YITP-14-14

Magnetars and the Chiral Plasma Instabilities
Akira Ohnishi1 and Naoki Yamamoto2, 3
1

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
2
Maryland Center for Fundamental Physics, Department of Physics,
University of Maryland, College Park, Maryland 20742-4111, USA
3
Department of Physics, Keio University, Yokohama 223-8522, Japan

We propose a possible new mechanism for a strong and stable magnetic field of compact stars due
to an instability in the presence of a chirality imbalance of electrons—the chiral plasma instability.
A large chirality imbalance of electrons inevitably occurs associated with the parity-violating weakprocess during core collapse of supernovae. We estimate the magnetic field due to this instability to
be of order 1016 G at the core. This mechanism naturally generates a large magnetic helicity fromthe chiral asymmetry, which ensures the stability of the large magnetic field.
PACS numbers: 97.60.Jd, 12.15.Ji, 52.35.-g

Introduction.—The origin of compact stars with themost powerful magnetic field (∼ 1015 G on the surface)in the Universe, called magnetars [1], is a mystery inastrophysics. Examples of the possible mechanisms include the fossil field or dynamo hypothesis among others[2, 3]. However, these mechanisms have an important
problem that a strong magnetic field produced cannot
be sustained for a long time scale [2, 3]. Indeed, a purely
poloidal magnetic field [depicted in Fig. 1(a)] typicallyconsidered is known to be unstable [4], as confirmed withnumerical magnetohydrodynamic (MHD) simulations [5].
For other issues in these mechanisms, see, e.g., Ref. [3].
It is suggested that if nonzero magnetic helicityZ
H = dx A · B(1)(where B and A are the magnetic field and vector potential) is produced at the initial configuration for somereason, it can make the magnetic field stable [3]. This isbecause H is proportional to the Gauss linking numberof the magnetic flux tubes and serves as an approximate
conserved quantity [6].1 For example, linking of poloidal
and toroidal magnetic fields [depicted in Fig. 1(c)] has anonzero magnetic helicity and can exist stably [7]. However, the origin of the magnetic helicity itself remains to
be understood as well (see also below).
In this paper, we propose a new mechanism for a strong
and stable magnetic field in magnetars due to a novel instability in the presence of an imbalance between rightand left-handed electrons—the chiral plasma instability

1 Magnetic helicity is a conserved quantity for a perfect conductor

and is conserved approximately with finite conductivity; see thediscussion below.

(a)

(b)

(c)

FIG. 1. Configurations of magnetic fields in magnetars: (a)poloidal, (b) toroidal, and (c) linked poloidal-toroidal magnetic fields.

(CPI). The CPI was recently found in the context of electromagnetic and quark-gluon plasmas [8] based on chiralkinetic theory [9]. A related instability had been previously argued for the electroweak theory [10, 11] andfor the primordial magnetic field in the early Universe[12]. (See also Refs. [13] for recent works.) This instability appears somewhat similar to the Rayleigh-Taylor
instability that occurs in the presence of a density imbalance of two fluids at an interface. However, the former
is remarkable in that it is a consequence of relativistic
and quantum effects related to quantum anomalies [14]unlike the latter.
Our mechanism for a strong and stable magnetic field
is based only on the chirality asymmetry of electrons
that is inevitably produced in the parity-violating weakprocess (electron capture) during core collapse of supernovae. The energy of the chirality imbalance is then converted to a large magnetic field by the CPI. Furthermore,
we show that it naturally generates a large magnetic helicity at the same time. Possible generation of magnetic
helicity by the parity-violating (modified) Urca process
after the birth of a neutron star was previously suggested
in Ref. [15] though the CPI was not considered. To the2best of our knowledge, the CPI is the first and only microscopic mechanism to create a magnetic helicity in magnetars at any stage during its evolution. Note that ourmechanism does not require any exotic hadron or quark
phases inside compact stars under discussion, such as
ferromagnetic nuclear matter [16], ferromagnetic quark
matter [17], pion domain walls [18], and so on, which areessential in some of previous suggestions for the origin ofmagnetars.
In the following, we ignore the electron mass me , aselectrons may be regarded as ultrarelativistic at high density, µe  me . We will discuss the possible effects of me
later on.
Chiral Plasma Instabilities.—Let us briefly review thephysical argument of the CPI (see Ref. [19] for the detail).
For simplicity, we here ignore the effect of dissipation
which is not essential to understand the instability itself.
Suppose there is a homogeneous chiral asymmetry between right- and left-handed electrons in the core of compact stars, which we parametrize by a chiral chemical potential µ5 ≡ (µR −µL )/2. (We shall give an estimate of µ5
just after the onset of core collapse of supernovae later.)Let us consider a perturbation of a small magnetic fieldBz with wavelength λ in a cylindrical coordinate (r, θ, z)(see Fig. 2). In the presence of µ5 , this magnetic fieldleads to an electric current in the z direction (called thechiral magnetic effect) [20]:jz =2αµ5 Bz ,
π(2)

where α is the fine structure constant. Intuitively, thiscurrent can be understood as follows: to minimize the energy of the system, the magnetic moment of an electronis aligned in the direction of Bz . Remembering the definition of chirality, momentum of a right- (left-)handed
fermions is in the same (opposite) direction as the spin;
hence, a net electric current flows in the direction of Bz
at finite µ5 .
Now Ampère’s law states that the current (2) leads toa magnetic field in the θ direction at a distance R ∼ λas Bθ = πλ2 jz /(2πR). This in turn induces the current
due to the chiral magnetic effect: jθ = (2α/π)µ5 Bθ . According to Ampère’s law again, this current gives rise toa magnetic field in the z direction as
Bz0 =



Z
dR jθ (R) ∼2αµ5 λ
π

2
Bz .

(3)

So if λ & (αµ5 )−1 , it follows that Bz0 > Bz : the original magnetic field gives a positive feedback to itself, andit grows exponentially. This is the chiral plasma insta-

Bz , jzBθ , jθFIG. 2. Physical picture of the chiral plasma instability.

bility. This unstable mode then reduces µ5 so that the
instability is attenuated [8, 19].
One also finds that this instability generates not onlya poloidal magnetic field in the z direction but also a
toroidal magnetic field in the θ direction; the resulting
configuration has a finite magnetic helicity. Later, we
will estimate the magnitude of magnetic helicity fromthe helicity conservation.
Estimate of the chirality imbalance of electrons.—Howlarge can a magnetic field be due to this mechanism inside compact stars? From now on, we shall provide its
estimate based on the neutron density generated duringcore collapse of supernovae. The core of a compact star isalmost “neutronized” via the parity-violating weak process at this stage, where the largest chiral asymmetry ofelectrons is created [21].
We note that our estimate for the magnetic field below
assumes a number of simplifications, so it should be regarded as schematic. Nonetheless, it turns out that themagnetic field due to the CPI can be of order 1016 G
at the core (with a possible small deviation due to theuncertainty of the prefactor), and it could potentially explain the large magnetic field ∼ 1015 G on the surface.
Our estimate can, in principle, be made more realistic
by including various complications that we will discusslater.
The chirality imbalance of electrons is produced via
electron capture inside a core,e
p + e−
L → n + νL ,(4)

where the subscript L stands for left-handedness. Here
only left-handed fermions are involved, as it is describedby the V−A type weak interaction. Note that neutrons
and the chirality imbalance produced by the process (4)are not washed out by its inverse process during core collapse (and thus forming a neutron star eventually). Thisis because the time scale of (4) is much shorter than theneutrino diffusion time τNS ∼ 10 s and their production
occurs in a nonequilibrium manner. Note also that the3other possible processes (thermal neutrino emission) [22]do not change the neutron number nor the chirality imbalance on average.
That huge neutrons are produced in these processes
means that huge left-handed electrons are “eaten” by
protons, leading to the Fermi surface imbalance, µR >µL , or a nonzero chiral chemical potential for electrons,µ5 ≡ (µR − µL )/2 > 0. Because the number of neutrons
produced in this process is equal to the number difference
between right- and left-handed electrons, N5 , we haven5 ≈ ∆nn ,(5)where n5 is the chiral density of electrons and ∆nn is
the increased neutron density by electron capture. Considering the neutron density inside a neutron star, it isreasonable to take ∆nn ∼ (0.1–1) fm−3 at the core. In
the natural units ~ = c = 1, ∆nn ∼ 0.1–1Λ3 , where we
introduced the mass scale Λ = 200 MeV for later convenience. In the following, we use the natural units unlessotherwise stated.
The chiral number density is expressed by the chemical
potentials as
n5 =

µ5 2
(µ + 3µ2 )3π 2 5(6)

at sufficiently low temperature, where µ ≡ (µR +µL )/2 isthe chemical potential associated with U(1) (vector-like)particle number. We here ignored the contribution of thetemperature T to the density. Recalling that the typicalelectron chemical potential at the core is µ . Λ, one findsµtotal∼ Λ.
5(7)

It should be remarked that µtotal
we obtained is the5
total (or time-integrated) chiral chemical potential produced during core collapse. In reality, the production ofµ5 by the process (6) occurs simultaneously with the reduction of µ5 by the CPI (and with the reduction by theelectron mass me , which we shall argue later). In general, the production of µ5 varies as functions of t and x,
and so does the reduction of µ5 due to the CPI; tracing
the evolution of µ5 (t, x) requires the knowledge of theneutrino production rate depending on the evolution of aneutron star and MHD simulations, which is beyond thescope of the present paper. Below we shall ignore theinhomogeneity of µ5 (and inhomogeneity of electromagnetic fields) to give a simple estimate of the magnetic field
produced by our mechanism with taking into account the
reduction of µ5 by the CPI itself. The assumption of thehomogeneous µ5 was also made in Refs. [12] in a different
context.

Evolution of the chirality imbalance.—Without the effects of the CPI, n5 would gradually increase toward thevalue (5). Let us consider, for simplicity, a specific casewhere n and n5 grow homogeneously at a constant speed
at the core: converting it in terms of µ and µ5 , it means
they would grow as µ ∝ t1/3 and µ5 ∝ t1/3 , so one
can set µ = c1 (Λ4 t)1/3 and (dµ5 /dt)+ = c2 (Λ4 /t2 )1/3
(where “+” stands for the production). Considering thatthe time scale for the formation of a neutron star isτNS ∼ 10 s ∼ 1024 Λ−1 , we take c1 ∼ 10−8 and c2 ∼ 10−8
as typical values.
In practice, however, n5 decreases due to the CPI, andits evolution is dictated by the anomaly relation [14]dn52α=E · B.
(8)dt
π
It is natural to take the typical scale characterizing themagnitude of E · B to be µ45 (t). Assuming µ(t)  µ5 (t),which is justified a posteriori, the reduction rate is givenby (dµ5 /dt)− = −c3 µ45 /µ2 with some constant c3 .
When combined altogether, one arrives at the following
evolution equation for µ5 :
 4 13 4 13  µ 4 
dµ5
µ45
Λ
Λ
5
− c3 2 = c2
= c2
1−C
,
dt
t2
µ
t2
Λ
(9)
where C = c3 /(c21 c2 ). Hence, µ5 saturates with the magnitude µsat
= C −1/4 Λ after the time t  τNS . This5
state will continue until the end of the neutronization of
the star when the production of µ5 becomes negligible.
We note that this result should be taken with care, because, as we mentioned above, electromagnetic fields andµ5 vary depending on (t, x) in reality, where such saturation may not occur. Rather, the estimate here is to beunderstood as a simple model to incorporate the effectsof the CPI that reduces µ5 in Eq. (7).
Estimate of magnetic fields and magnetic helicity.—As
explained above, the state with nonzero n5 is unstableand decays rapidly by converting it to a magnetic field
due to the CPI. Assuming this state will decay into astate with n5 ∼ 0 at the end, one can estimate the magnitudes of a resulting magnetic field and magnetic helicity
from the energy and helicity conservations [8]. Note that
the CPI will end before µ5 is reduced to µend
∼ 10−16 Λ
5−1
where the wavelength λ ∼ (αµ5 ) becomes macroscopic(∼ 1 km). As µend
 µsat5
5 , it occurs after the productionof µ5 stops.
The decrease of the energy density as a consequence ofthe reduction of µ5 by the CPI isZ
dn55µ5 ∼ c3 (µsat(10)∆ = dt5 ) τNS .
dt

4Here we used the anomaly relation (8) and the fact thatµ5 (t) = µsat
through most of the time during τNS . The
5energy conservation requires that it is equal to the energy
2of the magnetic field (1/2)∆Binst. One can thus estimate
the magnetic field generated by this instability as
5

−1

1

Binst ∼ (c21 c2 ) 8 c3 8 (Λ5 τNS ) 2 ,(11)

assuming no dissipation of energy and perfect conversion
efficiency (see the discussion below). The dependence of−1/8
Binst on c3 is weak, so one may take c3∼ 1. One thenfinds that typical magnetic fields produced by the CPI
are of order 1015 G – 1016 G in this model. Assuming
the resulting poloidal magnetic field is dipole-like andis dominant, we can translate Binst into the magnetic
3
field on the surface, Bsurface ≈ (Rcore /Rstar ) Bcore [23],where Rcore and Rstar are the radii of the core and theneutron star itself, respectively (here the “core” is meantby the region which has the density comparable to or
larger than the nuclear density); e.g., when Rcore /Rstar '0.5, Bsurface ∼ 1014 G – 1015 G.
On the other hand, the helicity conservation reads
Z
α 
d 
N5 + H = 0, N5 = dx n5 ,
(12)
dt
π
where N5 is the global chiral charge of electrons and H is
the magnetic helicity (also called Chern-Simons number
in particle physics and mathematics) defined in Eq. (1).
Note that this is the global version of the anomaly relation in Eq. (8). From the helicity conservation, one
obtains the magnetic helicity at the end as
Z
π
dn5
1
∆H = −dtdx∼ V c21 c2 Λ4 τNS
(13)
α
dt
α
3
with V ≈ 4πRcore/3; so a large magnetic helicity is naturally produced as a consequence of the CPI, which thenensures the stability of the strong magnetic field. The
detailed configuration of the magnetic field with a large
magnetic helicity is under study [24].
We note that any microscopic process concerning theelectromagnetic and strong interactions respects parity
and cannot generate a parity-odd magnetic helicity; microscopically, a parity-odd quantity can originate fromthe parity-violating weak interaction alone. However, theweak interaction violates parity in the fermionic sector(leptons), so it cannot generate magnetic helicity directly.
It is this CPI that converts the parity-odd chiral asymmetry in the fermionic sector to the parity-odd magnetic
helicity in the gauge sector.
Note also that the interior of a star could acquire themagnetic helicity macroscopically by accident, by losing
helicity through the surface in its evolution. However,no such a evidence was observed in MHD simulations fora specific initial configuration with H = 0 [5]. At least,
our mechanism seems the only to ensure the inevitable(rather than accidental) production of magnetic helicity
in magnetars from the microscopic point of view.
Discussion.—Let us discuss several possible effects wehave ignored above, which can modify our simple estimate (11). One potentially important effect is the electron mass me which also reduces the chiral asymmetry.
When µ5 (t)  µ(t), the chirality flipping rate due to meis Γmass ∼ α2 m2e /µ [12]. For µ5 (t) = µsat
5 in our model,
the time scale ∼ Γ−1canbecomecomparable
to thatmassof the CPI, τinst ∼ (α2 µ5 )−1 [8]. Another important effect is the conversion efficiency of the chirality imbalance
into the magnetic energy, which could be less than 100%.
For example, a finite conductivity σ dissipates the energy
and makes the magnetic field in Eq. (11) smaller. Also,
heavier degrees of freedom (ions) inside a star may interfere with the electron dynamics and could reduce themagnetic field. Although evaluation of these effects is
not easy, it is not entirely unreasonable to expect thatthe magnetic field induced by the CPI can occupy a nonnegligible fraction of the gigantic magnetic field of magnetars.
What is the final configuration of the magnetic field?
As µ5 and the chiral magnetic effect are expected to disappear after the CPI, one might wonder how a magnetic
field is kept without currents at the end. As the CPI
has the quantum origin, this magnetic field may be accounted for by a specific configuration of the spins ofelectrons and/or that of the wave function of hadrons.
In the context related to the former, recall that magnets,
which originate from parallel alignment of spins, generate magnetic fields without macroscopic currents. In thecontext related to the latter, a specific configuration ofpion fields is known to generate a magnetic field at finite
density [18]. At this moment we do not have an intuitive
picture for the final state after the CPI, but we stressthat the resulting strong magnetic field is a consequenceof the conservation laws and should be robust. It would
be an interesting future question to explore in detail [24].
Finally, we comment on the possible evolution of thelarge magnetic field after the birth of magnetars by ourmechanism. Remember that the magnetic helicity H is astrict conserved quantity without dissipation. In reality,
the medium has a conductivity σ so that H is conserved
approximately; it is the finiteness of σ that allows magnetic flux tubes to reconnect, which results in the decrease of H. Therefore, one expects that magnetic fields
decay slowly by dissipation and the reconnection which

5
could manifest themselves in the form of giant outbursts.
Conclusion.—In conclusion, we proposed a possible
new mechanism for a strong magnetic field with magnetic
helicity in magnetars due to the chiral plasma instability.
Our mechanism is based only on the chirality imbalanceof electrons that is necessarily produced in the processof electron capture during core collapse. The magnetic
field is estimated as ∼ 1016 G at the core (in our simple model) which may potentially explain the magnetic
field ∼ 1015 G on the surface. This large magnetic field

is a macroscopic consequence of the quantum effects tied
to quantum anomalies. More realistic calculations, e.g.,using magnetohydrodynamics for chiral plasmas, are necessary to reach a definite conclusion. We defer these calculations to future work.
We thank Y. Akamatsu, P. Bedaque, T. Enoto,
K. Fukushima, S. Mahmoodifar, S. Reddy, K. Sumiyoshi,
and Y. Suwa for useful discussions. N. Y. was supported
by JSPS Research Fellowships for Young Scientists when
this collaboration started.

[1] R. C. Duncun and C. Thompson, Astrophys. J. 392, L9(1992).
[2] A. K. Harding and D. Lai, Rept. Prog. Phys. 69, 2631
(2006).
[3] H. C. Spruit, AIP Conf. Proc. 983, 391 (2008).
[4] G. A. E. Wright, Mon. Not. R. astr. Soc. 162, 339 (1973);
P. Markey and R. J. Tayler, Mon. Not. R. astr. Soc. 168,505 (1974); E. Flowers and M. A. Ruderman, Astrophys.
J. 215, 302 (1977).
[5] J. Braithwaite and H. C. Spruit, Astron. Astrophys. 450,
1097 (2006).
[6] M. A. Berger, Plasma Phys. Controlled Fusion 41, B167
(1999).
[7] K. H. Prendergast, Astrophys. J. 123, 498 (1956).
[8] Y. Akamatsu and N. Yamamoto, Phys. Rev. Lett. 111,
052002 (2013).
[9] D. T. Son and N. Yamamoto, Phys. Rev. Lett. 109,
181602 (2012); Phys. Rev. D 87, 085016 (2013);
M. A. Stephanov and Y. Yin, Phys. Rev. Lett. 109,
162001 (2012); J. -W. Chen, S. Pu, Q. Wang and X. N. Wang, Phys. Rev. Lett. 110, 262301 (2013).
[10] A. N. Redlich and L. C. R. Wijewardhana, Phys. Rev.
Lett. 54, 970 (1985); K. Tsokos, Phys. Lett. B 157, 413(1985); A. J. Niemi and G. W. Semenoff, Phys. Rev. Lett.
54, 2166 (1985).
[11] V. A. Rubakov, Prog. Theor. Phys. 75, 366 (1986);
V. A. Rubakov and A. N. Tavkhelidze, Phys. Lett. B
165, 109 (1985).
[12] M. Joyce and M. E. Shaposhnikov, Phys. Rev. Lett. 79,
1193 (1997); A. Boyarsky, J. Frohlich and O. Ruchayskiy,
Phys. Rev. Lett. 108, 031301 (2012).
[13] Z. V. Khaidukov, V. P. Kirilin, A. V. Sadofyev andV. I. Zakharov, arXiv:1307.0138 [hep-th]; K. Jensen,
P. Kovtun and A. Ritz, JHEP 1310, 186 (2013); A. Avdoshkin, V. P. Kirilin, A. V. Sadofyev and V. I. Zakharov,
arXiv:1402.3587 [hep-th].
[14] S. Adler, Phys. Rev. 177, 2426 (1969); J. S. Bell andR. Jackiw, Nuovo Cimento A 60, 47 (1969).
[15] J. Charbonneau and A. Zhitnitsky, JCAP 1008, 010(2010).
[16] D. H. Brownell and J. Callaway, Nuovo Cimento B 60,
169 (1969); M. J. Rice, Phys. Lett. A 29, 637 (1969); S.
D. Silverstein, Phys. Rev. Lett. 23,139 (1969) [Erratum:ibid. 23, 453 (1969)]; K. Makishima, Prog. Theor. Phys.
Suppl. 151, 54 (2003).
[17] T. Tatsumi, Phys. Lett. B 489, 280 (2000).
[18] D. T. Son and M. A. Stephanov, Phys. Rev. D 77, 014021(2008); M. Eto, K. Hashimoto and T. Hatsuda, Phys.
Rev. D 88, 081701 (2013).
[19] Y. Akamatsu and N. Yamamoto, arXiv:1402.4174 [hepth].
[20] A. Vilenkin, Phys. Rev. D 22, 3080 (1980); H. B. Nielsen
and M. Ninomiya, Phys. Lett. B 130, 389 (1983);
A. Y. Alekseev, V. V. Cheianov, and J. Frohlich, Phys.
Rev. Lett. 81, 3503 (1998); D. T. Son and A. R. Zhitnitsky, Phys. Rev. D 70, 074018 (2004); K. Fukushima,
D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D78,
074033 (2008).
[21] In this paper, we do not consider the chiral asymmetryof electrons generated after the birth of a neutron star,as it is much smaller than that produced at the stage ofcore collapse; for the possible astrophysical consequencesof the chiral asymmetry at later stages, see Ref. [15].
[22] S. L. Shapiro and S. A. Teukolsky, 1983, Black Holes,
White Dwarfs, and Neutron Stars (Wiley, New York,
1983).
[23] It is also possible in our mechanism to make a quasistationary configuration with the toroidal magnetic fieldB ∼ 1/R2 when the poloidal one is dipole-like; in thiscase, the contribution of the toroidal magnetic field isdomonant at large R, and Bsurface ≈ (Rcore /Rstar )2 Bcore .
[24] Y. Akamatsu, A. Rothkopf, and N. Yamamoto, work inprogress.