#### Constraining axion-like-particles with hard X-ray emission from magnetars

HJE
Constraining axion-like-particles with hard X-ray emission from magnetars
Jean-Francois Fortin 0 2 3
Kuver Sinha 1 2 3
0 Departement de Physique, de Genie Physique et d'Optique , Universite Laval
1 Department of Physics and Astronomy, University of Oklahoma , USA
2 2325 Rue de l'Universite , Quebec, QC G1V 0A6 , Canada
3 440 W. Brooks St., Norman, OK 73019 , U.S.A
Axion-like particles (ALPs) produced in the core of a magnetar will convert to photons in the magnetosphere, leading to possible signatures in the hard X-ray band. We perform a detailed calculation of the ALP-to-photon conversion probability in the magnetosphere, recasting the coupled di erential equations that describe ALP-photon propagation into a form that is e cient for large scale numerical scans. We show the dependence of the conversion probability on the ALP energy, mass, ALP-photon coupling, magnetar radius, surface magnetic eld, and the angle between the magnetic eld and direction of propagation. Along the way, we develop an analytic formalism to perform similar calculations in more general n-state oscillation systems. Assuming ALP emission rates from the core that are just subdominant to neutrino emission, we calculate the resulting constraints on the ALP mass versus ALP-photon coupling space, taking SGR 1806-20 as an example. In particular, we take benchmark values for the magnetar radius and core temperature, and constrain the ALP parameter space by the requirement that the luminosity from ALPto-photon conversion should not exceed the total observed luminosity from the magnetar. The resulting constraints are competitive with constraints from helioscope experiments in the relevant part of ALP parameter space.
Cosmology of Theories beyond the SM; Beyond Standard Model; CP violation
1 Introduction 3 4 5
Results
2.1
2.2
3.1
3.2
4.1
4.2
Conclusion 2
ALP-to-photon conversion probability
General formalism
ALP-photon oscillations
2.3 Implications for conversion probability
Dependence of the conversion probability
Photon luminosity
Constraints on ALP parameter space
Photon luminosity
ALP parameter space constraints
A Neutrino and ALP production mechanisms
A.1 Neutrino emissivity A.2 Normalized ALP spectrum
exceed the quantum critical value Bc = me2=e = 4:414
1013 G. For recent reviews of
magnetars, we refer to [1{3].
The extreme magnetic eld of magnetars can potentially be exploited in the search for
new physics beyond the Standard Model. Clearly, the most fertile area to investigate is
the physics of axions, originally introduced as a solution to the strong CP problem [4{11].
Our interest in this paper will be in the broader class of axion-like particles (ALPs) which
arise generically in compacti cations in string theory, and have been applied to phenomena
ranging from in ation to baryogenesis [12{14]. In the presence of an external magnetic eld,
ALPs can convert into photons, and vice versa, via the Primako process. The relevant
coupling is given by the rst term in
L
g
4
aF
~
F
5N;
(1.1)
{ 1 {
where a denotes the ALP and the coupling g
ga has mass dimension
lies at the heart of many ALP searches and constraints. The second term in (1.1), which
is model-dependent, describes the coupling between the ALP a and nucleons N , and leads
(among other processes) to ALP production in the core of neutron stars. For reviews, we
refer to [15, 16].
Photons (ALPs) emitted by a magnetar will convert to ALPs (photons) via eq. (1.1).
It is therefore interesting to investigate whether observations of magnetars can constrain
the ALP mass and coupling strength g. This avenue has been pursued by several authors
recently [17{20].1 We brie y summarize the ndings of [17] and [18], which are germane
to the present work.
The authors of [17] and [18] studied the surface emission of photons from neutron
stars and their subsequent conversion into ALPs in the magnetosphere. Modi cations of
the spectral shape, light curves, and polarization signals that could potentially be identi ed
observationally were investigated after taking into account e ects of gravitational redshift
and light de ection. For a star that emits from its entire surface, these features include
deviations in the apparent radius and emission area from limits set by the neutron star
equations of state. Moreover, interesting polarization signals can also arise | for example,
an emission region that is observed phase-on will exhibit an inversion of the plane of
polarization compared to the case where photon-axion conversion is absent [18].
The
features studied in these papers concentrate on the soft X-ray emission from neutron stars.
We note that the general strategy of these papers, and our current work, is di erent
from usual cooling arguments that are used to place limits on ALPs from white dwarfs [23]
or supernovas [24]. While attempts have been made to use cooling simulations of neutron
stars to constrain ALPs [25], our work relies speci cally on the Primako
process in the
magnetosphere.
to them as follows:
The purpose of our paper is to understand the following question: can ALPs produced
inside the neutron star lead to observational signatures via ALP-to-photon conversion in
the magnetosphere? There are two aspects to this question, and we outline our approach
(i) The rst aspect concerns the probability of ALP-to-photon conversion in the
magnetosphere, and its dependence on the ALP parameters and the properties of the
magnetar. We have performed a careful analysis of the coupled di erential equations
describing the ALP-photon system as it propagates through the magnetosphere. We
have recast the equations into a form that is e cient for large scale numerical scans,
and checked that there is a gain of almost two orders of magnitude in computation
time. Along the way, we develop an analytic formalism to perform similar
calculations in more general n-state oscillation systems. We provide detailed scans of the
conversion probability as a function of the ALP energy ! in the 1{200 keV range,
mass ma, coupling g, surface magnetic eld strength B0 of the magnetar, magnetar
radius r0, and angle between the magnetic eld and the direction of propagation .
1Magnetars have also been studied recently by particle physicists in other contexts, for example to
constrain milli-magnetically charged particles [21] and millicharged fermions [22].
{ 2 {
HJEP06(218)4
(ii) The second aspect concerns the emission rate of ALPs from the magnetar. We assume
that ALPs are emitted uniformly from the core of the neutron star and escape into the
magnetosphere. Furthermore, we assume that the emission rate is just subdominant
to the neutrino emission rate for a given core temperature. This conservatively leaves
standard cooling mechanisms of neutron stars unaltered, while also giving the best
case scenario for signals of ALP-to-photon conversion in the magnetosphere.
We
take nucleon-nucleon bremsstrahlung N + N ! N + N + a as the main production
mechanism of ALPs, while noting that various other production mechanisms have
been considered in the literature [26{35]. We reserve a more detailed study of this
question and especially its relation to ALP-photon conversion for a future publication.
emission rate from the core, we study possible signatures of the resulting photon
3:3T where T is the magnetar core temperature, assuming degenerate nuclear matter in the interior [36].
With core temperatures T in the range
(0:6{3:0)
50{250 keV, the hard X-ray band 100{900 keV should be the natural
focus to investigate unique features of ALP-to-photon conversion.
Since ALPs are steadily produced by the core and converted to photons in the
magnetosphere, we will mainly be interested in quiescent magnetar emission.2 Both Soft-Gamma
Repeaters (SGRs) and Anomalous X-ray Pulsars (AXPs) exhibit quiescent soft X-ray
emission below 10 keV. This mostly thermal emission with luminosity L
1033{4
1035 erg s 1
is presumably powered by the internal magnetic energy [37]. The spectra of several AXPs
and SGRs also show hard non-thermal tails extending up to 150{200 keV [38{41] with
luminosities similar to those observed in the soft X-ray band. While the soft X-ray emission
is thermal and originates from the hot surface of the magnetar, the origin of the hard X-ray
spectrum is an area of active study. We refer to [42{44] for more details.
ALPs produced from the core and converting to photons in the magnetosphere can
contribute signi cantly to the spectral peak above 100 keV for optimal selections of ALP
parameters. The spectral peak given by !
3:3T can be anywhere up to almost 1 MeV
and for optimal selections of ma and g, one can match the observed luminosities. Of
course, a detailed analysis would be required before one can claim that this is the dominant
contribution or mechanism underlying hard X-ray emissions from magnetars.
Our far more conservative approach in this paper will be to constrain ALP parameters
using the observed hard X-ray emission. In other words, we will demand that the
luminosity coming from ALP-to-photon conversion be bounded by the observed luminosity in the
range 1{200 keV, while remaining agnostic about the physical processes that give rise to this
emission. We will take the benchmark value of the magnetic eld of SGR 1806-20 [45, 46],
although other candidates from the McGill Magnetar Catalog [47] can also be considered.
For the magnetar radius, we assume a typical value of r0 = 10 km, while for the core
temperature, we assume a range of values between T9 = 0:6 to T9 = 3:0 where T9 = T =(109 K).
2In particular, we will have nothing to say about the hard X-ray outbursts or giant supersecond ares
shown by some SGRs and AXPs.
{ 3 {
We then compute the predicted photon ux from ALP-to-photon conversion in the entire
energy range from ! = 1{200 keV and compare this against the observed average luminosity
of SGR 1806-20 in this band. This places limits on the parameter space of ALP masses ma
and ALP-photon coupling g. The limits we obtain are competitive with constraints from
helioscope experiments in the relevant part of ALP parameter space for high enough core
temperatures. We reserve a more detailed spectral analysis, as well as possible polarization
signals in the hard X-ray regime, for a future publication.
The rest of the paper is structured as follows. In section 2, we provide the calculation of
the ALP-to-photon conversion probability from a general formalism for n-state oscillation
problems. In sections 3 and 4, we apply our results to SGR 1806-20, and present the
ALP-to-photon conversion probability
This section develops a general formalism for oscillation in the weak-dispersion limit. We
rst give an analytic treatment of n-state oscillation problems using conservation of
probability. We then apply this formalism to ALP-photon oscillations in the magnetosphere of
magnetars. Our main goal here is to recast the standard evolution equations in the
weakdispersion limit of [17] into a formalism that is more amenable for numerical applications,
such as large scans over parameter space. We have checked that our formalism leads to
integration times that are faster by around two orders of magnitude.
2.1
General formalism
For a general n-state oscillation problem in the weak-dispersion limit [48], the coupled
system of di erential equations for the linearized wave equations is of the form
i
dai(x)
dx
n
j=1
=
X Aij (x)aj (x);
(2.1)
(2.2)
where ai(x) are the oscillating
elds and Aij (x) is some matrix dictated by the
oscillation problem at hand. Conservation of probability dx
Aji(x) = Aij (x).3 Using generalized spherical coordinates, one possible choice for the form
d Pin=1 jai(x)j2 = 0 implies that
of the solutions is
8n 1
: j=i
ai(x) = < Y sin[ j (x)]= cos[ i 1(x)]e i i(x);
with
0(x) = 0. This form satis es conservation of probability such that the total
probability is properly normalized, i.e. Pin=1 jai(x)j2 = 1.
3More generally, Aji(x) = Aij(x) also leads to conservation of probability. The remainder of the analysis
will be performed for a real-symmetric A(x) since A(x) is real-symmetric for ALP-photon conversion [17].
The generalization to hermitian A(x) is straightforward and is left to the reader.
9
;
{ 4 {
Di erentiating with respect to x gives
i
<X cot[ j(x)]
: j=i
d j(x)
dx
tan[ i 1(x)]
d i 1(x)
dx
i
d i(x) =
dx
9
;
ai(x);
(2.3)
and thus the coupled system of di erential equations becomes
The right-hand side can be expressed in terms of the new variables as 1
n
ai(x) j=1
X Aij(x)aj(x) = X Aij(x) nQkn=i1 sin[ k(x)]o cos[ i 1(x)]
j=1
n
nQkn=j1 sin[ k(x)]o cos[ j 1(x)]
e i[ j(x) i(x)]:
Hence, gathering real and imaginary parts, the coupled system of di erential equations
simpli es further to
d i 1(x)
Sij(x) = nQkn=i1 sin[ k(x)]o sin[ i 1(x)]
nQkn=j1 sin[ k(x)]o cos[ j 1(x)]
Cij(x) = nQkn=i1 sin[ k(x)]o cos[ i 1(x)]
nQkn=j1 sin[ k(x)]o cos[ j 1(x)]
sin[ j(x)
i(x)];
cos[ j(x)
i(x)]:
(2.5)
It is interesting to note that the evolution equations are not functions of the n individual
phases i(x), but only of the n
1 phase di erences
ij(x) =
i(x)
j(x). Including
the n
1 angles i(x), this leads to 2(n
1) real di erential equations instead of 2n real
di erential equations, as expected from conservation of probability.
Before proceeding, it is also important to mention that the general formalism
introduced here can be used for any oscillation problem. Hence, it might be useful in the study
of neutrino oscillations.
{ 5 {
altogether [17].
is given by
We now focus on ALP-photon oscillations in a magnetic eld [17], as for example the
magnetic
eld of a magnetar. As long as the magnetic eld space variations occur on
larger distances than the photon (and ALP) wavelength and the magnetic eld is not too
large, the ALP-to-photon conversion probability can be calculated in the weak-dispersion
limit and the general formalism developed above can be used directly. Moreover, since the
vacuum resonance and the ALP-photon resonance are well separated in the magnetized
plasma of a magnetar in the ALP parameter space of interest here, the perpendicular
photon electric eld mode decouples from the evolution equations and can be forgotten
HJEP06(218)4
Hence, the coupled system of di erential equations describing ALP-photon oscillations
where
k
and
M are functions of r,
a =
k
m2a
2!
;
k =
1
2
q! sin2 ;
1
2
M =
gB sin :
Here a(r) and E (r) are the ALP and parallel photon electric elds respectively, r is the
distance from the center of the magnetar, ! is the energy of the ALP and photon electric
elds, ma is the ALP mass, g is the ALP-photon coupling constant, is the angle between
the direction of propagation of the ALP-photon
eld and the magnetic eld, and q is a
dimensionless function of the magnetic eld B given by [17, 48]
7
45
q =
b2q^;
q^ =
1 + 1:2b
1 + 1:33b + 0:56b2
;
with b = B=Bc where Bc = me2=e = 4:414
1013 G is the critical QED eld strength.
With the help of the dimensionless variable x = r=r0 where r0 is the magnetar radius,
the coupled system of di erential equations (2.6) becomes
i
d
dx
a !
E
k
=
!r0 +
ar0
M r0
M r0
!r0 +
kr0
!
a !
E
k
=
A(x) D(x) !
D(x) B(x)
a !
E
k
:
The relevant matrix for ALP-photon oscillation is therefore real-symmetric with A(x)
Hence the general formalism developed above can be used directly.
However, for further convenience, we choose solutions of the form
a(x) = cos[ (x)]e i a(x);
Ek(x) = i sin[ (x)]e i E(x);
with (x), a(x) and E(x) real functions. The extra phase in (2.8) will simplify the initial
conditions for pure initial states. Thus the coupled system of di erential equations (2.7)
simpli es to
i
d
dr
a !
E
k
=
! +
a
M
M
! +
k
!
a !
E
k
;
d (x)
dx
d (x)
dx
+ i cot[ (x)]
i tan[ (x)]
{ 6 {
(2.6)
(2.7)
(2.8)
=
D(x) cos[ a(x)
E (x)];
= A(x)
D(x) tan[ (x)] sin[ a(x)
= B(x)
D(x) cot[ (x)] sin[ a(x)
E (x)];
E (x)]:
Again, the fact that both real parts lead to the same di erential equation con rms the
form (2.8) as expected from arguments about conservation of probability.
Since the focus is on the conversion probability and only the relative phase (x) =
a(x)
E (x) appears in the equations above, one gets to
d (x)
dx
further to
where (
1
) determines the initial state at the surface of the magnetar. To avoid singularities
for (
1
) = n =2 with n 2 Z, i.e. for pure initial states, the initial condition for
(
1
) must
satisfy
(
1
) = m
with m 2 Z. It is therefore possible to set
initial state4 and the ALP-photon conversion probability is simply Pa! (x) = sin2[ (x)].
(
1
) = 0 for a pure ALP
2.3
Implications for conversion probability
Unfortunately, (2.9) cannot be solved analytically for generic cases. Nevertheless, exact
solutions exist for some speci c cases. For example, in the no-mixing case, D(x) = 0 and
solutions to (2.9) are
(x) = (
1
);
(x) =
(
1
) +
dx0 [A(x0)
B(x0)];
(2.10)
over, there exist exact solutions to the evolution equations (2.9) in the case A(x) = B(x)
for
(
1
) = 0. Indeed, in this speci c case it is easy to see that
One can also write a general solution for (x) in terms of the functions D(x) and
(x) as
(x) = (
1
)
4This is possible thanks to the choice of solutions (2.8).
R1x dx0 jD(x0)j is bounded from above for all x
out that the case A(x) = B(x) in (2.11) saturates the upper bound (2.13) if D(x)
1. As a side note, it is interesting to point
Finally, the form of the evolution equations (2.9) leads to some qualitative
understanding for magnetars in the simple case of a dipolar eld B = B0(r0=r)3 for which
A(x)
B(x) = [ a
k(x)]r0 =
a
r0;
D(x) =
M r0 =
First, since for a magnetar D(x) ! 0 as x !
1, (x) is a constant independent of x for
x large as expected from (2.10), i.e. the conversion probability is a well-de ned quantity
with a
xed value far away from the magnetar. This behavior should occur on physical
ground since the magnetic eld decreases as the state propagates away from the magnetar,
leading to a vanishing mixing between ALPs and photons.
Moreover, from the upper bound (2.13) with a pure ALP initial state (
1
) = 0, the
conversion probability is constrained by
Pa! (x) = sin2[ (x)]
(sin2 R1x dx0 D(x0)
1
otherwise
if R1x dx0 D(x0) < 2 :
than P
= sin2(
M0r0=2), i.e. Pa! (
1
)
Hence, for a magnetar the conversion probability for a pure ALP initial state is constrained
if R11 dx D(x) < 2 . In the simple case of a dipolar eld, the conversion probability is smaller
P , if
M0r0 <
. Unfortunately this bound
is not relevant here since
M0r0
1. Indeed, for sample values such as ! = 100 keV,
ma = 10 8 keV, g = 10 15 keV 1, r0 = 10 km, B0 = 20
three rather di erent pure numbers controlling the solutions to the evolution equations
given by
ar0
2:5
demonstrating that the analytic bound cannot be used in the following to constrain the
ALP parameter space.5
3
Results
In this section, the evolution equations (2.9) are solved numerically. First, the
dependence of the conversion probability on the ALP and magnetar parameters is shown for a
benchmark point in the hard X-ray range. Then, the conversion probability in the (ma; g)
plane is given for several values of !. The conversion probability and photon luminosity
depend on properties of the magnetar, like its radius and magnetic eld. We choose the
magnetic eld B0 = 20
radius r0 = 10 km as our benchmark values.6
1014 G corresponding to SGR 1806-20 and a typical magnetar
5Here the plasma contribution is negligible for stellar magnetic
elds and hard X-ray frequencies [48].
6Although the magnetic
eld is quite large, its position dependence is strong enough to ensure the
weak-dispersion limit can be used where the conversion occurs, which is far away from the magnetar.
{ 8 {
curve respectively) as a function of the dimensionless distance x from the magnetar surface for the
benchmark point ! = 100 keV, ma = 10 8 keV, g = 10 15 keV 1, r0 = 10 km, B0 = 20
Dependence of the conversion probability
For a magnetar in the simple case of a dipolar magnetic eld, the conversion probability
of a pure ALP initial state to a photon state, given by Pa!
depends on six di erent parameters. The ALP parameters are the ALP energy !, the
ALP mass ma and the ALP-photon coupling constant g. The magnetar parameters are the
magnetar radius r0, the (dimensionless) magnetar magnetic eld at the surface b0 = B0=Bc
and the angle between the direction of propagation and the magnetic eld .
The evolution of (x) as well as cos2[ (x)] and sin2[ (x)] at the benchmark point
! = 100 keV, ma = 10 8 keV, g = 10 15 keV 1, r0 = 10 km, B0 = 20
= =2 is shown in gure 1. The benchmark point is chosen in the hard X-ray range with
Pa! (
1
) = sin2[ (
1
)],
appropriate magnetar parameters for SGR 1806-20.
The conversion probability Pa!
= sin2[ (
1
)] around the same benchmark point in
function of one of the ALP or magnetar parameter is shown in the corresponding panel
of gure 2.
Several comments are in order for this benchmark point. First, the conversion
probability peaks in function of the ALP energy ! in the X-ray range. This observation is
important since the normalized ALP spectrum from nucleon-nucleon bremsstrahlung
emission for a degenerate medium relevant to magnetars peaks in the hard X-ray range for our
benchmark model. Second, with respect to the ALP mass ma, the conversion
probability plateaus around a non-zero (zero) value for small (large) ALP masses, with a sharp
transition between the two regimes at around ma
5
10 8 keV. Hence the conversion
probability vanishes for large ALP masses. This behavior can be explained qualitatively
from (2.9). Indeed, since D(x)
1=x3, the conversion probability reaches a well-de ned
limit as x increases. Beyond some distance x from the magnetar, the conversion
probability is essentially
xed. On the one hand, if the ALP mass is small enough such that
jA(x )j
jD(x )j, then jA(x)j
jD(x)j 8 x 2 (1; x ) and thus the evolution equations
from the surface to x only have a negligible dependence on the ALP mass. On the other
hand, if the ALP mass is large enough such that jA(
1
)j
jD(
1
)j, then jA(x)j
jD(x)j
{ 9 {
for the benchmark point ! = 100 keV, ma = 10 8 keV, g = 10 15 keV 1, r0 = 10 km, B0 =
the benchmark point for one of the ALP or magnetar parameter.
8 x 2 (1; 1). Since A(x)
B(x) has a de nite sign, jA(x)
B(x)j
jD(x)j and
(x)
decreases rapidly in the evolution equations, leading to variations on (x) that average out,
implying a vanishing conversion probability for large ALP masses. Third, the dependence
on the ALP-photon coupling constant g is important, as the conversion probability is zero
for g . 2
10 16 keV 1 and increases signi cantly for larger g. The vanishing conversion
probability for small g is expected since ALP-photon oscillations are suppressed as g
decreases. E ects of the oscillatory nature of the problem can also be seen for larger g. Fourth
and fth, a change of the magnetar radius r0 or the dimensionless surface magnetic eld b0
leads to a variation of the conversion probability that is quite mild. Sixth, the dependence
of the conversion probability with respect to the angle
is also mild away from
= 0
where it vanishes, as expected from the ALP-photon coupling
M . Moreover, the
conversion probability is an even function with respect to the angle , i.e. Pa! (
This is expected since a solution to the evolution equations (2.9) with (
1
) = 0,
) = Pa! ( ).
(
1
) = 0
and is also a solution to the evolution equations with (
1
) = 0,
and
there is no physical di erence between
(
1
) = 0 and
(
1
) =
, the conversion
probability far away from the magnetar should not depend heavily on this particular choice.7
The conversion probability in the (ma; g) plane for ! = 1, 100 and 200 keV with
magnetar parameters r0 = 10 km and B0 = 20
and
=
=2 is shown in gure 3.
1014 G relevant to our benchmark model
Again, several comments are in order. Figure 3 shows that as the ALP energy !
increases, the locations of the transitions in ma and g both increase. Moreover, the
conversion probability is negligible everywhere except for small ALP mass and large ALP-photon
coupling constant, as expected from our previous discussion. In the latter region, the
oscillatory nature of the problem can clearly be seen.8
With the !- and -dependence of the conversion probability in the (ma; g) plane, it is
now straightforward to compute the ALP-to-photon luminosity for SGR 1806-20, assuming
r0 = 10 km and B0 = 20
gure 3, we therefore expect that the constraint on
the magnetar photon luminosity in the hard X-ray range will exclude the ALP parameter
space with small ALP mass and large ALP-photon coupling constant.
3.2
Photon luminosity
The photon luminosity from ALP-photon oscillations in the band !
computed from the ALP-to-photon conversion probability obtained above.
(!i; !f ) can be
Indeed, from the normalized ALP spectrum dNa=d! such that R01 d! dNa=d! = 1, the
ALP-to-photon luminosity in the band !
(!i; !f ) is given by
La!
=
Na Z 2
2
0
d
Z !f
!i
d! !
dNa
d!
Pa! (!; );
where Na is the total number of ALPs emitted by the magnetar. The -average in (3.1)
is necessary to obtain the magnetar luminosity. However, since the -average is
computationnally intensive, it is replaced by
where R is a conservative suppression factor computed for several (!; ma; g) points which
is numerically given by R
The total number of ALPs emitted by the magnetar is constrained by cooling models.
Indeed, since magnetar cooling is well understood in terms of neutrino cooling luminosity,
the ALP cooling luminosity should not overtake the neutrino cooling luminosity, hence
Z 1
0
dNa
d!
La = Na
d! !
L = 4
Z r0
0
dr r2q_ ;
where q_ is the neutrino emissivity. Assuming that the neutrino emissivity is constant
throughout the whole magnetar, (3.2) implies
(3.1)
(3.2)
Na
4 r03q_
3 R01 d! ! ddN!a ;
7The choice
(
1
) =
is as physically motivated as our initial choice of
(
1
) = 0.
8Moreover, contrary to [48], Pa! can be large since
ar0, kr0 and
M r0 depend di erently on x [17].
10?9
energy ! corresponding to ! = 1 keV (top panels), ! = 100 keV (middle panels) and ! = 200 keV
(bottom panels). The two panels (left and right) show the same conversion probability from di erent
points of view.
0.6
0.0
10?6
and the ALP-to-photon luminosity (3.1) which saturates the previous bound is9
In the following, the constraints on the ALP parameter space are computed from the
ALP-to-photon luminosity (3.3).
Usually, it is necessary to make some assumptions on the ALP production
mechanisms as well as the ALP-nucleon coupling constant gaN to determine the normalized ALP
spectrum and the total number of ALPs emitted by the magnetar. However, to obtain
conservative constraints on the ALP parameter space, we choose to take the normalized ALP
spectrum (A.2) from nucleon-nucleon bremsstrahlung emission for a degenerate medium
relevant to magnetars [48]. Moreover, as we have done above, we bound the total number
of ALPs emitted by the magnetar with respect to the neutrino cooling luminosity, but
using modi ed URCA emission (A.1) [49]. The nucleon-nucleon bremsstrahlung emission of
ALPs and modi ed URCA emission of neutrinos are the respective dominant production
mechanisms and their use is justi ed to stay as conservative as possible and to impose
as few assumptions as possible on the ALP and neutrino production mechanisms. Both
production mechanisms are discussed at greater length in the appendix.
4
Constraints on ALP parameter space
In this section, we put together all the ingredients to compute the total ALP-to-photon
lumnosity. We then compare our results with observations.
4.1
Photon luminosity
For the conversion probability, we choose the benchmark values of r0 and B0 used in the
rest of the paper. The ALP emissivity, on the other hand, is bounded by the neutrino
cooling luminosity via modi ed URCA emission. This is strongly dependent on the core
temperature of the magnetar.
In gure 4, we rst display the photon luminosity in the broad band from !i = 1 keV
to !f = 200 keV assuming an ALP emissivity that equals the modi ed URCA emission
of neutrinos for a core temperature of T = 109 K. The photon luminosity is obtained
from (3.3), (A.1), (A.2) and the conversion probability. The results are displayed on the
(ma; g) plane.
4.2
ALP parameter space constraints
While the surface temperature of the magnetar can be easily deduced from the thermal
emission, the relation between the surface and core temperatures depends on a variety of
factors that a ect the conduction of heat. These include the strength of the magnetic eld
in the blanketing envelope and its angle with respect to the radial direction, as well as the
in (3.3) must be replaced by (gaN =gaN )2La! where gaN is given by (A.4).
9As explained in the appendix, for speci c models with explicit dependence on gaN , the luminosity La!
1.5
L0.5
0.0
10?10
La??(ma, g) for T = 109 K
10?9
200 keV in the (ma; g) plane. The computations are done for SGR 1806-20 assuming r0 = 10 km
and B0 = 20
1014 G. The magnetar core temperature is assumed to be T = 109 K. The two
panels show the same conversion probability from di erent points of view.
10?13
10?14
1?) 10?15
V
e
k
(g 10?16
a given magnetar core temperature, all the ALP parameter space above the corresponding curve is
excluded. The computations are done for SGR 1806-20 assuming r0 = 10 km and B0 = 20
The magnetar core temperatures span the range from T9 = 0:6 to T9 = 3:0 where T9 = T =(109 K).
From top to bottom, the curves correspond to T9 = 0:6, 0:7, 0:8, 0:9, 1:0, 2:0 and 3:0 respectively
(i.e. the red, orange, yellow, green, blue, indigo and violet curve). For comparison, the exclusion
contour from CAST is shown in black.
chemical composition of the magnetar. An exploration of these e ects is beyond the scope
of our paper. We refer to [3] and references therein for a thorough discussion.
We instead display our results for several core temperatures between T = 6
109 K. Since the observed luminosity of SGR 1806-20 in this range is
1036 erg s 1 [2, 50],10 any point in the (ma; g) plane with La!
> Lobs is
excluded. The exclusion curves, where La!
the excluded region in the ALP parameter space corresponds to small ALP mass and
= Lobs, are shown in
gure 5. As expected,
10The observed luminosity in the hard X-ray band in [50] is quoted as Lobs = 3:6
1036 erg s 1 for an
assumed distance of 15 kpc. Since the distance is now believed to be 8:7 kpc [2], we modi ed the observed
luminosity accordingly.
large ALP-photon coupling constant. Furthermore, the excluded region is larger for higher
magnetar core temperature. In fact, due to the strong neutrino emissivity dependence on
the magnetar core temperature, the largest total number of ALPs emitted by the
magnetar allowed by the cooling argument also has a strong dependence on the magnetar core
temperature, which translates into exclusion contours in the (ma; g) plane even where the
conversion probability is negligible for high core temperature.
For comparison, the exclusion contour from the CAST helioscope experiment [51],
based entirely on the Primako
process, is also shown. From
gure 5, it is clear that the
magnetar constraints on the ALP parameter space are better than CAST only for high
magnetar core temperatures T & 2
109 K.
As mentioned before, it is also possible to use an opposite point of view. Indeed,
since the mechanism responsible for hard X-ray quiescent emission in magnetars is not
known, from the analysis presented here one can argue that ALPs exist and are produced
in magnetars with a subdominant luminosity to neutrinos such that magnetar cooling is
not disturbed. With magnetar core temperatures that satisfy T & 2
109 K in line with
the magnetar model [52], the magnetar hard X-ray emission (possibly with the appropriate
spectral feature) could be generated by ALP-to-photon conversion in the magnetosphere
without violating the bound from CAST. A detailed analysis of all important production
mechanisms for hard X-ray photons and axions must however be undertaken and the
evolution equations must then be solved with the appropriate initial conditions (for example,
with a mixed initial state if the amplitudes are comparable) before such conclusions can
be reached.11
5
Conclusion
Our goal in this paper has been to exploit the rapidly advancing eld of magnetar science
to study the physics of ALPs. Magnetars, with their extremely strong magnetic
elds,
form a natural arena for investigating ALPs.
Our basic idea was to consider the conversion of ALPs emitted from the core of the
neutron star into photons in the magnetosphere. We assumed that the emission rate for ALPs
is just subdominant to the neutrino emission rate for a given temperature. For
nucleonnucleon bremsstrahlung, we obtain a broad ALP spectrum peaked around !
3:3T . The
coupled di erential equations describing ALP-photon propagation in the magnetosphere
were converted into a form that is e cient for extensive scans over multiple parameters.
We then presented the conversion probability as a function of the ALP energy, mass,
coupling g, surface magnetic eld strength B0 of the magnetar, magnetar radius r0, and
angle between the magnetic eld and the direction of propagation . Along the way, we
developed an analytic formalism to perform similar calculations in more general n-state
oscillation systems.
11It is important to note that the ALP mass and ALP-photon coupling would not be determined from
this point of view, they would only have to satisfy the appropriate inequalities. A spectral analysis could
zero in on the right ALP parameters.
HJEP06(218)4
Taking benchmark values of the radius, magnetic eld, and core temperature of SGR
1806-20, we then constrained the ALP-photon coupling by requiring that the photon ux
coming from ALP conversion cannot exceed the observed luminosity of the magnetar. Our
results are depicted in gure 5.
There are several future directions that would be interesting to explore. Firstly, our
approach has been to consider the photon ux from ALP-to-photon conversion for the entire
energy range between ! = 1{200 keV, and compare that to the broad band spectrum of
the quiescent emission from SGR 1806-20 in the same range. It would be interesting to
perform a bin-by-bin spectral analysis, and presumably the constraints one would obtain
HJEP06(218)4
from such an analysis would be more stringent. Secondly, a polarization analysis along
the lines of [17, 18], but in the hard X-ray band, would be very interesting. Another
aspect of our work that merits further study is the incorporation of other ALP production
mechanisms | such as electron bremsstrahlung on the surface | and their relation to
ALP-photon conversion. Finally, our analytical treatment of the ALP-photon conversion
probability can be utilized in other contexts, apart from magnetar physics, for example, in
extra-galactic ALP-photon conversion ([53] and references therein).
We make a few comments about the observational and astrophysical aspects that a ect
our analysis. The core temperature is clearly the parameter that most strongly in uences
our results, and we refer to section 2 of [3] and references therein for a discussion of the
relevant astrophysical modelling. Our results also depend on the radius [through (3.3)]
and mass [through (A.1)] of the magnetar, for which we took standard benchmark values.
We refer to the recent review [54] for observational prospects of the mass-radius relation
and equation of state. The observed luminosity in the hard X-ray regime also signi cantly
a ects the limits presented in gure 5. Results from the Hard X-ray Modulation Telescope
(HXMT) will be very useful in further understanding the mechanism outlined in our paper.
We conclude with an intriguing speculation. In recent years, satellites like INTEGRAL,
RXTE, XMM-Newton, ASCA and NuSTAR have revealed that a considerable fraction
of the bolometric luminosity of magnetars falls in the hard, rather than the soft, X-ray
band [55]. While this has been observed for around nine magnetars, it is di cult to rule
out this phenomenon for non-detected sources [1]. The process of hard X-ray emission
considered in this paper | ALP production from the core followed by conversion in the
magnetosphere | produces a spectral peak in the correct range, and could be making an
appreciable contribution to the observed luminosity.
Acknowledgments
JFF is supported by NSERC and FRQNT. KS would like to thank Matthew Baring for a
very illuminating discussion on the current work, and possible future directions. He would
also like to thank Eddie Baron for useful discussions.
Neutrino and ALP production mechanisms
This appendix discusses the simplest production mechanisms for neutrinos and ALPs in
magnetars. For neutrino emission, the dominant production mode is the modi ed URCA
mechanism while for ALP emission, the dominant production mode is nucleon-nucleon
bremsstrahlung.
A.1
Neutrino emissivity
Several neutrino production mechanisms, like cyclotron emission of neutrino pairs by
electrons or neutrino bremsstrahlung in the Coulomb eld of ions, can lead to magnetar cooling.
However, the dominant neutrino cooling mode is the modi ed URCA process [3, 49],
q_ = (7
1020 erg s 1 cm 3
)
0
2=3
RM
T
where is the magnetar density, 0 = 2:8
1014 g cm 3 is the nuclear saturation density,
and RM
1 is a suppression factor that appears with the onset of proton and/or neutron
super uidity. Indeed, if super uidity is achieved, the dominant cooling mechanism becomes
Cooper pair cooling. Although in magnetars super uidity is not expected for protons, it
is theoretically possible for neutrons when core temperatures reach T & 108 K. The exact
critical temperature for neutron super uidity is however not known and this issue is thus
not yet settled. To stay conservative, we therefore focus only on the not controversial and
well-understood modi ed URCA process (A.1).
It is important to notice from (A.1) that the neutrino emissivity dependence on the
magnetar core temperature T is quite strong. In fact, the modi ed URCA process has the
strongest temperature dependence of all the neutrino production mechanisms mentioned
above. Hence, within our assumptions, a small variation in the core temperature leads to a
large variation in the neutrino emissivity. Finally, for numerical purposes, we assume that
= 0 and RM = 1 in the computation of the ALP-to-photon luminosity.
A.2
Normalized ALP spectrum
There are again several ALP production mechanisms in neutron stars. Some examples are
analogs of neutrino production mechanisms like cyclotron emission of ALPs by electrons or
ALP bremsstrahlung in the Coulomb eld of ions. The dominant production mechanism for
ALPs is however nucleon-nucleon bremsstrahlung emission in the degenerate limit [26{36],
which has a normalized ALP spectrum given by
dNa =
d!
x2(x2 + 4 2)e x
8( 2 3 + 3 5)(1
e x) ;
(A.2)
where x = !=kBT . The normalized ALP spectrum (A.2) satis es R01 d! dNa=d! = 1
and does not depend on the ALP-nucleon coupling constant gaN . Therefore, the
normalized ALP spectrum is useful since we do not need to specify the exact model leading to
ALP emission from nucleon-nucleon bremsstrahlung. Indeed, with the cooling argument
demanding that ALP luminosity does not overtake neutrino luminosity, we can stay quite
general with respect to the exact type of ALP we are constraining. For example, the
ALP-nucleon coupling constant, which is model-dependent, is not needed.
However, to ensure that the cooling argument presented here, where ALP emission from
nucleon-nucleon bremsstrahlung does not overcome neutrino emission from the modi ed
URCA process, is plausible, it is nevertheless necessary to compare ALP emissivity with
neutrino emissivitiy (A.1). Although this is model-dependent, there are bounds on gaN and
our constraint would not be as interesting if the cooling argument was already excluded by
these bounds. Following [26{36], the ALP emissivity is
q_a = (1:3
1025 erg s 1 cm 3
)
gaN
10 10 GeV 1
2
1=3
0
T
and thus the ALP emissivity (A.3) is larger than (smaller than) [equal to] the neutrino
emissivity (A.1) if gaN > gaN (gaN < gaN ) [gaN = gaN ] where gaN is given by
gaN
10 10 GeV 1
= 7:3
T
109 K
:
(A.4)
0:6
(T =109 K)
ing our model-independent approach.
Therefore, magnetar cooling by ALP emission is subdominant to cooling by neutrino
emission if the ALP-nucleon coupling constant gaN
gaN (A.4). For
=
0, RM = 1 and
3:0, the necessary value for gaN is well below the bound of [16],
validat
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