperature
56
( Fe)m m
H
B BMSQM ud
E E
ε
ρ ρ
< <
Zero external pressure conditions in the MIT bag model description of SQM:
2
0,
8
0
l H H i
i
H
t H
H
p B
p H
H
π
⎧
= −Ω = Ω = Ω + +⎪⎪
⎨
∂Ω⎪ = −Ω + =
⎪ ∂⎩
∑
0 ,
i
H i
i
M
H H μ
∂Ω ∂Ω⎛ ⎞
= ⇒ = − ⎜ ⎟
∂ ∂⎝ ⎠
∑
In order to satisfy this equation, the bag pressure should be field dependent.
Otherwise, this equation would mean that the response of the system to an external
magnetic field would considerably exceed the paramagnetic response, that is hard
to expect without spontaneous spin ordering in the system.
0
4
H B
M
Hπ
∂
− − =
∂
1
3
B f
f
ρ ρ= ∑,

Nuclear Energetics"
13October 18, 2017
In order to determine the absolute stability window:
1. To find the respective chemical potentials of all fermion species:
a) Charge neutrality
2 3 0u d s e
ρ ρ ρ ρ −− − − =
b) Chemical equilibrium with respect to weak processes
,
,
e e
e e
d u e u e d
s u e u e s
s u d u
ν ν
ν ν
− −
− −
→ + + + → +
→ + + + → +
+ ↔ +
,d u d se
μ μ μ μ μ−= + =
c) From the stability constraint:
56
( Fe) 930 MeVm
H
B MSQM
E
ε
ρ
= ≈
2. The upper (lower) bound on the bag pressure from the absolute stability window:
2
( )
, , ,
( , , )
( )
8
u l i
i u d s e
i u d e
H
B H
π=
=
= − Ω −∑
1) Upper bound on B:
56 19
( Fe) 930 MeV (for 10 G)m
H
B ud
E
Hε
ρ
= ≈ <2) Lower bound on B:

Nuclear Energetics"
14October 18, 2017
The maximum value of the upper bound Bu: Bmax=Bu(H=0)≈85 MeV/fm3 for ms=90 MeV,
Bmax≈80 MeV/fm3 for ms=120 MeV, Bmax≈75 MeV/fm3 for ms=150 MeV
Bu vanishes at Hmax≈ 3.3·1018 G for ms=90 MeV; at Hmax≈ 3.2·1018 G for ms=120 MeV
at Hmax≈ 3.1·1018 G for ms=150 MeV
Thus, in order magnetized SQM would be absolutely stable, the magnetic field
strength should satisfy the constraint H<Hmax. In fact, Hmax represents the
upper bound on the magnetic field strength which could be reached in a
magnetized strange quark star
mu = md = 5 MeV, and
ms = 90; 120; 150 MeV;
For the lower bound Bl: Bmax=Bl(H=0)≈57 MeV/fm3; Bl vanishes at Hl,0≈ 2.7·1018 G

Nuclear Energetics"
15October 18, 2017
The maximum values of the bounds ρB
u , ρB
l : ρB
u≈5.6ρ0, ρB
l≈4.4ρ0 for ms=90 MeV,
ρB
u≈5.4ρ0, ρB
l≈4.3ρ0 for ms=120 MeV, ρB
u≈5.2ρ0 , ρB
l≈4.1ρ0 for ms=150 MeV
The increase of the current mass ms leads to the decrease of the upper bound on
ρB
u: ρB
u(H=0) ≈ 2.4ρ0 for ms=90 MeV; ρB
u(H=0) ≈ 2.3ρ0 for ms=120 MeV; ρB
u(H=0) ≈
2.2ρ0 for ms=150 MeV
Magnetic fields H>~ 1018 G strongly affect the upper and lower bounds on the
baryon number density from the absolute stability window.
mu = md = 5 MeV, and
ms = 90; 120; 150 MeV;

Nuclear Energetics"
16October 18, 2017
Magnetized strange quark stars
For the parameters within the absolute stability window, magnetized strange
quark stars made up entirely of SQM and self-bound by strong interactions
can be formed.
Aim: to study thermodynamic properties of magnetized SQM under conditions
relevant to the interior of strange quark stars with account of the effects of the
pressure anisotropy.

Nuclear Energetics"
17October 18, 2017
SQM in inhomogeneous magnetic field
Magnetic field varies from the core to the surface. This variation can be modeled by
the dependence H(μB):
0
0
( ) 1
B B
B
B s cenH H H e
γ
μ μ
β
μ
μ
⎛ ⎞−
− ⎜ ⎟
⎝ ⎠
⎡ ⎤
⎢ ⎥= + −
⎢ ⎥
⎣ ⎦
In numerical calculations: Hs = 1015 G, the exponential parametrization with β=45, γ=3.
μB0 is the baryon chemical potential at the surface of a strange quark star,
Hcen and Hs– central and surface magnetic field strengths
The current quark masses: mu = md = 5 MeV, and ms = 150 MeV.
The bag pressure: B = 74 MeV/fm3 - smaller than the upper bound B ≈ 75 MeV/fm3
from the absolute stability window for these current quark masses.

Nuclear Energetics"
18October 18, 2017
SQM in inhomogeneous magnetic field
From the set of four equations one can find the chemical potentials of quarks and
electrons and then the baryon chemical potential at the surface of a strange
quark star:
In order to determine the baryon chemical potential μB0 at the surface of a strange
quark star for this set of parameters:
2 3 0u d s e
ρ ρ ρ ρ −− − − =
1) Charge neutrality
2) Chemical equilibrium with respect to weak processes
,d u d se
μ μ μ μ μ−= + =
3) Further we assume a spherically symmetric radial distribution of the magnetic field
inside a star. Then the longitudinal pressure pL should vanish at the surface of a star:
2
0
8
l i
i
H
p B
π
= − Ω − − =∑
0 0 0 0 927.4 MeVB u d sμ μ μ μ= + + ≈

Nuclear Energetics"
19October 18, 2017
Anisotropic pressure of SQM in inhomogeneous H
Transverse (the upper 3 curves) and longitudinal (the lower 3 curves) pressures in
magnetized SQM vs. μB at T=0 and variable central field Hcen. The full dots correspond to
the points where pL'(μB)=0.
States of MSQM with the central magnetic field, characterized by the appearance
pL'(μB)<0, are unstable: instability is developed along the magnetic field

Nuclear Energetics"
20October 18, 2017
Energy density of magnetized SQM vs. μB
The energy density E (upper curves except the lower curve) and its matter part
contribution Em=E-H2/8π (the lower curve) vs. μB.
The relative role of the matter Em and magnetic field Ef contributions to E=Em+ Ef:
the matter part dominates over the field part at such μB and H.

Nuclear Energetics"
21October 18, 2017
Anisotropic EoS of magnetized SQM
Because of the pressure anisotropy in strongly magnetized SQM, the equation of
state of the system is also highly anisotropic.

Nuclear Energetics"
22October 18, 2017
Conclusions
In summary, we have considered the behavior of SQM at zero temperature
under additional constraints of charge neutrality and beta equilibrium within
the framework of the MIT bag model in strong magnetic fields up to 1019 G.
It has been determined the absolute stability window of magnetized SQM in the
planes “H – B” and “H – ρB”, for which at zero external pressure:
1) magnetized SQM has the energy per baryon less than that of the most
stable 56Fe nucleus,
2) the energy per baryon of magnetized two-flavor quark matter is larger than
that of 56Fe nucleus.
It has been clarified that the upper bound on the bag pressure from the
absolute stability window vanishes at H~3· 1018 G. The magnitude of this field,
in fact, plays the role of the upper bound on the magnetic field strength which
can be reached in a strongly magnetized strange quark star.

Nuclear Energetics"
23October 18, 2017
Conclusions
For the parameters within the domain of absolute stability, magnetized strange
quark stars made up entirely of SQM and self-bound by strong interactions can
be formed. It has been studied the impact of a strong magnetic field, varying
with the baryon chemical potential, on thermodynamic properties of SQM in the
interior of a magnetized strange quark star:
• It is clarified that the central magnetic field strength is bound from above
by the value at which the derivative pL'(μB) vanishes first somewhere in the
interior of a strange quark star under varying the central field. Above this
upper bound, the instability along the magnetic field is developed in
magnetized SQM.
• The total energy density E, transverse pT and longitudinal pL pressures in
magnetized SQM have been calculated as functions of the baryon chemical
potential. Also, the highly anisotropic EoS has been determined in the
form of pT(E) and pL(E) dependences.

Nuclear Energetics"
24October 18, 2017
Thank you for attention

Isayev kipt khnu17

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