Photon Production From The Scattering of Axions Out of a Solenoidal Magnetic Field

Abstract

We calculate the total cross section for the production of photons from the scattering of axions by a strong inhomogeneous magnetic field in the form of a 2D delta-function, a cylindrical step function and a 2D Gaussian distribution, which can be approximately produced by a solenoidal current. The theoretical result is used to estimate the axion-photon conversion probability which could be expected in a reasonable experimental situation. The calculated conversion probabilities for QCD inspired axions are bigger by a factor of 2.67 (for the cylindrical step function case) than those derived by applying the celebrated 1D calculation of the (inverse) coherent Primakoff effect. We also consider scattering at a resonance $E_{axion} \sim m_{axion}$, which corresponds to the scattering from a delta-function and gives the most enhanced results. Finally, we analyze the results of this work in the astrophysical extension to suggest a way in which they may be directed to a solution to some basic solar physics problems and, in particular, the coronal heating problem. Comment: 19 pages, 1 figure, added analysis of our results in the astrophysical extension

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arXiv:0906.2537v3 [hep-ph] 17 Mar 2010
Photon Production From The Scattering of Axions Out of a
Solenoidal Magnetic Field
Eduardo I. Guendelman∗and Idan Shilon†
Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Giovanni Cantatore‡
Universit´ a and INFN Trieste, via valerio 2, 34127 Trieste, Italy
Konstantin Zioutas§
University of Patras, Patras, Greece
Abstract
We calculate the total cross section for the production of photons from the scattering of axions
by a strong inhomogeneous magnetic field in the form of a 2D δ-function, a cylindrical step function
and a 2D Gaussian distribution, which can be approximately produced by a solenoidal current.
The theoretical result is used to estimate the axion-photon conversion probability which could be
expected in a reasonable experimental situation. The calculated conversion probabilities for QCD
inspired axions are bigger by a factor of 2.67 (for the cylindrical step function case) than those
derived by applying the celebrated 1D calculation of the (inverse) coherent Primakoff effect. We
also consider scattering at a resonance Eaxion∼ maxion, which corresponds to the scattering from
a δ-function and gives the most enhanced results. Finally, we analyze the results of this work in
the astrophysical extension to suggest a way in which they may be directed to a solution to some
basic solar physics problems and, in particular, the coronal heating problem.
∗Electronic address: guendel@bgu.ac.il
†Electronic address: silon@bgu.ac.il
‡Electronic address: cantatore@trieste.infn.it
§Electronic address: Konstantin.Zioutas@cern.ch
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I.INTRODUCTION
The possible existence of a light pseudoscalar particle is a very interesting possibility. For
example, the axion [1] - [3], which was introduced in order to solve the strong CP problem in
QCD, has since then also been postulated as a candidate for the dark matter in the universe.
A great number of ideas and experiments for the direct detection of this particle have been
proposed in the past [4], [5]. For example, It was recognized by Sikivie that axion detection
exploiting axion to photon conversion in a magnetic eld was a possibility [6].
Related to that, in a series of recent publications by one of us [7], it was shown that an
axion-photon system displays a continuous axion-photon duality symmetry when an external
magnetic field is present and when the axion mass is neglected. This allows one to analyze
the behavior of axions and photons in external magnetic fields in terms of an axion-photon
complex field. For example, the deflection of light from magnetars has been recently studied
using these techniques [8]. It is important to note here that the same duality symmetry exists
also when considering massive photons, under the condition mγ= ma, that is the photon and
the axion masses are equal. These conditions can be achieved when conducting experiments
where the axion-photon conversion region is filled with a suitable refractive gas. In this
letter we show that the coupling of axion-photon complex particles to a localized magnetic
flux generated by a solenoid renders scattering solutions with a cross section which could
conceivably be measured.
To see this, let us write the Lagrangian describing the relevant light pseudoscalar coupling
to the photon,
L = −1
4FµνFµν+1
2∂µφ∂µφ −1
2m2
aφ2− −g
8φǫµναβFµνFαβ.
(1)
Following Ref. [9] (and references therein), we focus on the case where an electromagnetic
field with propagation along the x and y directions and a strong magnetic field pointing in
the z-direction are present. The magnetic field may have an arbitrary space dependence in
x and y, but it is assumed to be time independent.
For small electromagnetic perturbations around the static magnetic background (i.e, the
axion and the electromagnetic wave), we consider only small quadratic terms in the La-
grangian for the axion and the electromagnetic fields. By choosing a static magnetic field
pointing in the z direction and having an arbitrary x and y dependence and specializing
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to x and y dependent electromagnetic field perturbations and axion fields, the interaction
between the background magnetic field and the axion and photon fields reduces to
LI= −βφEz, (2)
where β(x,y) = gB(x,y). Choosing the temporal gauge for the electromagnetic field and
considering only the z-polarization for the electromagnetic waves (since only this polarization
couples to the axion) we get the following 2+1 dimensional effective Lagrangian
L2=1
2∂µA∂µA +1
2∂µφ∂µφ −1
2m2
aφ2+ βφ∂tA ,(3)
where A is the z-polarization of the photon, so that Ez= −∂tA.
Without assuming any particular x and y dependence for β, but insisting that it will be
static, we see that neglecting the axion mass ma(the validity of this assumption will be
discussed at the end of this work), we discover a continuous axion photon duality symmetry.
This is due to a rotational O(2) symmetry in the axion-photon field space, allowed by the
axion and photon kinetic terms and by expressing the interaction term, LI, in an O(2)
symmetric way by dropping a total time derivative from it:
LI=1
2β(φ∂tA − A∂tφ) .(4)
Defining now the axion-photon complex field, Ψ, as
Ψ =
1
√2(φ + iA)(5)
and plugging this into the Lagrangian results in
L = ∂µΨ∗∂µΨ −i
2β(Ψ∗∂tΨ − Ψ∂tΨ∗) , (6)
where Ψ∗is the charge conjugation of Ψ. From this we obtain the equation of motion for Ψ
∂µ∂µΨ + iβ∂tΨ = 0 .(7)
We therefore have the magnetic field, or β/2 (the U(1) charge), coupled to a charge density.
Introducing the charge conjugation [10] , that is
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Ψ → Ψ∗, (8)
shows that the free part of the action is indeed invariant under (8). When acting on the free
vacuum the A and φ fields give rise to a photon and an axion respectively, but in terms of the
particles and antiparticles (defined in terms of Ψ), we see that a photon is an antisymmetric
combination of particle and antiparticle and an axion a symmetric combination, since
φ =
1
√2(Ψ∗+ Ψ) and A =
1
i√2(Ψ − Ψ∗) .(9)
Hence, the axion is even under charge conjugation, while the photon is odd. These two
eigenstates of charge conjugation will propagate without mixing as long as no external
magnetic field in the perpendicular direction to the eigenstates (i.e axion and photon) spatial
dependence is applied. The interaction with the external magnetic field is not invariant under
(8). In fact, under (8) we can see that
SI→ −SI, (10)
where SI =
?LIdxdydt. Therefore, these symmetric and antisymmetric combinations,
corresponding to axion and photon, will not be preserved in the presence of B in the analog
QED language, since the ”analog external electric potential” breaks the symmetry between
particle and antiparticle and therefore will not keep in time the symmetric or antisymmetric
combinations. In fact, if the corresponding external electric potential is taken to be a
repulsive potential for particles, it will be an attractive potential for antiparticles, so the
symmetry breaking is maximal.
Even at the classical level these two components suffer opposite forces, thus under the
influence of an inhomogeneous magnetic field both a photon or an axion will be decomposed
through scattering into their particle and antiparticle components, each of which is scattered
in a different direction, since the corresponding electric force is related to the gradient of
the effective electric potential, i.e., the gradient of the magnetic field, times the U(1) charge
which is opposite for particles and antiparticles. If we look at the scattering amplitudes for
particles and antiparticles, we see that they have opposite signs. Calling S the scattering
amplitude for a particle, the amplitude for an antiparticle is then −S. Therefore, an axion
[i.e.the symmetric combination of particle antiparticle (1,1)] goes under scattering to
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(1,1) + (S,−S). So the amplitude for axion going into photon (1,−1) is S. Hence, we
conclude that the amplitude for axion-photon conversion is equal to the particle scattering
amplitude.
For this effect to have meaning, we have to work at least in a 2+1 formalism [11]. The 1+1
reduction [7], [10] which allows motion only in a single spatial direction, is unable to produce
such separation, since in order to separate particle and antiparticle components we need at
least two dimensions to obtain a final state with particles and antiparticles propagating in
slightly different directions.
This is in a way similar to the Stern-Gerlach experiment in atomic physics [12], where
different spin orientations suffer a different deflection force proportional to the gradient of
the magnetic field in the direction of the spin. Here, instead of spin we have that the photon
is a combination of two states with different U(1) charge and each of these components
will suffer opposite force under the influence of the external inhomogeneous magnetic field.
Notice also that since particle and antiparticles are distinguishable, there are no interference
effect between the two processes.
Therefore an original beam of photons will be decomposed through scattering into two
different elementary particle and antiparticle components (and also, of course, the photons
that were not scattered). These two beams are observable, since they both have photon
components, so the observable consequence of the axion-photon coupling will be the splitting
of a photon, or axion, beam by a magnetic field of the configuration considered here, whereas
in the normal Primakoff effect analysis there is no explicit recognition of a splitting. This
effect is, moreover, of first order in the axion-photon coupling (g), unlike the “light shining
through a wall phenomena” which depend on the coupling constant squared (g2).
II.FIRST APPROXIMATION: MAGNETIC FIELD OF AN INFINITELY THIN
SOLENOID
To apply the results of the previous section to some specific system with magnetic field,
we write separately the time and space dependence of the axion-photon field as Ψ(? r,t) =
e−iωtψ(? r).
As a first model, we are considering an inhomogeneous magnetic field of the form B =
Φδ2(x,y). This kind of a potential can not, of course, be realized in the lab, however, we
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