The following is a list of materials I would recommend, if someone wishes to understand the theory underlying axions better. The focus here is on things not already mentioned in the rest of the text. Your mileage may vary, of course.

First of all, at least have a short look at the original papers by Roberto Peccei and Helen Quinn (Peccei and Quinn 1977a, 1977b), as well as the 'responses' by Wilczek and Weinberg (Wilczek 1978; Weinberg 1978).

If your interest is in understanding the reasoning behind the origin of the strong CP problem via the QCD vacuum structure and the \(\mathrm{U}(1)\) problem, it may be worthwhile to look at (some of) the original papers. See Weinberg's paper introducing it (Weinberg 1975) and 't Hooft's paper on symmetry breaking via the Adler-Bell-Jackiw anomaly (’T Hooft 1976) as the next logical step. In (Hooft 1986) 't Hooft then shows how instantons solve the \(\mathrm{U}(1)\) problem. In 1999 't Hooft also wrote a 'historical review' (’T Hooft 1999) about renormalization of gauge theories, which touches on the \(\mathrm{U}(1)\) problem as well. It may be easier to follow (as these things so often are in hindsight).

In the vein of review papers, Roberto Peccei wrote a nice review paper in 2008 (Peccei 2008b) covering the connection between the \(\mathrm{U}(1)\) problem, the QCD vacuum structure, the strong CP problem and axions. Similarly, Jihn Kim (the K in the KSVZ model) wrote another very nice review in 2010 (Kim and Carosi 2010). I would probably recommend to read these two for a good overview. They give enough references to dive deeper, if desired.

On the side of axion detection, you should of course look into the two papers by Pierre Sikivie, introducing all our modern axion detection experiment techniques (Sikivie 1983, 1985). Sikivie wrote another review in 2020 (Sikivie 2021) covering search methods for invisible axions.

Another great overview, covering both theory and experimental parts of the axion, are Igor Irastorza's notes here: (Irastorza 2022). And there is also the text book 'Axions' (Kuster, Raffelt, and Beltrán 2007), which may be worth a look as a summary of different aspects of axion physics.

If you only want to look into a single document covering all aspects of axions, from theory to experimental searches, look no further than the 'landscape of QCD axion models' (Luzio, Giannotti, et al. 2020), also mentioned in the main text.

Finally, in my master thesis (Schmidt 2016) I attempted to introduce the relevant physics for the axion (and chameleon) to a level satisfactory to me at the time. Whether I succeeded or not, you can be the judge. This may (or may not) be a good introduction if you are a student, for whom courses in QFT are fresh in their mind.

Fig. 1 is an illustration of the three discrete transformations part of the \cpt symmetry in a 1 dimensional spacetime. The three transformations are:

• \(C\) - charge conjugation: replaces each particle by its anti particle and thus reversing the charge \(q ↦ -q\)
• \(P\) - parity transformation: mirrors all positions at some origin, \(\vec{x} ↦ -\vec{x}\).
• \(T\) - time reversal: reverses the arrow of time, \(t ↦ -t\)

The \cpt symmetry of the Standard Model states that a hypothetical mirror universe to ours - obtained by the three transformations together - follows the exact same physical laws and thus 'evolves' identically (due to the time reversal 'evolution' in quotation marks).

Kim-Shifman-Vainshtein-Zakharov model

The so called KSVZ model (Kim 1979; Shifman, Vainshtein, and Zakharov 1980) is the simplest invisible axion model. It adds a scalar field \(σ\) and a superheavy quark \(Q\), which \(σ\) couples to via a Yukawa coupling. The main problem with the standard axion is that its energy scale is the electroweak scale \(v_F \approx \SI{250}{GeV}\) resulting in too strong interactions. The KSVZ model effectively achieves a symmetry breaking scale \(f_a \gg v_F\) and thus results in an 'invisible axion'. It contains a tree-level axion-photon coupling \(g_{aγ}\), but no axion-electron couplings. The latter can be found at one-loop level (Srednicki 1985).

Dine-Fischler-Srednicki-Zhitnitskii model

The DFZS axion model (Dine, Fischler, and Srednicki 1981; Zhitnitskii 1980) is another axion model, in which the scalar field \(σ\) couples to two Higgs doublet fields, \(H_u\) and \(H_d\). It does not require an extra superheavy quark, in this case the coupling of the scalar to the doublets achieves the decoupling of the axion symmetry breaking scale \(f_a\) from the electroweak scale \(v_F\). The end result is the same, an 'invisible axion' which is very light and has only light interactions with other matter. However, in contrast to the KSVZ model axion-lepton couplings appear at tree level!

Generalizations

Further generalizations of axion models over the KSVZ and DFSZ models are possible, resulting in more flexible couplings. From a practical standpoint of an axion experiment it is usually better to consider axion interactions via effective couplings, as the limiting factor is detecting something rather than determining its properties. In particular because the two models above yield very small expected coupling constants in regions of axion masses easily accessible via laboratory experiments. Therefore, an effective Lagrangian like the following

\begin{equation} \label{eq:theory:axion:general_axion_couplings} \mathcal{L}_{a,\text{eff}} = \frac{1}{2} \partial^{μ} a \partial_{μ} a - \frac{1}{2} m_a^2 a^2 - \frac{g_{aγ}}{4} \widetilde{F}^{μν} F_{μν} a - g_{ae} \frac{\partial_{μ} a}{2 m_e} \overline{ψ}_e γ^5 γ^{μ} ψ_e, \end{equation}

which contains an axion-photon coupling \(g_{aγ}\) and an axion-electron coupling \(g_{ae}\) is useful for experimental searches. Limits given on one of these parameters can always be converted to the specific couplings of one of the existing models if needed.

In principle other couplings exist, for example the axion-nucleon coupling \(g_N\). For brevity we ignore these as they are not relevant in the context of this thesis. Future helioscopes like IAXO may be able to be sensitive to at least \(g_N\) though (Luzio et al. 2022).

Starting with the Lagrangian in eq. \eqref{eq:theory:axion:general_axion_couplings} and extending it by the Lagrangian for a free photon \(A_ν\),

\[ \mathcal{L} = \mathcal{L}_{a,\text{eff}} + \mathcal{L}_γ = \mathcal{L}_{a,\text{eff}} - \frac{1}{4} F_{μν} F^{μν}, \]

where \(F_{μν} = ∂_μ A_ν - ∂_ν A_μ\) is the electromagnetic field strength tensor. And noting that the axion interaction term of eq. \eqref{eq:theory:axion:general_axion_couplings} can be rewritten as,

\[ \mathcal{L}_{aγ} = \frac{1}{4}g_{aγ} \tilde{F}^{μν} F_{μν} a = -g_{aγ} a \vec{E}·\vec{B}, \]

we can apply the Euler Lagrange equations to both the axion \(a\) and photon \(A_ν\) to derive a modified Klein-Gordon equation for the axion,

\[ \left(\Box + m_a²\right) a = \frac{1}{4}g_{aγ} F_{μν} \tilde{F}^{μν}, \]

which has a photon source term. Similarly, for the photon equation of motion we derive the homogeneous Maxwell equations with an axion source term,

\[ ∂_μ F^{μν} = g_{aγ} (∂_μ a) \tilde{F}^{μν}. \]

Without going into too much detail, let's shortly sketch how one derives the axion-photon conversion probability from here.

By choosing a suitable gauge and specifying directions of electric and magnetic fields in a suitable coordinate system, we can then derive the mixing between photon and axion states. For example if the propagation of particles is along the \(z\) axis and we fix the two degrees of freedom of \(A_ν\) by the Lorenz gauge (\(∂_μ A^μ = 0\)) and Coulomb gauge (\(\vec{\nabla}·\vec{A} = 0\)) we can derive a single equation of motion for the axion and \(A_ν\) field by starting with a plane wave approach. We obtain

\[ \left[ (ω² + ∂²_z) \mathbf{1} - \mathbf{M} \right] \vektor{A_{\perp}(z) \\ A_{\parallel}(z) \\ 0} = 0, \]

with the parallel and orthogonal polarization of the photon \(A_{\parallel}\) and \(A_{\perp}\), respectively and matrix \(\mathbf{M}\):

\[ \mathbf{M} = \mtrix{ m²_γ & 0 & 0 \\ 0 & m²_γ & -ω g_{aγ} B_T \\ 0 & -ω g_{aγ} B_T & m²_a \\ } \text{ where } m²_γ = ω²_p. \]

Here effects of quantum electrodynamic (QED) vacuum polarization and other polarization effects are ignored. \(ω_p\) refers to the plasma frequency, \(ω_p = 4πα n_e / m_e\), where \(n_e\) the electron density and \(m_e\) the electron mass. The mass \(m_γ\) refers to an effective photon mass that can appear in media, \(ω\) is the frequency and \(B_T\) the transverse magnetic field. The constant magnetic field \(B_T\) appears, because we assume for our purpose the magnetic field is constant along \(z\), the propagation direction of the photon.

By recognizing that the orthogonal component \(A_{\perp}\) is decoupled from the other two, the problem reduces to a 2 dimensional equation. Note that a side effect of this decoupling is that photons produced from an incoming axion in a magnetic field are always linearly polarized in the direction of the external magnetic field! Further, this equation can be linearized in the ultra relativistic limit \(m_γ² \ll ω²\) to

\[ \left[ \left( ω + i∂_z \right) \mathbf{1} - \frac{\mathbf{M}}{2ω} \right] \vektor{ A_{\parallel}(z) \\ a(z) } = 0. \]

As the mass matrix \(\mathbf{M}\) is non diagonal, the fields \(A_{\parallel}\) and \(a\) are interaction eigenstates and not propagation eigenstates. If we wish to compute the axion to photon conversion probability, we need the propagation eigenstates however. Transforming from one to the other is done by a regular rotation matrix \(\mathbf{R}\), which diagonalizes \(\mathbf{M}/2ω\). In the basis of the propagation eigenstates the fields \(A'_{\parallel}\) and \(a'\) are then decoupled and can be easily solved by a plane wave solution. The fields we can measure in an experiment are those of the interaction eigenstates of course. ⁵

The interaction eigenstates after a distance \(z\) can therefore be expressed by

\[ \vektor{ A_{\parallel}(z) \\ a(z) } = \mathbf{R}^{-1} \mathbf{M_{\text{diag}}} \mathbf{R} \vektor{ A_{\parallel}(0) \\ a(0) }, \]

where \(\mathbf{M_{\text{diag}}}\) is the diagonalized mass matrix,

\[ \mtrix{ e^{-i λ_+ z} & 0 \\ 0 & e^{-λ_- z} }, \]

where \(λ_{+, -}\) are its eigenvalues and coefficients in the exponential of the plane wave solutions of the propagation eigenstate fields \(A'_{\parallel}\) and \(a'\) given by

\[ λ_{+,-} = \pm \frac{1}{4ω} \sqrt{ \left(ω²_P - m²_a\right)² + \left(2 ω g_{aγ} B_T\right)²}. \]

Then finally, one can compute the conversion probability by starting from initial conditions where no electromagnetic field is present, \(A_{\parallel}(0) = 0, a(0) = 1\). Computing the resulting \(A_{\parallel}(z)\) with these conditions yields the expression which needs to be squared for the probability to measure a photon at distance \(z\) when starting purely from axions in an external, transverse magnetic field \(B_T\)

with \(q = \frac{m²_γ - m²_a}{2ω}\) and we dropped additional terms \(∝ g_{aγ} B_T\) as arguments to \(\sinc = \sin(x)/x\), because they are extremely small compared to \(q z\) for reasonable axion masses, magnetic fields and coupling constants.

If further the coherence condition \(qL < π\) is such that \(qL \ll π\), the \(\sinc\) term approaches 1 and the relevant conversion probability is finally:

where we dropped the \(T\) suffix for the transverse magnetic field and replaced \(z\) by the more apt \(L\) for the length of a magnet. This is the case for long magnets and/or low axion masses, which is generally applicable in this thesis. The point at which this condition does not strictly hold anymore is the axion mass at which helioscope experiments start to lose sensitivity.

Note that the above conversion probability is given in natural units, specifically Lorentz-Heaviside units, \(c = \hbar = ε_0 = 1\) (meaning \(α = e²/4π \approx 1/137\)). Arguments (\(B, L\)) need either be converted to natural units as well or the missing factors need to be added. The same equation in SI units is given by:

\[ P_{a↦γ, \text{vacuum}} = ε_0 \hbar c^3 \left( \frac{g_{aγ} B L}{2} \right)^2. \]

A detailed derivation for the above can be found in (Masaki, Aoki, and Soda 2017). ⁶ An initial derivation for the first axion helioscope prototype is found in (Van Bibber et al. 1989) based on (Raffelt and Stodolsky 1988). Sikivie gives expected rates in his groundbreaking papers about axion experiments, (Sikivie 1983, 1985) but is extremely short on details. Another source in the form of (Raffelt 1996) in which G. Raffelt covers a very large number of topics relevant to axion searches.

The effective Lagrangian as shown in eq. \eqref{eq:theory:axion:general_axion_couplings} allows for multiple different axion interactions, which result in the production of axions in the Sun. For KSVZ-like axion models (models with only \(g_{aγ}\)) the only interaction allowing for axion production in the Sun is the Primakoff effect ⁹ for axions. For DFSZ models with an axion-electron coupling \(g_{ae}\) multiple other production channels are viable.

One of the first papers to look at the implications of the axion in terms of astrophysical phenomena is (Mikaelian 1978) by K. Mikaelian. G. Raffelt expanded on this later in (Raffelt 1986) with calculations for the Compton and Bremsstrahlung production rates for DFSZ axion models. In (Raffelt 1988) he further calculates the production rate for the Primakoff effect (and later reviews the physics (Raffelt 1996)). J. Redondo combined all production processes in (Redondo 2013) to compute a full solar axion flux based on the axion-electron coupling \(g_{ae}\) using numerical calculations of the expected metallicity contents at different points in the Sun. This is done by making use of the opacities for different elements at different pressures and temperatures as tabulated by the 'Opacity Project' (Team 1995; Hummer and Mihalas 1988; Seaton 1987, 2005; Seaton et al. 1994; Badnell et al. 2005), based on the values provided by the numerical AGSS09 (Asplund et al. 2009) solar model. All these considered axion production channels are:

• (\(P\)) Primakoff production via \(g_{aγ}\) in both KSVZ and DFSZ axion models \[ γ + γ ↦ a \]
• (\(\text{ff}\)) electron ion bremsstrahlung (in radio astronomy low energetic cases with photons are also called free-free radiation) \[ e + Z \longrightarrow e + Z + a \]
• (\(ee\)) electron electron bremsstrahlung (Raffelt 1986), \[ e + e \longrightarrow e + e + a \]
• (\(\text{fb}\)) electron capture (also called recombination or free-bound electron transitions) \[ e + Z \longrightarrow a + Z^- \]
• (\(C\)) Compton scattering (Raffelt 1986) \[ e + \gamma \longrightarrow e + a \]
• (\(\text{bb}\)) and de-excitation (bound-bound electron transitions) via an axion \[ Z^* \longrightarrow Z + a \]

See fig. 2 for the corresponding Feynman diagrams.

With these production channels we can write down the expected axion flux on Earth, based on the production rate per volume in the Sun as the integral

where

are all contributing axion production channels. The superscripts correspond to the bullet points above. Most important without going into the details of how each \(Γ\) can be expressed, is that they scale with the coupling constant squared. \(g²_{aγ}\) for \(Γ^P_{e}\) and \(g²_{ae}\) for the others. In (Redondo 2013) these production rates are expressed by relating them to the corresponding photon production rates for these processes, which are well known.

For a detailed look, see (Redondo 2013) and the master thesis of Johanna von Oy (Von Oy 2020) in which she – among other things – reproduced the calculations done by Redondo. Her work is used as part of this thesis to compute the axion production as required for the expected axion flux in the limit calculation (and provides the data for the plots in this section). The code responsible for computing the emission rates for different axion models is (Von Oy 2023), in particular the readOpacityFile program. It also uses the Opacity Project as the basis to compute the opacities for different elements.

Fig. 3(a) shows the differential axion flux arriving on Earth based on the different contributing interactions, in this case using \(g_{ae} = \num{1e-13}, g_{aγ} = \SI{1e-12}{GeV⁻¹}\). We see that the total flux (blue line) peaks at roughly \(\SI{1}{keV}\), with the small spikes from atomic interactions visible. The Primakoff flux has been amplified by a factor of \(100\) here to make it more visible, as for this choice of coupling constants its contribution is negligible.

As a result of the expected extremely low mass of the axion, the expected solar axion spectrum for Primakoff only models is essentially a blackbody spectrum corresponding to the temperatures near the core of the Sun, \(\mathcal{O}(\SI{15e6}{K})\), see fig 3(b) and compare it to the Primakoff flux on the left ¹⁰. Partially for this reason, in many cases analytical expressions are given to describe the Primakoff axion flux, which are solutions obtained for specific solar models. One such recent result is from (Dafni et al. 2019),

where \(Φ_{P10} = \SI{6.02e10}{keV⁻¹.cm⁻².s⁻¹}\) and \(E_a\) the energy of the axion in \(\si{keV}\). (Dafni et al. 2019) also contains analytic expressions for the Compton and Bremsstrahlungs components. For the atomic processes this is not possible.

Fig. 4(a) shows how the solar axion flux (for DFSZ models) depends both on the energy and the relative radius in the Sun. We can see clearly that the major part of the axion flux comes from a region between \(\SIrange{7.5}{17.5}{\%}\) of the solar radius. The reason is the cubic scaling of the associated volumes per radius on the lower end and dropping temperatures and densities at the upper end. Interesting substructure due to the details of the axion-electron coupling is visible. The radial component alone comparing it to KSVZ models is seen in fig. 4(b), where we can see that the DFSZ flux drops off significantly at a specific radius resulting in the net flux from slightly smaller radii.

The used solar model is an important part of the input for the calculation of the emission rate and thus differential flux. (Hoof, Jaeckel, and Thormaehlen 2021) conclude – based on a similar code (Hoof and Thormaehlen 2021) – that the statistical uncertainty of the solar models is \(\mathcal{O}(\SI{1}{\%})\), while the systematic uncertainty can reach up to \(\mathcal{O}(\SI{5}{\%})\). This is an important consideration for the systematic uncertainties later on.

Chameleons are a different type of hypothetical particle. A scalar particle arising from extensions to general relativity, which acts as a "fifth force" and can be used to model dark energy. We will not go into detail about the underlying theory here. Refer to (Waterhouse 2006) for an in-depth introduction to chameleon gravity and (Brax and Davis 2015) on how they differ from other modified gravity models.

Suffice it to say, chameleonic theory also yields a coupling to photons, \(β_γ\), which can be utilized for conversion into X-rays in transverse magnetic fields. Similarly, they can therefore be produced in the Sun. However, the chameleon field has a peculiar property in that its mass is dependent on the density of the surrounding medium. This means chameleons cannot be produced in the solar core and escape. A suitable location to produce chameleons, is the solar tachocline region. A region of differential rotation at the boundary between the inner radiative zone and outer convective zone, at around \(0.7 · R_{\odot}\) in the Sun. This leads to strong magnetic fields at low enough densities. Due to the much lower temperatures present in the solar tachocline compared to the solar core, the peak of the solar chameleon spectrum is below \(\SI{1}{keV}\). See (Brax, Lindner, and Zioutas 2012) and (Anastassopoulos et al. 2015) for details about the chameleon production in the tachocline region and resulting spectrum.

Their possible detection with CAST was proposed in 2012 (Brax, Lindner, and Zioutas 2012), with actual searches being performed initially with a Silicon Drift Detector (SDD) (Anastassopoulos et al. 2015) and later with a GridPix detector using data taken in 2014/15 (Christoph Krieger 2018; Anastassopoulos et al. 2019). In addition, the KWISP (Kinetic WISP) detector (Cuendis et al. 2019; Baier, n.d.) was deployed at CAST, attempting search for the chameleon via their chameleon-matter coupling \(β_m\), which should lead to a chameleons interacting with a force sensor (without conversion to X-rays in the CAST magnet).

As the chameleon flux is highly dependent on details of solar physics that are understood to a much lesser extent than the temperature and composition in the solar core (required for axions), uncertainty on chameleon results is much larger. See also (Zanzi and Ricci 2015) for a discussion of chameleon fields in the context of solar physics.

The conversion probability for back conversion into X-rays in a magnetic field, assuming coherent conversion, is given by (Brax, Lindner, and Zioutas 2012):

Here, \(m_{\text{pl}}\) is the reduced Planck mass,

\[ m_{\text{pl}} = \frac{M_{\text{pl}}}{\sqrt{8 π}} = \sqrt{ \frac{ \hbar c }{ 8 π G } }, \]

i.e. using natural units with \(G = \frac{1}{8π}\) instead of \(G = 1\), used in cosmology because it removes the \(8π\) term from the Einstein field equations. Similar to the axion-photon conversion probability, it is in natural units and \(B\) and \(L\) should be converted or the missing constants reinserted. The equation is valid in the chameleon-matter coupling range \(1 \leq β_m \leq \num{1e6}\), which corresponds to non-resonant production in the Sun as well as evades significant chameleon interaction with materials on its path to the CAST magnet.

Note that in many cases \(M_γ = \frac{m_{\text{pl}}}{β_γ}\) is introduced. If however, one defines its inverse, \(g_{βγ} = \frac{β_γ}{m_{\text{pl}}}\), eq. \eqref{eq:theory:chameleon_conversion_prob} takes the form of eq. \eqref{eq:theory:conversion_prob} for the axion, including an effective coupling constant of units \(\si{GeV⁻¹}\).

The field of axion searches is expanding rapidly in recent years, especially in haloscope experiments. A thorough overview of all the different possible ways to constrain axion couplings and best limits in each of them is not in the scope of this thesis. We will give a succinct overview of the general ideas and reference the best current limits on the relevant coupling constants in the regions of interest for this thesis. A great, frequently updated overview of the current best axion limits is maintained by Ciaran O'Hare, available here (O’HARE 2020). Generally, axion couplings can be probed by three main avenues:

Pure, indirect astrophysical constraints

Different astrophysical phenomena can be used to study and constrain axion couplings. One example is the cooling rate of stars. If axions were produced inside of stars and generally manage to leave the star without interaction, they carry energy away. Similar to neutrinos they would therefore contribute to stellar cooling. From observed cooling rates and knowledge of solar models, constraints can be set on potential axion contributions. Many other astrophysical sources can be probed in similar ways. In all cases these constraints are indirect in nature. Which coupling can be constrained depends on the physical processes considered.

Direct astrophysical constraints

Certain types of laboratory experiments attempt to measure axions directly and produce constraints due to non-detection. Solar helioscopes attempt to directly measure axions produced in the Sun; more on these in chapter 5. Haloscope experiments utilize microwave cavities in an attempt to tune to the frequency resonant with the axion mass of cold axions part of the dark matter halo. While relying on astrophysically produced axions, the intent is direct detection. Recent interest in axions also means data from WIMP experiments like the XENON collaboration (Aprile et al. 2017) is being analyzed for axion signatures. Haloscopes and helioscopes depend on the axion-photon coupling \(g_{aγ}\), while WIMP experiments may consider the axion-electron \(g_{ae}\) or axion-nucleon \(g_{aN}\) couplings. For the astrophysical production mechanism an additional dependency on the coupling producing the axion source is added, which may result in only being able to give constraints on products of different couplings.

Direct production constraints

The final approach is full laboratory experiments, which both attempt to first produce axions and then detect them. This idea is commonly done in so called 'light shining through the wall' (LSW) experiments like the ALPS experiment at DESY (Bähre et al. 2013). Here, a laser cavity in a magnetic field is intended as an axion production facility. These produced axions would leave the cavities, propagate through some kind of wall (e.g. lead) and enter a second set of equivalent cavities, just without an active laser. Produced axions could convert back into photons in the second set of cavities. The disadvantage is that one deals with the \(g_{aγ}\) coupling both in production and reconversion. ¹¹

The bounds of interest for this thesis are the axion-photon coupling \(g_{aγ}\) for masses below around \(\SI{100}{meV}\) ¹² and the product of the axion-photon and axion-electron couplings \(g_{ae}·g_{aγ}\) in similar mass ranges. The latter for the case of dominant axion-electron production \(g_{ae}\) in the Sun and detection via \(g_{aγ}\) in the CAST magnet.

The current best limit on the axion-photon coupling from laboratory experiments is from the CAST Nature paper in 2017 (Collaboration and others 2017), providing a bound of

\[ g_{aγ, \text{Nature}} < \SI{6.6e-11}{GeV^{−1}} \text{ at } \SI{95}{\%} \text{ CL}. \]

Similarly, for the product on the axion-electron and axion-photon couplings, the best limit from a helioscope experiment is also from CAST in 2013 (Barth et al. 2013). This limit is

\[ g_{ae}·g_{aγ} \lesssim \SI{8.1e-23}{GeV^{-1}} \text{ for } m_a \leq \SI{10}{meV} \]

and acts as the main comparison to the the results presented in this thesis.

From astrophysical processes the brightness of the tip of the red-giant branch (TRGB) stars is the most stringent way to restrict the axion-electron coupling \(g_{ae}\) alone. This is because axion production would induce more cooling, which would lead to a larger core mass at helium burning ignition, resulting in brighter TRGB stars. (Capozzi and Raffelt 2020) calculate a limit of \(g_{ae} = \num{1.3e-13} \text{ at } \SI{95}{\%} \text{ CL}\). A similar limit is obtained in (Straniero et al. 2020). (Bertolami et al. 2014) compute a comparable limit from the White Dwarf luminosity function. However, for purely astrophysical coupling constraints the strong assumptions needing to be made about the underlying physical processes imply these limits are by themselves not sufficient. In fact there is reason to believe at least some of the current astrohpysical bounds are overestimated. In (Dennis and Sakstein 2023) the authors use a machine learning (ML) model to predict the brightness of TRGB stars to allow for much faster simulations of the parameter space relevant for such bounds. Using Markov Chain Monte Carlo models based on the ML output, they show values up to \(g_{ae} = \num{5e-13}\) are not actually excluded if the full uncertainty of stellar parameters is included. As their calculations only went up to such values, even larger couplings may still be allowed.

Finally, observations from X-ray telescope can also be used for limits on the product of \(g_{ae}·g_{aγ}\). This has been done in (Dessert, Long, and Safdi 2019) based on data from the Suzaku mission and followed up on by the same authors in (Dessert, Long, and Safdi 2022) using data from the Chandra mission. In the latter they compute an exceptionally strong limit of

\[ g_{ae} · g_{aγ} < \SI{1.3e-25}{GeV^{-1}} \text{ at } \SI{95}{\%} \text{ CL}, \]

valid for axion masses below \(m_a \lesssim \SI{5e-6}{eV}\).

Chameleons

The current best bound on the chameleon-photon coupling was obtained using a previous GridPix detector with data taken at CAST in 2014/15. The observed limit obtained by C. Krieger in (Christoph Krieger 2018; Anastassopoulos et al. 2019) is \[ β_{γ, \text{Krieger}} = \num{5.74e10} \text{ at } \SI{95}{\%} \text{ CL}. \]