decay $\tilde{t} \to b\ell qq$, setting lower stop mass limits of 890 GeV (e) or 1000 GeV ($\mu$) [222]. The fully hadronic R-parity violating top squark decays $\tilde{t} \to b\bar{s}$, $\tilde{t} \to d\bar{s}$, and $\tilde{t} \to b\bar{d}$, involving $\lambda''$, have been searched for by ATLAS [202, 208, 223–225], and CMS [226, 227]. Other recent searches target top squarks decaying through cascades into several b- and light-quarks, for example as in [228]. The most recent results set lower top squark mass limits up to 1.35 TeV in top-squark mass [214] if decays include top- and b-quarks [214] lower top squark mass limits are up to 1.35 TeV in top-squark mass. Constraints on masses up to 670 GeV are found if the top squarks decays include top- and light-quarks [229].

Various searches for multi-lepton and lepton plus jets events are interpreted in a model with RPV decays of charginos and neutralinos (including the LSP) involving a non-zero $\lambda$ [166, 201, 205] or $\lambda'$ [202, 210, 230, 231] coupling. Neutralino decays involving non-zero $\lambda''$ lead to fully hadronic final states, and searches for multi-jet events and jet-pair resonances are used to set limits, typically on the production of colored particles like top squarks or gluinos, which are assumed to be the primary produced sparticles in these interpretations, as discussed earlier. For instance, the ATLAS search [214] in events with one lepton and high jet multiplicity excludes up to 320 (365) GeV in higgsino (wino) masses. If top squarks arise in the decay of the neutralino, lepton-enriched final state events are also relevant, see for example preliminary results in [144]. The multilepton search in [232] set limits on the production of charginos and neutralinos for a minimal SUSY model with an approximate $B-L$ symmetry. In this case, charginos and neutralinos with masses between 100 GeV and 1.1 TeV are excluded depending on the assumed decay BR into a lepton plus a W, Z or H boson. Finally, recent results from ATLAS [144] set the first experimental constraints on bilinear RPV models (bRPV) with degenerate higgsino masses exploiting same-sign/three-lepton signatures, excluding masses smaller than 440 GeV.

Limits also exist on sleptons in RPV models, both from LEP and the Tevatron experiments. From LEP, lower limits on $\tilde{\mu}_R$ and $\tilde{e}_R$ masses in such models are 97 GeV, and the limits on the stau mass are very close: 96 GeV [233]. CMS has searched for resonant smuon production in a modified CMSSM scenario [234], putting limits on $\lambda'_{211}$ as a function of $m_0, m_{1/2}$. Production of pairs of sneutrinos in R-parity violating models has been searched for at LEP [233]. Assuming fully leptonic decays via $\lambda$-type couplings, lower mass limits between 85 and 100 GeV are set. At the Tevatron [235, 236] and at the LHC [234, 237–239], searches have focused on scenarios with resonant production of a sneutrino decaying to $e\mu$, $\mu\tau$ and $e\tau$ final states. Limits have been set on sneutrino masses as a function of the value of relevant RPV couplings. As an example, the LHC experiments exclude a resonant tau sneutrino with a mass below 3.9 TeV for $\lambda_{312} = \lambda_{321} > 0.07$ and $\lambda'_{311} > 0.11$ [240].

Formulating a general assessment of the existing bounds on RPV SUSY models is becoming increasingly important in order to evaluate loopholes and potential new search directions. A classification of all potential RPV signatures at the LHC for small couplings has been recently published [241] to evaluate the coverage of the most general RPV-MSSM setup, without making any assumptions about the sparticle spectrum details. Small RPV couplings imply that sparticles are pair-produced at the LHC as in the RPC-MSSM case. Figure 89.10 shows a summary of the 95% C.L. limits on sparticle masses in RPV models for two types of $\lambda$ couplings, compared to the direct production mass bound when the sparticle under consideration is the LSP. This is done to quantify the impact of possible assumptions made on the nature, mass and coupling choices for the LSP which in turn might lead to different leptons, jets and top-quark multiplicities. As underlined in [241], changes in exclusion limits on sparticle masses are at most around 20%.

Finally, RPV signatures are often similar to signatures of Stealth SUSY [242–244]. In these scenarios, squarks can decay to a quark and a chargino (neutralino), which can subsequently decay to a singlino $\tilde{S}$ and a $W^{\pm}$ (photon), with the $\tilde{S}$ decaying to two gluons and a soft gravitino $\tilde{G}$. Dedicated CMS searches [245, 246] exclude squark masses up to 1.85 TeV in the photon channel and up to 550 GeV in the charged lepton channel. Gluino masses up

to 2.1 TeV are excluded in the di-photon channel [246]. Limits have also been placed on top squarks in stealth SUSY scenarios, where the top squark decays to a top quark, a gluon and a $\tilde{S}$, and the $\tilde{S}$ subsequently decays to a $\tilde{G}$ and two gluons. In such a scenario, top squark masses up to 870 GeV have been excluded by CMS [229].

## 89.9 Exclusion limits on long-lived sparticles

Long-lived sparticles arise in many different SUSY models. In particular in co-annihilation scenarios, where the NLSP and LSP are nearly mass-degenerate, this is rather common in order to obtain the correct Dark Matter relic density. Prominent examples are scenarios featuring stau co-annihilation, or models of SUSY breaking, e.g. minimal anomaly-mediated SUSY breaking (AMSB), in which the appropriate Dark Matter density is obtained by co-annihilation of the LSP with an almost degenerate long-lived wino. R-parity violating SUSY models might also lead to non-prompt decays. Searches for events with a displaced hadronic vertex, with or without a matched lepton, are for instance interpreted in a model with RPV neutralino decays involving a non-zero $\lambda'$ coupling [231]. However, in general, other sparticles can also be long-lived and it is desirable to establish a comprehensive search program for these special long-lived cases, which lead to distinct experimental search signatures. Past experiments have performed dedicated searches for long-lived SUSY signatures, but given the absence of any experimental evidence for SUSY so far, more effort and focus has gone into such searches at the LHC recently. Signatures interpreted in terms of SUSY models include disappearing tracks, identification of tracks with atypical properties or unusual ionization, small and localized deposits of energy inside of the calorimeters without associated tracks, or stopped particles that decay out of time with collisions. Some examples are reported below with interpretations provided for gluinos, squarks and electroweak SUSY particles in a variety of SUSY models.

If the decay of gluinos is suppressed, for example if squark masses are high, gluinos may live longer than typical hadronization times. It is expected that such gluinos will hadronize to long-living strongly interacting particles known as R-hadrons. In particular, if the suppression of the gluino decay is strong, as in the case that the squark masses are much higher than the TeV scale, these R-hadrons can be (semi-)stable in collider timescales. Searches for such R-hadrons exploit the typical signature of stable charged massive particles in the detector. R-hadrons decaying in the detector are searched for using $dE/dx$ measurements and searches for displaced vertices. As shown in the left plot of Fig. 89.11, the ATLAS experiment excludes semi-stable gluino R-hadrons with masses below 1.9 – 2.3 TeV for all lifetimes in a simplified model where such gluinos always form R-hadrons, and decay into jets and a light neutralino, by combining a number of analyses [208, 247–250]. A combination of CMS searches for long-lived particles, as shown in Fig. 89.12, reaches similar limits [251–255].

Alternatively, since such R-hadrons are strongly interacting, they may be stopped in the calorimeter or in other material, and decay later into energetic jets. These decays are searched for by identifying the jets [256–258] or muons [258] outside the time window associated with bunch-bunch collisions. As shown in Fig. 89.12, the CMS collaboration sets limits on such stopped R-hadrons over 13 orders of magnitude in gluino lifetime, up to masses of 1390 GeV [258]. Recent results from ATLAS [259] sets constraint on the mass of gluino R-hadrons using large out-of-time energy deposits in the calorimeters. Also in this case, masses of up to 1.4 TeV are excluded for gluino lifetimes of $10^{-5} - 10^3$ s.

Top squarks can also be long-lived and hadronize to a R-hadron, for example in the scenario where the top squark is the next-to-lightest SUSY particle, with a small mass difference to the LSP. Searches for massive stable charged particles are sensitive to such top squarks. Tevatron limits are approximately $m_{\tilde{t}} > 300$ GeV [260, 261]. ATLAS sets a limit of 1340 GeV on such top squarks [249], the CMS limits are comparable [253]. Intermediate lifetimes of top squarks decaying through RPV coupling into a quark and a lepton are also targeted by ATLAS and CMS. Limits on top squarks decaying into a quark and a muon
<figure><img src="image_374.png" /></figure>

August 2023

<figure><img src="image_375.png" /></figure>

March 2022

<figure><img src="image_376.png" /><figcaption>Figure 89.11: Limits at 95% C.L. on the gluino mass in R-hadron models (left), and on the chargino mass in a model where the wino-like chargino is almost degenerate with the LSP (right), as a function of gluino or chargino lifetime, as obtained by ATLAS.</figcaption></figure>

Selection of observed exclusion limits at 95% C.L. (theory uncertainties are not included). The y-axis tick labels indicate the studied long-lived particle.

<caption>Figure 89.12: Excluded regions, at 95% C.L., in the lifetimes of long-lived particles in several models, as obtained by CMS.</caption>

are set by ATLAS [262] using events that pass a muon or missing-transverse-momentum trigger and contain a displaced muon track and a displaced vertex. Masses up to 1.7 TeV are excluded for a lifetime of 0.1 ns, and masses below 1.3 TeV are excluded for lifetimes between 0.01 ns and 30 ns. CMS [263] utilizes events with two leptons with transverse impact parameter values between 0.01 and 10 cm not required to form a common vertex to exclude top squarks with masses between 100 and at least 460 GeV for $0.01 < cτ_0 < 1000$ cm, where $cτ_0$ is the proper decay length.

In addition to colored sparticles, sparticles like charginos may also be long-lived, especially in scenarios with compressed mass spectra. Charginos decaying in the detectors away from the primary vertex could lead to signatures such as kinked-tracks, or apparently disappearing tracks, since, for example, the pion in $\tilde{\chi}_1^\pm \to \pi^\pm \tilde{\chi}_1^0$ might be too soft to be reconstructed. At the LHC, searches have been performed for such disappearing tracks,

and interpreted within anomaly-mediated SUSY breaking models [264–267]. The right plot of Fig. 89.11 shows constraints for different ATLAS searches on the chargino mass-vs-lifetime plane for an AMSB model ($\tan \beta = 5, \mu > 0$) in which a wino-like $\tilde{\chi}_1^\pm$ decays to a soft pion and an almost mass-degenerated wino-like $\tilde{\chi}_1^0$ [249, 250, 265–268]. The disappearing track search [267] provides constraints also for higgsino-like models. Results from [268] exploiting anomalously large specific ionisation losses $dE/dx$ are extrapolated to the stable regime. For a similar AMSB-like model, CMS excludes $cτ$ values between 0.15 and 18 m for a chargino mass of 505 GeV [264], see Fig. 89.12. Their most recent results [269] expand and improve the constraints on masses of charginos as well as of colored sparticles by making use of electron and muon plus disappearing track signatures, measurements of the $dE/dx$ ionization energy loss of candidate tracks, and by re-analyzing the fully hadronic signatures using ML techniques.
<figure><img src="image_377.png" /><figcaption>**Figure 89.13:** The plot on the left shows the survival probability of a pMSSM parameter space model in the gluino-neutralino mass plane after the application of the relevant CMS search results. The plot on the right shows a graphical representation of the ATLAS exclusion power in a pMSSM model. Each vertical bar is a one-dimensional projection of the fraction of models points excluded for each sparticle by ATLAS analyses. The experimental results are obtained from data taken at $\sqrt{s} = 7$ and 8 TeV.</figcaption></figure>

Charginos with a lifetime longer than the time needed to pass through the detector appear as charged stable massive particles. Limits have been derived by the LEP experiments [270], by D0 at the Tevatron [261], and by the LHC experiments [249, 271], and such charginos with mass below 1090 GeV are excluded.

In gauge mediation models, NLSP neutralino decays need not be prompt, and experiments have searched for late decays with photons in the final state. CDF has searched for delayed $\tilde{\chi}_1^0 \to \gamma\tilde{G}$ decays using the timing of photon signals in the calorimeter [272]. CMS has used the same technique at the LHC [273]. Results are given as exclusion contours in the neutralino mass versus lifetime plane, and for example in a GMSB model with a neutralino mass of 300 GeV, $\tau_r$ values between 10 and 2000 cm are excluded [273]. D0 has looked at the direction of showers in the electromagnetic calorimeter with a similar goal [274], and ATLAS has searched for photon candidates that do not point back to the primary vertex, as well as for delayed photons [275].

Charged slepton decays may be kinematically suppressed, for example in the scenario of a NLSP slepton with a very small mass difference to the LSP. Such a slepton may appear to be a stable charged massive particle. Interpretation of searches at LEP for such signatures within GMSB models with a stau NLSP or slepton co-NLSP exclude masses up to 99 GeV [270]. Searches for stable charged particles at the Tevatron [260, 261] and at the LHC [249, 253] are also interpreted in terms of limits on stable charged sleptons. The limits obtained at the LHC exclude stable staus with masses below 430 GeV when produced directly in pairs, and below 660 GeV when staus are produced both directly and indirectly in the decay of other particles in a GMSB model. Recent results from ATLAS and CMS set constraints on long-lived charged slepton with intermediate lifetimes. ATLAS searches for charged leptons with large impact parameters and exclude selection, smuon and stau masses up to 720 GeV, 680 GeV, and 340 GeV, respectively, in case of lifetimes of 0.1 ns. They also exclude smuons with lifetimes down to 1 ps for a smuon mass of 100 GeV, and smuon masses up to 520 GeV are excluded for a proper lifetime of 10 ps [276]. CMS [193] targets long-lived stau with $\tau_r$ of about 0.1 mm excluding masses between 115 and 220 GeV for the case that the LSP is nearly massless.

## 89.10 Global interpretations

Apart from the interpretation of direct searches for sparticle production at colliders in terms of limits on masses of individual SUSY particles, model-dependent interpretations of allowed SUSY parameter space are derived from global SUSY fits. Typically these fits combine the results from collider experiments with indirect constraints on SUSY as obtained from low-energy experiments, flavor physics, high-precision electroweak results, and astrophysical data.

In the pre-LHC era these fits were mainly dominated by indirect constraints. Even for very constrained models like the CMSSM, the allowed parameter space, in terms of squark and gluino masses, ranged from several hundreds of GeV to a few TeV. Furthermore, these global fits indicated that squarks and gluino masses in the range of 500 to 1000 GeV were the preferred region of parameter space, although values as high as a few TeV were allowed with lower probabilities [277–284].

With ATLAS and CMS now probing mass scales around 1 TeV and beyond, the importance of the direct searches for global analyses of allowed SUSY parameter space has increased. For example, imposing the new experimental limits on constrained supergravity models pushes the most likely values of first generation squark and gluino masses significantly beyond 2 TeV, typically resulting in overall values of fit quality much worse than those in the pre-LHC era [155–157, 185, 285–292]. The measured value of $m_h$ pushes the sparticle masses upwards. Although these constrained models are not yet ruled out, the extended experimental limits impose very tight constraints on the allowed parameter space.

For this reason, the emphasis of global SUSY fits has shifted towards less-constrained SUSY models. Interpretations in the pMSSM [183–187, 271, 285, 293] as well as in simplified models, have been useful to generalize SUSY searches, for example to redesign experimental analyses in order to increase their sensitivity for compressed spectra, where the mass of the LSP is much closer to squark and gluino masses than predicted by the CMSSM. As shown in Table 89.1, for neutralino masses above 0.5–1 TeV the current set of ATLAS and CMS searches, interpreted in simplified models, cannot exclude the existence of squarks or gluinos with masses only marginally above the neutralino mass. However, as these exclusion limits are defined in the context of simplified models, they are only valid for the assumptions in which these models are defined.

Fig. 89.13 shows graphically the LHC exclusion power in the pMSSM based on searches performed at $\sqrt{s} = 7$ and 8 TeV as reported by ATLAS [183] and CMS [184]. The plot on the left shows the survival probability in the gluino-neutralino mass plane, which is a measure of the parameter space that remains after inclusion of the relevant CMS search results. As can be seen, gluino masses below about 1.2 TeV are almost fully excluded. This result agrees well with the typical exclusion obtained at 8 TeV in simplified models for gluino production. However, as shown in the right plot of Fig. 89.13, when a similar analysis for other sparticles is performed it becomes apparent that exclusions on the pMSSM parameter can be significantly less stringent than simplified model limits might suggest. This is especially apparent for the electroweak sector, where even at rather low masses several of the pMSSM test points still survive the constraint of ATLAS searches at $\sqrt{s} = 7$ and 8 TeV. This again indicates that care
tivity for colored sparticles will increase somewhat as well, the
expanded data set will be particularly beneficial for electroweak
gaugino searches, and for the more difficult final states presented
by compressed particle spectra, stealth SUSY, long-lived sparti-
cles, or R-parity violating scenarios.

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# 90. Axions and Other Similar Particles

Revised August 2023 by L. J Rosenberg (U. Washington), G. Rybka (U. Washington) and B. Safdi (UC Berkeley).

## 90.1 Introduction

In this section, we list coupling-strength and mass limits for light neutral scalar or pseudoscalar bosons that couple weakly to normal matter and radiation. Such bosons may arise from the spontaneous breaking of a global U(1) symmetry, resulting in a massless Nambu-Goldstone (NG) boson. If there is a small explicit symmetry breaking, either already in the Lagrangian or due to quantum effects such as anomalies, the boson acquires a mass and is called a pseudo-NG boson. Typical examples are axions (a) [1-4] and majorons [5, 6], associated, respectively, with a spontaneously broken Peccei-Quinn PQ and lepton-number symmetry. Axions may also arise in extra dimension constructions as the zero-modes of higher-dimensional gauge fields compactified on internal manifolds; in this case, the absence of a local contributions to the axion mass is due to the higher-dimensional gauge symmetry [7, 8].

A common feature of these light bosons $\phi$ is that their coupling to Standard-Model particles is suppressed by the energy scale that characterizes the symmetry breaking, i.e., the decay constant $f$. For example, in models where the axion arises as a Goldstone boson, the interaction Lagrangian at energy scales below $f$ is

$$ \mathcal{L} = f^{-1} J^{\mu} \partial_{\mu} \phi, \qquad (90.1) $$

where $J^\mu$ is the Noether current of the spontaneously broken global symmetry. If $f$ is very large, these new particles interact very weakly. Detecting them would provide a window to physics far beyond what can be probed at accelerators.

Axions are of particular interest because the PQ mechanism is a compelling scheme to preserve CP-symmetry in QCD. Moreover, the cold dark matter (CDM) of the Universe may well consist of axions and they are searched for in dedicated experiments with a realistic chance of discovery.

Originally it was assumed that the PQ scale $f_a$ was related to the electroweak symmetry-breaking scale $v_{EW} = (\sqrt{2}G_F)^{-1/2} = 246.22$ GeV. However, the associated “standard” and “variant” axions were quickly excluded—we refer to the Listings for detailed limits. Here we focus on “invisible axions” with $f_a \gg v_{EW}$ as the main possibility.

Axions have a characteristic two-photon vertex, inherited from their mixing with $\pi^0$ and $\eta$. This coupling allows for the main search strategy based on axion-photon conversion in external magnetic fields [9], an effect that also can be of astrophysical interest. While for axions the product “$a\gamma\gamma$ interaction strength $\times$ mass” is essentially fixed by the corresponding $\pi^0$ properties, one may consider a more general class of axion-like particles (ALPs) where the two parameters (coupling and mass) are independent. A number of experiments explore this more general ALP parameter space. ALPs populating the latter are predicted to arise generically, in addition to the axion, in low-energy effective field theories emerging from string theory [7, 8, 10–16]. The latter may contain also very light Abelian vector bosons under which the Standard-Model particles are not charged: so-called hidden-sector photons, dark photons or paraphotons. They share a number of phenomenological features with the axion and ALPs, notably the possibility of hidden photon-to-photon conversion and of hidden photon dark matter [17–19]. Their physics cases and the current constraints are compiled in Refs. [20–23].

## 90.2 Theory

### 90.2.1 Peccei-Quinn mechanism and axions

The QCD Lagrangian includes a CP-violating term $\mathcal{L}_\Theta = -\bar{\Theta}(\alpha_s/8\pi)G^{\mu\nu a}\tilde{G}^a_{\mu\nu}$, where $-\pi \le \bar{\Theta} \le +\pi$ is the effective $\Theta$ parameter after diagonalizing quark masses, $G^a_{\mu\nu}$ is the color field strength tensor, and $\tilde{G}^{a,\mu\nu} \equiv \epsilon^{\mu\nu\lambda\rho}G^a_{\lambda\rho}/2$, with $\epsilon^{0123} = 1$, its dual. This term induces an electric dipole moment (EDM) of the neutron of strength $d_n = C_{EDM}e\bar{\Theta}$. The coefficient $C_{EDM}$ is calculated using QCD sum rules as $C_{EDM} = 2.4(1.0) \times 10^{-16}$ cm [24], while lattice QCD computations find $C_{EDM} = 1.48(14)(31) \times 10^{-16}$ cm [25] including statistical and

systematic uncertainties. Experimental upper bounds on the latter, $|d_n| < 1.8 \times 10^{-26}$ e cm [26, 27], imply $|\bar{\Theta}| \lesssim 10^{-10}$ even though $\bar{\Theta} = \mathcal{O}(1)$ is otherwise completely satisfactory. Note that the proton has a similar EDM as the neutron, but it is less strongly constrained at present; this could change in the future [28]. The proton EDM may be written as $d_p = C_{EDM}^p e\bar{\Theta}$, with $C_{EDM}^p = -3.8(11)(8)$ [25]. The opposite signs of the neutron and proton EDMs may be understood from the opposite signs of their magnetic moments. The spontaneously broken global PQ symmetry $U(1)_{PQ}$ was introduced to solve the “strong CP problem” [1, 2] of small $\bar{\Theta}$, the axion being the pseudo-NG boson of $U(1)_{PQ}$ [3, 4]. This symmetry is broken due to the axion's anomalous triangle coupling to gluons,

$$ \mathcal{L} = \left(\frac{a}{f_a} - \bar{\Theta}\right) \frac{\alpha_s}{8\pi} G^{\mu\nu a} \tilde{G}^a_{\mu\nu}, \qquad (90.2) $$

where $a$ is the axion field and $f_a$ the axion decay constant. Color anomaly factors have been absorbed in the normalization of $f_a$ which is defined by this Lagrangian. Thus normalized, $f_a$ is the quantity that enters all low-energy phenomena [29]. Non-perturbative topological fluctuations of the gluon fields in QCD induce a potential for $a$ whose minimum is at $a = \bar{\Theta}f_a$, thereby canceling the $\bar{\Theta}$ term in the QCD Lagrangian and thus restoring CP symmetry.

The resulting axion mass, in units of the PQ scale $f_a$, is identical to the square root of the topological susceptibility in QCD, $m_a f_a = \sqrt{\bar{\chi}}$. The latter can be evaluated further [30, 31], exploiting the chiral limit (masses of up and down quarks much smaller than the scale of QCD), yielding $m_a f_a = \sqrt{\bar{\chi}} \approx f_\pi m_\pi$, where $m_\pi \approx 135$ MeV and $f_\pi \approx 92$ MeV. In more detail one finds, by including $\mathcal{O}(\alpha)$ QED corrections and next-to-next-to-leading order (NNLO) corrections in chiral perturbation theory [32],

$$ m_a = 5.691(51) \left( \frac{10^9 \text{ GeV}}{f_a} \right) \text{meV}. \qquad (90.3) $$

A direct calculation of the topological susceptibility via QCD lattice simulations finds almost the same central value, albeit with an about five times larger error bar [33].

The QCD axion acquires a variety couplings to low-energy Standard Model observables, as we discuss in detail in this review. The most model-independent couplings are those of the axion to the neutron and proton EDM operators. While the axion removes the static neutron and proton EDMs, displacements of the axion field $a$ from its vacuum expectation value induce field-dependent EDMs, which are characterized through the operator

$$ \mathcal{L}_{aN\gamma} = -\frac{i}{2} g_{aN\gamma} a \bar{\Psi}_N \sigma_{\mu\nu} \gamma_5 \Psi_N F^{\mu\nu}, \qquad (90.4) $$

with $N = n, p$ for neutrons or protons, respectively. The coupling $g_{aN\gamma}$ is related to $C_{EDM}$ by $g_{aN\gamma} = eC_{EDM}/f_a$ and similarly for $g_{ap\gamma}$. Direct probes of the axion-EDM operators are difficult, though by studying the thermal production of axions in the early universe one can broadly constrain, using Planck and Baryon Acoustic Oscillation (BAO) data, $m_a \lesssim 0.3$ eV [34]. In the presence of an oscillatory relic axion DM field, this operator induces an oscillating neutron EDM; this effect forms the basis of the CASPEr experiment [35, 36] that is discussed more later in this review. Non-linearly realized discrete-symmetries may be used to suppress the QCD axion mass below the prediction in Eq. (90.3), in which case a variety of cosmological and terrestrial constraints on the operator in Eq. (90.4) are also relevant [37–44]

Axions with $f_a \gg v_{EW}$ evade almost all current experimental limits. One generic class of models invokes “hadronic axions” where new heavy quarks carry $U(1)_{PQ}$ charges, leaving ordinary quarks and leptons without tree-level axion couplings. The prototype is the KSVZ model [45, 46], where in addition the heavy new quarks are electrically neutral. Another generic class of axion models requires at least two Higgs doublets and ordinary quarks and leptons carry PQ charges, the prototype being the DFSZ model [47, 48]; in this case the quarks and leptons may
have tree-level axion couplings. All of these models contain at least one electroweak singlet scalar that acquires a vacuum expectation value and thereby breaks the PQ symmetry. The KSVZ and DFSZ models are frequently used as benchmark examples, but other models exist where both heavy quarks and Higgs doublets carry PQ charges (see [49] for an extensive review). In supersymmetric models, the axion is part of a supermultiplet and thus inevitably accompanied by a spin-0 saxion and a spin-1/2 axino, which both also have couplings suppressed by $f_a$ and are expected to have large masses due to supersymmetry breaking [50]. In extra-dimension constructions the axion emerges from a higher-dimensional gauge field; the other components of the gauge field, however, tend to have masses proportional to the Kaluza-Klein mass scale of the extra dimensions.

### 90.2.2 Model-dependent axion couplings

Although the generic axion interactions scale approximately with $f_π/f_a$ from the corresponding $π^0$ couplings, there are non-negligible model-dependent factors and uncertainties. The axion's two-photon interaction plays a key role for many searches,

$$ \mathcal{L}_{a\gamma\gamma} = -\frac{g_{a\gamma\gamma}}{4} a F_{\mu\nu} \tilde{F}^{\mu\nu} = g_{a\gamma\gamma} a \mathbf{E} \cdot \mathbf{B}, \quad (90.5) $$

where $F$ is the electromagnetic field-strength tensor and $\tilde{F}^{\mu\nu} \equiv e^{\mu\nu\lambda\rho}F_{\lambda\rho}/2$, with $\epsilon^{0123} = 1$, its dual. The coupling constant is [51]

$$ g_{a\gamma\gamma} = \frac{\alpha}{2\pi f_a} \left(\frac{E}{N} - 1.92(4)\right) = \left(0.203(3)\frac{E}{N} - 0.39(1)\right) \frac{m_a}{\text{GeV}^2}, \quad (90.6) $$

where $E$ and $N$ are the electromagnetic and color anomalies of the axial current associated with the axion. In grand unified models, and notably for DFSZ [47, 48], $E/N = 8/3$, whereas for KSVZ [45, 46] $E/N = 0$ if the electric charge of the new heavy quark is taken to vanish. In general, a broad range of $E/N$ values is possible [49, 52–54], as indicated by the diagonal yellow band in Fig. 90.1, whose width is roughly determined by the boundary values $E/N = 44/3$ and $E/N = 5/3$, respectively. However, this band still does not exhaust all the possibilities. In fact, there exist classes of QCD axion models whose photon couplings populate the entire still-allowed region outside the yellow band in Fig. 90.1 [55–60], motivating axion search efforts over a wide range of masses and couplings.

The two-photon decay width is

$$ \Gamma_{a \to \gamma\gamma} = \frac{g_{a\gamma\gamma}^2 m_a^3}{64 \pi} \approx 1.1 \times 10^{-24} \text{ s}^{-1} \left(\frac{m_a}{\text{eV}}\right)^5. \quad (90.7) $$

The second expression uses Eq. (90.6) with $E/N = 0$. Axions decay faster than the age of the Universe if $m_a \gtrsim 20$ eV.

The interaction with fermions $f$ has derivative form and is invariant under a shift $a \to a + a_0$ as behooves a NG boson,

$$ \mathcal{L}_{aff} = \frac{C_f}{2f_a} \partial_\mu a \bar{\Psi}_f \gamma^\mu \gamma_5 \Psi_f. \quad (90.8) $$

Here, $\Psi_f$ is the fermion field, $m_f$ its mass, and $C_f$ a model-dependent coefficient. The dimensionless combination $g_{aff} \equiv C_f m_f/f_a$ plays the role of a Yukawa coupling and $\alpha_{aff} \equiv g_{aff}^2/4\pi$ of a “fine-structure constant.” The often-used pseudoscalar form $\mathcal{L}_{aff} = -i(C_f m_f/f_a) a \bar{\Psi}_f \gamma_5 \Psi_f$ need not be equivalent to the appropriate derivative structure, for example when two NG bosons are attached to one fermion line as in axion emission by nucleon bremsstrahlung [169–171].

In hadronic axion models, $C_e$ vanishes at tree level, but is then generated radiatively at one loop [172, 173],

$$ C_e \approx \frac{3\alpha^2}{4\pi^2} \left[ \frac{E}{N} \log \left( \frac{f_a}{m_e} \right) - 1.92 \log \left( \frac{\Lambda_\chi}{m_e} \right) \right], \quad (90.9) $$

where $\Lambda_\chi \simeq 1$ GeV is the chiral symmetry breaking scale. In the DFSZ model [47, 48], the tree-level coupling coefficient to electrons is [172]

$$ C_e = \frac{\sin^2 \beta}{3}, \quad (90.10) $$

<figure><img src="image_378.png" /><figcaption>**Figure 90.1:** Exclusion plot for the ALP-photon coupling. Figure courtesy of Ciaran O'Hare [61], contains exclusions from refs. [62–68]. Constraints in red are from terrestrial experiments. ABRA 10 cm, SHAFT, BASE, ADMX SLIC, ADMX, RBF+UF, CAPP, QUAX, and ORGAN search for axion DM and assume the axion in 100% DM; CROWS. ALPS-I, OSQAR, PVLAS are light-shining-through-walls experiments that are purely terrestrial, while CAST searches for axions produced in the Sun with no DM assumption. Those shown in green are astrophysical but make no assumption about axions being DM. See text for details regarding the orange band expectation for the QCD axion and assumptions for blue high mass axion astrophysical limits.</figcaption></figure>

where $\tan\beta$ is the ratio of the vacuum expectation values of the two Higgs doublets giving masses to the up- and down-type quarks, respectively: $\tan\beta = v_u/v_d$. As a cautionary note, many of the original axion references, such as [172], defined $\tan\beta = v_d/v_u$, but the choice $\tan\beta = v_u/v_d$ adopted here and in more modern axion works is a reflection of the choice taken in the two-Higgs-doublet literature [174]. The resulting prediction for the axion-electron coupling, as a function of the axion mass, for the KSVZ axion ($E/N = 0$), hadronic axions, with $5/3 \le E/N \le 44/3$, and the DFSZ axion is displayed in Fig. 90.2. For the DFSZ range we have taken into account the constraint $0.28 \lesssim \tan\beta \lesssim 140$ [175] arising from the requirement of perturbative unitarity of the Yukawa couplings of Standard Model fermions.

For nucleons, $C_{p,n}$ have been determined as [51]

$$ \begin{aligned} C_p &= -0.47(3) + 0.88(3)C_u - 0.39(2)C_d - 0.038(5)C_s \\ &\quad - 0.012(5)C_c - 0.009(2)C_b - 0.0035(4)C_t, \\ C_n &= -0.02(3) + 0.88(3)C_d - 0.39(2)C_u - 0.038(5)C_s \\ &\quad - 0.012(5)C_c - 0.009(2)C_b - 0.0035(4)C_t, \end{aligned} \quad (90.11) $$

in terms of the corresponding model-dependent quark couplings $C_q, q = u, d, s, c, b, t$. For hadronic axions, the latter vanish at tree-level, which means that $C_n$ is expected to be much smaller than $C_p$. In the DFSZ model, $C_u = C_c = C_t = \frac{1}{3} \cos^2 \beta$ and $C_d = C_s = C_b = \frac{1}{3} \sin^2 \beta$, and $C_p$ and $C_n$, as functions of $\beta$,

$$ \begin{aligned} C_p &= -0.435 \sin^2 \beta + (-0.182 \pm 0.025), \\ C_n &= 0.414 \sin^2 \beta + (-0.160 \pm 0.025). \end{aligned} \quad (90.12) $$

The resulting prediction for the axion-neutron coupling, as a function of the axion mass, for the KSVZ axion and the DFSZ axion is displayed in Fig. 90.3. Note, however, that these couplings receive in-medium corrections, which may be important when considering compact stars [176].

The axion-pion interaction is given by the Lagrangian [173, 177]

$$ \mathcal{L}_{a\pi} = \frac{C_{a\pi}}{2f_{\pi}f_a} \partial_{\mu} a (\pi^0 \pi^+ \partial_{\mu} \pi^- + \pi^0 \pi^- \partial_{\mu} \pi^+ - 2\pi^+ \pi^- \partial_{\mu} \pi^0), \quad (90.13) $$
where $C_{a\pi} = 2(C_p - C_n)/(3g_A)$ and $g_A$ is the nucleon axial charge $g_A = 1.2723(23)$. Additionally, there are axion-pion-nucleon interactions of the form

$$ \mathcal{L}_{a\pi N} = \frac{C_{a\pi N}}{2f_{\pi}f_{a}} \partial_{\mu}a (i\pi^{+}\bar{p}\gamma^{\mu}n - i\pi^{-}\bar{n}\gamma^{\mu}p), \quad (90.14) $$

$$ \text{where } C_{a\pi N} = (C_{ap} - C_{an})/(\sqrt{2}g_A). $$

## 90.3 Laboratory Searches

### 90.3.1 Light shining through walls

Searching for “invisible axions” is extremely challenging due to their extraordinarily feeble coupling to normal matter and radiation. Currently, the most promising approaches rely on the axion-two-photon interaction, allowing for axion-photon conversion in external electric or magnetic fields [9]. For the Coulomb field of a charged particle, the conversion can be viewed as a scattering process, $\gamma + Ze \leftrightarrow Ze + a$, called Primakoff effect [178]. In the other extreme of a macroscopic field, usually a large-scale B-field, the momentum transfer is small, the interaction is coherent over a large distance, and the conversion can be viewed as an axion-photon oscillation phenomenon in analogy to neutrino flavor oscillations [179].

Photons propagating through a transverse magnetic field, with incident $E_\gamma$ and magnetic field $B$ parallel, may convert into axions. For $m_a^2 L/2\omega \ll 2\pi$, where $L$ is the length of the $B$ field region and $\omega$ the photon energy, the resultant axion beam is coherent with the incident photon beam and the conversion probability is $\Pi \sim (1/4)(g_{a\gamma\gamma}BL)^2$. One such realization uses a laser beam propagating down the bore of a superconducting dipole magnet (like, say, the bending magnets in high-energy accelerators). If another magnet is in line with the first, but shielded by an optical barrier, then photons may be regenerated from the pure axion beam [180, 181]. The overall probability is $P(\gamma \rightarrow A \rightarrow \gamma) = \Pi^2$.

The first such Light-Shining-through-Walls (LSW) experiment was performed by the BFRT (Brookhaven-Fermilab-Rochester-Trieste) collaboration. It utilized two magnets of length $L = 4.4$ m and $B = 3.7$ T and found $|g_{a\gamma\gamma}| < 6.7 \times 10^{-7}$ GeV$^{-1}$ at 95% CL for $m_a < 1$ meV [182, 183]. More recently, several such experiments were performed (see Listings [98, 101, 184–188]). The current best limit, $|g_{a\gamma\gamma}| < 3.5 \times 10^{-8}$ GeV$^{-1}$ at 95% CL for $m_a \lesssim 0.3$ meV (see Fig. 90.1), has been achieved by the OSQAR (Optical Search for QED Vacuum Birefringence, Axions, and Photon Regeneration) experiment, which exploited two 9 T LHC dipole magnets and an 18.5 W continuous wave laser emitting at the wavelength of 532 nm [101]. The ALPS I (Any Light Particle Search I) experiment achieved a similar sensitivity [98], see Fig. 90.1. Some of these experiments have also reported limits for scalar bosons where the photon $E_\gamma$ must be chosen perpendicular to the magnetic field $B$.

The concept of resonantly enhanced photon regeneration may open unexplored regions of coupling strength [189–191]. In this scheme, both the production and detection magnets are within Fabry-Perot optical cavities and actively locked in frequency. The $\gamma \rightarrow a \rightarrow \gamma$ rate is enhanced by a factor $FF'/\pi^2$ relative to a single-pass experiment, where $F$ and $F'$ are the finesses of the two cavities. The resonant enhancement could be of order $10^{(10-12)}$, improving the $g_{a\gamma\gamma}$ sensitivity by $10^{(2.5-3)}$. The experiment ALPS II (Any Light Particle Search II) is based on this concept and aims at an improvement of the current laboratory bound on $g_{a\gamma\gamma}$ by a factor $\sim 10^3$ [192]. ALPS II began operations in 2023.

Resonantly enhanced photon regeneration has already been exploited in experiments searching for “radiowaves shining through a shielding” [193–196]. For $m_a \lesssim 10^{-5}$ eV, the upper bound on $g_{a\gamma\gamma}$ established by the CROWS (CERN Resonant Weakly Interacting sub-eV Particle Search) experiment [100] is slightly less stringent than the one set by OSQAR, see Fig. 90.1.

### 90.3.2 Photon polarization

An alternative to regenerating the lost photons is to use the beam itself to detect conversion: the polarization of light propagating through a transverse $B$ field undergoes dichroism and birefringence [197]. Dichroism: The $E_\parallel$ component, but not $E_\perp$, is

depleted by axion production, causing a small rotation of linearly polarized light. For $m_a^2 L/2\omega \ll 2\pi$, the effect is independent of $m_a$. For heavier axions, it oscillates and diminishes as $m_a$ increases, and it vanishes for $m_a > \omega$. Birefringence: This effect occurs because there is mixing of virtual axions in the $E_\parallel$ state, but not for $E_\perp$. Hence, linearly polarized light will develop elliptical polarization. Higher-order QED also induces vacuum magnetic birefringence (VMB). A search for these effects was performed in the same dipole magnets of the BFRT experiment mentioned before [198]. The dichroic rotation gave a stronger limit than the ellipticity rotation: $|g_{a\gamma\gamma}| < 3.6 \times 10^{-7}$ GeV$^{-1}$ at 95% CL, for $m_a < 5 \times 10^{-4}$ eV. The ellipticity limits are better at higher masses, as they fall off smoothly and do not terminate at $m_a$.

### 90.3.3 Long-range forces

New bosons would mediate new long-range forces, which are severely constrained by “fifth force” experiments [199]. Those looking for new mass-spin couplings provide significant constraints on pseudoscalar bosons [200–205], see for example in Fig. 90.3 the limit on the axion-neutron coupling [206] from torsion balance tests of the gravitational inverse square law [207]. Presently, the most restrictive limits are obtained from combining long-range force measurements with stellar cooling arguments [208, 209]. For the moment, any of these limits are far from realistic values expected for the QCD axion. Still, these efforts provide constraints on more general low-mass bosons.

In Ref. [210], a method was proposed that can extend the search for axion-mediated spin-dependent forces by several orders of magnitude. By combining techniques used in nuclear magnetic resonance and short-distance tests of gravity, this method appears to be sensitive to the QCD axion in the $\mu$eV – meV mass range, independent of the cosmic axion abundance, if axions have a CP-violating interaction with nuclei as large as the current experimental bound on the electric dipole moment of the neutron allows. (Note that the CP-violation already present in the Standard Model is sufficient for generating an axion-mediated force, though in this case the signal appears too small to be detectable for the QCD axion in the foreseeable future [210].) The ARIADNE (Axion Resonant InterAction DetectiON Experiment) is under development and employs this approach to search for axion-mediated spin-dependent short-range interactions between a hyper-polarized 3-He sample and an unpolarized tungsten source mass [211]. The method relies on superconducting magnetic shielding to screen the sample from ordinary magnetic field noise. Experimental tests to demonstrate the requirements of ARIADNE, including characterization of the magnetic field backgrounds, are under way [212, 213].

## 90.4 Axions from Astrophysical Sources

### 90.4.1 Stellar energy-loss limits

Axions are produced in hot astrophysical plasmas, and can thus transport energy out of stars. The coupling strength of these particles with normal matter and radiation is bounded by the constraint that stellar lifetimes or energy-loss rates are not in conflict with observation [214, 215].

We begin this discussion with our Sun and concentrate first on the axion-photon coupling. In the Sun axions are produced by the Primakoff process $\gamma + Ze \rightarrow Ze + a$. Integrating this process over a standard solar model yields the axion luminosity [216]

$$ L_a = g_{10}^2 \times 1.85 \times 10^{-3} L_{\odot}, \quad (90.15) $$

where $g_{10} = |g_{a\gamma\gamma}| \times 10^{10}$ GeV. The maximum of the spectrum is at 3.0 keV, the average at 4.2 keV, and the number flux at Earth is $g_{10}^2 \times 3.75 \times 10^{11}$ cm$^{-2}$ s$^{-1}$. The solar photon luminosity is fixed, so energy losses due to the Primakoff process require enhanced nuclear energy production and thus enhanced neutrino fluxes. The all-flavor measurements by SNO (Sudbury Neutrino Observatory), together with a standard solar model, imply $L_a \lesssim 0.10 L_\odot$, corresponding to $g_{10} \lesssim 7$ [217], mildly superseding a similar limit from helioseismology (sound speed, surface helium and convective radius) [218]. In Ref. [147], this limit was improved to $g_{10} < 4.1$ (at $3\sigma$), see Fig. 90.1, exploiting a new statistical analysis that combined helioseismology and solar neutrino observations, including
theoretical and observational errors, and accounting for tensions between input parameters of solar models, in particular the solar element abundances. Going beyond the axion-photon coupling, Ref. [217] considered also a non-zero axion-electron coupling and obtained the bound on the latter displayed in Fig. 90.2.

<figure><img src="image_379.png" /><figcaption>**Figure 90.2:** Exclusion plot for ALP-electron coupling as described in the text. Figure courtesy of Ciaran O'Hare [61], contains data from refs. [168, 217, 219–228]. The hadronic QCD axion-electron coupling arises at one loop and is given in Eq. (90.9); varying $E/N$ between 5/3 and 44/3 gives the indicated hadronic axion band. The KSVZ axion has $E/N = 0$, while the DFSZ axion-electron coupling has the tree-level coupling given in Eq. (90.10) that depends on the unknown but bounded value of $\tan \beta$ (see text for details).</figcaption></figure>

A more restrictive limit derives from globular-cluster (GC) stars that allow for detailed tests of stellar-evolution theory. The stars on the horizontal branch (HB) in the color-magnitude diagram have reached helium burning with a core-averaged energy release of about 80 erg g⁻¹ s⁻¹, compared to Primakoff axion losses of $g_{10}^2$ 30 erg g⁻¹ s⁻¹. The accelerated consumption of helium reduces the HB lifetime by about 80/(80 + 30 $g_{10}^2$). In contrast, the red giant (RG) branch stars are not strongly affected by Primakoff losses, since unlike for the HB stars the plasma frequency in the RG stellar cores is below the temperature. Thus, Primakoff production reduces the R-parameter, which is the number of observed HB branch stars to RG branch stars in a GC. Measurements of the R-parameter in a large sample of 39 Galactic GCs

give a weak indication of non-standard losses which may be accounted by Primakoff-like axion emission, if the photon coupling is in the range $|g_{a\gamma\gamma}| = (2.9 \pm 1.8) \times 10^{-11} \text{ GeV}^{-1}$ [122,229]. Still, the upper bound found in this analysis,

$$|g_{a\gamma\gamma}| < 6.6 \times 10^{-11} \text{ GeV}^{-1} \ (95\% \text{ CL}), \qquad (90.16)$$

represents one of the strongest limits on $g_{a\gamma\gamma}$ for a wide mass range. More recently, the R₂-parameter has emerged as a potentially more sensitive probe of axions; the R₂-parameter is defined as the ratio of the asymptotic giant branch (AGB) stars to the HB stars in a GC. AGB stars have more efficient Primakoff production than HB stars, meaning that R₂ is expected to decrease for increasing $|g_{a\gamma\gamma}|$. Ref. [123] used an R₂ measurement based off of 48 GCs as observed by the Hubble Space Telescope [230] to constrain

$$|g_{a\gamma\gamma}| < 4.7 \times 10^{-11} \text{ GeV}^{-1} \ (95\% \text{ CL}). \qquad (90.17)$$

The constraint above on $g_{a\gamma\gamma}$ may be translated to $f_a > 4.8 \times 10^7 \text{ GeV} \ (m_a < 0.12 \text{ eV})$, using $E/N = 0$ as in the KSVZ model, or to $f_a > 1.8 \times 10^7 \text{ GeV} \ (m_a < 0.32 \text{ eV})$, for the DFSZ axion model, with $E/N = 8/3$, see Fig. 90.1.

If axions couple directly to electrons, the dominant emission processes are atomic axio-recombination and axio-deexcitation, axio-bremsstrahlung in electron-ion or electron-electron collisions, and Compton scattering [231]. Stars in the RG branch of the

color-magnitude diagram of GCs are particularly sensitive to these processes: they would lead to an extension of the latter to larger brightness. In fact, the tip of the RG branch (TRGB) – the brightest point of the RG branch – provides the currently most sensitive method to test the axion coupling to electrons. The strongest bounds on it are derived from analyses of the TRGB in several GCs [232] and in the Galactic GC $\omega$ Centauri [226]. The two analyses lead to very similar results,

$$|g_{aee}| < 1.48 \times 10^{-13} \ (95\% \text{ CL}) \quad \text{and} \quad |g_{aee}| < 1.3 \times 10^{-13} \ (95\% \text{ CL}), \qquad (90.18)$$

respectively, see Fig. 90.2. Reference [232] finds also a small hint for extra cooling, corresponding to $|g_{aee}| = 0.60^{+0.32}_{-0.58} \times 10^{-13}$, while Ref. [226] does not find any evidence for exotic cooling.

Bremsstrahlung is also efficient in white dwarfs (WDs), where the Primakoff and Compton processes are suppressed by the large plasma frequency. A comparison of the predicted and observed luminosity function of WDs can be used to put limits on $|g_{aee}|$ [233, 234]. An analysis based on detailed WD cooling treatment and data on the WD luminosity function (WDLF) of the Galactic disk found that electron couplings above $|g_{aee}| \gtrsim 3 \times 10^{-13}$ are disfavoured [235]. Lower couplings cannot be discarded from the current knowledge of the WDLF of the Galactic disk [235–237]. These probes may be improved by the Rubin Observatory Large Synoptic Survey Telescope (LSST), which is expected to increase the sample of WDs in the Galactic halo to hundreds of thousands [238]. For pulsationally unstable WDs (ZZ Ceti stars), the period decrease $\dot{P}/P$ is a measure of the cooling speed. The corresponding observations of a handful pulsating WDs imply additional cooling that can be interpreted in terms of axion energy losses [239–243], though with axion-electron coupling values likely incompatible with other constraints [244].

<figure><img src="image_380.png" /><figcaption>**Figure 90.3:** Exclusion plot for ALP-neutron coupling as described in the text. Figure courtesy of Ciaran O'Hare [61], includes data from refs. [40, 42, 206, 245–255]. The hadronic axion model prediction is given in Eq. (90.11) with vanishing quark couplings, while the DFSZ model prediction depends on $\tan \beta$ as is found in Eq. (90.12), giving the shaded yellow region above. Note that for a fine-tuned value of $\tan \beta$ $g_{an}$ can be taken to zero. On the other hand, the neutron star cooling constraints [254] also probe the axion-proton coupling $g_{ap}$ at a comparable level (not shown), and both $g_{an}$ and $g_{ap}$ cannot simultaneously be taken to zero in the DFSZ model.</figcaption></figure>

Analogous constraints derive from neutron stars (NS) cooling. Broadly speaking, there are two classes of NS cooling searches for axions. One approach uses the proto-NS formed after SN 1987A to constrain the axion luminosity produced during the first seconds of the proto-NS when the neutrino luminosity is measured with terrestrial experiments. The second class of NS cooling constraints consider older NSs, typically much older than ~100 yr, and constrains the axion luminosity using the surface temperature as measured with photons versus the age of NS as measured
through e.g. kinematic considerations. We begin with a discussion of the limits from SN 1987A.

Numerical simulations reveal that the energy-loss rate due to axions in a nuclear medium at the density $3 \times 10^{14}$ g cm$^{-2}$ and temperature 30 MeV should not exceed about $1 \times 10^{19}$ erg g$^{-1}$ s$^{-1}$ to be consistent with the neutrino signal observed from SN 1987A [256]. The energy-loss rate from nucleon bremsstrahlung, $N+N \rightarrow N+N+a$, is $(C_N/2f_a)^2(T^4/\pi^2m_N)F$. Here $F$ is a numerical factor that represents an integral over the dynamical spin-density structure function because axions couple to the nucleon spin. For realistic conditions, even after considerable effort, one is limited to a heuristic estimate leading to $F \approx 1$ [215]. The SN 1987A limits are of particular interest for hadronic axions where the bounds on $|g_{aee}|$ are moot. Using a proton fraction of 0.3, $g_{ann} = 0$, $F = 1$, and $T = 30$ MeV, one finds $f_a > 4 \times 10^8$ GeV and $m_a < 16$ meV [215]. A more detailed numerical calculation [257] with state of the art SN models, again assuming $g_{ann} = 0$, found that a coupling larger than $|g_{app}| \gtrsim 6 \times 10^{-10}$, would shorten significantly the timescale of the neutrino emission. This result is, not surprisingly, rather close to the estimate in Ref. [215]. Improving the calculation of axion emission via nucleon-nucleon bremsstrahlung beyond the basic one-pion exchange approximation appears to loosen the bound [258, 259]. The latter analysis finds a reduction of the axion emissivity by an order of magnitude if one takes into account the non-vanishing mass of the exchanged pion, the contribution from two-pion exchange, effective in-medium nucleon masses and multiple nucleon scattering, leading to the result [259]

$$g_{ann}^2 + 0.61 g_{app}^2 + 0.53 g_{ann} g_{app} < 8.26 \times 10^{-19}, \quad (90.19)$$

which is consistent with the original estimate in [215]. This analysis, however, still neglects another very efficient mechanism for axion production in a SN: the pion production process, $\pi^- + p \rightarrow a+n$, also induced by the axion coupling to nucleons. Pion production was recognized long ago as being competitive with nucleon bremsstrahlung [260–262]. However, only recently it was shown that the pion abundance in the early stages of a SN is much larger than previously expected [263]. It was shown in Ref. [264] that pion induced processes dominate over bremsstrahlung and so strengthen the bound on the axion-nucleon coupling. On the other hand, given the modeling uncertainties and the sparse data from SN 1987A, the constraint on the axion-nucleon couplings from SN 1987A should be considered more as indicative than as a sharp bound [257]. Furthermore, note that if axions interact sufficiently strongly they are trapped. Only about three orders of magnitude in $g_{aNN}$, where $N=n,p$, or $m_a$ are excluded. For even larger couplings, the axion flux would have been negligible, yet it would have triggered additional events in the detectors, excluding a further range [265].

Further bounds on the axion-nucleon coupling can be derived from observations of the cooling of older NSs. Recently, strong constraints were obtained by modeling the cooling of the “Magnificent Seven” (M7) NSs in the presence of axions [254]. Axion emission would accelerate NS cooling. Thus, if the age and temperature of a NS are independently known, then by modeling axion-induced cooling in conjunction with the standard NS cooling mechanisms, in particular cooling via neutrino emission from the bulk of the NS and thermal emission from the NS surface, one is able to constrain the axion’s coupling strength. Ref. [254] used a sample of four of the M7 along with PSR J0659 to constrain $m_a < 16$ meV for the KSVZ QCD axion at 95% CL and $m_a \lesssim 5$ meV ($m_a \lesssim 30$ meV) in the DFSZ scenario for asymptotically large (small) $\tan \beta$. For axions that couple exclusively to neutrons (protons) Ref. [254] constrains $|g_{ann}| < 1.3 \times 10^{-9}$ ($|g_{app}| < 1.5 \times 10^{-9}$), as illustrated in Fig. 90.3. The five NSs used in that work are unique in that their ages, all in the range 0.1–1 Myr, are known by kinematic measurements (e.g., tracing back to the SN remnant), while their surface temperatures are measured in the X-ray band. Beyond statistical uncertainties on these measurements, there are a number of confounding systematic uncertainties that must be accounted for in NS cooling analyses, including uncertainties related to the core-surface temperature relation, which involves the chemical composition of the

NS envelope. The NS equation of state and the superfluid critical temperatures of the dense nuclear matter are also sources of uncertainty. Note that axions are predominantly produced by nuclear bremsstrahlung [266,267] except for temperatures near the superfluid critical temperature, when Cooper pair breaking and formation (PBF) processes may efficiently produce axions [268,269]. Given the large densities it is crucial to account for medium dependent couplings when computing the axion luminosities [270].

The young NS in the supernova remnant Cassiopeia A (Cas A) was previously though to strongly constrain axions, at a level similar to the NS cooling bound described above, due to PBF axion production [271]. This is because it was observed that Cas A was cooling rapidly, which may be explained by neutrino cooling with neutron superfluidity and proton superconductivity [272, 273]. However, it is possible that the excessive cooling of Cas A is at least in part instrumental in origin [274]. Fully accounting for uncertainties on both the NS properties and the data, constraints from young NSs such as Cas A and the NS in the supernova remnant HESS J1731-347 [275] appear to be subdominant compared to the limits from SN 1987A and the M7 [254].

Finally, let us note that if $m_a \sim$ meV, SNe would lose a large fraction of their energy as axions. This would lead to a diffuse SN axion background in the Universe with an energy density comparable to the extra-galactic background light [276]. However, there is no apparent way of detecting it or the axion burst from the next nearby SN. On the other hand, neutrino detectors such as IceCube, Super-Kamiokande or a future mega-ton water Cherenkov detector will probe exactly the mass region of interest by measuring the neutrino pulse duration of the next Galactic SN [257].

### 90.4.2 Searches for solar axions and ALPs

Instead of using stellar energy losses to derive axion limits, one can also search directly for these fluxes, notably from the Sun. The main focus has been on ALPs with a two-photon vertex. They are produced by the Primakoff process with a flux given by Eq. (90.15) and an average energy of 4.2 keV, and can be detected at Earth with the reverse process in a macroscopic B-field (“axion helioscope”) [9]. In order to extend the sensitivity in mass towards larger values, one can endow the photon with an effective mass in a gas, $\omega_{plas}$, and tune the mass to match the axion’s dispersion relation ($\omega_{plas} = m_a$) [277].

An early implementation of these ideas used a conventional dipole magnet, with a conversion volume of variable-pressure gas with a xenon proportional chamber as x-ray detector [278]. The conversion magnet was fixed in orientation and collected data for about 1000 s/day. Axions were excluded for $|g_{a\gamma\gamma}| < 3.6 \times 10^{-9}$ GeV$^{-1}$ for $m_a < 0.03$ eV, and $|g_{a\gamma\gamma}| < 7.7 \times 10^{-9}$ GeV$^{-1}$ for $0.03 < m_a < 0.11$ eV at 95% CL.

Later, the Tokyo axion helioscope used a superconducting magnet on a tracking mount, viewing the Sun continuously. They reported $|g_{a\gamma\gamma}| < 6 \times 10^{-10}$ GeV$^{-1}$ for $m_a < 0.3$ eV [279,280]. This experiment was recommissioned and a similar limit for masses around 1 eV was reported [281].

The most recent helioscope CAST (CERN Axion Solar Telescope) uses a decommissioned LHC dipole magnet on a tracking mount. The hardware includes grazing-incidence x-ray optics with solid-state x-ray detectors, as well as x-ray Micromegas position-sensitive gaseous detectors. Exploiting an IAXO (see below) pathfinder system, CAST has established the limit

$$|g_{a\gamma\gamma}| < 6.6 \times 10^{-11} \text{ GeV}^{-1} \quad (95\% \text{ CL}), \quad (90.20)$$

for $m_a < 0.02$ eV [99]. To cover larger masses, the magnet bores are filled with a gas at varying pressure. The runs with $^4$He cover masses up to about 0.4 eV [282], providing the high-mass component of the CAST limits shown in Fig. 90.1. To cover yet larger masses, $^3$He was used to achieve a larger pressure at cryogenic temperatures. Limits up to 1.17 eV allowed CAST to “cross the axion line” for the KSVZ model [283–285], see Fig. 90.1.

Sensitivity to significantly smaller values of $g_{a\gamma\gamma}$ can be achieved with a next-generation axion helioscope with a much larger magnetic-field cross section. Realistic design options for this “International Axion Observatory” (IAXO) have been studied in some detail [286] and its physics potential has been reviewed
recently [287]. Such a next-generation axion helioscope may also push the sensitivity in the product of couplings to photons and to electrons, $g_{a\gamma\gamma}g_{aee}$, into a range beyond stellar energy-loss limits and test the hypothesis that WD, RG, and HB cooling is dominated by axion emission [288,289]. As a first step towards IAXO, an intermediate experimental stage called BabyIAXO is currently under preparation at DESY [290].

Direct detection experiments searching for dark matter (DM) consisting of weakly interacting massive particles have also the capability to search for solar axions and ALPs. For low masses, $m_a \lesssim 100$ eV, the XENONnT experiment [291] has provided the most stringent bound among those experiments on the axion-electron coupling constant,

$$|g_{aee}| < 1.9 \times 10^{-12} \quad (90\% \text{ CL}), \qquad (90.21)$$

see Fig. 90.2, by exploiting the axio-electric effect in liquid xenon. Slightly less stringent limits were set by PandaX-II [292] and by LUX [293]. However, this technique has not reached the sensitivity of energy-loss considerations in stars.

### 90.4.3 Conversion of astrophysical axion fluxes

Large-scale B fields exist in astrophysics that can induce axion-photon oscillations. In practical cases, the strength of B is much smaller than in the laboratory, whereas the conversion region L is much larger. Therefore, while the product BL can be large, realistic sensitivities are usually restricted to very low-mass particles, far away from the “QCD axion band” in a plot like Fig. 90.1.

One example of this is SN 1987A, which would have emitted a burst of ALPs due to the Primakoff production in its core. They would have partially converted into γ-rays in the Galactic B-field. The lack of a gamma-ray signal in the GRS instrument of the SMM satellite in coincidence with the observation of the neutrinos emitted from SN 1987A therefore provides a strong bound on their coupling to photons [294,295]. This bound has been revisited and the underlying physics has been brought to the current state-of-the-art, as far as modeling of the supernova and the Milky-Way magnetic field are concerned, resulting in the limit [151, 152] $|g_{a\gamma\gamma}| < 5.2 \times 10^{-12} \text{ GeV}^{-1}$ for $m_a \lesssim 5 \times 10^{-10}$ eV.

Reference [119] reports no evidence of a γ-ray burst in observations of extragalactic SNe with the Fermi Large Area Telescope (LAT). Under the assumption that at least one SN was contained within the LAT field of view, the authors derive an upper bound on the axion photon coupling which is about a factor of 5 weaker than the one from SN 1987A, see Fig. 90.1.

The cumulative emission of ALPs from all past core-collapse SNe would lead to a diffuse gamma-ray flux peaked at energies ~50–100 MeV which can convert in the Galactic magnetic field into photons. Using Fermi-LAT measurements of the diffuse γ-ray flux, Ref. [296] obtains a conservative bound on the photon coupling, $|g_{a\gamma\gamma}| < 5 \times 10^{-10} \text{ GeV}^{-1}$, for $m_a \lesssim 10^{-11}$ eV, see Fig. 90.1, which can be decreased by nearly three orders of magnitude, if an ALP-nucleon coupling of maximal phenomenologically allowed strength, $|g_{aNN}| \sim 10^{-9}$, is allowed.

Hot, young stars, such as Wolf-Rayet stars, efficiently produce ALPs with energies ~10–100 keV via the Primakoff effect. Large numbers of those stars are hosted by the Galactic Quintuplet and Westerlund 1 super star clusters (SSCs). The non-observation of hard X-rays originating from axion-photon conversion in the Galactic magnetic field in archival NuSTAR (Nuclear Spectroscopic Telescope Array) data from these SSCs leads to a bound on the axion-photon coupling [159], $|g_{a\gamma\gamma}| < 3.6 \times 10^{-12} \text{ GeV}^{-1}$, for $m_a \lesssim 5 \times 10^{-11}$ eV, see Fig. 90.1. A somewhat weaker bound in the same mass range was established exploiting NuSTAR data on Betelgeuse [105].

A hard X-ray excess in data from the nearby M7 isolated NSs [297] may be explained by ALPs produced in the cores of those stars and converted in the surrounding magnetic fields, with $|g_{aNN}g_{a\gamma\gamma}| \in (2 \times 10^{-21}, 10^{-18}) \text{ GeV}^{-1}$, for $m_a \lesssim 10^{-5}$ eV [298]. The non-observation of an X-ray excess from the magnetic WD RE J0317-853 [299] by Chandra yields the constraint $|g_{aee}g_{a\gamma\gamma}| \lesssim 1.3 \times 10^{-25} \text{ GeV}^{-1}$, for $m_a \ll 10^{-5}$ eV [128], which provides a non-trivial constraint on the ratio $g_{aee}/g_{aNN}$ for axion models explaining the M7 X-ray excess.

In addition to being produced in NS cores, axions may also be produced in NS magnetospheres. In particular, it has recently been shown that axions could be copiously produced in the polar-cap regions of NS magnetospheres due to time-varying, un-screened electric fields [145]. Those axions could then convert to photons in the outer regions of the NS magnetosphere, leading to observable, broadband radio flux. The absence of such radio flux sets competitive constraints on ALPs, as indicated by the “pulsars” constraint in Fig. 90.1.

### 90.4.4 Conversion of astrophysical photon fluxes

Magnetically induced oscillations between photons and ALPs can modify the photon fluxes from distant sources in various ways, featuring (i) frequency-dependent dimming, (ii) modified polarization, and (iii) avoiding absorption by propagation in the form of axions. For example, dimming of SNe Ia could influence the interpretation in terms of cosmic acceleration [300], although it has become clear that photon-ALP conversion could only be a subdominant effect [301].

Polarization measurements are particularly powerful probes of photon-to-axion conversion since only photons polarized parallel to astrophysical magnetic fields may convert into axions, meaning that in the presence of large-scale ordered magnetic fields axions tend to leave distinct polarization signatures in passing radiation. Searches for linear polarization from radio galaxies (see, e.g., Ref. [302]) and quasars [303] may approach in sensitivity $|g_{a\gamma\gamma}| \sim \text{few} \times 10^{-13} \text{ GeV}^{-1}$, albeit with uncertainties related to the underlying modeling assumptions. Recently, robust limits at the level $|g_{a\gamma\gamma}| \le 5.4 \times 10^{-12} \text{ GeV}^{-1}$ at 95% CL have been derived from optical polarization measurements of nearby magnetic WDs [129] (see also [304]), see Fig. 90.1. The magnetic fields of magnetic WDs may be well-measured through the Zeeman splitting of bound-bound transitions in the WD atmospheres.

Remarkably, it appears that the Universe could be too transparent to TeV γ-rays that should be absorbed by pair production on the extra-galactic background light [109, 305–308]. The situation is not conclusive at present [309–312], but the possible role of photon-ALP oscillations in TeV γ-ray astronomy is tantalizing [313–315]. On the other hand, much of this parameter space is excluded by the previously-discussed WD polarization probes. Fortunately, the region in ALP parameter space, $g_{a\gamma\gamma} \sim 10^{-12} – 10^{-10} \text{ GeV}^{-1}$ for $m_a \lesssim 10^{-7}$ eV [316], required to explain the anomalous TeV transparency of the Universe, could be conceivably probed by the next generation of laboratory experiments (ALPS II) and helioscopes (IAXO) mentioned above.

This parameter region can also be probed by searching for an irregular behavior of the gamma ray spectrum of distant active galactic nuclei (AGN), expected to arise from photon-ALP mixing in a limited energy range. In this type of studies, the uncertainty in the magnetic field around the source needs to be taken into account. This typically leads to a range of limits on the ALP-photon coupling that depend on the modeling assumptions. The H.E.S.S. collaboration has set a limit of $|g_{a\gamma\gamma}| \lesssim 2.1 \times 10^{-11} \text{ GeV}^{-1}$, for $1.5 \times 10^{-8} \text{ eV} \lesssim m_a \lesssim 6.0 \times 10^{-8} \text{ eV}$, from the non-observation of an irregular behavior of the spectrum of the AGN PKS 2155-304 [317]. The Fermi-LAT collaboration has put an even more stringent limit on the ALP-photon coupling [118] from observations of the gamma ray spectrum of NGC 1275, the central galaxy of the Perseus cluster, see Fig. 90.1. A similar analysis has been carried out in Ref. [318], using Fermi-LAT data of PKS 2155-304, and in Ref. [131], using ARGO-YBJ and Fermi-LAT data of Mrk 421, see Fig. 90.1. However, these constraints were obtained under certain theoretical assumptions about magnetic fields surrounding the sources, not confirmed yet by direct astronomical observations in these particular directions; this introduces large systematic uncertainties in the reported constraints [319].

Evidence for spectral irregularities has been reported in Galactic sources, such as pulsars and supernova remnants, and has been interpreted as hints for ALPs [334, 335] (see also discussion in Ref. [336] and references therein). However, the inferred ALP parameters, $|g_{a\gamma\gamma}| \sim 10^{-10} \text{ GeV}^{-1}$, $m_a \sim \text{neV}$, are in tension with the CAST helioscope bounds. Reference [336] updated the analysis of the pulsar signal region including astrophysical nuisance parameters and correctly deriving confidence intervals on ALP
<figure><img src="image_381.png" /><figcaption>**Figure 90.4:** QCD axion mass predictions for obtaining the correct DM abundance assuming the PQ symmetry is and is not restored after inflation and, in the case the PQ symmetry is restored after inflation, depending on the domain wall number $N$. Figure courtesy of Ciaran O'Hare [61], includes data from refs. [18,33,320–333]. The lower-end of the possible mass range is bounded by BH superradiance constraints and the theoretical requirement that $f_a$ be below the Planck scale, while the upper end is bounded by the somewhat model-dependent NS cooling constraints (not shown), which exclude QCD axion masses above roughly ~15-30 meV, depending on the UV completion. At even higher masses there are strong constraints from the production of DM in the early universe, which behaves as hot DM and contributes to $\Delta N_{\text{eff}}$.</figcaption></figure>

parameters by means of Monte Carlo simulations. The tension with the CAST bounds can be evaded if environmental effects in matter, which would suppress the ALP production in dense astrophysical plasmas like the solar interior, are invoked. If this explanation is correct, the claimed ALP signal would be in reach of the next-generation LSW experiment ALPS II. Other ways to make the CAST bound compatible with photon-ALP conversions in the low-density Galactic medium are invoking photon-ALP-dark photon oscillations [337] or the existence of a number of ‘hidden’ ALPs [338].

At smaller masses, $m_a \lesssim 10^{-12}$ eV, galaxy clusters become highly efficient at interconverting ALPs and photons at X-ray energies. Constraints on spectral irregularities in the spectra of luminous X-ray sources (Hydra A, M87, NGC 1275, NGC 3862, Seyfert galaxy 2E3140; taken by Chandra and XMM-Newton) located in or behind galaxy clusters then lead to stringent upper limits on the ALP-photon coupling [110–112, 339–341], (for the limits exploiting spectra from Hydra and M87 from Refs. [110] and [111], respectively, see Fig. 90.1). Reference [112] recently performed the most sensitive x-ray searches for ALPs to date by employing Chandra’s High-Energy Transmission Gratings that allow for an unsurpassed spectral resolution. New observations of the AGN NGC 1275 then led to the bound $|g_{a\gamma\gamma}| < 8 \times 10^{-13} \text{ GeV}^{-1}$ at 99.7% C.L. for ultra-light ALPs, as illustrated in Fig. 90.1. On the other hand, magnetic field modeling uncertainties in the clusters may not be fully accounted for in these analyses [319].

### 90.4.5 Finite density effects for the QCD axion

If the QCD axion sector has a discrete, $Z_N$, shift symmetry, its potential is much shallower, and its mass, in units of its decay constant, $m_a f_a \simeq (3\pi)^{-1/4} N^{3/4} 2^{-N/2} m_\pi f_\pi$, is smaller than the one of the canonical QCD axion by an exponential factor $\propto 2^{-N/2}$ [57, 58]. This opens up the parameter space over which one should search for a QCD axion towards the left of the yellow canonical QCD axion band in Fig. 90.1, Fig. 90.2, and Fig. 90.3. Novel bounds apply to this exceptionally light QCD axion due to finite density effects [342–344]. In fact, in dense stellar media, the minimum of the axion potential may be shifted to $\pi$. This has a number of phenomenological consequences that span from the modification of nuclear processes in stars due to $\theta = O(\pi)$ to modifications in the orbital decay of binary systems (and subsequently in the emitted gravitational waves). Recently strong constraints have been set using the modification of the WD mass-radius relation due to such axions [345].

In fact, $\theta \sim 1$ in the solar core would lead to an increased proton-neutron mass difference, which would prohibit the neutrino line corresponding to the $^7\text{Be}-^7\text{Li}$ mass difference observed by Borexino [346].

Moreover, the fact that the position of the minimum of the axion potential depends on the nuclear density of the medium may also source a long-range force between dense stars [342]. This new long-range force sourced by the axion can be constrained by the measurement of the orbital decay of double pulsar or NS-pulsar binaries [342]. The gravitational wave signal of NS-NS mergers or black hole (BH) - NS mergers would also be modified by these axionic long-range forces [342, 343]. There is also a corresponding bound [344] exploiting the gravitational waves observation from the binary NS inspiral GW170817 detected by LIGO (Laser Interferometer Gravitational-Wave Observatory) and Virgo [347].

### 90.4.6 Superradiance of black holes

Light bosonic fields such as axions or ALPs can affect the dynamics and gravitational wave emission of rapidly rotating astrophysical BHs through the superradiance mechanism. When the boson’s Compton wavelength is of order of the size of the BH’s ergoregion, they form gravitational bound states around the BH. Their occupation number grows exponentially by extracting energy and angular momentum from the black hole, forming a coherent axion or ALP bound state emitting gravitational waves. When accretion cannot replenish the spin of the BH, superradiance dominates the BH spin evolution; this is true for both superr passive and stellar mass BHs. The existence of destabilizing light bosonic fields thus leads to exclusion regions in the mass versus spin plot of rotating BHs. Stellar BH spin measurements exploiting well-studied binaries and two independent techniques exclude a mass range – up to self-interaction caveats discussed below – $6 \times 10^{-13} \text{ eV} < m_a < 2 \times 10^{-11} \text{ eV}$ at $2\sigma$, which for the axion excludes $3 \times 10^{17} \text{ GeV} < f_a < 1 \times 10^{19} \text{ GeV}$ [12, 348, 349], see Fig. 90.4. These bounds apply when gravitational interactions dominate over the axion self-interaction, which is largely true for the QCD axion in this mass range. On the other hand, Mehta et al. [350, 351] considered superradiance bounds for string theory ALPs with a quartic self-coupling (of either sign) given by $\lambda = m_a^2/f_a^2$, using the bosenova model of Ref. [349], and found that for $m_a \sim 10^{-12} \text{ eV}$ the superradiant cloud, and thus the constraints, are destroyed for $f_a \lesssim 10^{11} - 10^{12} \text{ GeV}$, depending precisely on the mass and the source.

Baryakhtar et al. [352] used a more advanced model for the ef-
fect of self-interactions on superradiant evolution, and a different (frequentist) statistical methodology, leading to more conservative bounds from stellar mass BHs (region near $10^{-12}$ eV). In particular, Baryakhtar et al. [352] found that QCD axions are only excluded for $m_a \lesssim 6 \times 10^{-12} M_\odot$ instead of the $2 \times 10^{-11} M_\odot$, as claimed in [350,351]. Baryakhtar et al. [352] also vetoed supermassive BHs (SMBHs) with spins measured at low significance, and thus found no ALP bounds in the SMBH region (near $10^{-18}$ eV); Mehta et al. [350, 351, 353], in constrast, included SMBHs in a (quasi-)Bayesian analysis with a flat prior on the spin and found non-trivial constraints. In summary, it appears likely that the lower end of the QCD axion mass range is excluded by superradiance, with the requirement $m_a \gtrsim 10^{-11}$ eV, but the exact location of this limit is still debated. It is also too early to claim that ultra-light ALP parameter space is excluded by SMBHs, though this may change with future work.

Long lasting, monochromatic gravitational wave signals, which can be distinguished from ordinary astrophysical sources by their clustering in a narrow frequency range, are expected to be produced by axions or ALPs annihilating to gravitons. Gravitational waves could also be sourced by axions/ALPs transitioning between energy levels of the effective gravitational atom. Accordingly, the gravitational wave detector Advanced LIGO should be sensitive to the axion in the $m_a \lesssim 10^{-10}$ eV region. LIGO measurements of BH spins in binary merger events could also provide statistical evidence for the presence of an axion [354, 355]. Similar signatures could arise for supermassive and intermediate mass BHs for particles with masses $\lesssim 10^{-15}$ eV. Gravitational waves from such sources could be detected at lower-frequency observatories such as LISA (Laser Interferometer Space Antenna). First attempts to use LIGO/Virgo data to search for monochromatic signatures of axion superradiance were recently made in [356, 357].

## 90.5 Cosmic Axions

### 90.5.1 Cosmic axion populations

There are two distinct populations of cosmic relic axion populations: a non-thermal one behaving as CDM, and a thermal one comprising a HDM component, in analogy to massive neutrinos. For $m_a \lesssim 0.01$ eV, thermal axions are dominantly produced by processes involving quarks and gluons [358, 359], while, for larger masses, the dominant thermalization process is $\pi + \pi \leftrightarrow \pi + a$ [173, 360, 361]. A number of evaluations exploiting cosmological precision data have found restrictive constraints on a possible HDM fraction that translate into an upper bound on the axion mass, $m_a \lesssim 1$ eV, if a LO axion-pion chiral effective field theory (EFT) analysis of the axion-pion thermalization rate is used [362–365]. Recently, it was found that the NLO contribution exceeds the LO contributions for masses below $m_a \lesssim 1.2 (0.12/C_{a\pi})$ eV [366]. Therefore, in order to assess the reach in sensitivity of future cosmological data sets, one has to improve the EFT description or to compute axion-pion scattering via lattice QCD techniques. Alternatively, recently Ref. [367] used pion-pion scattering data to overcome the EFT description issues and found the constraint $m_a \lesssim 0.24$ eV, see Fig. 90.4.

For $m_a \gtrsim 10$ eV, axions may decay faster than the age of the Universe (see Eq. (90.7)), depending on the decay constant, which would remove the cosmic axion population while injecting photons. If the axions were in thermal equilibrium in the early Universe, which depends on the inflationary reheat temperature and assumptions about the pre-BBN cosmology [368, 369], this excess radiation provides additional limits on the axion- or ALP-photon coupling up to very large masses [368, 369]. Such photons could contribute to the extragalactic background light (EBL), could ionize primordial hydrogen and thus contribute to the optical depth after recombination, or could spoil the agreement of big bang nucleosynthesis (BBN) with observations. The corresponding up-to-date bounds from Refs. [166, 167] are displayed in Fig. 90.1. It has recently been shown that there is an *irreducible axion background* [168] that arises from axion freeze-in before BBN; decays of that axion population provide strong constraints on axions with masses above roughly 100 eV (see Fig. 90.1). For high enough decay constants, high-mass axions ($m_a \gtrsim 10$ eV) can account for the

DM abundance through the misalignment mechanism, described in more detail below, though additional entropy dilution or fine-tuning is needed to avoid overproducing the DM abundance [370] (see also [371] for heavy axion production through lepton-flavor-violating decays). The decay of such high-mass axions to photon pairs could leave narrow spectral features in e.g. optical or X-ray backgrounds [126, 138–140, 161, 163, 372, 373] (see Fig. 90.1).

The main cosmological interest in axions at present derives from their possible role as CDM. In addition to thermal processes, axions are abundantly produced by the misalignment (MIS) mechanism [374–376]. The axion DM abundance crucially depends on the cosmological history. Let us first consider the so called *pre-inflationary PQ symmetry breaking scenario*, in which the PQ symmetry is broken before and during inflation and not restored afterwards. After the breakdown of the PQ symmetry, the axion field relaxes somewhere in the bottom of the “wine-bottle-bottom” potential. Near the QCD epoch, topological fluctuations of the gluon fields such as instantons explicitly break the PQ symmetry. This tilting of the “wine-bottle-bottom” drives the axion field toward the *CP*-conserving minimum, thereby exciting coherent oscillations of the axion field that ultimately represent a condensate of CDM. The fractional cosmic mass density in this homogeneous field mode, created by the MIS mechanism, is [33, 377–379],

$$ \begin{aligned} \Omega_a^{\text{MIS}} h^2 &\approx 0.12 \left( \frac{f_a}{9 \times 10^{11} \text{ GeV}} \right)^{1.165} F \Theta_i^2 \\ &\approx 0.12 \left( \frac{6 \text{ meV}}{m_a} \right)^{1.165} F \Theta_i^2, \end{aligned} \quad (90.22) $$

where $h$ is the present-day Hubble expansion parameter in units of 100 km $s^{-1}$ Mpc$^{-1}$, and $-\pi \le \Theta_i \le \pi$ is the initial “misalignment angle” relative to the *CP*-conserving position attained in the causally connected region which evolved into today’s observable Universe. $F = F(\Theta_i, f_a)$ is a factor accounting for anharmonicities in the axion potential. For $F\Theta_i^2 = \mathcal{O}(1)$, $m_a$ should be above $\sim 6$ meV in order that the cosmic axion density does not exceed the observed CDM density, $\Omega_{\text{CDM}}h^2 = 0.12$. However, much smaller axion masses (much higher PQ scales) are still possible if entropy is diluted for example by the late decay of a scalar condensate (see for example [380] and references therein) or the initial value $\Theta_i$ just happens to be small enough in today’s observable Universe (“anthropic axion window” [381]). In the latter cosmological scenario, however, quantum fluctuations of the axion field during inflation are expected to lead to isocurvature density fluctuations which get imprinted to the temperature fluctuations of the CMB [382, 383]. Their non-observation puts severe constraints on the Hubble expansion rate $H_I$ during inflation [384–389], which read, in the simplest cosmological inflationary scenario,

$$ H_I \lesssim 5.7 \times 10^8 \text{ GeV} \left( \frac{5 \text{ neV}}{m_a} \right)^{0.4175}, \quad (90.23) $$

if axions represent all of DM. In Ref. [390] an alternative to the MIS mechanism was proposed: the so-called kinetic MIS mechanism. In this case, the axion field is assumed to have an initial velocity which may be generated, e.g., by a hypothesized explicit breaking of the axion shift symmetry in the early universe. The amount of QCD axion dark matter generated by the kinetic MIS mechanism can fit the observed dark matter abundance for any $f_a \lesssim 1.5 \times 10^{11}$ GeV, down to the minimum value allowed by the SN 1987A constraint.

In the above prediction of the fractional cosmic mass density in axions, the exponent, 1.165, arises from the non-trivial temperature dependence of the topological susceptibility $\chi(T) = m_A^2(T) f_A^2$ at temperatures slightly above the QCD quark-hadron phase transition. Lattice QCD calculations of this exponent [33, 323, 325, 391–393], but also Ref. [324], found it to be remarkably close to the prediction of the dilute instanton gas approximation [394] which was previously exploited. Therefore, the state-of-the-art prediction of the axion mass relevant for DM for a fixed initial misalignment angle $\Theta_i$ differs from the previous prediction by just a factor of order one.

In the *post-inflationary PQ symmetry breaking scenario*, on the other hand, $\Theta_i$ will take on different values in different patches of
the present Universe. The average contribution is [33,377–379]

$$ \Omega_a^{\text{MIS}} h^2 \approx 0.12 \left( \frac{30 \ \mu\text{eV}}{m_a} \right)^{1.165} . \quad (90.24) $$

The decay of cosmic strings and domain walls gives rise to a further population of CDM axions, whose abundance suffers from significant uncertainties [321, 327–329, 378, 379, 395–403] which arise from the difficulty in understanding the energy loss process of topological defects and the generated axion spectrum in a quantitative way. In fact, in the present state-of-the-art it is still possible that the CDM contribution from the decay of topological defects is subdominant [327] or overwhelmingly large [328] in comparison to the one from the MIS mechanism. Correspondingly, the plausible range of axion masses providing all of CDM in scenarios with postinflationary PQ symmetry breaking is still rather large, namely [327, 328]

$$ m_a \approx 26 \ \mu\text{eV} - 0.5 \ \text{meV}, \quad (90.25) $$

for models with short-lived (requiring unit color anomaly $N=1$) domain walls, such as the KSVZ model. On the other hand, the most recent adaptive mesh refinement axion string simulations, which are able to evolve over larger dynamic ranges than static lattice simulations, suggest the mass range $m_a \approx 40–180 \ \mu\text{eV}$ [321]. For models with long-lived ($N>1$) domain walls, such as an accidental DFSZ model [404], where the PQ symmetry is broken by higher dimensional Planck suppressed operators, the mass is predicted to be significantly higher [287, 329, 404–406],

$$ m_a \approx (0.58 – 130) \ \text{meV}. \quad (90.26) $$

However, the upper part of the predicted range is in conflict with stellar energy-loss limits on the axion. The pre- and post-inflationary mass predictions are summarized in Fig. 90.4.

In this post-inflationary PQ symmetry breakdown scenario, the spatial axion density variations are large at the QCD transition, and they are not erased by free streaming. Gravitationally bound “axion miniclusters” form before and around matter-radiation equality [407–409]. A significant fraction of CDM axions can reside in these bound objects [400, 410]. Remarkably, the minicluster fraction can be bounded by gravitational lensing [411–413], although more simulations are required to understand whether miniclusters are dense enough and survive in sufficient quantities for lensing bounds to apply.

The non-thermal production mechanisms attributed to axions are generic to light bosonic weakly interacting particles such as ALPs [18]. The relic abundance is set by the epoch when the axion mass becomes significant, $3H(t) \approx m_a(t)$, and ALP field oscillations begin. For ALPs to contribute to the DM density this epoch must precede that of matter radiation equality. For a temperature independent ALP mass this leads to the bound:

$$ m_a \gtrsim 7 \times 10^{-28} \ \text{eV} \left( \frac{\Omega_m h^2}{0.15} \right)^{1/2} \left( \frac{1+z_{\text{eq}}}{3.4 \times 10^3} \right)^{3/2} . \quad (90.27) $$

ALPs lighter than this bound are allowed if their cosmic energy density is small, but they are quite distinct from other forms of DM [414]. Ignoring anharmonicities in the ALP potential, and taking the ALP mass to be temperature independent, the relic density in DM ALPs due to the MIS mechanism is given by

$$ \Omega_{\text{ALP}}^{\text{MIS}} h^2 = 0.12 \left( \frac{m_a}{4.7 \times 10^{-19} \text{ eV}} \right)^{1/2} \left( \frac{f_a}{10^{16} \text{ GeV}} \right)^2 \times \left( \frac{\Omega_m h^2}{0.15} \right)^{3/4} \left( \frac{1+z_{\text{eq}}}{3.4 \times 10^3} \right)^{-3/4} \Theta_i^2 . \quad (90.28) $$

An ALP decay constant near the GUT scale gives the correct relic abundance for ultralight ALPs (ULAs) with $m_{\text{ULA}} \approx 10^{-19} \ \text{eV}$ [415, 416]. ULAs encompass the entire Earth in a single Compton wavelength, and for large occupation numbers are modelled as a coherent classical field. The coherence time is determined by the mass and virial velocity in the Milky Way,

$\tau_{\text{coh}} \sim 1/m_{\text{ULA}}v_{\text{vir}}^2$, with the detailed properties described by a stochastic model with an approximately Rayleigh Jeans distribution [417]. Natural models for ULAs can be found in string and M-theory compactifications [7, 8, 10–15], in field theory with accidental symmetries [418, 419], or new hidden strongly coupled sectors [420, 421].

In addition to the gravitational potential energy, the ULA field also carries gradient energy. On scales where the gradient energy is non-negligible, ULAs acquire an effective pressure and do not behave as CDM. The gradient energy opposes gravitational collapse, leading to a Jeans scale below which perturbations are stable [422]. The Jeans scale suppresses linear cosmological structure formation relative to CDM [423–425]. The Jeans scale at matter-radiation equality in the case that ULAs make up all of CDM is:

$$ k_{J,\text{eq}} = 8.7 \ \text{Mpc}^{-1} \left( \frac{1 + z_{\text{eq}}}{3.4 \times 10^3} \right)^{-1/4} \left( \frac{h^2 \Omega_{\text{ALP}}^{\text{MIS}}}{0.12} \right)^{1/4} \times \left( \frac{m_{\text{ULA}}}{10^{-22} \ \text{eV}} \right)^{1/2} . \quad (90.29) $$

On non-linear scales the gradient energy leads to the existence of a class of pseudo-solitons known as oscillators, or axion stars [426]. Axion stars are expected to form in all cosmological scenarios, and for all types of ALPs, including the QCD axion [427–429]. Some axion stars may explode into radio photons, heating the interstellar medium [104].

Cosmological and astrophysical observations are consistent with the CDM model, and departures from it are only allowed on the scales of the smallest observed DM structures with $M \sim 10^{6–8} M_{\odot}$. The CMB power spectrum and galaxy autocorrelation power spectrum limit the ULA mass to $m_{\text{ULA}} \gtrsim \mathcal{O}(\text{few}) \times 10^{-24} \ \text{eV}$ from linear theory of structure formation [414, 430]. Analytic models [431] and N-body simulations [432] for non-linear structures show that halo formation is suppressed in ULA models relative to CDM. This leads to constraints on the ULA mass of $m_{\text{ULA}} > 10^{-22} \ \text{eV}$ from observations of high-z galaxies [432–434], and $m_{\text{ULA}} > 2 \times 10^{-20} \ \text{eV}$ from the Lyman-alpha forest flux power spectrum [435]. Including the effects of anharmonicities on structure formation with ALPs can weaken these bounds if the misalignment angle $\Theta_i \approx \pi$ [436]. A comprehensive study of Milky Way satellites by the DES collaboration resulted in the bound $m_{\text{ULA}} > 2.9 \times 10^{-21} \ \text{eV}$ [437].

Cosmological simulations that treat gradient energy in the ULA field beyond the N-body approximation have just recently become available [427, 438–440], and show, among other things, evidence for the formation of axion stars in the centres of ULA halos (various consequences of axion stars are considered in Refs. [441]). These central axion stars have been conjectured to play a role in the apparently cored density profiles of dwarf spheroidal galaxies, and other central galactic regions [127, 427, 442–446]. However, the relationship between the halo mass and the axion star mass [447] leads to problems with this scenario in some galaxies [448–450]. It should be emphasised that many of the conclusions about the role of ULA axion stars in galactic dynamics are based on use of simulation results that do not contain baryons and feedback could be important [451–453].

Inside DM halos the axion gradient energy causes coherence on the de Broglie wavelength and fluctuations on the coherence time [427, 454]. These fluctuations can be thought of as short-lived quasiparticles and lead to relaxation processes that can be described statistically [416, 455] (this relaxation processes also leads to the gravitational condensation of axion stars [428]). The typical relaxation time is:

$$ t \sim 10^{10} \text{ years} \left( \frac{m_{\text{ULA}}}{10^{-22} \text{ eV}} \right)^3 \left( \frac{v}{100 \text{ km s}^{-1}} \right)^2 \left( \frac{r}{5 \text{ kpc}} \right)^4 , \quad (90.30) $$

where $v$ and $r$ are the velocity and radius of the orbit in the host DM halo.

Relaxation processes such as these are not observed in galaxies, though there are some circumstances where they may be desirable [416]. An absence of observed relaxation can be used to set
limits on the ULA mass. An absence of Milky Way disk thickening excludes $m_{\text{ULA}} > 0.6 \times 10^{-22}$ eV [456], while stellar streams give the stronger bound $m_{\text{ULA}} > 1.5 \times 10^{-22}$ eV [457]. The survival of the old star cluster in Eridanus II [458] excludes the range of masses $10^{-21}$ eV $\lesssim m_{\text{ULA}} \lesssim 10^{-19}$ eV [459]. Recently Ref. [460] finds that ULA DM is excluded for $m_{\text{ULA}} \lesssim 3 \times 10^{-19}$ eV at 99% CL because such DM would kinematically heat the stars in the Segue I and Segue II ultra-faint dwarf galaxies to a level inconsistent with their observed dynamics.

Finally, one should note that the beyond-CDM physics of ULAs (Jeans scale, relaxation, axion star formation) of course also applies to the QCD axion on smaller length scales. This is of particular interest inside axion miniclusters [407, 408, 428, 429].

## 90.5.2 Electron recoil searches

In a DM direct detection experiment, a DM ALP featuring a coupling to the electron can be absorbed by the target material, leading to a mono-energetic electronic recoil signal peaked at $m_a$. This mechanism allowed the EDELWEISS [461], PandaX [221], SuperCDMS [222] and XENON1T [462] collaborations to put the bounds on the axion-electron coupling displayed in Fig. 90.2 in the keV mass range. As discussed in the last sub-section, additionally dynamics beyond the standard misalignment mechanism are needed to explain the axion DM abundance for keV-scale axion masses (e.g., [370, 371]). Moreover, such models are subject to stringent constraints from axion decays to photons [370].

## 90.5.3 Telescope searches

The two-photon decay is extremely slow for axions with masses in the CDM regime, but could be detectable for eV masses. The signature would be a quasi-monochromatic emission line from galaxies and galaxy clusters. The expected optical line intensity for DFSZ axions is similar to the continuum night emission. A search for optical line emission in two Abell clusters using spectra from the VIMOS (Visible Multi-Object Spectrograph) integral field unit at the Very Large Telescope (VLT) excludes axions and ALPs with a two photon coupling bigger than $\sim 5 \times 10^{-12} \text{ GeV}^{-1}$ in the mass range between 4.5 and 5.5 eV [163], see Fig. 90.1. Spectral data on the dwarf spheroidal galaxy Leo T from the Multi Unit Spectroscopic Explorer (MUSE) at the VLT improve these constraints by more than an order of magnitude for ALP masses between 2.7 and 5.3 eV [162], see Fig. 90.1.

Very low-mass axions in halos produce a weak quasi-monochromatic radio line. Virial velocities in undisrupted dwarf galaxies are very low, and the axion decay line would therefore be extremely narrow. A search with the Haystack radio telescope on three nearby dwarf galaxies gave a limit $|g_{a\gamma\gamma}| < 1.0 \times 10^{-9} \text{ GeV}^{-1}$ at 96% CL for $298 < m_a < 363 \text{ µeV}$ [161]. However, this combination of $m_a$ and $g_{a\gamma\gamma}$ does not exclude plausible axion models.

A monochromatic signal is also produced in the conversion of DM axions in the background of slowly varying galactic B-fields [463]. The signal is, however, sensitive to magnetic field power on the scale of the axion mass [464]. Present and future radio telescopes appear to be able to probe ALP DM in the mass range $0.1 - 100 \text{ µeV}$ for couplings $g_{a\gamma\gamma} \gtrsim 10^{-13} \text{ GeV}^{-1}$ [464] – unfortunately not reaching down to the benchmark QCD axion sensitivity.

Resonant conversion of QCD axion or ALP DM in NS magnetospheres may give a detectable signal from individual NSs and populations of NSs for axion masses in the $\mu$eV range [465, 466]. Recent analyses of radio data in the frequency range 1-40 GHz from several NSs have found no evidence for a narrow peak predicted from this mechanism and therefore put bounds on the axion-photon coupling around $10^{-11} \text{ GeV}^{-2}$ for $m_a \sim 10 \text{ µeV}$ [133–136], see Fig. 90.1. However, these limits may still suffer from uncertainties because the line intensity is difficult to predict in detail in the complicated environments of NSs [135, 467–470]. Still, next-generation radio telescopes such as the Square Kilometer Array (SKA) may reach sensitivity to QCD axion-strength couplings [136]. Stimulated ALP decays in high radiation environments may be detectable, by e.g. the future SKA, down to $g_{a\gamma\gamma} \gtrsim 10^{-11} \text{ GeV}^{-1}$, for masses between $\mu$eV and 0.1 meV [471]. Furthermore, in condensed ALP dark matter structures such as

ALP stars, a parametric instability may lead to an exponential enhancement of the photon flux from ALP-photon conversion by a factor $\sim \exp[|g_{a\gamma\gamma}| \int ds \rho_a^{1/2}(s)$], where the integral is along any photon trajectory through the ALP overdensity [472, 473].

Photon propagation on an ULA DM background can induce birefringence that can be compared with upper limits from the CMB [106, 141, 474] and may also be probed with other sources such as pulsars [475].

## 90.5.4 Microwave cavity experiments

Over a large part of the plausible $m_a$ range for CDM, Galactic halo axions may be detected by their resonant conversion into a quasi-monochromatic microwave signal in a high-Q electromagnetic cavity permeated by a strong, static B field [9, 476, 477]. The cavity frequency is tunable, and the signal is maximized when the cavity resonant frequency is the total axion energy, rest mass plus kinetic energy, of $\nu = (m_a/2\pi) [1 + O(10^{-6})]$, the width above the rest mass representing the axions' virial distribution in the galaxy near Earth. The frequency spectrum may also contain finer structure from axions more recently fallen into the galactic potential and not yet completely virialized [478, 479] or otherwise incompletely thermalized.

<figure><img src="image_382.png" /><figcaption>**Figure 90.5:** Exclusion plot for ALP-photon coupling with closeup on the parameter space of RF cavity experiments as described in the text. This is a zoom-in of Fig. 90.1 that highlights regions probed by axion DM direct detection experiments. In particular, note that all of the terrestrial experiments shaded in red, except that of CAST, assume that the axion is 100% of the local DM. Figure courtesy of Ciaran O'Hare [61], including data from Refs. [43, 64–68, 76, 78, 81–84, 86, 87, 91, 93, 99, 122, 123, 145, 216, 480–484]</figcaption></figure>

The feasibility of this technique was established in early experiments (RBF (Rochester-Brookhaven-Florida) and UF (University of Florida)) of relatively small sensitive volume, $O(1$ liter), with HFET-based microwave amplifiers, setting limits in the range $4.5 < m_a < 16.3 \text{ µeV}$ [91, 95, 482], but lacking by 2–3 orders of magnitude the sensitivity required to detect realistic axions, see Fig. 90.5.

Later, the first generation of ADMX ($B \sim 8$ T, $V \sim 200$ liters) achieved sensitivity to KSVZ axions, assuming they saturate the local DM density and are well virialized, over the mass range $1.9–3.3 \text{ µeV}$ [485]. Should halo axions have a significant narrow component not yet virialized, ADMX was sensitive to DFSZ axions over the entire mass range [486]. The corresponding 90% CL exclusion regions shown in Fig. 90.5 assume a local CDM density of $7.5 \times 10^{-25} \text{ g cm}^{-3}$ (450 MeV cm$^{-3}$).

Somewhat later, the ADMX experiment commissioned an upgrade [64] that replaced the microwave HFET amplifiers by near-quantum-limited low-noise dc SQUID microwave amplifiers [487] and, more recently, with Josephson parametric amplifiers (related
to the earlier dc SQUID amplifiers), also with noise near-quantum-limited [481]. With its large volume and low noise, ADMX has achieved an unprecedented axion DM sensitivity in the mass range between 2.7 and 4.2 μeV [66–68, 480], down to the DFSZ benchmark axion-photon coupling over much of the range, see Fig. 90.5. This apparatus is also sensitive to certain other hypothetical light bosons, such as hidden photons or chameleons, over a limited parameter space [18, 488, 489].

ADMX continues to scan at DFSZ sensitivity, and has since been joined by an array of haloscopes. One of the CAPP experiments has also achieved DFSZ sensitivity over a small range of frequencies near 4.55 μeV [76]. A number of other experiments from the CAPP group have operated at higher frequencies, many with KSVZ or near KSVZ sensitivity. [72–74, 74, 75] The HAYSTAC experiment also operates at or near KSVZ sensitivity and was the first to demonstrate near-quantum limited noise using a Josephson parametric amplifier [82, 490], and demonstrated ‘state-squeezing’ to increase scan speed by nearly a factor of 2. [83, 84] Other haloscopes operating at or near KSVZ over narrow bands include TASEH [93] and QUAX-αγ [90]. The ORGAN experiment is the highest frequency haloscope, operating with a small cavity and sensitive to axion-like-particle masses up to 67.1 μeV, though not yet with sensitivity to the KSVZ model. [86, 87] There are also a number of other haloscopes sensitive to axion-like-particles with couplings above those predicted by KSVZ. [65, 81, 88, 89, 483] Fig. 90.5 shows the exclusions on axion-photon coupling from these haloscopes to date.

### 90.5.5 New concepts for axion DM direct detection

Other new concepts for searching for axion DM are also being investigated. “Plasma haloscopes” are under development by the ALPHA consortium [491]. This concept uses a metamaterial consisting of a parallel-wire array, thereby inducing an effective photon mass which is tunable by adjusting wire separation, and sensitive to axion masses 35–400 μeV. Another detector concept exploits the fact that a magnetized mirror would radiate photons in the background of axion DM, which could be collected like in a dish antenna [492]. The proposed MADMAX experiment will place a stack of dielectric layers in a magnetic field in order to resonantly enhance the photon signal, aiming a sensitivity to probe the mass range $40 \mu eV \lesssim m_a \lesssim 200 \mu eV$ [493, 494]. Optical dielectric haloscopes with single photon signal detection have been proposed to search for axions in the 50 meV–10 eV mass range [495]. Other semi-resonant or nonresonant axion haloscope ideas to read high mass axions that are based around similar principles but novel configurations include BREAD [496] and CADEX [497].

Absorption of axions on molecular transitions can be sensitive to the axion pseudoscalar coupling $g_{aNN}$ to nucleons or the pseudoscalar coupling $g_{aee}$ to electrons in the 0.5–20 eV range [498]. It has also been pointed out that the coupling induced by an axion field between photonic modes [96, 499] or atomic states [500] separated by an energy equal to the axion mass may be leveraged for axion detection, though the implementation may be challenging. Certain magnetic topological insulators are conjectured to contain axion-polaritons, quasiparticles composed of magnons and the electric field, with magnetically tunable mass in the meV to eV range [501–503]. Axion dark matter leads to resonant photon emission from such materials, which can be detected [504, 505].

In the intermediate mass region, neV $\lesssim m_a \lesssim 0.1$ μeV, one may exploit a cooled LC circuit and precision magnetometry to search for the oscillating electric current induced by DM axions in a strong magnetic field [506, 507]. A number of small-scale pathfinder experiments have implemented this approach: ABRA-CADABRA [62, 63] used a nonresonant configuration in a toroidal magnet for a broadband search, ADMX-SLIC [70] used a resonant configuration in a solenoid magnet for a narrowband search, SHAFT [92] used a nonresonant configuration in a toroidal magnet enhanced with a ferromagnetic core, BASE [80] used antiprotons in a Penning trap as the LC circuit, and DANCE [508] used a laser interferometer. The exclusion limits from these experiments are shown in Fig. 90.1; none are yet sensitive to the QCD axion. Future work in this direction is expected to have much more competitive limits [509, 510].

The oscillating Galactic DM axion field induces oscillating nu-

clear electric dipole moments (EDMs) [35],

$$d_N(t) = g_{aN\gamma} \sqrt{2\rho_{aDM}} \cos(m_a t)/m_a, \quad (90.31)$$

where $g_{aN\gamma}$ is the coupling of the axion to the nucleon EDM operator, defined in Eq. (90.4). An analysis of the ratio of spin-precession frequencies of stored ultracold neutrons and $^{199}$Hg atoms measured by neutron EDM experiments for an axion-induced oscillating neutron EDM revealed no signal consistent with axion DM, excluding a sizeable region of parameter space in the mass region $10^{-24}$ eV $\le m_a \le 10^{-17}$ eV [40], which surpass the limits on anomalous energy loss of SN 1987A by more than seven orders of magnitude and are competitive with the ones from the requirement of successful BBN established in [38].<fnref n="1" /> In fact, the oscillating dark matter axion field could increase the neutron-proton mass difference at neutron freeze-out and thus result in the underproduction of $^4$He during BBN. The oscillating EDMs cause also the precession of nuclear spins in a nucleon spin polarized sample in the presence of an electric field. The resulting transverse magnetization can be searched for by exploiting magnetic-resonance (MR) techniques, which are most sensitive in the range of low oscillation frequencies corresponding to sub-neV axion masses. The aim of the corresponding Cosmic Axion Spin Precession Experiment (CASPER) [36] is to probe axion DM in the anthropic window, $f_a \gtrsim 10^{15}$ GeV ($m_a \lesssim$ neV), motivated from Grand Unification [511–514]. There are two interactions probed through MR: the electric dipole coupling as searched for by CASPER-electric (with first result presented in [39]) and an interaction between the axion field gradient and the nuclear spin, which is aimed for by CASPER-Gradient. In the meantime, the latter has been explored through comagnetometry between a variety of nuclei: H-$^{12}$C in CASPER-ZULF [246, 247], Xe-Rb in NASDUCK [515], and K-$^3$He [251, 252]. The exclusions from these searches are shown in Fig. 90.3. Sub-μeV ALP masses can also be probed by using the storage ring EDM method proposed in Ref. [516] which exploits a combination of B and E-fields to produce a resonance between the g-2 spin precession frequency and the DM ALP field oscillation frequency. This method, however, does not reach the sensitivity to probe the QCD axion prediction for $g_{aN\gamma}$. An eventually non-zero axion electron coupling $g_{aee}$ will lead to an electron spin precession about the axion DM wind [517, 518]. The QUAX a-e experiment exploits MR inside a magnetized material [519]. Because of the higher Larmor frequency of the electron, it is sensitive in the classic window. Its first run established the bound in the mass range 42.4 to 43.2 μeV [227], slightly better than the solar v bound, shown in Fig. 90.2.

## 90.6 Conclusions

There is a strengthening physics case for very weakly coupled light particles beyond the Standard Model. The elegant solution of the strong CP problem proposed by Peccei and Quinn yields a particularly strong motivation for the axion. In many theoretically appealing ultraviolet completions of the Standard Model axions and ALPs occur automatically. Moreover, they are natural CDM candidates. Perhaps the first hints of their existence have already been seen in the anomalous excessive cooling of stars and the anomalous transparency of the Universe for very high energy gamma rays. Interestingly, a significant portion of previously unexplored, but phenomenologically very interesting and theoretically very well motivated axion and ALP parameter space can be explored in the foreseeable future by a number of terrestrial experiments searching for axion/ALP DM, for solar axions/ALPs, and for light apparently shining through a wall.

## 90.7 Acknowledgements

It is a pleasure to thank Masha Baryakhtar, Francesca Calore, Luca Di Luzio, Andrew Geraci, Maurizio Giannotti, Igor Irastorza, David J.E. Marsh, Alessandro Mirizzi, Ciaran O’Hare, Georg Raffelt, Ken’ichi Saikawa, Yannis Semertzidis, Guenter

<footnote n="1">¹These limits are still to the left of the canonical QCD axion line. However, as already mentioned in non-minimal models where the QCD axion is a mediator between strongly interacting hidden sectors enjoying a discrete, $Z_N$, shift symmetry, the QCD axion line can be shifted to the left in discrete steps, without any fine tuning [57, 58, 421].</footnote>
Sigl, Sergey Troitsky, and Lindley Winslow for discussions, suggestions, and comments on this review. Particular thanks to Ciaran O'Hare for maintaining a publicly accessible collection of axion plots and providing updates to accommodate this review [61].

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# 91. Searches for Quark and Lepton Compositeness

Revised October 2023 by K. Hikasa (Tohoku U.), M. Tanabashi (Nagoya U.; KMI, Nagoya U.), K. Terashi (ICEPP, Tokyo U.) and N. Varelas (U. of Illinois at Chicago).

## 91.1 Limits on contact interactions

If quarks and leptons are made of constituents, then at the scale of constituent binding energies (compositeness scale) there should appear new interactions among them. At energies much below the compositeness scale (Λ), these interactions are suppressed by inverse powers of Λ. The dominant effect of the compositeness of fermion ψ should come from the lowest dimensional interactions with four fermions (contact terms), whose most general flavor-diagonal color-singlet chirally invariant form reads [1, 2]

$$ \mathcal{L} = \mathcal{L}_{LL} + \mathcal{L}_{RR} + \mathcal{L}_{LR} + \mathcal{L}_{RL}, $$

with

$$ \begin{aligned} \mathcal{L}_{LL} &= \frac{g_{\text{contact}}^2}{2\Lambda^2} \sum_{i,j} \eta_{LL}^{ij} (\bar{\psi}_L^i \gamma_{\mu} \psi_L^i) (\bar{\psi}_L^j \gamma^{\mu} \psi_L^j), \\ \mathcal{L}_{RR} &= \frac{g_{\text{contact}}^2}{2\Lambda^2} \sum_{i,j} \eta_{RR}^{ij} (\bar{\psi}_R^i \gamma_{\mu} \psi_R^i) (\bar{\psi}_R^j \gamma^{\mu} \psi_R^j), \\ \mathcal{L}_{LR} &= \frac{g_{\text{contact}}^2}{2\Lambda^2} \sum_{i,j} \eta_{LR}^{ij} (\bar{\psi}_L^i \gamma_{\mu} \psi_L^i) (\bar{\psi}_R^j \gamma^{\mu} \psi_R^j), \\ \mathcal{L}_{RL} &= \frac{g_{\text{contact}}^2}{2\Lambda^2} \sum_{i,j} \eta_{RL}^{ij} (\bar{\psi}_R^i \gamma_{\mu} \psi_R^i) (\bar{\psi}_L^j \gamma^{\mu} \psi_L^j), \end{aligned} \quad (91.1) $$

where i, j are the indices of fermion species. Color and other indices are suppressed in Eq. (91.1). Chiral invariance provides a natural explanation why quark and lepton masses are much smaller than their inverse size (inverse radius) Λ. Note $\eta_{\alpha\beta}^{ij} = \eta_{\beta\alpha}^{ji}$, thus $\mathcal{L}_{LR}$ and $\mathcal{L}_{RL}$ are identical. Therefore, in order to specify the contact interaction among the same fermion species i = j, it is enough to use $\eta_{LL}$, $\eta_{RR}$ and $\eta_{LR}$. We will suppress the indices of fermion species hereafter. We may determine the scale Λ unambiguously by using the above form of the effective interactions; the conventional method [1] is to fix its scale by setting $g_{\text{contact}}^2/4\pi = g_{\text{contact}}^2(\Lambda)/4\pi = 1$ for the new strong interaction coupling and by setting the largest magnitude of the coefficients $\eta_{\alpha\beta}$ to be unity. In the following, we denote

$$ \Lambda = \Lambda_{LL}^{\pm}, \Lambda_{RR}^{\pm}, \Lambda_{VV}^{\pm}, \Lambda_{AA}^{\pm}, \Lambda_{V-A}^{\pm}, \quad (91.2) $$

for $(\eta_{LL}, \eta_{RR}, \eta_{LR}) = (\pm 1, 0, 0)$, $(0, \pm 1, 0)$, $(\pm 1, \pm 1, \pm 1)$, $(\pm 1, \pm 1, \mp 1)$, $(0, 0, \pm 1)$. Such interactions can arise by interacting constituents (when the fermions have common constituents), and/or by exchanging the binding quanta (whenever binding quanta couple to constituents of both particles).

Fermion scattering amplitude induced from the contact interaction in Eq. (91.1) interferes with the Standard Model (SM) amplitude destructively or constructively [2]. The sign of interference depends on the sign of $\eta_{\alpha\beta}$ ($\alpha, \beta = L, R$). For instance, in the parton level $qq \to qq$ scattering cross section in the $\Lambda_{LL}^{\pm}$ model, the contact interaction amplitude and the SM gluon exchange amplitude interfere destructively for $\eta_{LL} = +1$, while they interfere constructively for $\eta_{LL} = -1$. In models of quark compositeness, the quark scattering cross sections induced from the contact interactions receive sizable QCD radiative corrections. The exact next-to-leading order (NLO) QCD corrections to the quark scatterings cross sections induced by the contact interactions are given in [3].

Over the last three decades experiments at the CERN SpS [4, 5], the Fermilab Tevatron [6, 7], and the CERN LHC [8–12] have searched for quark contact interactions, characterized by the four-fermion effective Lagrangian in Eq. (91.1), using jet final states. These searches have been performed primarily by studying the angular distribution of the two highest transverse momentum, $p_T$, jets (dijets), and the inclusive jet $p_T$ spectrum. The variable $\chi = \exp((y_1 - y_2))$ is used to measure the dijet angular distribution, where $y_1$ and $y_2$ are the rapidities of the two jets with

the highest transverse momenta. For collinear massless parton scattering, $\chi$ is related to the polar scattering angle $\theta^*$ in the partonic center-of-mass frame by $\chi = (1+|\cos\theta^*|)/(1-|\cos\theta^*|)$. The choice of $\chi$ is motivated by the fact that the angular distribution for Rutherford scattering, which is proportional to $1/(1-\cos\theta^*)^2$, is independent of $\chi$. In perturbative QCD the $\chi$ distributions are relatively uniform and only mildly modified by higher-order QCD or electroweak corrections. Signatures of quark contact interactions exhibit more isotropic angular distribution than QCD and can be identified as an excess at low values of $\chi$. In the inclusive jet cross section measurement, quark contact interaction effects are searched for as deviations from the predictions of perturbative QCD in the tails of the high-$p_T$ jet spectrum [11].

<figure><img src="image_383.png" /><figcaption>Figure 91.1: Normalized dijet angular distributions in several dijet mass (mjj) ranges. The data distributions are compared to PYTHIA8 predictions with NLO and electroweak corrections applied (solid line) and with the predictions including a contact interaction (CI) term in which only left-handed quarks participate of compositeness scale $\Lambda_{LL}^{\pm} = 15$ TeV (dashed line) and $\Lambda_{LL}^{-} = 22$ TeV (dotted line). The theoretical uncertainties and the total theoretical and experimental uncertainties in the predictions are displayed as shaded bands around the SM prediction. Figure adopted from [9].</figcaption></figure>

Recent results from the LHC, using data collected at proton-proton center-of-mass energy of $\sqrt{s} = 13$ TeV, extend previous limits on quark contact interactions. Figure 91.1 shows the normalized dijet angular distributions for several dijet mass ranges measured in ATLAS [9] at $\sqrt{s} = 13$ TeV. The data distributions are compared with SM predictions, estimated using PYTHIA8 [13] with GEANT4-based [14] ATLAS detector simulation and corrected to NLO QCD calculation provided by NLO Jet++ [15] including electroweak corrections [16]. Predictions including a contact interaction term in which only left-handed quarks participate at compositeness scale $\Lambda_{LL}^{\pm} = 15$ TeV ($\Lambda_{LL}^{-} = 22$ TeV) with destructive (constructive) interference are also shown. Over a wide range of $\chi$ and dijet mass the data are well described by the SM predictions. Using the dijet angular distributions measured at high dijet masses and $\sqrt{s} = 13$ TeV, the ATLAS [9] and CMS [12] Collaborations have set 95% confidence level (C.L.) lower limits on the contact interaction scale $\Lambda$, ranging from 9.2 to 29.5 TeV for different quark contact interaction models that correspond to various combinations of $(\eta_{LL}, \eta_{RR}, \eta_{LR})$, as summarized in Figure 91.2. The contact interaction scale limits extracted using the dijet angular distributions include the exact NLO QCD corrections to dijet production induced by contact interactions [3]. In proton-
proton collisions, the $\Lambda_{LL}^{\pm}$ and $\Lambda_{RR}^{\pm}$ contact interaction models result in identical tree-level cross sections and NLO QCD corrections and yield the same exclusion limits. For $\Lambda_{VV}^{\pm}$ and $\Lambda_{AA}^{\pm}$, the contact interaction predictions are identical at tree level, but exhibit different NLO QCD corrections and yield different exclusion limits.

<figure><img src="image_384.png" /><figcaption>**Figure 91.2:** Observed (solid lines) and expected (dashed lines) 95% C.L. lower limits on the contact interaction scale $\Lambda$ for different contact interaction models from ATLAS [9] and CMS [12] using the dijet angular distributions. The contact interaction models used for the dijet angular distributions include the exact NLO QCD corrections to dijet production. The shaded band for the $\Lambda_{LL/RR}^{+}$ model indicates the range of contact interaction scale that was not excluded in ATLAS [9], due to an observed excess in the data that produced a weaker limit. The observed 95% C.L. lower limit on the $\Lambda_{LL}^{-}$ from CMS [17] is shown as the second solid line in brown.</figcaption></figure>

If leptons ($l$) and quarks ($q$) are composite with common constituents, the interaction of these constituents will manifest itself in the form of a $llqq$-type four-fermion contact interaction Lagrangian at energies below the compositeness scale $\Lambda$. The $llqq$ terms in the contact interaction Lagrangian can be expressed as

$$ 
\begin{aligned}
\mathcal{L}_{LL} &= \frac{g_{\text{contact}}^2}{\Lambda^2} \sum_{i,j} \eta_{LL}^{ij} (\bar{q}_L^i \gamma_\mu q_L^i) (\bar{l}_L^j \gamma^\mu l_L^j), \\
\mathcal{L}_{RR} &= \frac{g_{\text{contact}}^2}{\Lambda^2} \sum_{i,j} \eta_{RR}^{ij} (\bar{q}_R^i \gamma_\mu q_R^i) (\bar{l}_R^j \gamma^\mu l_R^j), \\
\mathcal{L}_{LR} &= \frac{g_{\text{contact}}^2}{\Lambda^2} \sum_{i,j} \eta_{LR}^{ij} (\bar{q}_L^i \gamma_\mu q_L^i) (\bar{l}_R^j \gamma^\mu l_R^j), \\
\mathcal{L}_{RL} &= \frac{g_{\text{contact}}^2}{\Lambda^2} \sum_{i,j} \eta_{RL}^{ij} (\bar{q}_R^i \gamma_\mu q_R^i) (\bar{l}_L^j \gamma^\mu l_L^j).
\end{aligned} 
\quad (91.3) $$

Searches on quark-lepton compositeness have been reported from experiments at LEP [18–21], HERA [22, 23], the Tevatron [24, 25], and recently from the ATLAS [26, 27] and CMS [28, 29] experiments at the LHC. The most stringent searches for $llqq$ contact interactions are performed by the LHC experiments using high-mass oppositely-charged lepton pairs produced through the $q\bar{q} \to l^+l^-$ Drell-Yan process. The contact interaction amplitude of the $u\bar{u} \to l^+l^-$ process ($l = e$ or $\mu$) interferes with the corresponding SM amplitude constructively (destructively) for $\eta_{\alpha\beta}^{ul} = -1$ ($\eta_{\alpha\beta}^{ul} = +1$). The ATLAS Collaboration, using a 139 fb$^{-1}$ dataset at $\sqrt{s} = 13$ TeV, has extracted limits on the $llqq$ contact interaction for the right-right ($\eta_{RR} = \pm 1$, $\eta_{LL} = \eta_{LR} = \eta_{RL} = 0$), left-left ($\eta_{LL} = \pm 1$, $\eta_{RR} = \eta_{LR} = \eta_{RL} = 0$), and left-right ($\eta_{LR} = \eta_{RL} = \pm 1$, $\eta_{RR} = \eta_{LL} = 0$) models. In

Ref. [27], ATLAS used a background estimate directly obtained from data. Combining the dielectron and dimuon channels, the 95% C.L. lower limits on the $llqq$ contact interaction scale $\Lambda$ are 36 TeV (27 TeV) for the right-right model, 36 TeV (26 TeV) for the left-left model, and 33 TeV (29 TeV) for the left-right model, each with constructive (destructive) interference [27]. The CMS Collaboration, using a $140 \text{ fb}^{-1}$ dataset at 13 TeV, has set 95% C.L. exclusion limits on the $llqq$ contact interaction scale that range from $\Lambda_{LL} > 23.9$ TeV for the destructive interference to $\Lambda_{RR} > 36.4$ TeV for the constructive interference, for the left-left and the right-right models, respectively [29].

Note that the contact interactions arising from the compositeness of quarks and leptons in Eq. (91.1) can also be regarded as a part of more general dimension-six operators in the context of low-energy standard-model effective field theory (SMEFT). For a complete list of SM gauge-invariant dimension-six operators, see [30, 31]. A computation of the one-loop anomalous dimension matrix for SMEFT operators are found in Refs. [32–34]. See also Refs. [35–37] for recent reviews. Ref. [17] sets limits on the Wilson coefficients of the contact interaction operators in the SMEFT framework, using the CMS data of the inclusive jet production cross sections together with the deep inelastic scattering data from HERA and the CMS $t\bar{t}$ cross section data. The results are translated into a 95% C.L. exclusion limit on the quark compositeness $\Lambda_{LL}^{-} > 24$ TeV, which is shown as the second solid line in brown in Figure 91.2. In models where the SM fermions get their masses through the mixing with the composite states, the top-quark is expected to show compositeness properties [38] resulting in the $t\bar{t}t$ contact interaction operators in the SMEFT context. An enhancement of the four-top-quark production cross section is expected at hadron colliders in these models [39]. The ATLAS Collaboration has extracted 95% C.L. observed (expected) limits on the $t\bar{t}t$ contact interaction operator $|C_{4t}|/\Lambda^2 < 1.9 \text{ TeV}^{-2}$ (1.6 TeV$^{-2}$) using their $36.1 \text{ fb}^{-1}$ data at $\sqrt{s} = 13$ TeV [40]. Consistent SMEFT reinterpretations of top-quark data, and high energy non-resonant dijet and dilepton data are provided, e.g., in Refs. [41, 42].

Interactions of hypothetical dark matter candidate particles with SM particles through mediators can also be described as contact interactions at low energy. See “Searches for WIMPs and Other Particles” in this volume for limits on the interactions involving dark matter candidate particles.

## 91.2 Limits on excited fermions

Another typical consequence of compositeness is the appearance of excited leptons and quarks ($l^*$ and $q^*$). Phenomenologically, an excited lepton is defined to be a heavy lepton which shares a leptonic quantum number with one of the existing leptons (an excited quark is defined similarly). For example, an excited electron $e^*$ is characterized by a nonzero transition-magnetic coupling with electrons. The smallness of the lepton mass and the success of QED prediction for $g-2$ suggest chirality conservation, i.e., an excited lepton should not couple to both left- and right-handed components of the corresponding lepton [43–45].

Excited leptons may be classified by $SU(2) \times U(1)$ quantum numbers. Typical examples are:

1. Sequential type

$$ \begin{pmatrix} \nu^* \\ l^* \end{pmatrix}_L, \quad [\nu_R^*], \quad l_R^* $$

$\nu_R^*$ is necessary if we assume $\nu^*$ has a Dirac mass.

2. Mirror type

$$ [\nu_L^*], \quad l_L^*, \quad \begin{pmatrix} \nu^* \\ l^* \end{pmatrix}_R. $$

3. Homodoublet type

$$ \begin{pmatrix} \nu^* \\ l^* \end{pmatrix}_L, \quad \begin{pmatrix} \nu^* \\ l^* \end{pmatrix}_R. $$

Similar classification can be made for excited quarks.