Physics Letters B 765 (2017) 109–112
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Skyrme model study of proton and neutron properties in a strong
magnetic field
Bing-Ran He
Department of Physics, Nanjing Normal University, Nanjing 210023, PR China
a r t i c l e i n f o a b s t r a c t
Article history:
Received
29 September 2016
Received
in revised form 27 November 2016
Accepted
4 December 2016
Available
online 12 December 2016
Editor: J.-P.
Blaizot
The proton and neutron properties in a uniform magnetic field are investigated. The Gell–Mann–Nishijima
formula is shown to be satisfied for baryon states. It is found that with increasing magnetic field strength,
the proton mass first decreases and then increases, while the neutron mass always increases. The ratio
between magnetic moment of proton and neutron increases with the increase of the magnetic field
strength. With increasing magnetic field strength, the size of proton first increases and then decreases,
while the size of neutron always decreases. The present analysis implies that in the core part of the
magnetar, the equation of state depend on the magnetic field, which modifies the mass limit of the
magnetar.
© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Recently , experiments have observed that there exists a strong
magnetic field when baryons collide with each other, and astro-
physicists
have observed that the strong magnetic field exists in
magnetars [1–3]. The baryon states have electric charge distribu-
tion,
thus the interaction between baryons and magnetic fields
modifies the properties of baryons. The Skyrme model [4], which
identifies the soliton solution from mesons theories as the baryon,
has been widely accepted, and also have lots of applications to
hadron physics, astrophysics and also condensed matter physics.
The study of Skyrmion in a uniform magnetic field shows that, in
the leading order of large N
C
, i.e., O(N
C
), the mass and shape of
Skyrmion depend on the strength of magnetic field [5].
In
this letter, the O(N
−1
C
) effects are introduced in the semi-
classical
quantization approach [6], and then the physical baryon
states, i.e., proton and neutron, in a uniform magnetic field are
studied. The Gell–Mann–Nishijima formula for baryon states is
shown
to be satisfied. The semi-classical quantization of Skyrmion
introduces time dependence to (eB) terms of the model. Because
the wave functions for baryon states are different, the magnetic
response of baryon states is different. It is found that with the
increase of the magnetic field strength, the effective proton mass
first decreases and then increases, consequently, the proton size
first increases and then decreases. On the other hand, the effective
neutron mass always increases, and consequently, the neutron size
E-mail address: hebingran@njnu.edu.cn.
always decreases. Furthermore, the ratio between magnetic mo-
ment
of proton and neutron increases with the increase of the
magnetic field strength. Finally, since both the mass and size of
proton and neutron depend on the strength of the magnetic field,
the equation of state for magnetar is modified.
2. The model
The action of the model contains two parts:
=
d
4
xL +
WZW
, (1)
where L is expressed as
L =
f
2
π
16
Tr
(D
μ
U
†
D
μ
U ) +
1
32g
2
Tr([U
†
D
μ
U , U
†
D
ν
U ]
2
)
+
m
2
π
f
2
π
16
Tr
(U +U
†
−2). (2)
Here f
π
is the pion decay constant, m
π
is the pion mass, and g
is
a dimensionless coupling constant. The covariant derivative for
U is expressed as D
μ
U = ∂
μ
U −iL
μ
U + iU R
μ
, where L and R
are the external fields expressed as L
μ
=R
μ
=eQ
B
V
B
μ
+eQ
E
H
μ
for the present purpose. Here e is the unit electric charge, Q
B
=
1
3
1 is the baryon number charge matrix, Q
E
=
1
6
1 +
1
2
τ
3
is the
electric charge matrix, 1 is the rank 2unit matrix, τ
3
is the third
Pauli matrix, and V
B
μ
is the external gauge field of the U (1)
V
baryon number. In the symmetric gauge, the magnetic field H
μ
is
http://dx.doi.org/10.1016/j.physletb.2016.12.019
0370-2693/
© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.