-
Axion Constraints from Quiescent Soft Gamma-ray Emission from
Magnetars
Sheridan J. Lloyd,∗ Paula M. Chadwick,† and Anthony M.
Brown‡
Centre for Advanced Instrumentation, Dept. of Physics,University
of Durham, South Road, Durham, DH1 3LE, UK
Huai-Ke Guo§ and Kuver Sinha¶
Department of Physics and Astronomy, University of Oklahoma,
Norman, OK 73019, USA(Dated: December 15, 2020)
Axion-like-particles (ALPs) emitted from the core of a magnetar
can convert to photons in itsmagnetosphere. The resulting photon
flux is sensitive to the product of (i) the ALP-nucleon couplingGan
which controls the production cross section in the core and (ii)
the ALP-photon coupling gaγγwhich controls the conversion in the
magnetosphere. We study such emissions in the soft-gamma-ray range
(300 keV to 1 MeV), where the ALP spectrum peaks and astrophysical
backgrounds fromresonant Compton upscattering are expected to be
suppressed. Using published quiescent soft-gamma-ray flux upper
limits in 5 magnetars obtained with CGRO COMPTEL and
INTEGRALSPI/IBIS/ISGRI, we put limits on the product of the
ALP-nucleon and ALP-photon couplings. Wealso provide a detailed
study of the dependence of our results on the magnetar core
temperature.We further show projections of our result for future
Fermi-GBM observations. Our results motivatea program of studying
quiescent soft-gamma-ray emission from magnetars with the
Fermi-GBM.
I. INTRODUCTION
The axion arises as a solution to the strong CP problemof QCD
and is a plausible cold dark matter candidate[1–5]. The search for
axions, and more generally axion-like-particles (ALPs) (for which
the relationship betweenparticle mass and the Peccei-Quinn scale is
relaxed), nowspans a vast ecosystem including helioscopes,
haloscopes,interferometers, beam dumps, fixed target
experiments,and colliders [6].
This paper concerns indirect detection of ALPs, specif-ically
their conversion into photons in the magneto-spheres of neutron
stars with strong magnetic fields(magnetars) [7–9]. The mechanism
is as follows 1:relativistic ALPs (a) emitted from the core by
nu-cleon (N) bremsstrahlung (from the Lagrangian termL ⊃
Gan(∂µa)N̄γµγ5N) escape into the magnetosphere,where they convert
to photons (from the Lagrangian term
L ⊃ − 14gaγγaFµν F̃µν) in the presence of the neutron
star magnetic field B. The ALP emission rate stronglydepends on
the core temperature, Tc, as T
6c [12 and
∗ [email protected]† [email protected]‡
[email protected]§ [email protected]¶ [email protected] The
conversion of relativistic ALPs near neutron stars begins
with [10] where the probability of conversion was
overestimated,followed by the classic paper [11] which correctly
accounted fornon-linear QED and the photon mass in the ALP-photon
con-version equations. In [11] an order of magnitude calculation
ofthe conversion probability near the magnetar surface
concludedthat it was too small to produce observable signals (the
pho-ton mass term dominates over the ALP-photon mixing term atthe
surface). However, the conversion becomes appreciable awayfrom the
surface, due to the different scaling of the photon mass(∼ 1/r6)
compared to the ALP-photon mixing (∼ 1/r3).
13] while the conversion rate generally increases withstronger
B, making magnetars, with their high Tc ∼ 109K and strong B ∼ 1014
G, a natural target for thesestudies.
The purpose of this paper is to initiate an investiga-tion of
the signals resulting from ALP-photon conversionsin the quiescent
soft-gamma-ray spectrum (300 keV−1MeV) from magnetars, similar to
probes in the X-rayband in magnetars [7 and 8] and in pulsars [14].
Sincethe peak of photon energies arising from ALP-photonconversion
lies in the soft-gamma-ray band, this is an es-pecially important
regime to explore. Moreover, whilesearches for new physics in the
soft and hard X-ray emis-sion from magnetars must contend with
background fromthermal emission and resonant Compton
upscatteringrespectively, the astrophysical background in the
soft-gamma-ray regime is relatively suppressed as we discussin
Sec.VII.
Starting with the photon polarization tensor, we pro-vide in
Sec.VI expressions for the photon refractive in-dices in the strong
and weak magnetic field regimes forphoton energies ω . 2me, where
me is the electronmass2. In Sec.III, the coupled ALP-photon
propaga-tion equations are then solved numerically using the
ap-propriate refractive indices. In Sec.IV, the productionin the
magnetar core is discussed: this proceeds viabremsstrahlung from
neutrons ψn: ψn + ψn ↔ ψn +ψn + a. Combining all of the above
ultimately yieldsthe photon luminosity coming from ALP-photon
con-versions La→γ , as well as the spectral energy distribu-tion.
These quantities are obtained for a selection of 5magnetars: 1E
2259+586, 4U 0142+61, 1E 1048.1-5937,
2 In the soft-gamma-ray regime, one has to start directly from
thephoton polarization tensor and take appropriate limits,
insteadof starting with the Euler-Heisenberg Lagrangian.
arX
iv:2
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14
Dec
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mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]
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2
1RXS J170849.0-400910 and 1E 1841-045. Using pub-lished
quiescent soft-gamma-ray flux upper limits (ULs),constraints are
then put on the product of couplingsGan × gaγγ using a spectral
analysis whose details areshown in Sec.X.
The main message of our paper is that quiescent soft-gamma-ray
emission from magnetars is a fertile target toinvestigate the
physics of ALPs. The Fermi-GBM is avery useful instrument to
determine the UL soft-gamma-ray fluxes of the 23 confirmed
magnetars and such a study
could yield very restrictive constraints on Gan × gaγγ .
II. PHENOMENOLOGY
In this section, we discuss the predicted luminosityfrom
ALP-photon conversion in the magnetosphere. Weassume a dipolar
magnetic field defined by
B = Bsurf
(r0r
)3. (1)
ALPs propagating radially outwards from a magnetarobey the
following evolution equations derived in [11]
id
dx
aE‖E⊥
= ωr0 + ∆ar0 ∆Mr0 0∆Mr0 ωr0 + ∆‖r0 0
0 0 ωr0 + ∆⊥r0
aE‖E⊥
, where (2)
∆a = −m2a2ω
, ∆‖ = (n‖ − 1)ω, ∆⊥ = (n⊥ − 1)ω, ∆M =1
2gaγγB sin θ. (3)
The parallel and perpendicular electric fields are denotedby
E‖(x) and E⊥(x), respectively, while a(x) denotes theALP field. The
distance from the magnetar is given bythe rescaled dimensionless
parameter x = r/r0, wherer is the distance from the magnetar and r0
its radius.The energy of the photon is given by ω, the ALP massby
ma, and the ALP-photon coupling by gaγγ . θ is theangle between the
direction of propagation and the B-field.
The refractive indices n‖ and n⊥ are obtained fromthe photon
polarization tensor, which can be worked outat one-loop level in
various limits of the photon energyω and the strength of the
magnetic field B relative tothe quantum critical magnetic field Bc,
given by Bc =m2e/e = 4.413 × 1013 G. Here e =
√4πα and the fine
structure constant α ≈ 1/137.Near the surface, the B-field of
the magnetars we con-
sider typically exceeds Bc, so that ω . 2me and B > Bc.The
corresponding refractive indices are given in Eq. C15.Given the
spatial dependence from Eq. 1, the magneticfield decreases to below
the critical strength at a distance∼ 3r0. Beyond that, we are in a
regime where ω . 2meand B � Bc, with
(ω
2me
)2 (BBc
)2� 1. The correspond-
ing refractive indices are given in Eq. 43. For furtherdetails,
see Sec.VI.
After calculating the parallel refractive index n‖,
theprobability of conversion can be obtained as a functionof gaγγ
and the mass ma by numerically solving Eq. 2.The interesting regime
for conversion is r = ra→γ ∼O(1000) r0 (the “radius of
conversion”), where the con-version probability becomes large. This
arises from the
ALP-photon mixing becoming maximal when ∆M ∼ ∆‖.Far away from
the surface, ∆M ∼ 1/r3, while ∆‖ ∼ 1/r6from Eq. 43, with the two
becoming equal around ra→γ .
Along with the probability of conversion, we requirethe
normalized ALP spectrum and the number of ALPsbeing produced from
the magnetar core. Integrating theproduct of these quantities over
the ALP energy rangeω ⊂ (ωi, ωf ) = (300 keV, 1000 keV) gives us
the finalpredicted luminosity from ALP-photon conversions.
Ourmaster equations for the final predicted theory photonluminosity
are Eq. 12 - Eq. 17, which we solve numer-ically. A semi-analytic
calculation following [8] is alsoperformed to validate our results.
We provide furtherdetails in Sec.III and Sec.IV.
III. ALP-PHOTON PROBABILITY OFCONVERSION
In this section, we provide details of the propagationof the
ALP-photon system through the magnetosphere,with the aim of
deriving the probability of conversionPa→γ(ω, θ). Our treatment
largely follows the frameworkdeveloped by one of the authors in [7,
8]. For later workthat followed these initial calculations, we
refer to [14].We note that [21] performed detailed numerical
compu-tations of the conversion probability in the soft
X-raythermal emission band, and our results agree with theirsin the
appropriate limit. We note in passing that ALPdecays can be
neglected.
The propagation of the system is governed by Eq. 2and Eq. 3,
while the relevant refractive indices will be
-
3
Magnetar Distance Surface B Age UL Fluxkpc Field kyr 300 keV−1
MeV
1014 G 10-10 erg cm-2 s-1
1E 2259+586 3.2+0.2−0.2 [15] 0.59 230 1.17 [16]4U 0142+61
3.6+0.4−0.4 [17] 1.3 68 8.16 [18]
1RXS J170849.0-400910 3.8+0.5−0.5 [15] 4.7 9 1.92 [16]1E
1841-045 8.5+1.3−1.0 [19] 7 4.6 2.56 [16]
1E 1048.1-5937 9.0+1.7−1.7 [17] 3.9 4.5 3.04 [16]
TABLE I. Magnetar sample with sum of UL flux in the 300 keV−1
MeV band. UL fluxes and distances are from the referencesshown,
surface B field and age are from the online27 version of the McGill
magnetar catalog [20].
presented in Sec.VI. It is clear from the structure of themixing
matrix in Eq. 2 that E⊥ does not mix with theALP; we will thus not
consider it any further. It is con-venient to reparametrize the
other fields as follows:
a(x) = cos[χ(x)]e−iφa(x),
E‖(x) = i sin[χ(x)]e−iφE(x), (4)
where χ(x), φa(x) and φE(x) real functions. The propa-gation
equations then simplify to
dχ(x)
dx= −D(x) cos[∆φ(x)],
d∆φ(x)
dx= A(x)−B(x) + 2D(x) cot[2χ(x)] sin[∆φ(x)],
(5)where χ(1) is the initial state at the surface of the
mag-netar, and we have defined the relative phase ∆φ(x) =φa(x) −
φE(x). For pure initial states, the initial condi-tion for ∆φ(1)
satisfies ∆φ(1) = mπ with m ∈ Z. Fora pure ALP initial state it is
therefore possible to set∆φ(1) = 0. The ALP-photon conversion
probability isthen simply
Pa→γ(x) = sin2[χ(x)] . (6)
Our results for the conversion probability will be basedon a
full numerical solution to the evolution equations.The probability
of conversion Pa→γ is thus obtained bynumerically solving the
propagation equations in Eq. 2.For the calculations, we need the
refractive indices thatappear in Eq. 3. These refractive indices
are derived inSec.VI.
We now outline a semi-analytic solution that agreesvery well
with our full numerical solution. The semi-analytic solution can be
obtained by analogy with time-dependent perturbation theory in
quantum mechanics,leading to [11]
Pa→γ(x) =
∣∣∣∣∣∫ x
1
dx′∆M (x′)r0
×exp
{i
∫ x′1
dx′′ [∆a −∆‖(x′′)]r0
}∣∣∣∣∣2
= (∆M0r0)2
∣∣∣∣∣∫ x
1
dx′1
x′3exp
[i∆ar0
(x′ −
x6a→γ5x′5
)]∣∣∣∣∣2
.
(7)
These equations are accurate for small enough values of g,which
fall in the regime we are interested in. The secondexpression
utilized the dimensionless conversion radius,where the probability
of conversion becomes maximal
xa→γ =ra→γr0
=
(7α
45π
)1/6(ω
ma
B0Bc| sin θ|
)1/3. (8)
This is valid when the conversion radius is much largerthan the
radius of the magnetar. In that limit q̂‖ → 1 andthe integral in
the exponential can be trivially calculated.The conversion
probability becomes
Pa→γ(x) =
(∆M0r
30
r2a→γ
)2 ∣∣∣∣∣∫ ∞
r0ra→γ
dt1
t3
×exp[i∆ara→γ
(t− 1
5t5
)] ∣∣∣∣∣2
, (9)
where the norm of the integral in (9) is order one for
ourbenchmark points. We can further simplify the expres-sion in the
large |∆ara→γ | regime by using the methodof steepest descent, and
the small |∆ara→γ | regime witha change of variables:
Pa→γ(x) =
(∆M0r
30
r2a→γ
)2
×
π
3|∆ara→γ |e6∆ara→γ
5 |∆ara→γ | & 0.45Γ( 25 )
2
565 |∆ara→γ |
45
|∆ara→γ | . 0.45.(10)
We display the function χ(x) and the probability ofconversion as
a function of the radial distance in Fig. 1.These plots are
obtained from a full numerical solutionto the evolution
equations
Before closing this section, we also provide an heuristicway of
studying the conversion probability. The mixingangle between E‖(x)
and the ALP a(x) is given by
tan 2θmix =∆M
(∆a −∆‖)/2∼ gB
(1− n‖)ω. (11)
For benchmark values of the magnetic field and other pa-rameters
relevant for this work, one can check that at thesurface of the
magnetar the mixing is negligible. How-ever, the mixing (and hence
the probability of conversion)
-
4
102
103
104
105
0.00
0.05
0.10
0.15
100 1000 104
105
0.0
0.2
0.4
0.6
0.8
1.0
FIG. 1. We show χ(x) (left panel) and cos2[χ(x)] and sin2[χ(x)]
(right panel, blue and red curve respectively) as a functionof the
dimensionless distance x from the magnetar surface, obtained from a
full numerical solution to the evolution equations.The benchmark
point is taken to be ω = 500 keV, ma = 10
−8 keV, gaγγ = 10−9GeV−1, r0 = 10 km, B0 = 0.59 × 1014 G and
θ = π/2 for Magnetar 1E-2259+586.
actually increases away from the surface. This can be
un-derstood from the fact that the photon mass term ∆‖ in
the denominator in Eq. 11 goes as ∆‖ ∼ 1/r6, whereasthe
ALP-photon mixing term in the numerator goes as∆M ∼ 1/r3. There is
a point around r ∼ O(1000r0)where the numerator and denominator
become compara-ble, resulting in a large mixing angle. The
probability ofconversion becomes large at this position, which we
callthe radius of conversion ra→γ . Beyond ra→γ , the mixingangle
again becomes small since the ALP mass term ∆ain the denominator of
Eq. 11 dominates over both ∆‖ aswell as ∆M .
We note that a phase resolved analysis will require
theintroduction of a viewing angle and a time-dependentrotational
phase that is related to the magnetar angularvelocity. If one
assumes that the emission region is lo-calized on the magnetar
surface, an opening angle willalso be introduced. The spectrum will
therefore be func-tions of these extra parameters, and it is
possible thata careful investigation of phase-resolved data will
yieldconstraints stronger than the ones we are achieving inthe
current work. We leave this analysis for future work.
We briefly comment on the subsequent propagation ofunconverted
ALPs after they leave the magnetosphere.ALPs with masses . 10−12 eV
emanating from the mag-netars in our sample can convert to photons
in the mag-netic field of the Milky Way and this may yield
con-straints on gaγγ . Such constraints depend on
severalastrophysical parameters, such as the coherent and ran-dom
magnetic fields, electron density, the distance of thesource, the
exact value of the Galactic magnetic field,the clumpiness of the
interstellar medium, and the warm
ionized medium and the warm neutral medium. A fullstudy of these
effects may be interesting. We refer to[22, 23] for further details
of these topics.
IV. NUCLEON BREMSSTRAHLUNG AND ALPPRODUCTION
In this section, we outline our calculation of the pre-dicted
photon luminosity coming from ALP-photon con-versions, which we
denote by La→γ . The observed lu-minosity of photons produced by
the conversion processcan be schematically written as
La→γ = (production of a) × Pa→γ , (12)
where Pa→γ is the conversion probability calculated
ear-lier.
The production in the magnetar core proceeds viabremsstrahlung
from neutrons ψn: ψn + ψn ↔ ψn +ψn + a. The coupling term in the
Lagrangian is L =Gan∂µaψ̄nγ
µγ5ψn [13]. The interaction between thespectator nucleon and the
nucleon emitting the axion ismodeled by one-pion exchange (OPE)
with LagrangianLnπ = i(2mn/mπ)fγ5π0ψ̄nψn, where f ≈ 1. We referto
[24], [25] and references therein for more details. Therelevant
tree-level Feynman diagrams are given in [13].
The photon luminosity from axion conversion is [26]:
La→γ =
∫ ∞0
dω1
2π
∫ 2π0
dθ · ω · dNadω· Pa→γ(ω, θ), (13)
where Na is the axion emission rate (number per time)
-
5
and dNa/dω is the axion energy spectrum:
dNadω
=NaT
x2(x2 + 4π2)e−x
8(π2ζ3 + 3ζ5)(1− e−x), (14)
where x = ω/T and is a dimensionless quantity. Thetotal emission
rate of ALPs Na can be obtained from thefollowing emissivity
formula [26]
Q = 1.3× 1019erg · s−1 · cm−3(
Gan
10−10GeV−1
)2×(ρ
ρ0
)1/3(T
109K
)6, (15)
which is the axion emission rate per volume. Here ρ isthe
magnetar density and ρ0 = 2.8× 1014 g · cm−3 is thenuclear
saturation density. For a magnetar with radiusr, the axion emission
rate is then given by∫ ∞
0
dω ωdNadω
= Q× 43πr3, (16)
which is proportional to G2an. For the range of ALP-photon
couplings gaγγ we are interested in, we can use thesemi-analytic
expression for the conversion probabilitygiven in Eq. 10. Then, it
is clear that Pa→γ ∝ g2aγγ .It then follows that La→γ ∝ G2ang2aγγ .
Assuming thedistance of the magnetar is d, then the νFν spectrum
isgiven by
νFν(ω) = ω2 1
4πd21
ω
dLa→γdω
, (17)
and we choose the unit MeV2cm−2s−1MeV−1.
V. ALP EMISSIVITY IN MEAN FIELDTHEORY
In this section, we discuss the steps involved in the
cal-culation of the ALP emissivity Q from a magnetar core inmean
field theory, following the results recently obtainedin [27].
Although we do not use this more sophisticatedtreatment for the
production process in this paper, weinclude this discussion for
completeness and for use infuture work.
To be specific, the discussion will model the nuclearmatter
inside a neutron star with the NLρ EoS [28],which is a relativistic
mean field theory where nucleonsinteract by exchanging the scalar σ
meson and the ω andρ vector mesons. Our EoS supports a neutron star
ofmass 2M� with pressure consistent with GW170817 andNICER data for
posterior distributions of the pressure at0.5, 1, 2, 3 times
nuclear saturation density.
In the mean field approximation, we can take the neu-tron and
proton as free particles with effective Diracmasses given by m∗ =
m− gσσ and with effective chem-ical potentials µ∗n = µn−Un and µ∗p
= µp−Up. Here, Uiare the nuclear mean fields
Un = gωω0 −1
2gρρ03, (18)
Up = gωω0 +1
2gρρ03. (19)
The chemical potentials µi and µ∗i are relativistic and
contain the rest mass of the particle. The energy disper-sion
relations are given by
En =√p2 +m2∗ + Un, (20)
Ep =√p2 +m2∗ + Up. (21)
Note that they have been modified by the presence of thenuclear
mean field and that the ρ meson distinguishes theneutron from the
proton by creating a difference in meanfield experienced by the
respective particles.
The formalism for calculating the rate of particle pro-cesses is
given in [29], which uses parameter set I of themodel in [28]. For
the calculations, the energies in the
matrix element should use E∗ ≡√p2 +m2∗, while the
energy factors in the delta functions and Fermi-Dirac fac-tors
should use E = E∗+Un. The emissivity is given by[13]
Q =
∫d3p1
(2π)3
d3p2
(2π)3
d3p3
(2π)3
d3p4
(2π)3
d3ω
(2π)3
S∑|M|2
25E∗1E∗2E∗3E∗4ωω
× (2π)4 δ4(p1 + p2 − p3 − p4 − ω)f1f2 (1− f3) (1− f4) .(22)
Here, the pi and Ei are the momenta of the nucleonsparticipating
in the Feynman diagram. The Fermi-Diracfactors are given by fi =
(1+e
(Ei−µn)/T )−1. The matrixelement is given by
S∑spins
|M|2 = 2563
f4m4nG2an
m4π
[k4
(k2 +m2π)2
+l4
(l2 +m2π)2 +
k2l2 − 3 (k · l)2
(k2 +m2π) (l2 +m2π)
], (23)
where k and l are three-momentum transfers k = p2−p4and l =
p2−p3. The symmetry factor for these diagramsis S = 1/4.
We outline four different regimes in which we computeQ. The
first is relativistic matter with arbitrary degener-acy in the
Fermi surface approximation, when neutronsare strongly degenerate,
in which case only neutrons nearthe Fermi surface participate in
the bremsstrahlung pro-cess. The axion emissivity is
QFS =31
2835π
f4G2anm4n
m4πpFnF (y)T
6, (24)
-
6
where
F (y) = 4− 11 + y2
+2y2√
1 + 2y2arctan
(1√
1 + 2y2
)
−5y arcsin
(1√
1 + y2
), (25)
with y = mπ/(2pFn).
The Fermi surface approximation extends the lowerendpoint of
integration of neutron energy to −∞. Animprovement to the Fermi
surface approximation can beobtained, which keeps the neutron
energy bounded bym∗ + Un < En < ∞. The ALP emissivity in this
im-
proved approximation is [27]
QFS,improved =2
3π7f4G2anm
4n
m4πpFnF (y)T
6K2(ŷ), (26)
where
K2(ŷ) =
∫ ∞−2ŷ
du1
1− euln
{cosh (ŷ/2)
cosh [(u+ ŷ)/2]
}×∫ u+2ŷ
0
dww2
1− ew−uln
{cosh [(u+ ŷ − w)/2]
cosh (ŷ/2)
}.
(27)
The third approximation we discuss assumes non-relativistic
neutrons. The full momentum dependence ofthe matrix element in Eq.
23 is retained when evaluatingthe emissivity from Eq. 22. The
expression obtained inthis case is [27]
Qnon−rel =32√
2
3π8f4m4nG
2an
m4πm
1/2∗ T
6.5
∫ ∞0
du dv
∫ v0
dw
∫ 1−1dr ds
∫ 2π0
dφu1/2v3/2w3/2(v − w)2
×(α4(r2 + 3
)− 6α2
(r2 − 1
)(v + w)− 3
(r2 − 1
) (2(1− 2r2
)vw + v2 + w2
))[2w (α2 − 2r2v + v) + (α2 + v)2 + w2
]2×[(1 + eβ(E1−µn))(1 + eβ(E2−µn))(1 + e−β(E3−µn))(1 +
e−β(E4−µn))
]−1, (28)
where α = mπ/√
2m∗T . This integral can be performednumerically.
The final approximation we discuss involves a calcula-
tion of the fully relativistic phase-space integration inEq. 22,
performed with a constant matrix element inEq. 23. The result is
(we refer to [27] for a full derivation)
Qrel =
(1− β
3
)f4m4nG
2an
8π7m4π
(1 +
m2πk2typ
)−2 ∫ ∞m∗
dq0
∫ ∞0
dq
∫ √q20−m2∗0
dk
∫ q0m∗
dl0
∫ √l20−m2∗0
dl
∫ ω+(l0,l)ω−(l0,l)
dω
× kω(q0 −√k2 +m2∗)
θ(2kq − |q20 − q2 − 2q0√k2 +m2∗|)θ(2ql − |m2∗ + q2 + l2 − q20 −
l20 + 2q0l0|)√
k2 +m2∗
√k2 +m2∗ + q
20 − 2q0
√k2 +m2∗
(29)
×[(1 + e(
√k2+m2∗−µ
∗n)/T )(1 + e(
√k2+m2∗+q
20−2q0
√k2+m2∗−µ
∗n)/T )(1 + e−(q0−l0−µ
∗n)/T )(1 + e−(l0−ω−µ
∗n)/T )
]−1.
This integral can also be performed numerically.
The emissivities resulting from the four approxima-tions
described were compared in [27], and it was foundthat they show
remarkable convergence for temperaturesT . 10 MeV, which is the
regime we are mainly in-terested in for the magnetar core. Using
these results,one can calculate the normalized ALP spectrum andALP
emissivity to yield a constraint on the product
Gan × gaγγ. We leave this for future work.
VI. CALCULATION OF REFRACTIVE INDICES
In this section, we provide general expressions for thephoton
refractive indices in the parallel and perpendic-ular directions.
We are interested in several different
-
7
regimes of the photon frequency and the strength of theexternal
magnetic field:
(i) ω � 2me and B � Bc: soft X-rays in an exter-nal magnetic
field that is much weaker than the criticalstrength. This regime is
relevant for the conversion of lessenergetic ALPs into photons at
the radius of conversion∼ 500r0, where B ∼ 10−5Bc. Since the photon
energiesare much smaller than me, the Euler-Heisenberg
approx-imation can be used to calculate the refractive indices.
(ii) ω . 2me and B � Bc, with(
ω2me
)2 (BBc
)2� 1:
hard X-rays and soft gamma-rays in an external mag-netic field
that is much weaker than the critical strength.This regime is
relevant for the conversion of energeticALPs with ω ∼ O(100) keV -
O(1) MeV into photonsat the radius of conversion ∼ 500r0, where B ∼
10−5Bc.This regime is relevant for the observational
signaturesconsidered in this paper.
(iii) ω . 2me and B > Bc: hard X-rays and softgamma-rays in
an external magnetic field that is strongerthan the critical
strength. This regime is relevant for theconversion of energetic
ALPs with ω ∼ O(100) keV -O(1) MeV into photons from the magnetar
surface to adistance of ∼ 3r0.
We now turn to a discussion of the refractive indices inregimes
(ii) and (iii), which are relevant for this paper.
Quantum corrections to the photon propagator can bestudied using
the photon polarization tensor Πµν , definedin the following way
[30]
L ⊃ −12
∫x′Aµ(x)Π
µν(x, x′)Aν(x′) , (30)
where Aµ is the propagating photon. To evaluate Πµν ,
we can consider the perpendicular and parallel compo-nents of
the momentum four-vector kµ. We note thatthese components are
defined with respect to the ex-
ternal magnetic field ~B, which we take to point in thedirection
~e1: k
µ = kµ‖ + kµ⊥, k
µ‖ = (ω, k
1, 0, 0), and
kµ⊥ = (0, 0, k2, k3). The metric tensor can likewise be
decomposed into the parallel and perpendicular direc-tions: gµν
= gµν‖ + g
µν⊥ , where g
µν‖ = diag(−1,+1, 0, 0)
and gµν⊥ = diag(0, 0,+1,+1).
We will assume a pure and homogeneous external mag-netic field
to work out the photon polarization tensor,since taking into
account the spatial variation of themagnetic field would be
significantly more complicated.This is justified, since the dipolar
magnetic field variesat a scale given by the magnetar radius, while
the pho-ton wavelength is much smaller in the soft gamma-rayregime.
At one loop, the polarization tensor is given by[31–34],
Πµν(k) =α
2π
∫ 1−1
dν
2
∞−iη∫0
ds
s
{e−iΦ0s
[−N0kµkν
+(N1 −N0)(gµν‖ k
2‖ − k
µ‖ k
ν‖
)+(N2 −N0)
(gµν⊥ k
2⊥ − k
µ⊥k
ν⊥)]
+(1− ν2)e−i(m2e−i�)skµkν
}, (31)
where Φ0 = m2e − i�+ n1k2‖ + n2k
2⊥, s is the proper time,
ν governs the loop momentum distribution, and � and ηare
parameters that tend to 0+. The external magneticfield appears in
the scalar functions N0, N1, N2, n1 andn2. In terms of the variable
z = eBs, these functions aregiven by
N0(z) =z
sin z(cos νz − ν sin νz cot z) , n1(z) =
1− ν2
4,
N1(z) = z(1− ν2) cot z , n2(z) =cos νz − cos z
2z sin z,
N2(z) =2z (cos νz − cos z)
sin3 z. (32)
The polarization tensor is most compactly expressed interms of
the projection operators Pµν‖ and P
µν⊥ , defined
in the following way
Pµν‖ = gµν‖ −
kµ‖ kν‖
k2‖and Pµν⊥ = g
µν⊥ −
kµ⊥kν⊥
k2⊥,
(33)
in terms of which the tensor can be re-expressed as [31]
Πµν(k) = Pµν‖ Π‖ + Pµν⊥ Π⊥ , (34)
where
{Π‖Π⊥
}=
α
2π
∫ 1−1
dν
2
∞−iη∫0
ds
s
[e−iΦ0s
{k2‖N1 + k
2⊥N0
k2‖N0 + k2⊥N2
}].
(35)The expression in Eq. 35 is amenable to a
perturbativeexpansion, which we now explore.
A. ω . 2me and B � Bc, with(
ω2me
)2 (BBc
)2� 1
We first note that a perturbative expansion of Πpertp(where p
=‖,⊥) in powers of the magnetic field can beobtained by an
expansion in powers of (eB)2n:
Πpertp =
∞∑n=0
Π(2n)p , (36)
-
8
with the even powers being due to Furry’s theorem, and
Π(2n)p =(eB)2n
n!
[(∂
∂(eB)2
)nΠp
]eB=0
, (37)
Since the limit z → 0 does not admit any poles in thecomplex s
plane for the integrands, the integration overs can be performed on
the real positive axis. This yields
the following expressions for the Π(2n)p [35]:{
Π(2n)‖
Π(2n)⊥
}=
α
2π
∫ 1−1
dν
2
∫ ∞0
ds
se−iφ0s
z2n
n!
×[(
∂
∂z2
)n({ k2‖N1 + k2⊥N0k2‖N0 + k
2⊥N2
}e−isk
2⊥ñ2
)]z=0
,(38)
where ñ2 = n2 − 1−ν2
4 = O(z2).
The integral over s in Eq. 38 can be performed explic-itly.
Using the expressions in Eq. 32, one obtains
Π(2n)p =α
2π
∫ 1−1
dν
2
n−1∑l=0
(2n+ l − 1)!(−1)n+l
[k2‖c‖(n,l)p (ν
2)
+k2⊥c⊥(n,l)p (ν
2)
] (eB
m2e
)2n(k2⊥m2e
)l, (39)
where the coefficients c‖(n,l)p (ν2) and c
⊥(n,l)p (ν2) can be
obtained explicitly from expanding Eq. 32.We note that a
perturbative expansion can be obtained
when both expansion parameters in Eq. 39 are small:
eB
m2e≡ BBc� 1 and
(B
Bc
)2ω2 sin2 θ
m2e� 1 ,
(40)where we have introduced the angle θ between the mag-netic
field and the photon propagation direction. Theleading order tensor
is{
Π(2)‖
Π(2)⊥
}= − α
12π
∫ 1−1
dν
2
(eB
φ0
)2 (1− ν2
)2×
[{ −21−ν2
1
}k2‖ +
{1
5−ν22(1−ν2)
}k2⊥
].(41)
The integration over ν finally yields [36 and 37]{Π
(2)‖
Π(2)⊥
}= − α
2π
(B
Bc
)2ω2 sin2 θ
2
45
{74
}, (42)
The corresponding indices of refraction are given bynp = 1− 12ω2
Bc
This regime is relevant for the conversion of energeticALPs with
ω ∼ O(100) keV - O(1) MeV into photonsfrom the magnetar surface to
a distance of ∼ 3r0. Weonly quote the final answer here, referring
to [38] for afull derivation:{
n‖n⊥
}= 1 +
α
4πsin2 θ
[(2
3
B
Bc− Σ
){10
}
−[
2
3+BcB
ln
(BcB
)]{1−1
}+O
(1eB
)+O(ω2)
].
(44)
Here, Σ ∼ O(1) is a constant.
VII. SOFT GAMMA-RAY BACKGROUND
Magnetars exhibit thermal X-ray emission below 10keV and a hard
pulsed non-thermal X-ray emission withpower law tails above 10 keV.
This hard X-ray emissioncan extend to between 150 − 275 keV [39–41]
and ap-pears to turn over above 275 keV due to ULs being ob-tained
with INTEGRAL SPI (20−1000 keV) and CGROCOMPTEL (0.75−30 MeV) [18].
A spectral break above1 MeV is also inferred by the non-detection
of 20 magne-tars using Fermi -LAT above 100 MeV [42]. The hard
X-ray emission is most likely caused by resonant
Comptonupscattering (RCU) of surface thermal X-rays by non-thermal
electrons moving along the magnetic field linesof the
magnetosphere. The initial modeling of [43], us-ing B field
strengths typical of magnetars, at three timesthe quantum critical
field strength Bc produces flat dif-ferential flux spectra with
sharp cut-offs at energies di-rectly proportional to the electron
Lorentz factor (γe)and places the maximum extent of the Compton
resonas-phere within a few stellar radii of the magnetar
surface.
In [44], Monte-Carlo models of the RCU of soft ther-mal photons,
incorporating the relativistic QED reso-nant cross section,
produces flat spectra up to 1 MeV forhighly relativistic electrons
(γe=22), whilst mildly rela-tivistic electrons (γe=1.7) demonstrate
spectral breaks at316 keV. In [45], an analytic model of RCU,
consideringrelativistic particle injection (γe>>10) and
decelerationwithin magnetic loops predicts a spectral peak at ∼
1MeV and a narrow annihilation line at 511 keV (both asyet
unobserved). This model also places the active fieldloops emitting
photons at 3−10 stellar radii for a surfaceB field of ∼ 1015 G.
The analysis of [43] is recently extended in [46], allow-ing for
a QED Compton cross scattering section whichincorporates
spin-dependent effects in stronger B fields.Electrons with
energies
-
9
at some magnetar rotational phases and viewing angleswhich
violates COMPTEL ULs, however the model ne-glects the effects of
Compton cooling and attenuationprocesses such as photon absorption
due to magneticpair creation (γ → e+e-) and photon splitting (⊥
→‖‖).Also, the effect of electron Compton cooling is expectedto
steepen the cut-offs seen in the predicted hard X-rayspectral tails
and allow the models to then be in agree-ment with the COMPTEL ULs.
The emission region isplaced at 4 − 15 and 2.5−30 stellar radii for
γe values of10 and 100 respectively.
The attenuation processes of magnetic pair creationand photon
splitting which act to suppress photon emis-sion in RCU are
considered in detail in [47] for typicalmagnetar surface B fields
of 10 Bc. In this case, the pho-ton splitting opacity alone
constrains the emission regionof observed 250 keV emission in
magnetars to be outsidealtitudes of 2-4 stellar radii and photons
emitted from themagnetar surface at magnetic co-latitudes 1 MeV for
typical magnetar surfaceB fields of 10 Bc. Also the emission of
photons from fieldloops at 5 stellar radii guarantee escapeof 1 MeV
photons at nearly all co-latitudes. The pho-ton opacity caused by
pair creation is shown to be muchless restrictive and does not
impact the 1 MeVis permitted at some but not all rotational phases
in themeridional case and that in most cases the RCU emissionwill
vary with rotational phase in the 300 keV − 1 MeVband.
Therefore, the RCU process may produce a back-ground to the
signal we wish to measure and this back-ground might be expected to
produce pulsed emissionwhen photon opacity due to photon splitting
is takeninto account. On the other hand, a spectral turn over
ispossible if the electrons in the magnetosphere field loopsare
mildly relativistic. In addition, pulsed emission inmagnetars has
not been observed in the 300 keV − 1MeV band which would be
suggestive of an RCU emis-sion mechanism. We also note that photon
splitting /pair creation opacity will not attenuate photon
emission10 stellar radii [47]. As axion to photonconversion will
occur at ∼ 300 stellar radii, photon opac-ity processes can be
disregarded. In addition, the 440magnetar bursts observed with the
Fermi -GBM over 5years have been spectrally soft with typically no
emis-sion above 200 keV[48].
A reasonable assumption resulting from the above dis-cussion
would be that there is no RCU background and
that all emission in the 300 keV − 1 MeV band resultsfrom ALP to
photon conversion. We instead opt fora slightly more conservative
approach and require thatany emission from ALP-photon conversion be
boundedby the observed emission. This results in ULs on
theALP-photon coupling.
VIII. MAGNETAR CORE TEMPERATURES
We now summarise the need for a magnetar heatingmechanism over
and above that found in conventionalpulsars and discuss temperature
modelling which sup-ports the range of values we have chosen for
the magnetarcore temperature (Tc).
The quiescent X-ray luminosity of magnetars of1034−1035 erg s-1
exceeds the spin down luminosity of1032−1034 erg s-1, thus
excluding rotation spin downas the sole magnetar energy source.
Furthermore, thelack of Doppler modulation in X-ray pulses arising
frommagnetars indicates a lack of binary companions, whichcombined
with the slow periods of magnetars (2−12 s)excludes an accretion
powered interpretation [49 and 50].
In reference [51], the authors show the need for heatingby
theoretical cooling curves for neutron stars of mass 1.4M�, with
and without proton superfluidity in the core,which yield effective
surface temperatures below thoseobserved in seven magnetars
(including four in our se-lection, namely: 1E 1841-045, 1RXS
J170849.0-400910,4U 0142+61 and 1E 2259+586). They then use a
gen-eral relativistic cooling code which accounts for thermallosses
from neutrino and photon emission and allows forthermal conduction
to show that magnetars are hot in-side with Tc=10
8.4 K at age 1000 yr and temperaturesof 109.1 K in the crust,
where the heat source shouldbe located for efficient warming of the
surface, to offsetneutrino heat losses from the core.
The authors of [52] consider the case of magnetars bornwith
initial periods of ≤ 3 ms combined with a strong in-ternal toroidal
B field of ≥ 3 × 1016 G and an exteriordipole B field of ≤ 2 × 1014
G. In this case, efficient heat-ing of the core can occur via
ambipolar diffusion whichhas a time varying decay scale as a
function of Tc andB field strength. As the core cools, an
equilibrium is es-tablished between increasing B field decay and
reducingneutrino emission, leading to reduced cooling which cankeep
Tc at 10
8.9 K 2250 yr after magnetar creation.The magnetar temperature
modeling of [53] considers
heating throughout the magnetar core arising from mag-netic
field decay and ambipolar diffusion, together withthe cooling
caused by the neutrino emission of the mod-ified URCA process and
Cooper pairing of nucleons. Inthis case, the authors find that
strong core heating can-not account for the observed surface
temperatures andconclude that, as in the case of [51], high surface
tem-peratures require heating of the crust, rather than thecore,
with the crust and the core being thermally decou-pled from one
another. However the authors of [53] show
-
10
10-2
10-1
100
101
10-6
10-5
10-4
10-3
4U 0142+61
10-2
10-1
100
101
10-6
10-5
10-4
10-3
1E 2259+586
10-2
10-1
100
101
10-6
10-5
10-4
10-3
1E 1048.1-5937
10-2
10-1
100
101
10-6
10-5
10-4
10-3
1RXS J170849.0-400910
10-2
10-1
100
101
10-6
10-5
10-4
10-3
1E 1841-045
FIG. 2. The spectral energy distributions of the five magnetars
from Refs. [16, 18]. The ULs within the analysis band, 300keV− 1
MeV (grey shaded region, with legend showing relevant experiments)
or immediately adjacent to it, are used in theanalysis. For each
magnetar, the gray line is an example spectrum from axion
conversion for the mass and coupling labelledout, which falls on
the limit curve in Fig. 3 of the main paper, with the corresponding
UL responsible for it denoted by thegreen thick line.
that Tc at 104 yr can vary between 0.8 × 108 K with no
heating of the superfluid core, 1.4 × 108 K with heatingof the
crust and 5 × 108 K with core heating. At 103 yr,with heating of
the superfluid core, Tc can reach 7 × 108K.
The strong B field of magnetars can produce stronglyanisotropic
thermal conductivity in the neutron star
crust whilst also allowing the synchrotron neutrino pro-cess to
become a predominant cooling mechanism whileother contributions to
the neutrino emissivity are farmore weakly suppressed. These
effects allow the temper-ature at the base of the crust heat
blanketing envelope toreach 109.6 K while the surface temperature
remains at105 and 106.7 K [54], for a B field parallel and radial
to
-
11
the neutron star surface respectively. This is compatiblewith
the observed surface temperatures of 106.5 − 106.95K for the seven
magnetars in [51] and could allow Tc toexceed 109 K.
Finally, the quiescent luminosity of magnetars1034−1035 erg s-1
implies a Tc of (2.7 − ≥ 8.0) × 108 Kfor a magnetar with an
accreted iron envelope and (1.0− 5.5) × 108 K for an accreted light
element envelope[55].
There are no published Tc values for the magnetars inour
selection. We therefore study the dependence of ourresults on a
range of core temperatures.
IX. MAGNETAR SELECTION AND ULSOFT-GAMMA-RAY FLUX DETECTION
We select 5 magnetars which have published ULsfor differential
energy fluxes between 300 keV−1 MeV(Table I). These are obtained
from the INTEGRALSoft-Gamma-Ray imager (ISGRI) detector, its
Imageon Board instrument (IBIS) and spectrometer (SPI);and from the
non-contemporaneous observations of theCOMPTEL instrument on the
Compton Gamma-RayObservatory CGRO [16 and 18].
We extract UL fluxes from the spectral energy distri-butions of
[16 and 18] using an energy resolved analysisas described in
Sec.X.
X. SPECTRAL ANALYSIS
For the five magnetars, the experimental ULs are takenfrom Ref.
[16, 18] and are shown in Fig. 2. We select theULs which fall
within or overlap with the range range300 keV−1 MeV. For each
magnetar and for each ax-ion mass, we require that the spectrum
does not exceedany of the ULs on the log-log νFν plot in Fig. 2.
Themaximal coupling Gangaγγ satisfying above criterion ischosen as
the exclusion UL for the coupling product. Incomparing the spectrum
with each UL in energy bin, say,“i”, we compare the averaged
spectrum within that binwith the experimental UL there. More
precisely speak-ing, for each magnetar and each ALP mass, we find
thelargest coupling Gangaγγ compatible with the
followingcondition:∫
ω−i
-
12
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-22
10-21
10-20
10-19
10-18
10-17
10-16
10-15
10-14
10-13
1E 2259+586
4U 0142+61
1RXS J170849.0-400910
1E 1841-045
1E 1048.1-5937
FIG. 3. The 95% CL upper limits on the coupling Gan × gaγγ for
our sample of 5 magnetars, obtained for emissions fallingwithin
experimental exclusion bins which overlap with the range 300 keV−1
MeV assuming Tc = 5 × 108K.
◼
◼
◼
◼
◼
◼
◼
◆
◆
◆
◆
◆◆
◆◆
◆ ◆
*
*
*
*
*
*
*
●
●
●
●
●
●
▲
▲
▲
▲
▲
▲
▲
1 2 3 4 5 6 7 8 9 10
10-20
10-19
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
◼ 1E 2259+586◆ 4U 0142+61
* 1RXS J170849.0-400910● 1E 1841-045
▲ 1E 1048.1-5937
FIG. 4. The 95% CL UL on the coupling product Gan ×gaγγ as Tc is
varied, for the magnetars in our study. TheALP mass is fixed at
10−7 eV and the luminosity from ALP-photon conversion is assumed to
saturate UL luminosity listedin Table I.
Magnetar UL Luminosity Gangaγγ (GeV−2)
at 300−500 keV1035 erg s-1
1E 2259+586 6.9 1.24 × 10−184U 0142+61 8.8 1.17 × 10−18
1RXS J170849.0-400910 9.8 1.06 × 10−181E 1841-045 49.0 2.16 ×
10−18
1E 1048.1-5937 55.0 2.41 × 10−18
TABLE III. Predicted 3σ UL on Gangaγγ (GeV−2) for our
original magnetar sample for future GBM observations atma =
10
−7eV.
XII. DISCUSSION: PROPOSED MAGNETAROBSERVATIONS WITH THE GBM
The GBM is a non-imaging instrument with a widefield of view.
However, it is possible to assign detectedevents to individual
pulsars using the Earth OccultationTechnique (EOT) or pulsar timing
models. EOT usesa catalogue of sources which exhibit step like
changes inphoton count rate as seen by the GBM, when the sourcesare
eclipsed by or rise above the Earth limb. In 3 years,EOT has
detected 9 of 209 sources between 100−300 keV[67].
The orbital precession of Fermi can be used to applyEOT without
a predefined source catalogue. By imag-ing with a differential
filter using the Earth occultationmethod (IDEOM), the Earth limb is
projected onto thesky and used to determine count rates from
600,000 vir-tual sources with a 0.25° spacing [68], identifying 17
newsources.
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13
Magnetar Surface B Age Distance UL Luminosity GangaγγField kyr
kpc at 300−500 keV (GeV−2)
1014 G 1035 erg s-1
SGR 1806-20 19.6 0.2 8.7 [57] 51.4 1.84 × 10−181E 1547.0-5408
3.18 0.7 4.5 [58] 13.7 1.29 × 10−18SGR 1900+14 7 0.9 12.5 [59]
106.0 3.13 × 10−18
CXOU J171405.7-381031 5.01 0.9 13.2 [60] 118.2 3.53 × 10−18SGR
1627-41 2.25 2.2 11.0 [61] 82.1 3.39 × 10−18
PSR J1622-4950 2.74 4.0 9.0 [62] 55.0 2.65 × 10−18SGR J1745-2900
2.31 4.3 8.3 [63] 46.7 2.51 × 10−18
Swift J1834.9-0846 1.42 4.9 4.2 [64] 12.0 1.39 × 10−18XTE
J1810-197 2.1 11.3 3.5 [65] 8.3 1.09 × 10−18SGR 0501+4516 1.87 15.4
2.0 [66] 2.7 6.35 × 10−19
TABLE IV. Predicted 3σ UL Gangaγγ (GeV−2) values obtainable for
proposed future observations of a wider magnetar sample,
at ma = 10−7eV, assuming a GBM UL energy flux at 300−500 keV of
3.5 × 10-4 MeV cm-2 s-1 yielding the UL luminosities
shown. Distances are from the references given, surface B field
and age are from the online27 version of the McGill magnetarcatalog
[20].
The Fermi -GBM Occultation project now monitors248 sources in
the energy range 8 keV−1 MeV with themajority of the signal seen
between 12−50 keV 4.
In contrast, the author of [69] uses a pulsar timingmethod
instead. The GBM CTIME data is used to pro-vide photon counts for 4
magnetars, 1RXS J170849.0-400910, 1E 1841-045, 4U 0142+61 and 1E
1547.0-5408.The photon counts are attributed to the peak
pulsedemission of each magnetar by epoch folding and usingtiming
models (obtained with the Rossi X-ray TimingExplorer), which tags
each event by pulsar phase. Thiscount rate is converted to an
energy flux for 7 energychannels between 11 keV−2 MeV by
determining theGBM effective area as a function of photon
direction,energy and probability of detection of a photon with
agiven energy. This yields pulsed ULs, just above thoseobtained by
COMPTEL for J170849.0-400910, 1E 1841-045 and 4U 0142+61.
The GBM is thus a very useful instrument to deter-mine the UL
soft-gamma-ray fluxes of the 23 confirmedmagnetars 5 in the McGill
Magnetar Catalog [20], mostof which have no ULs defined in the 300
keV−1 MeVband of interest. We project the possible UL values ofGan
× gaγγ that can potentially be obtained using ouroriginal sample of
magnetars, as well as a wider magne-tar sample in Table III and IV
respectively. We use a 3 σUL flux sensitivity of 118 mCrab
(equivalent to 3.5×10-4MeV cm-2 s-1 assuming the Crab spectrum in
[70]), be-tween 300−500 keV determined from 3× the error of 3yr of
GBM EOT observations of 4 sources including theCrab [67].
4 https://gammaray.msfc.nasa.gov/gbm/science/earth_occ.
html accessed on 25th November 20195
http://www.physics.mcgill.ca/~pulsar/magnetar/main.html
accessed on 25th November 2019
XIII. CONCLUSIONS
In this paper, we have explored constraints on theproduct of the
ALP-nucleon and ALP-photon couplings.The constraints are obtained
from the conversion of ALPsproduced in the core of magnetars into
photons in themagnetosphere. When interpreting our results in
Figure3, the following caveats apply: since the magnetars in
ourselection have no published values of Tc, the results
aredisplayed for a benchmark Tc of 5 × 108K. We furthershow the
limits that can be obtained by varying Tc inFigure 4 for a fixed
ALP mass of 10−7eV. We also notethat a more stringent limit can be
obtained by a com-bined analysis of the upper limits from all
magnetars.
Our results motivate a program of studying quies-cent
soft-gamma-ray emission from magnetars in the 300keV−1 MeV band
with Fermi -GBM. The GBM will beable to determine the UL
soft-gamma-ray fluxes of con-firmed magnetars, most of which have
no ULs definedin soft gamma-rays. With these ULs, it is possible
thateven more stringent constraints on the product of theALP
couplings may be obtained.
ACKNOWLEDGEMENTS
HG and KS are supported by DOE Grant DE-SC0009956. KS would like
to thank Jean-Francois Fortinfor many discussions on ALP-photon
conversions nearmagnetars, to be incorporated into a forthcoming
pub-lication [71]. He would also like to thank KITP SantaBarbara
for hospitality during part of the time that thiswork was
completed. AMB and PMC acknowledge thefinancial support of the UK
Science and Technology Fa-cilities Council consolidated grant
ST/P000541/1.
https://gammaray.msfc.nasa.gov/gbm/science/earth_occ.htmlhttps://gammaray.msfc.nasa.gov/gbm/science/earth_occ.htmlhttp://www.physics.mcgill.ca/~pulsar/magnetar/main.html
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14
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Axion Constraints from Quiescent Soft Gamma-ray Emission from
Magnetars AbstractI IntroductionII PhenomenologyIII ALP-Photon
Probability of ConversionIV Nucleon Bremsstrahlung and ALP
ProductionV ALP Emissivity in Mean Field TheoryVI Calculation of
Refractive IndicesA 2me and B Bc, with (2me)2(BBc)2 1B 2me and B
> Bc
VII Soft Gamma-ray BackgroundVIII Magnetar Core TemperaturesIX
Magnetar Selection and UL Soft-Gamma-Ray Flux DetectionX Spectral
AnalysisXI ResultsXII Discussion: Proposed Magnetar Observations
with the GBMXIII Conclusions Acknowledgements References
