Photon propagation in a cold axion background with and without magnetic field
A cold relic axion condensate resulting from vacuum misalignment in the early universe oscillates with a frequency , where is the axion mass. We determine the properties of photons propagating in a simplified version of such a background where the sinusoidal variation is replaced by a square wave profile. We prove that previous results that indicated that charged particles moving fast in such a background radiate, originally derived assuming that all momenta involved were much larger than , hold for long wavelengths too. We also analyze in detail how the introduction of a magnetic field changes the properties of photon propagation in such a medium. We briefly comment on possible astrophysical implications of these results.
pacs:14.80.Va, 96.50.S-, 95.35.+d.
Axions, originally introduced to solve the strong CP problem(1), are to this date a viable candidate to constitute the dark matter of the universe(2). Their contribution to the mass density results from the energy stored in the collective oscillations around the minimum of the axion potential
with a frequency that is given by the axion mass . We know that this mass must be somewhere in the range(3)
The coupling of axions to photons takes place through the
where is the dual electromagnetic tensor. The dimensionful quantity is the axion decay constant – the equivalent of as axions are assumed to be the pseudo Goldstone bosons associated to the breaking of the Peccei-Quinn symmetry (1). On we have a range of bounds: GeV coming from direct experimental searches of axions coupling directly to matter(5); GeV from (somewhat weaker) astrophysical constraints(6), largely mass independent; or for coming from the phase II of the CAST experiment(7). For some reviews of the experimental/observational search for axions see (3).
The constant is model dependent, but it is typically of order 1 in most axion models(8). The axion, being a pseudo Goldstone boson, satisfies the relation , thus constraining the basic parameters of the theory. However, the results presented below apply also to other light pseudoscalar particles, sometimes termed axion-like particles (ALP). The coupling between ALP and photons could in principle be stronger, since it is not related to their mass.
Integrating by parts, we can write the term coupling axions or ALP to photons like
The Lagrangian for a photon in the cold axion background is then
and the relevant quantity to determine the physical effect of this coupling is
Now we can proceed to quantizing the photon field in such a background. This has been previously done in (11) in the case where is assumed to be a constant, . It was found that in this case the two physical photon polarizations get their dispersion relations modified in the following way
As a consequence processes that are forbidden on Lorentz-invariance grounds such as or have a non-vanishing probability if certain kinematical constraints are fulfilled. The interested reader can see (9) for possible observable consequences. If measured, these effects would constitute prima facie evidence that not only axions or ALP exist but they do constitute the primary ingredient of the dark matter of the universe.
It was argued in (9); (10) that taking as a constant was a good approximation if the momenta of all particles involved in the process were larger than , the period of oscillations. However, if the wavelength of some of the particles are comparable or lower than the period of oscillation one must necessarily deal with a time-dependent external potential. Thus it seems to us quite important to establish the basic principles of photon propagation in a time dependent axion background. For this reason in this paper we solve the problem of photon propagation in an oscillatory, but spatially constant, axion background exactly. We shall also include an external magnetic field to see how the combined effect modifies the properties of photons moving in such an environment. We will discuss at the end of the paper some possible physical consequences.
To keep the paper technically simple we have approximated the sinusoidal time dependence of the background by a square wave with the same period. A sinusoidal wave involves Mathieu special functions complicating the calculation enormously. We base this approximation on the similarity of the present effect with the emergence of the band structure in periodic potentials(12), exchanging time and space, and momenta and energies. It is well known in solid state physics that even such a simple model fully captures the esentials of metallic conductors and semi-conductors. Therefore we firmly believe that the physics of the problem being discussed remains unaltered by our technical simplification.
Ii Solving for the eigenmodes and eigenvalues
We introduce a Fourier transform with respect to the spatial coordinates only and write the photon field as
The equation for is
We now define
which can also be written as
To solve the equations of motion we introduce the polarization vectors defined in (11)
These vectors satisfy
and do not depend on , so
As mentioned we will approximate the sine function in by a square wave function:
The relevant parameters are
There is an equation for each polarization. However, they are related. To recover one from the other we can just make the replacement . Also, because changes sign after a time in the square wave approximation one solution is a time-shifted copy of the other: . In what follows we will work in the case . It is obvious that the conclusions also apply to the other physical polarization,
Since is piecewise-defined, we will solve the equation in two regions:
– Region 1: ,
– Region 2: ,
We impose that both functions coincide at and we do the same for their derivatives
We now write and demand that have the same periodicity as
For these conditions to be fulfilled, the coefficients have to solve the linear system
The linear system can be expressed as
The problem being discussed here is formally similar to the solution of the Kronig-Penney(12) one-dimensional periodic potential, except the periodicity is now in time rather than in space.
In order to find a non-trivial solution one has to demand the condition of vanishing determinant of , which is
with and given by (22) and (24) respectively. In order to get analytical expressions we will work in the limit of long wavelengths , which is just the one that is potentially problematic as discussed in the introduction. Expanding both sides:
If the determinant vanishes the system to solve is
In the limit , ,
Finally, imposing the usual normalization,
This completes the determination of the eigenvectors.
ii.1 Exact determination of the eigenvalues
We can also solve (32) exactly, without having to assume the long-wavelength limit as above, but this can be done only numerically. The solution only depends on and through the dimensionless combination . There are values of for which there is no solution, as seen in figure 1. However, these gaps get narrower when the product decreases. In practice, the largest possible physical value for this quantity is and then the gaps are practically nonexistent and certainly totally irrelevant for the purposes of the present paper.
It is interesting to investigate whether complex solutions exist for in the forbidden narrow bands. We note that the R.H.S. of the equation (32) is necessarily real, thus must necessarily be purely real or purely imaginary. In the latter case the L.H.S. is replaced by a having as argument the imaginary part of . For this to have a solution, the R.H.S. must be positive and larger than one. Inspection of this term reveals that it is larger that one in the forbidden zones but actually alternates sign. Therefore not even an imaginary solution exists for the first, third,… forbidden regions.
ii.2 Calculation of the transition
In order to make the photon field hermitian, we add (9) and its conjugate. Introducing creation and annihilation operators for each one of the proper modes we get (both polarizations are included)
where . Now we want to compute for an initial state of one electron of momentum and a final state of an electron of momentum and a photon of momentum .
If we take constant, and we have
In the square wave approximation (19), the time integration yields
Equation (51) holds for any value of . The symbol indicates the use of (41). It turns out that at the leading order in the expansion this expression agrees exactly with the one obtained in (10) assuming that was constant except for the fact that for each value of the polarization only one of the two delta functions that are not suppressed by terms of the form can be simultaneously satisfied; namely the one that implies that or equals , contributing with a factor with respect to what is found for constant to the amplitude. Thus in the transition reduced matrix element one gets for each polarization exactly one-half of what is obtained if is constant. But in the present case both polarizations contribute so finally we get of the result obtained with constant .
As a consequence the predictions concerning the radiation yield of a high energy charged particle propagating in the cold axion background(9) are confirmed.
Iii Propagation in a magnetic field
We will now compute the propagator of the photon field with two backgrounds: a cold axion background and a constant magnetic field. To do so, we take (3) and write the axion and photon fields as a background term plus a dynamical field. We get two relevant terms
where is the axion field, is the photon field and corresponds to a magnetic field: , . The first term is just (4). Here we will take to be constant; therefore the results that follow are valid only if the distance travelled by the photon, , verifies .
The vertices and Feynman rules corresponding to these terms are shown in figure 2.
With the first vertex we can compute the propagator in an axion background, see figure 3. The successive interactions with the axion background can be summed up and the result is the propagator
The physical polarizations, projected out by , exhibit poles at as expected. The projectors are defined in (14) and . Of course the same result can be obtained by direct inversion of the photon equation of motion (10).
We now compute the propagator in the presence of a magnetic field, using the second term in (54). In order to do that we use the propagator just found, represented by a double-wavy line and include the interactions with the external magnetic field. The dashed line corresponds to the axion propagator.
Summing all the diagrams we get
In order to simplify the result we shall assume that , which may correspond to an experimentally relevant situation. Then we get
and finally, defining ,
iii.1 Particular case: no axion background
As a relevant particular case we now set in the previous expression, i.e. we consider only the influence of the magnetic background, and get
where now stands for the usual photon propagator, obtained after setting in (55).
This propagator has poles when and also for
If we assume that is a small parameter and expand in powers of it, these poles in the frequency plane lie at
Physically this pole structure corresponds to the perpendicular polarization vector
propagating unchanged, while the parallel polarization and the would-be
longitudinal polarization change their propagation
For completeness we give the full propagator without the assumption
where is the usual photon propagator and
Let us now restore the condition that is helpful in simplifying the formulae. In order to write the propagator in a more compact form we introduce a four-vector
Note the rather involved structure of the dispersion relation implied by (64). We consider the propagation of plane waves of well defined frequency and moving in the direction. The Fourier transform with respect to the spatial component will describe the space evolution of a photon state emitted at with polarization given by the vector . We decompose
where and are the roots of the denominator
in agreement with (66), and
Even for the largest magnetic fields conceivable the product is rather small compared to the range of acceptable values of the axion mass and it appears justified to neglect . The space Fourier transform of the propagator is then
Let us now contract the propagator with the initial and final polarization vectors
where . Its squared modulus is
This quantity, once properly normalized, describes the quantum mechanical probability of measuring the polarization represented by the vector at a distance from the origin, where it was created with a polarization represented by . Since we restrict ourselves to the case and assume that the polarization vectors are orthogonal to the direction of propagation, we can write
The extrema of (III.1), for a given initial angle , are at
corresponding to the values of the angle where the probability of finding the polarization vector is maximum or minimum. The mean value of the angle is