Photon propagation in a cold axion background with and without magnetic field

Photon propagation in a cold axion background with and without magnetic field

Abstract

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.
1

I Introduction

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

 a(t)=a0cos(mt), (1)

with a frequency that is given by the axion mass . We know that this mass must be somewhere in the range(3)

 1 eV>m>10−6 eV. (2)

The coupling of axions to photons takes place through the universal term2

 Laγγ=gaγγα2πafaFμν~Fμν, (3)

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

 Laγγ=12ημAν~Fμν, (4)

with

 ημ=η(t)δμ0 ,η(t)=η0sinmt. (5)

The Lagrangian for a photon in the cold axion background is then

 L=−14FμνFμν+12ημAν~Fμν, (6)

and the relevant quantity to determine the physical effect of this coupling is

 η0=2gaγγαπa0mfa. (7)

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

 ω±=√→k2±η0|→k|. (8)

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

 Aμ(t,→x)=∫d3k(2π)3ei→k⋅→x^Aμ(t,→k). (9)

The equation for is

 [gμν(∂2t+→k2)−iϵμναβηαkβ]^Aν(t,→k)=0. (10)

We now define

 Sν λ=ϵμναβηαkβϵμλρσηρkσ, (11)

which can also be written as

 Sμν= [(η⋅k)2−η2k2]gμν+k2ημην (13) +η2kμkν−(η⋅k)(ημkν+ηνkμ),

and

 Pμν±=SμνS∓i√2Sϵμναβηαkβ ,S=Sμ μ=2η2→k2. (14)

The properties of these quantities are discussed in (11). Note that the time dependence (due to ) in cancels. With the help of these projectors we can write (10) as

 [gμν(∂2t+→k2)+√S2(Pμν+−Pμν−)]^Aν(t,→k)=0. (15)

To solve the equations of motion we introduce the polarization vectors defined in (11) and write3

 ^Aν(t,→k)=∑λ=+,−fλ(t)εν(→k,λ). (16)

These vectors satisfy

 Pμν±εν(→k,±)=εμ(→k,±),Pμν±εν(→k,∓)=0 (17)

and do not depend on , so

 [∂2t+→k2±η(t)|→k|]f±(t)=0. (18)

As mentioned we will approximate the sine function in by a square wave function:

 η(t)={+η02nT

The relevant parameters are

 η0=2gaγγαπa0mfa,T=πm. (20)

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: ,

 d2f1(t)dt2+(→k2+η0|→k|)f1(t)=0, (21)
 f1(t)=A′eiαt+Ae−iαt ,α2=→k2+η0|→k|. (22)

– Region 2: ,

 d2f2(t)dt2+(→k2−η0|→k|)f2(t)=0, (23)
 f2(t)=B′eiβt+Be−iβt ,β2=→k2−η0|→k|. (24)

We impose that both functions coincide at and we do the same for their derivatives

 f1(0)=f2(0),f′1(0)=f′2(0). (25)

We now write and demand that have the same periodicity as

 g1(t)=eiωtf1(t)=A′ei(ω+α)t+Aei(ω−α)t, (26) g2(t)=eiωtf2(t)=B′ei(ω+β)t+Bei(ω−β)t, (27) g1(T)=g2(−T),g′1(T)=g′2(−T). (28)

For these conditions to be fulfilled, the coefficients have to solve the linear system

 A′+A =B′+B αA′−αA =βB′−βB ei(ω+α)TA′+ei(ω−α)TA =e−i(ω+β)TB′+e−i(ω−β)TB
 (ω+α) ei(ω+α)TA′+(ω−α)ei(ω−α)TA =(ω+β)e−i(ω+β)TB′+(ω−β)e−i(ω−β)T. (29)

The linear system can be expressed as

 ^M⎛⎜ ⎜ ⎜⎝A′AB′B⎞⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜⎝0000⎞⎟ ⎟ ⎟⎠, (30)

with

 ^MT=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1αei(ω+α)T(ω+α)ei(ω+α)T1−αei(ω−α)T(ω−α)ei(ω−α)T−1−β−e−i(ω+β)T−(ω+β)e−i(ω+β)T−1β−e−i(ω−β)T−(ω−β)e−i(ω−β)T⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (31)

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

 cos(2ωT)=cos(αT)cos(βT)−α2+β22αβsin(αT)sin(βT), (32)

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:

 ω2−13ω4T2+...=→k2−(13→k4−112η20→k2)T2+..., (33)

which means

 ω2≈(1+η20T212)→k2. (34)

If the determinant vanishes the system to solve is

 ⎛⎜ ⎜⎝11−101−12(1−βα)001⎞⎟ ⎟⎠⎛⎜⎝A′AB′⎞⎟⎠=⎛⎜ ⎜⎝112(1+βα)h(α,β,T)⎞⎟ ⎟⎠B, (35)

where

 h(α,β,T)=−α−βα+βeiαT−e−i2ωTeiβTeiαT−e−i2ωTe−iβT, (36)

 A′B = [1−α−βα+βeiαTei2ωT−eiβTeiαTei2ωT−e−iβT (38) −12(1+βα)+12(1−βα)α−βα+βeiαTei2ωT−eiβTeiαTei2ωT−e−iβT] AB = [12(1+βα)−12(1−βα)α−βα+βeiαTei2ωT−eiβTeiαTei2ωT−e−iβT] (39) B′B = [−α−βα+βeiαTei2ωT−eiβTeiαTei2ωT−e−iβT]. (40)

In the limit , ,

 A′B≈−B′B≈14η0|→k|,AB≈1−η02|→k|. (41)

Finally, imposing the usual normalization,

 ∫fk(t)f∗k′(t)=2πδ(|→k|−|→k′|), (42)

we get

 B = ⎡⎢ ⎢⎣√→k2+η0|→k|2|→k|+η0(∣∣∣AB∣∣∣2+∣∣∣A′B∣∣∣2) (43) +√→k2−η0|→k|2|→k|−η0(1+∣∣∣B′B∣∣∣2)⎤⎥ ⎥⎦−1/2≈(1+η04|→k|).

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 e→eγ

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)

 Aμ(t,→x) = ∫d3k(2π)3∑λ[a(→k,λ)g(t,→k,λ)εμ(→k,λ)e−ikx (46) +a†(→k,λ)g∗(t,→k,λ)ε∗μ(→k,λ)eikx],

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 .

 ⟨f|S|i⟩ = ieε∗μ(→k,λ)¯uqγμup(2π)3δ(3)(→k+→q−→p) (48) ×∫dtg∗(t,→k,λ)ei(ω+Eq−Ep)t

If we take constant, and we have

 ⟨f|S|i⟩=ieε∗μ(→k,λ)¯uqγμup(2π)4δ(4)(k+q−p). (49)

In the square wave approximation (19), the time integration yields

 ⟨f|S|i⟩ = ie¯uqγμupε∗μ(→k,λ)(2π)3δ(→k+→q−→p)π{Aδ(α+Eq−Ep) (51) +Bδ(β+Eq−Ep)+A′δ(−α+Eq−Ep)−B′δ(−β+Eq−Ep)} ≈ ie¯uqγμupε∗μ(→k,λ)(1+η04|→k|)(2π)3δ(→k+→q−→p)π{(1−η02|→k|)δ(α+Eq−Ep) (53) Missing or unrecognized delimiter for \left

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

 Laγγ→12ϵμναβημAν∂αAβ+2gaγγαπfaa∂μAν~Fμν, (54)

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

 Dμν=−i(gμν−Xμνk2+Pμν+k2−η0|→k|+Pμν−k2+η0|→k|). (55)

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

 Dμν=Dμν+fμhν−ig2k2−m2+ig2K, (56)

where

 fμ=Dμα~Fαλkλ, hν=~FσϕkϕDσν, (57) g=2gaγγαπfa, K=~FβρkρDβγ~Fγξkξ. (58)

In order to simplify the result we shall assume that , which may correspond to an experimentally relevant situation. Then we get

 fμ=ik0giμk2Bi−iη0(→B×→k)i(k2−η0|→k|)(k2+η0|→k|) (59)
 hν=ik0gjνk2Bj+iη0(→B×→k)j(k2−η0|→k|)(k2+η0|→k|) (60)
 K=ik20→B2k2(k2−η0|→k|)(k2+η0|→k|), (61)

and finally, defining ,

 Dμν = Dμν+ik20gjμglν⎧⎪⎨⎪⎩bjbl(k4−η20→k2)(k2−m2)−k20k2→b2 (62) +iη0k2[bj(→b×→k)l−bl(→b×→k)j]−η20→b2→k2Xjl(k4−η20→k2)[(k4−η20→k2)(k2−m2)−k20k2→b2]⎫⎪ ⎪⎬⎪ ⎪⎭.

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

 Dμν=Dμν+ik20gjμglνbjblk4(k2−m2)−k20k2→b2, (64)

where now stands for the usual photon propagator, obtained after setting in (55).

This propagator has poles when and also for

 k20=12(2→k2+m2+→b2±√m4+→b4+2m2→b2+4→b2→k2). (65)

If we assume that is a small parameter and expand in powers of it, these poles in the frequency plane lie at

 k20 ≃ →k2⎛⎝1+→b2m2⎞⎠+m2+→b2, (66) k20 ≃ →k2⎛⎝1−→b2m2⎞⎠. (67)

Physically this pole structure corresponds to the perpendicular polarization vector propagating unchanged, while the parallel polarization and the would-be longitudinal polarization change their propagation4.

For completeness we give the full propagator without the assumption

 Dμν=Dμν+fμhνik2−m2+K (68)

where is the usual photon propagator and

 fμ=hμ=1k2(gμ0→b⋅→k+gμjbjk0) (69)
 K=1k2[(→b⋅→k)2−→b2k20] (70)

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

 Dμν(k)=−igμνk2+ik20bμbνk2[k2(k2−m2)−k20→b2]. (71)

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

 1k2[k2(k2−m2)−k20→b2]=A→k2−k20+B→k2−F+C→k2−G, (72)

where and are the roots of the denominator

 G = k20−m22−12√m4+4k20→b2≈⎛⎝1−→b2m2⎞⎠k20−m2, F = k20−m22+12√m4+4k20→b2≈⎛⎝1+→b2m2⎞⎠k20, (73)

in agreement with (66), and

 A = 1k20−F1k20−G≈−1k20→b2 (74) B = −1k20−F1F−G≈1k20→b2 (75) C = 1k20−G1F−G≈1m4+3k20→b2≈1m4. (76)

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

 Dμν(k0,x) = −gμν2k0eik0|x|−k0bμbν2Aeik0|x| (77) −k20bμbν2B√Fei√F|x|.

Let us now contract the propagator with the initial and final polarization vectors

 ϵμDμν(k0,x)ϵν0 ≈ →ϵ⋅→ϵ02k0eik0|x|+(→ϵ⋅^b)(→ϵ0⋅^b)2k0eik0|x| (79) −(→ϵ⋅^b)(→ϵ0⋅^b)2k0eik0|x|ei→b22m2k0|x|,

where . Its squared modulus is

 |ϵμDμν(k0,x)ϵν0|2= 14k20⎡⎣E21+4(E1E2+E22)sin2⎛⎝→b24m2k0|x|⎞⎠⎤⎦, (80)

where

 E1=→ϵ⋅→ϵ0,E2=(→ϵ⋅^b)(→ϵ0⋅^b). (81)

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

 ^k=^x,^b=^y,→ϵ=cosα^y+sinα^z,→ϵ0=cosβ^y+sinβ^z, (82)

so that

 E1=cos(α−β),E2=cosαcosβ. (83)

The extrema of (III.1), for a given initial angle , are at

 tan2α(x)=[1+2f(x)]sin2β4f(x)+[1+4f(x)]cos2β, (84)

where

 f(x)=sin2⎛⎝→b24m2k0|x|⎞⎠, (85)

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