Axion–Photon Mixing in Magnetic Fields

The axion is a hypothetical particle originally proposed to solve the strong CP problem in quantum chromodynamics. It has since become a prominent dark matter candidate and a target for numerous experimental searches. One of the most striking properties of the axion is its predicted coupling to two photons through the term $g{\mathrm{a\gamma }}_{}aF\mathrm{\backslash tilde\{F\}}$, where $a$ is the axion field and $F$ and $\mathrm{\backslash tilde\{F\}}$ are the electromagnetic field strength and its dual. In the presence of a static magnetic field, this interaction allows axions and photons to interconvert via the Primakoff effect. The probability of conversion depends on the axion–photon coupling, the strength and extent of the magnetic field, the axion mass, and the photon energy. Experiments such as light-shining-through-walls setups, helioscopes searching for solar axions, and haloscopes probing dark matter axions all rely on quantifying this conversion probability to interpret their results.

In a homogeneous magnetic field of magnitude $B$ over a length $L$, the two-state system of axions and photons evolves analogously to neutrino oscillations. The conversion probability after traversing the region is given, in natural units, by

$P={\frac{g{\mathrm{a\gamma }}_{}BL}{2}}^2\times {\frac{\sin \frac{qL}{2}}{\frac{qL}{2}}}^2$

where $q$ is the momentum transfer between axion and photon. For relativistic axions and photons, $q$ is approximately $\frac{m^{a_{}}}{}2E$. If $\mathrm{qL}$ is much less than one, the sine term approaches its argument and the conversion probability scales as the square of the product $g{\mathrm{a\gamma }}_{}BL$. In this "coherent" regime, longer magnets dramatically enhance the probability. When $\mathrm{qL}$ becomes large, the sine term oscillates rapidly and the probability is suppressed by the factor $\frac{1}{q^2}$, reflecting the loss of coherence due to the mismatch in axion and photon momenta.

The coupling constant $g{\mathrm{a\gamma }}_{}$ is model-dependent. In the original Peccei–Quinn axion models, its value is inversely proportional to the axion decay constant and typically lies in the range 10⁻¹³ to 10⁻¹⁰ GeV⁻¹. Experimental searches often quote limits in terms of this coupling. The magnetic field and region length are determined by the apparatus—for instance, the CAST helioscope uses a 9 Tesla, 9.26-meter LHC dipole magnet. The axion mass controls the coherence condition; lighter axions maintain coherence over longer distances for a given energy. Photon energy is relevant because it appears in the denominator of $q$, so higher energies reduce the momentum mismatch, improving conversion.

To make the probability formula more explicit, one can write

$P={\frac{g}{{\mathrm{a\gamma }}_{}}2}^2\times {\frac{\sin }{\frac{m^{a_{}}L}{4E}}}^2$

This expression is implemented in our calculator with all quantities converted to consistent natural units. The coupling $g{\mathrm{a\gamma }}_{}$ is supplied in GeV⁻¹. The magnetic field in Tesla is converted to GeV² using 1 T ≈ 1.95×10⁻² GeV². The region length in meters is converted to GeV⁻¹ with 1 m ≈ 5.07×10¹⁵ GeV⁻¹. Axion mass and photon energy in eV are converted to GeV. After these conversions, the probability can be evaluated directly using the natural unit formula, producing a dimensionless result.

Axion–photon conversion is central to several experimental strategies. Light-shining-through-walls experiments send a laser beam through a magnetic field to generate axions, which pass through an opaque barrier and reconvert to photons in a second magnetic region. The conversion probability squared determines the event rate, so maximizing $g{\mathrm{a\gamma }}_{}BL$ is key. Helioscopes like CAST and the upcoming IAXO point powerful magnets at the Sun to detect solar axions converting into X-rays. Haloscopes such as ADMX place resonant cavities in strong magnetic fields to induce axion dark matter to convert into microwave photons, exploiting the coherent enhancement in resonant structures. Each of these approaches relies on precise estimates of conversion probabilities to interpret null results or potential signals.

As an illustration, consider a hypothetical helioscope with B = 10 T, L = 20 m, g_aγ = 10⁻¹¹ GeV⁻¹, m_a = 10⁻⁴ eV, and E = 1000 eV. Plugging these values into the calculator yields a probability on the order of 10⁻¹⁹, highlighting the experimental challenges involved. Nevertheless, with large magnetic volumes and long observation times, such small probabilities can translate into detectable event rates if the axion flux is sufficient.

The table below compares the conversion probability for three different parameter sets, illustrating how the probability scales with the product BL and the axion mass.

B (T)	L (m)	gaγ (GeV−1)	ma (eV)	E (eV)	P
5	5	1×10−10	1×10−5	1	2.4×10−19
10	20	1×10−11	1×10−4	1000	1.0×10−19
3	1	5×10−11	1×10−6	0.01	7.1×10−15

These examples show that increasing B and L generally boosts the probability, but coherence effects controlled by m_a and E can suppress it. For extremely light axions or high photon energies, the coherent approximation holds, and the probability scales simply with (g_aγBL/2)². For heavier axions or lower energies, oscillatory suppression becomes significant, necessitating careful optimization of experimental parameters.

When using the calculator, ensure that all inputs are realistic for the scenario under consideration. The coupling should be entered in GeV⁻¹; common limits are around 10⁻¹⁰ GeV⁻¹. The magnetic field should reflect the strength achievable in the laboratory or astrophysical environment. Length corresponds to the extent of the field region. Axion mass and photon energy must be specified in electron volts. The script computes the dimensionless quantity qL to assess coherence, labeling the result "Coherent" if qL < 1 and "Incoherent" otherwise. This classification aids in quickly interpreting whether extending the magnet length would significantly increase sensitivity or if the mass-induced decoherence dominates.

Axion searches continue to push the boundaries of sensitivity, employing innovative techniques such as dielectric haloscopes, interferometric detection, and resonant enhancement. Each method ultimately relies on the same basic physics encoded in the conversion probability. By providing a fast and transparent way to compute this probability, the calculator supports experimental design, phenomenological studies, and educational explorations. It underscores the interplay between theoretical parameters and practical considerations, illustrating why detecting axions remains a formidable yet tantalizing challenge in modern physics.