{'fraction_non_alphanumeric': 0.03905145219188513, 'fraction_numerical': 0.024873998332064252, 'mean_word_length': 4.388823759280969, 'pattern_counts': {'":': 0, '<': 17, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Axions - Motivation, limits and searches

The axion solution of the strong CP problem provides a number of possible windows to physics beyond the standard model, notably in the form of searches for solar axions and for galactic axion dark matter, but in a broader context also inspires searches for axion-like particles in pure laboratory experiments. We briefly review the motivation for axions, astrophysical limits, their possible cosmological role, and current searches for axions and axion-like particles.


Introduction
Quantum chromodynamics is a CP-violating theory, implying that the neutron should have a large electric dipole moment, far in excess of experimental limits [1]. One elegant solution of this "strong CP problem" was proposed by Peccei and Quinn (PQ) who postulated a new global U(1) symmetry that is spontaneously broken at some large energy scale, allowing for the dynamical restoration of the CP symmetry [2,3]. Weinberg [4] and Wilczek [5] realized that an inevitable consequence of this mechanism is a new pseudoscalar boson, the axion, which is the Nambu-Goldstone boson of the PQ symmetry. The axion solution would open a new low-energy window to high-energy physics. String theory provides another motivation in that axion-like fields typically appear so that plausibly one of them could play the role of the QCD axion [6].
The PQ symmetry is explicitly broken at low energies by instanton effects so that the axion acquires a small mass, where f a is the axion decay constant or PQ scale that governs all axion properties, f π = 93 MeV is the pion decay constant, and z = m u /m d is the mass ratio of up and down quarks. We will follow the previous axion literature and assume the canonical value z = 0.56 [7,8], although it could vary in the range 0.3-0.6 [9]. Axions were not found in early searches, ruling out "standard axions" where f a would have been related to the electroweak scale f EW . "Invisible" axion models with f a ≫ f EW were quickly constructed [10,11,12,13,14] and quickly recognized to be far from inconsequential because they can provide the dark matter of the universe [15,16,17,18] and can be searched in realistic experiments [19,20,21,22].
The properties of axions are closely related to those of neutral pions. One generic property is a two-photon interaction that plays a key role for most searches, Here, F is the electromagnetic field-strength tensor,F its dual, and E and B the electric and magnetic fields, respectively. The coupling constant is where, E and N, respectively, are the electromagnetic and color anomaly of the axial current associated with the axion field. In grand unified models, e.g. the DFSZ model [12,13], axions couple to ordinary quarks and leptons, implying E/N = 8/3, whereas in the KSVZ model [10,11] E/N = 0. While these cases are often used as generic examples, in general E/N is not known so that for fixed f a a broad range of g aγ values is possible [23]. Still, barring fine-tuned cancelations, g aγ scales from the corresponding pion interaction by virtue of Eq. (3). Taking the model-dependent factors to be of order unity defines the "axion line" in the m a -g aγ plane.
Axions transform into photons and vice versa in external B fields [19], a phenomenon similar to neutrino oscillations [24]. This effect can have intriguing applications, although the relevant range of g aγ and m a typically is far away from the axion line because for a given g aγ the implied m a usually suppresses conversion effecs by energy-momentum conservation. Therefore, beginning with "arions" [25,26], the axion relation between g aγ and m a was often taken to be relaxed, postulating nearly massless axion-like particles for which "ALPs" has recently become standard coinage. While CAST [27] (solar axions) and ADMX [21] (galactic dark matter) are the only experiments that probe ALPs on the axion line, we will briefly review other searches, notably the puzzling signal in the PVLAS experiment [28]. Section 2 is devoted to searches for axions and axion-like particles based on their two-photon interaction and the relevant astrophysical limits. In Sec. 3 other astrophysical limits are summarized. Section 4 is devoted to the possible cosmological role of axions before turning to a summary in Sec. 5.

Searches for solar axions
Particles with a two-photon vertex, including neutral pions and gravitons besides the hypothetical axions, can transform into photons in external electric or magnetic fields, an effect first discussed by Primakoff in the early days of pion physics [29]. Therefore, stars produce these particles from thermal photons in the fluctuating electromagnetic fields of the stellar plasma [30]. Calculating the solar axion flux is straightforward except for the proper inclusion of screening effects [31,32]. The transition rate for a photon of energy E into an axion of the same energy (recoil effects are neglected) is [33] where T is the temperature (natural units withh = c = k B = 1 are used). The screening scale in the Debye-Hückel approximation is where n e is the electron density and n j that of the j-th ion of charge Z j . Near the solar center κ s ≈ 9 keV and (κ s /T ) 2 ≈ 12 is nearly constant throughout the Sun. The axion flux at Earth, based on a standard solar model, is well approximated by dΦ a dE = g 2 10 6.0 × 10 10 cm −2 s −1 keV −1 E 2.481 e −E/1.205 , where E is in keV and g 10 = g aγ /(10 −10 GeV −1 ). The integrated flux parameters are Φ a = g 2 10 3.75 × 10 11 cm −2 s −1 and L a = g 2 10 1.85 × 10 −3 L ⊙ .
The maximum of the distribution is at 3.0 keV, the average energy is 4.2 keV. This flux can be searched with the inverse process where an axion converts into a photon in a macroscopic B field, the "axion helioscope" technique [19]. One would look at the Sun through a "magnetic telescope" and place an x-ray detector at the far end. The conversion can be coherent over a large propagation distance and is then pictured as a particle oscillation effect [24]. The conversion probability is where L is the path length and q the axion-photon momentum difference; in vacuum it is q = m 2 a /2E. For qL ∼ < 1 the oscillation length exceeds L. For qL ≪ 1 we have P a→γ = (g aγ BL/2) 2 , implying an x-ray flux of For qL ∼ > 1 this rate is reduced by the momentum mismatch. A low-Z gas would provide a refractive photon mass m γ so that q = |m 2 γ − m 2 a |/2E. For m a ≈ m γ the maximum rate can thus be restored [34].
The first helioscope that can actually reach the "axion line" is the CERN Axion Solar Telescope (CAST), using a refurbished LHC test magnet with B = 9.0 T and two pipes of length L = 9.26 m. The magnet is mounted with ±8 • vertical movement, allowing for observation of the Sun for 1.5 h at both sunrise and sunset. CAST operated for about 6 months in 2003. The non-observation of a signal above background in all three detectors leads to the exclusion range shown in Fig. 1. In the mass range m a ∼ < 0.02 eV the new limit is [27] g aγ < 1.16 × 10 −10 GeV −1 (95% CL) .
The data taken in 2004 do not show any apparent signal above background, leading to limits comparable to the sensitivity prospected in Fig. 1.
In the ongoing Phase II, a variable-pressure helium filling provides an effective photon mass. The vapor pressure of He 4 at 1.8 K, the magnet's operating temperature, allows one to reach m a up to 0.26 eV. In the next phase, He 3 will be used because its higher vapor pressure allows one to reach 0.8 eV. Yet higher masses require an isolating gas cell in the bore where He 3 at 5.4 K would allow one to reach m a of 1.4 eV.
For m a ∼ < 10 −4 , a competitive method to search for solar axions could be to look with an x-ray satellite at the Sun when it is shadowed by the Earth and search for x-rays from the axion-photon conversion in the Earth's magnetic field [42]. A related idea is to look for high-energy γ-rays, e.g. with the GLAST satellite, that "pass through the Sun" by magnetic conversion and back conversion [43].  [27]. The shaded band represents typical theoretical models ("axion line"). Also shown is the future sensitivity foreseen in the CAST proposal.

Do axions escape from the Sun?
CAST can detect axions only if they actually escape from the Sun. Their mean free path (mfp) against the Primakoff process is the inverse of Eq. (4). As an example we consider 4 keV axions and note that at the solar center T ≈ 1.3 keV and κ s ≈ 9 keV. The axion mfp is then λ a ≈ g −2 10 6 × 10 24 cm ≈ g −2 10 8 × 10 13 R ⊙ , or about 10 −3 of the radius of the visible universe. Therefore, g aγ would have to be more than 10 7 larger than the CAST limit for axions to be re-absorbed in the Sun. Even in this extreme case they are not harmless because they would carry the bulk of the energy flux that otherwise is carried by photons. The mfp of low-mass particles that are trapped in the Sun should be shorter than that of photons (about 10 cm near the solar center) to avoid a dramatic modification of the solar structure [44]. This requirement is so extreme that for anything similar to axions the possibility of re-absorption is not a serious possibility.

Globular cluster stars
The most restrictive astrophysical limit on g aγ arises from globular cluster stars. A globular cluster is a gravitationally bound system of stars that formed at the same time and thus differ primarily in their mass. A globular cluster provides a homogeneous population of stars, allowing for detailed tests of stellar-evolution theory. The stars surviving since formation have masses somewhat below 1 M ⊙ . In a color-magnitude diagram, where one plots essentially the surface brightness vs. the surface temperature, stars appear in characteristic loci, allowing one to identify their state of evolution.
The stars on the horizontal branch (HB) have reached helium burning where their core (about 0.5 M ⊙ ) generates energy by fusing helium to carbon and oxygen with a core-averaged energy release of about 80 erg g −1 s −1 . On the other hand, the average Primakoff energy loss is approximately g 2 10 30 erg g −1 s −1 . The main effect would be an acceleration of the consumption of helium and thus a reduction of the HB lifetime by a factor 80/(80 + 30 g 2 10 ), i.e., by about 30% for g 10 = 1. Number counts in 15 globular clusters [45] reveal typically 100 HB stars in the used fields of view. This is compared with the number of red giants (evolutionary phase before helium ignition, but after coreexhaustion of hydrogen) and shows that the HB lifetime in any one cluster is established within about 20-40%. Note that the Primakoff losses are much smaller for red giants so that their lifetime is not reduced by Primakoff emission.
Compounding the results of all 15 clusters, the helium-burning lifetime agrees with expectations to within about 10% [46,47]. Of course, with modern data these results likely could be improved. Either way, a reasonably conservative limit is ( Figs. 1 and 2) g aγ < 10 −10 GeV −1 .
It is comparable to the CAST limit Eq. (10), but applies for higher masses because the relevant temperature is about 10 keV so that significant threshold effects only begin at about m a ∼ > 30 keV. For QCD axions the coupling increases with mass so that the limit reaches to even larger masses.
In the helium-burning core, convection and semi-convection dredges helium fuel to the burning site near the core's center so that 25-30% of all helium is burnt during the HB phase. Therefore, while the standard theoretical predictions depend on a phenomenological treatment of convection, there is limited room for additional energy supply, even if the treatment of convection were grossly incorrect.

ALP-photon conversion in astrophysical magnetic fields
Large-scale magnetic fields are ubiquitous in astrophysics so that photon-axion conversions could be of interest. In practice, the energy-momentum mismatch typically prevents this effect from being important anywhere near the "axion line." On the other hand, "axion-like particles" (ALPs) with smaller masses may well exist and could then have intriguing consequences. Examples are the polarization of radio galaxies [48] and quasars [49,50], the diffuse x-ray background [51], or ultra-high energy cosmic rays if they have a photon component [52,53].
One interesting cosmological consequence is the dimming of distant sources by photon-ALP conversion in intergalactic B fields [54,55,56,57,58,59,60,61,62], for a review see Ref. [63]. The apparent dimming of distant SNe Ia is usually attributed to accelerated cosmic expansion. All things considered, the required dimming cannot be caused by photon-ALP conversion, but this effect may still figure for a detailed interpretation of the cosmic equation of state implied by current and future SN Ia data.
Preventing excessive suppression of photon-ALP conversion in this context requires m a ∼ < 10 −16 -10 −15 eV, i.e., the ALP mass should be smaller than the photon plasma frequency in the intergalactic medium. For such small masses, the absence of γ-rays from SN 1987A implies g aγ ∼ < 1 × 10 −11 GeV −1 , valid for m a ∼ < 10 −9 eV [64,65]. The ALPs would have been emitted in SN 1987A by the Primakoff effect and converted in the galactic magnetic field. A similar argument applied to the nearby red giant Betelgeuse yields a slightly less restrictive limit [66], but of course depends on fewer assumptions. ALP-photon conversion in stellar magnetic fields is also possible, notably in the magnetic fields of Sun spots [67] or the strong magnetic fields of pulsars [68,69], although at present no new limit or positive signature is available from these considerations.

Dichroism and birefringence in laboratory magnetic fields
Magnetic fields induce vacuum birefringence in that electromagnetic radiation with polarization parallel ( ) or perpendicular (⊥) to B has a different refractive index induced by a one-loop QED correction to Maxwell's equations [70,71]. This vacuum Cotton-Mouton effect has never been measured, but can be searched in the laboratory by propagating a linearly polarized laser beam in a strong B field and measure a small ellipticity that develops [72].
ALPs, axions, neutral pions, and other particles with a two-photon vertex also contribute to this effect in that the photon mixes with them in a transverse B field [73,24]. If sufficiently light, these particles also cause vacuum dichroism, i.e., the and ⊥ polarization states are absorbed differently due to photon-ALP conversion. The observable effect is a rotation of the plane of polarization. A huge effect of this sort has been reported by the PVLAS collaboration [28], where the amplitude of the observed dichroism is roughly 10 4 times larger than the expected QED ellipticity amplitude. An instrumental origin has not been identified, but further tests and modifications of the experiment will be performed.
In PVLAS, the superconducting magnet is rotated to modulate the signal because a static dichroism or birefringence is overwhelmed by residual instrumental effects. It has been proposed that the B field rotation mixes sidebands to the primary laser frequency with a separation corresponding to the rotation frequency [74]. While such effects would in principle occur, the quantitative treatment in this paper vastly overestimates the effect; not even the use of electromagnetic units is correct. The slow magnet rotation does not cause a 10 4 fold enhancement of the QED effect.
With this coupling strength, the Sun's ALP luminosity would exceed its photon emission by more than a factor of 10 6 and thus could live only for 1000 years. A number of models have been proposed to circumvent this problem in that the ALP properties and interactions could strongly depend on the environment in the Sun or on energy [76,77,78,79,80,81]. It has become clear that to square the particle interpretation of PVLAS with the astrophysical limits is a tall order. Therefore, barring the identification of an instrumental effect that causes the PVLAS signature, the best hope is to test this hypothesis directly, notably in a new generation of "shining light through walls" experiments [82,83,84,85,86,87,88,89].

Axion-Electron Interaction
Besides their generic interaction with photons, axions also interact with fermions Ψ by a derivative interaction of the form where C is a model-dependent numerical factor and m the fermion mass. The pseudoscalar form is equivalent when only a single axion is attached to the fermion line in a given Feynman graph. Subtleties for axion-nucleon interactions in a nuclear medium arise because of the axion-pion mixing [90]. The pseudoscalar form suggests to quantify the interaction in terms of a dimensionless Yukawa coupling g = Cm/f a and associated "fine structure constant" α a = g 2 /4π. In stellar plasmas, axions can be emitted by the Compton process γ + e → e + a, bremsstrahlung e + Ze → Ze + e + a, bound-free and free-bound transitions, or pair annihilation e + + e − → γ + a. For the Sun, L a = α a 6.0 × 10 21 L ⊙ with 25% due to the Compton process [31]. Helioseismology conservatively implies that a new energy-loss channel cannot exceed 20% of L ⊙ [41], implying α a < 3 × 10 −23 . Avoiding excessive white-dwarf cooling [91] or a delay of helium ignition in globular-cluster stars [46,47] implies α a ∼ < 1 × 10 −26 . In the DFSZ model we have C e = cos 2 β/3, assuming three families of fermions. For cos 2 β = 1 the exclusion range is shown in Fig. 2. Near the limit, axion emission can affect the cooling speed and thus the period decrease of ZZ Ceti stars (pulsating white dwarfs), although a previously measured anomalous period decrease of the star G117-B15A has disappeared in the light of more recent analyses [92].

Axion-Nucleon Interaction
The axion-electron interaction provides extremely restrictive limits, but in hadronic axion models this coupling does not exist at tree level. Axions are a QCD phenomenon, so their interactions with pions, nucleons and photons are far more generic than those with electrons. Unfortunately, the astrophysical arguments leading to the electron coupling cannot be trivially recycled because axions couple to the nucleon spin and thus at low energies do not interact with α particles. Therefore, processes of the sort γ +α → α+a do not occur, although they provide limits on the scalar or vector coupling of new bosons [93,94].
However, the Sun largely consists of hydrogen so that axions are produced by the Compton-like process γ + p → p + a. The cross section is σ = (4π/3) αα a E 2 /m 4 with m the fermion mass, i.e., the energy loss rate caused by protons relative to electrons is suppressed by a factor (m e /m p ) 4 = 0.9 × 10 −13 . Taking the proton density in the Sun to be 1/2 that of electrons, using L a = α a 1.5 × 10 21 L ⊙ for the solar axion luminosity from the electron Compton process, and observing that helioseismology allows at most a 20% exotic energy loss, we find α a ∼ < 3 × 10 −9 for the axion-proton interaction.
Much more restrictive limits arise from the observed neutrino burst of supernova (SN) 1987A [95,96,97,98,99,100,101,102,103,104]. The burst lasted for about 10 s, in agreement with the usual picture of neutrino trapping and slow diffusive energy loss of the collapsed SN core. Excessive axion losses would have shortened the burst duration. On the other hand, if axions interact so strongly that their mfp is similar to that of neutrinos, they will not much affect the neutrino signal. The results from different numerical simulations and treatments agree reasonably well [46,47] and lead to the "consolidated" exclusion range shown in Fig. 2.
Early papers found more restrictive limits, but they overestimated the emission rate. The main process is nucleon-nucleon bremsstrahlung. Axions couple to the nucleon spin so that only spin-nonconserving interactions contribute, i.e., the nuclear tensor force. If it is modeled by one-pion exchange, the interaction rate is so large that axions emitted from different collisions would destructively interfere, leading to a suppression of the Landau-Pomeranchuk-Migdal type [102]. Later it was noted that one-pion exchange may be a poor model for the relevant conditions and the bremsstrahlung rate was calculated on the basis of measured nuclear phase shifts, leading to a much smaller emission rate [104]. However, this result applies only in the low-energy limit. Either way, the "naive" bremsstrahlung rate used in early studies is reduced. Since the SN 1987A limits suffer from these and other nuclear-physics uncertainties, but also from the sparse experimental data, it is not easily possible to quantify an objective error estimate.
If axions interact so strongly that they are trapped more efficiently than neutrinos, still a significant flux will be emitted. It interacts with oxygen in the water Cherenkov detectors that have recorded the SN 1987A signal and could have mimicked some of the signal [105]. This argument excludes another range of axion parameters (Fig. 2).

Thermal axions and hot dark matter
Axions are produced in the hot thermal plasma of the early universe and, since they have a mass, contribute to the cosmic dark matter. Before the QCD confinement transition at T = Λ QCD , the relevant processes involve quarks and gluons, e.g., the gluonic Primakoff effect caused by the axions's generic two-gluon vertex [106,107]. After confinement, the most generic interaction process involves pions, π + π ↔ π + a [108]. Note that thermal pions begin disappearing only at T ∼ < 30 MeV, long after the QCD epoch. Axions decouple after the QCD epoch if f a ∼ < 3 × 10 7 GeV, i.e., for m a ∼ > 0.2 eV.
Thermal axions provide a cosmic hot dark matter component fully analogous to neutrinos. Therefore, the usual neutrino mass limits from cosmic large-scale structure data can be adapted to the axion case [109,110]. Relying only on the pion process for thermalization, for hadronic axions m a < 1.05 eV (95% CL), although this limit probably could be somewhat improved with a more recent set of cosmological data. For comparison, we note that this limit corresponds to m ν < 0.65 eV (95% CL).
Axions or axion-like particles with a two-photon vertex decay into two photons with a rate where the first expression is general, the second applies to axions, and the numerical one assumes z = 0.56 and the hadronic case E/N = 0. Comparing with the age of the universe of 4.3 × 10 17 s reveals that axions decay on a cosmic time scale for m a ∼ > 20 eV. The decay photons would cause a variety of observable consequences [111], depending on the axion mass, so that cosmology alone entirely rules out axions with m a > 1 eV (Fig. 2). The low-mass end is covered by the hot dark matter argument, seamlessly connecting to high masses to avoid excess radiation from axion decays [111]. Hot dark matter axions would be partially trapped in galaxies and galaxy clusters. Searching for a decay line [112,113] provides direct limits on a range of axion masses marked "Telescope" in Fig. 2.

Non-thermal axions and cold dark matter
The main cosmological interest in axions derives, of course, from their possible role as the dominant cold dark matter component. Almost immediately after "invisible" axion models had been constructed it was recognized that they can be abundantly produced by the "misalignment mechanism" [15,16,17]. After the spontaneous breaking of the PQ symmetry at some high energy scale, the axion field relaxes to a location somewhere in the Mexican hat potential. Near the QCD epoch, instanton effects explicitly break the PQ symmetry, the very effect that causes the dynamical PQ symmetry restoration.
This "tilting of the Mexican hat" causes the axion field to roll toward the CP-conserving minimum, but this motion does not stop there. Rather, coherent oscillations of the axion field are excited that ultimately represent a condensate of cold dark matter. The cosmic matter density in this homogeneous field mode is [114] Ω a h 2 ≈ 0.7 f a 10 12 GeV where −π ≤ Θ i ≤ π is the initial "misalignment angle" relative to the CP-conserving position. If the PQ symmetry breaking takes place after inflation, Θ i will take on different values in different patches of the universe. The approximate average contribution is [114,115] Comparing with the cosmic cold dark matter density of Ω CDM h 2 ≈ 0.13 this implies that axions with m a ≈ 10 µeV would provide the dark matter (Fig. 2). However, this number sets only a crude scale of the expected mass for axion dark matter. Apart from the overall particle-physics uncertainties entering this result, the cosmological sequence of events crucially matters. Assuming axions make up the cold dark matter of the universe, significantly smaller masses are possible if inflation took place after the PQ transition and the initial value Θ i was relatively small. Conversely, if the PQ transition takes place after inflation, there are additional sources for nonthermal axion production, notably the formation and decay of cosmic strings and domain walls associated with the breaking of the PQ symmetry [18,114]. Therefore, the mass of cosmic dark matter axions could be significantly larger than the 10 µeV scale. For a recent review of axion cosmology we refer to Ref. [114].
If axions are the dark matter of our galaxy, they are accessible to direct searches by the axion "haloscope" technique. A microwave resonator is placed in a strong magnetic field and thus can be driven by the galactic axion field due to the twophoton coupling [19,20]. After a series of pilot experiments, the ADMX experiment in Livermore has reached the axion line in a narrow range of masses [21]. The project will be upgraded in several stages so that it should be able to cover the range 1 µeV ≤ m a ≤ 100 µeV within the upcoming decade. Therefore, axion dark matter can be detected in a realistic range of parameters [116].
In the exciting event that dark matter axions are found, intriguing opportunities for observational axion cosmology open up. The fine energy resolution that is possible with a microwave resonator allows one to study the detailed axion velocity distribution that may reveal narrow peaks from the phase-space folding that occurred during galaxy formation [22,114].

Summary
Axions remain a well motivated solution of the strong CP problem. Ongoing and nearfuture searches for solar axions by CAST and for galactic cold dark matter axions by ADMX cover realistic parameters and thus have a chance to detect these elusive particles. If CAST were to detect solar axions, they would contribute a small hot dark matter component of the universe so that these two experimental directions are complementary. There remains a range of axion parameters, roughly corresponding to 0.1 meV ∼ < m a ∼ < 10 meV, where axions are not excluded, yet no detection strategy has been proposed. Perhaps we will not have to worry about this gap if axions show up in one of the accessible search ranges.
Unfortunately, the PVLAS signature likely has nothing to do with the strong CP problem or dark matter. The particle interpretation is in severe conflict with astrophysical arguments so that plausible theoretical models are hard to construct. However, the effect is so large that it can be tested relatively easily at several forthcoming "shining light through walls" experiments. Even if no new particle is confirmed, a strong push is made towards measuring the QED-implied birefringence of a magnetized vacuum. Such a measurement would be an astounding achievement in itself.