Axions: From Magnetars and Neutron Star Mergers to Beam Dumps and BECs
Jean-François Fortin, Huai-Ke Guo, Steven P. Harris, Doojin Kim, Kuver Sinha, Chen Sun
MMI-TH-214INT-PUB-21-004
Axions:From Magnetars and Neutron Star Mergers to Beam Dumps and BECs
Jean-François Fortin ∗ Département de Physique, de Génie Physique et d’Optique,Université Laval, Québec, QC G1V 0A6, Canada
Huai-Ke Guo † and Kuver Sinha ‡ Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA
Steven P. Harris § Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA
Doojin Kim ¶ Mitchell Institute for Fundamental Physics and Astronomy,Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA
Chen Sun ∗∗ School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel (Dated: February 26, 2021)We review topics in searches for axion-like-particles (ALPs), covering material that is complemen-tary to other recent reviews. The first half of our review covers ALPs in the extreme environments ofneutron star cores, the magnetospheres of highly magnetized neutron stars (magnetars), and in neu-tron star mergers. The focus is on possible signals of ALPs in the photon spectrum of neutron starsand gravitational wave/electromagnetic signals from neutron star mergers. We then review recentdevelopments in laboratory-produced ALP searches, focusing mainly on accelerator-based facilitiesincluding beam-dump type experiments and collider experiments. We provide a general-purposediscussion of the ALP search pipeline from production to detection, in steps, and our discussionis straightforwardly applicable to most beam-dump type and reactor experiments. We end with aselective look at the rapidly developing field of ultralight dark matter, specifically the formation ofBose-Einstein Condensates (BECs). We review the properties of BECs of ultralight dark matterand bridge these properties with developments in numerical simulations, and ultimately with theirimpact on experimental searches. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] a r X i v : . [ h e p - ph ] F e b ONTENTS
I. Introduction 3II. Astrophysical Searches: Axions and Neutron Stars 6A. Axion conversions in large-scale magnetic fields versus near neutron stars 6B. ALP production in neutron star cores 71. Superfluidity in neutron stars 72. ALP emissivity 83. ALP spectrum 12C. Conversion of ALPs to photons near neutron stars 131. ALP-photon conversion in dipolar magnetic fields 132. X-rays and gamma-ray signatures of ALPs: current constraints 143. Polarized X-rays and gamma-rays from magnetars 17D. Photon-to-ALP conversion in galactic and extra-galactic magnetic fields 18III. Neutron Star Mergers 20A. Dynamics of a neutron star merger 20B. Ultralight axions in the neutron star inspiral 22C. Axions in merger remnants 231. Axion mean free path in hot, dense matter 232. Axion cooling of merger remnants 243. Transport from trapped dark sector particles 27IV. Laboratory-Produced ALP Searches 28A. Beam-dump and reactor neutrino experiments 281. Production of ALP 292. Transportation of ALP 313. Detection of ALP 32B. Collider searches 36V. Bose-Einstein Condensates 37A. Theoretical estimate 381. Cosmological evolution 382. Stable self-gravitating structures 39B. Simulations of the ultra-light dark matter 41C. Experimental probes 41VI. Conclusions 42VII. Acknowledgements 4444
I. INTRODUCTION
Since its introduction more than four decades ago [1–6], the axion has come to occupy a central rolein several important directions of research. Indeed, the term “axion”, originally introduced by Wilczekto denote the particle associated with the Peccei-Quinn solution to the strong- CP problem, has nowtranscended the original context in which it was introduced. In quantum field theory, the term can meangeneric pseudoscalar Goldstone bosons described by a two-parameter model ( m a , f ) , where m a is the massof the particle and f is the scale of a spontaneously broken chiral symmetry. In string theory, the axionmay refer to general pseudoscalar matter fields (open string axions) or to fields that are components ofthe complex degrees of freedom associated with compact extra dimensions (closed string axions). We willuse the terms “Axion-like-particle” (ALP) and “axion” interchangeably in this review, generally remainingagnostic about the specific context in which the particle arises.Aspects of the physics of axions have been reviewed extensively by different authors over the years.Its connection to the strong- CP problem was reviewed by Peccei in [7] and Kim et al. in [8]. Axionsin the context of string theory have been explored in [9–12]. The topic of axion inflation was reviewedin [13]. Many reviews on axion detection methods have also been published, based on the pioneeringideas of Sikivie (direct detection) [14] and Raffelt (indirect detection) [15, 16]. The search for axionshas certainly entered a golden age. Building on these original ideas, numerous proposals to detect ALPshave been explored recently. These proposals span an astonishing range of experimental settings, fromlaboratory-based facilities to astroparticle physics.The last few years have seen a proliferation of reviews on axions and ALPs. For the benefit of thereader, we briefly summarize the areas covered by them and comment on where the current review falls.On the more theoretical end of the spectrum are the reviews by Choi et al. [17], Hook’s TASI lectures[18], Di Luzio et al. [19] and Marsh [20]. Hook’s lectures provide a thorough and intuitive introductionto the strong- CP problem and the role of axions in solving it. Choi et al.’s review covers relatively newdevelopments at the interface of string phenomenology and axions, such as the connection between theWeak Gravity Conjecture and axion field ranges [21, 22]. The review by Di Luzio et al. covers the QCDaxion in great detail, while the one by Marsh covers all aspects of axion cosmology, as well as manyaspects of astrophysical searches for axions. On the more experimental end of the spectrum are severalreviews by theorists, including Graham et al. [23], Irastorza et al. [24], and most recently Sikivie [25].Combined, these reviews provide an in-depth look at almost every aspect of current experimental searchesfor axions.The current review is placed firmly at the experimental end of the spectrum. We will focus on a set oftopics that are complementary to those covered by [23–25], covering the following topics: (1) Axions in extreme astrophysical environments:
The first half of the review, Sect. II andSect. III, will cover axions in the extreme environments of neutron star cores, the magnetospheres ofhighly magnetized neutron stars (magnetars), and in neutron star mergers. The focus will be on possiblesignals of axions in the photon spectrum of neutron stars and gravitational wave/electromagnetic signalsfrom neutron star mergers. (1 a ) For neutron stars, we will cover in detail the production of axions from the strongly degeneratenuclear matter in the core of the star. This is a subject with a long history, going back to the 1980’s[26–28] and becomes important in deriving axion constraints from supernovae, cooling neutron stars, and– as we will emphasize – in the emerging body of work studying the photon spectrum from neutronstars to constrain axions. In addition to reviewing the original literature, we will also discuss the effectsof nucleon superfluidity on the production rate. When the nuclear matter is not superfluid, axions areproduced in the neutron star core via nucleon bremsstrahlung processes. When one or both nucleon speciesare superfluid, the rate of the production due to bremsstrahlung is diminished, but a new productionmechanism appears, where axions are created by the thermally-induced formation of nucleon Cooperpairs. These issues will be discussed in detail. We will also discuss several other issues, including changesin the emissivity coming from corrections to the one-pion-exchange approximation.One of the newer features that we will emphasize is the spectral features of the axions produced fromthe core. The question of spectral features is not directly relevant for limits derived from cooling, whichdepend only on the integrated luminosity. However, they are relevant for constraints on axions stemmingfrom studying the actual X-ray and gamma-ray spectra of neutron stars. The way that axions can beprobed by studying such spectra is as follows. Once created, the axions escape from the interior of theneutron star and have a probability of converting to X-ray and gamma-ray photons in the magnetic fieldof the magnetosphere. The resulting axion-converted photons constitute an exotic source of emissionfrom neutron stars different from any putative standard astrophysical processes. Features of the observedspectrum can thus be used to constrain the product of the coupling of axions to nucleons and photons.The focus of our attention will be on a class of highly magnetized neutron stars called magnetars, althoughthe discussion would apply to other categories of neutron stars as well.Throughout, we will compare and contrast the conversion of axions to photons in large-scale magneticfields, which is the standard method of probing them in astroparticle physics, versus the conversionprocess in the vicinity of neutron stars. We will provide a lightning review of some of the main trendsin the former topic. It is hoped that the review will help the reader navigate, on the one hand, the fieldof particle physics in nuclear environments, and on the other hand, the fundamental physics of axions,especially in the context of X-ray and soft gamma-ray astronomy of neutron stars. (1 b ) We will review in detail neutron star mergers and the possibility of using them as a laboratoryfor the physics of axions, and perhaps hidden sectors in general. One of the distinguishing features ofmergers is multi-messenger astronomy, and the hope is that gravitational wave, electromagnetic, andcorrelated data from mergers can be leveraged to investigate axions. This calls for collaboration betweenbeyond-Standard-Model physicists, nuclear physicists, and experts on merger physics and simulations.Our hope is that this review will be a small step in furthering this collaboration.When two neutron stars merge, the constituent nuclear matter - already quite dense - reaches tem-peratures of tens of MeV, comparable to those reached in a core-collapse supernova. At these temper-atures, the nuclear matter is no longer superfluid, and thus axions are primarily produced by nucleonbremsstrahlung. We review the calculation of the axion mean free path and find that it is sufficientlylong that axions would not be trapped anywhere in a neutron star merger. Axions produced during themerger will free-stream through the nuclear matter and will take energy away from the remnant, coolingit. We review the calculation of the cooling timescale due to axion emission and the results of a mergersimulation which incorporated this cooling.While ultralight axions are not the focus of this part of our review, we will briefly indicate how aparticular type of such species with a kilometer-scale Compton wavelength can impact the dynamics ofthe inspiral phase of a neutron star merger. (2)
Axions at accelerator-based experiments:
We review recent developments in laboratory-produced axion or ALP searches in Sect. IV, focusing mainly on accelerator-based facilities includingbeam-dump type experiments and collider experiments. While astrophysical considerations afford excel-lent avenues to look into the axion parameter space, laboratory-based searches can constrain models ofaxions in the most model-independent (hence conservative) fashion. These searches are not predicatedupon assumptions (e.g., stellar models) that astrophysical searches may take. Indeed, in the context ofthe (ultimately transient) PVLAS anomaly [29] for which the preferred parameter region was alreadyexcluded by CAST [30], and in the wake of the more recent EDGES [31] and Xenon1T [32] anoma-lies, laboratory-based probes have received particular attention as they can provide the most stringentguidelines in constructing axion or ALP models. (2 a ) The first half of Sect. IV will be devoted to searches at beam-dump type and reactor experiments,including next-generation ones and related phenomenological studies. Many such experiments featuringhighly intensified neutrino fluxes have begun their operation or are being seriously planned. Much of theliterature has pointed out that they are capable of probing the ALP parameter space, and the associatedexperimental data will surge in the near future. Key specifications of these experiments are collectedand tabulated in our review. We provide a general-purpose discussion of the ALP search pipeline fromproduction to detection, in steps, and our discussion is straightforwardly applicable to most of the beam-dump type and reactor experiments. For detection of ALPs, we cover not only the traditional searchscheme that depends on ALP decays but newly proposed search channels (e.g., axion scattering andconversion channels). We review the existing and future expected limits in all three channels, showingthe complementarity among them. (2 b ) The other half of Sect. IV is reserved for a review of recent developments in collider ALP searches.As colliders feature relatively large center-of-mass energy, they have played an important role in thesearch for MeV-to-TeV mass-range axions or ALPs. Moreover, models of ALPs interacting with SMheavy resonances, e.g., massive gauge bosons and Higgs, can be exclusively probed at colliders, especiallythrough on-shell production of such resonances. As future proposed colliders are expected to producethem even more copiously, they will provide particular opportunities and richer phenomenology in theassociated channels. All these opportunities available at existing and future colliders have been activelyexploited over the past decade. We review this series of efforts and assort future energy-frontier colliderstogether with their key parameters. (3)
Galactic and Stellar Bose-Einstein Condensates:
We will end this review with a selective lookat the rapidly developing field of ultralight dark matter (DM), specifically the formation of Bose-EinsteinCondensates (BECs). Being scalars, ultralight dark matter exhibits unique collective properties thatare different from heavy dark matter, such as behaving more like waves than point particles. This inturn leads to interesting dynamics and profiles, as well as constraints such as that from wave functionstability. In Sect. V we look into these properties of BECs of ultralight dark matter and bridge themwith numerical simulations, and ultimately with their impact on experimental searches. We break thisaspect of the review into two connected parts. (3 a ) We start with the requirement of consistency. We estimate the critical condition for BEC toform, which gives an upper limit on the ultralight dark matter mass. We briefly review the cosmologicalevolution of an ultralight scalar field and show the Jean’s scale due to its quantum pressure. In addition,we also estimate the effect on the Jean’s scale if the scalar field has a sizable quartic self-interaction.After that, we discuss the stability requirement of the BEC system, which puts an upper limit on thesize of the BEC structure. (3 b ) We briefly go through the recent simulations of BEC dark matter. We show that the simulationsindicate an interesting mass relation between the BEC core and the dark matter halo. In concluding thissection, we quote a few constraints on ultralight dark matter on the galactic scale, including the resultsof cosmic microwave background (CMB) weak lensing, the Lyman- α , stellar stream and strong lensingconstraints on the halo mass function, Milky Way satellite counting, UV luminosity function, as well asthe empirical BEC-halo mass relation. On the stellar scale, we point to a few references where novelsignals could be used to probe boson stars.Since much of the review is on search strategies, we will not cover any issues of the underlying theoryor model-building aspects of axions. Rather, we will simply start with the parts of the axion Lagrangianthat are directly relevant for the search strategies we pursue: namely, its coupling to photons, electrons,and nucleons. L int ⊃ g aγγ aF µν ˜ F µν + ig aee a ¯ ψ e γ ψ e + G an ( ∂ µ a ) ¯ N γ µ γ N (1)where a denotes the ALP field, F µν the electromagnetic field strength tensor, ˜ F µν = (1 / (cid:15) µνρσ F ρσ thedual electromagnetic field strength tensor, ψ e the electron field, and N the nucleon field. The reader isdirected to the excellent theoretical reviews cited above for more details.Before proceeding, we mention some of the numerous topics that will not be covered in this review.Sect. II will only scratch the surface of the vast topic of axion conversion in large-scale magnetic fields.Sect. IV will skip over most of the classic laboratory-based techniques of axion detection, which arecovered in detail by the recent review of Sikivie [25]. Likewise, Sect. V will omit many topics in the fieldof ultralight axions and dark matter, foremost among them the idea of superradiance. II. ASTROPHYSICAL SEARCHES: AXIONS AND NEUTRON STARS
This section will be structured as follows. In Sect. II A, we will first contrast the conversion of axionsin large-scale magnetic fields to their conversion in the localized magnetic fields near neutron stars. InSect. II B, we will review the production of axions from the core of neutron stars. In Sect. II C, we willthen describe their fate in the magnetosphere, by describing the evolution equations of the axion-photonsystem and deriving expressions for the probability of conversion. We will use these results to showrecently obtained constraints on axion couplings and discuss the possibility of using polarization to probeaxions. Finally, in Sect. II D we will provide a lightning review of the conversion of axions in large-scalemagnetic fields.
A. Axion conversions in large-scale magnetic fields versus near neutron stars
The conversion of ALPs to photons or vice versa in astrophysical magnetic fields constitutes a majorsearch strategy for these particles. This conversion could happen in large-scale magnetic fields (a long-standing topic of study) or localized magnetic fields near compact objects like neutron stars and magnetars(a relatively newer topic of study). We discuss these topics in turn. ( i ) Galactic and extra-galactic case: In the traditional large-scale conversion scenario,the general theme is as follows: Photons are emitted from a distant source and travel to the earththrough intervening magnetic fields, where they undergo conversion to ALPs; this conversion resultsin spectral features of the source that may be discerned over background emission due to standardastrophysical processes, resulting in constraints on ALPs. The sources that have been studied in thisframework are diverse: active galactic nuclei [33] including blazars [34] and quasars, supernovae [35]and in particular SN 1987A [36], red supergiant stars like Betelgeuse [37], etc. Much of this programhas centered on ALP-photon inter-conversion in the large-scale astrophysical magnetic fields that theALP-photon system must traverse through in order to reach the earth. Depending on the location ofthe source where the ALP-photon system originates, these intervening environments could include themagnetic fields of the host galaxy [33], extragalactic space [38], the Milky Way [39] and the specificenvironment at the origin, for example the field within the blazar jet [40]. The accurate modeling ofthese intervening large-scale magnetic fields – in galactic and extragalactic environments and in activegalactic nuclei, BL Lac jets, etc. – presents a serious challenge in these endeavors. The literature hasgenerally resulted in leading constraints on g aγγ for ALP masses below ∼ O (10 − ) eV. ( ii ) Localized case: ALP-photon production and inter-conversion near localized sources, especially inthe dipolar magnetic fields of neutron stars and magnetars, will be the main focus of this section. In thesescenarios, ALPs present either in the dark matter halo surrounding the neutron star (non-relativistic case)or emitted from the core of the neutron star (relativistic case) convert to photons in the dipolar magneticfield near the neutron star, yielding hard X-rays or gamma-rays in the relativistic case and radiowavesin the dark matter case, respectively. In contrast to the large-scale conversion case, the magnetic fieldis nearly critical in strength ( ∼ G), is much more accurately known, and the distance traversed bythe ALP during the conversion is small (typically a few thousand kilometers). This is a fundamentallydifferent method of probing ALPs, in that the conversion occurs in the vicinity of the neutron star.The magnetic field is well-approximated by a dipole, in contrast to the galactic-scale conversion case,where the modeling of the field morphology is highly non-trivial. Due to the differences in the spatialdependence of the magnetic field between the two cases, the ALP-photon propagation equations havesolutions whose parametric behaviors in the two cases are also different. It therefore turns out that thetwo methods probe fundamentally different mass and coupling regimes of ALPs, with the localized casebeing sensitive to ALP masses below ∼ O (10 − ) eV.We now go on to a discussion of first the production of axions from the cores of neutron stars, andthen their conversion in the magnetosphere. B. ALP production in neutron star cores
The dense matter in the core of neutron stars is strongly degenerate nuclear matter, consisting of Fermiseas of neutrons, protons, electrons, and muons [41, 42]. At the highest densities encountered in neutronstars, exotic phases of matter may appear (see the reviews and textbooks [41–47]), but we do not discussthis possibility here. As one moves from the core to the edge of the star, the density of the nuclear matterdecreases and eventually the nuclear matter transitions to a solid crust, containing a Coulomb lattice ofnuclei and a degenerate Fermi gas of electrons [46, 48, 49].Axions can be produced both in the core and the crust of the neutron star. In the uniform nuclearmatter of the neutron star core, the dominant production channels are the three nucleon bremsstrahlungprocesses N + N (cid:48) → N + N (cid:48) + a , where N and N (cid:48) are either neutrons or protons. In the case wherethe nucleons are superfluid, these three processes can still proceed and produce axions, but the rate isBoltzmann suppressed, as we will discuss later in this section. In addition, the presence of superfluidnucleons creates a new axion production process, from the formation of Cooper pairs.In the solid crust of a neutron star, electron scattering off of nuclei e − + ( Z, A ) → e − + ( Z, A ) + a islikely to be the dominant axion production mechanism [26, 53]. However, free neutrons in the neutronstar crust exist and could be superfluid [49, 58], so the possibility of other axion production channelsexists (for inspiration, see the neutrino pair production processes in the crust [59]). As it is expected tobe subdominant to axion production in the core [53], we will neglect axion production in the crust forthe rest of this review. Axion production processes are reviewed in [16, 28, 60].
1. Superfluidity in neutron stars
With the exception of right after their birth in a core collapse supernova [61] or during a neutronstar merger (see Sect. III), neutron stars are expected to be at sufficiently low temperature that theneutrons or the protons, or perhaps both, are in the superfluid phase. Superfluid protons give rise tosuperconductivity, as protons are electrically charged. The presence of superfluidity in the core of neutronstars has dramatic consequences for neutron stars, including their specific heat and rate of neutrinoproduction. Of consequence for this review, superfluidity can significantly alter the axion production ratein neutron star cores.The nucleon-nucleon interaction is attractive under certain conditions, and thus the possibility existsfor nucleons near the Fermi surface in degenerate nuclear matter to form Cooper pairs, as a consequenceof the Cooper theorem [62]. Cooper pairing in a particular spin-angular momentum channel can occurbetween nucleons N and N (cid:48) as long as the interaction is attractive at the energy scale of interest. However,pairing between a neutron and a proton is unlikely in neutron stars due to the sizeable difference in theFermi momentum between the two species [63]. When the formation of Cooper pairs becomes possible,a gap at the Fermi energy appears in the single-particle energy spectrum of the paired species [64]. Thesize of the energy gap varies with density, and at sufficiently high temperature the gap vanishes and the Nucleon bremsstrahlung is also the dominant way of producing axions in supernovae [27, 28, 50–52]. Electron-nucleus scattering is also the dominant axion production mechanism in white dwarfs [54–57] and red giant stars[55]. The Fermi momentum ratio p Fn /p Fp = ( n n /n p ) / = [(1 − x p ) /x p ] / ≈ if the proton fraction x p ≡ n p /n B = 0 . . particle species is no longer in the superfluid phase (the non-superfluid phase is often called “ungapped”).A combination of nucleon-nucleon scattering data and nuclear theory calculations lead to our currentunderstanding that the dominant pairing channels in neutron star matter are S and P . Superfluidityin nuclear matter and its consequences in neutron stars are reviewed in [58, 59, 65–69].Nucleon pairing can be constrained by studying the cooling of neutron stars, which, at least for the firsthundred thousand years of the lifetime of the star, occurs predominantly through neutrino emission fromthe core [70]. Superfluidity strongly suppresses the rate of neutrino emission and thus slows the coolingof the neutron star. In these studies, any possible contribution of axions to cooling is neglected. Theauthors of [71] conducted simulations of neutron star cooling and compared the results to data from thesupernova remnant HESS J1731-347. Combining the results of their Markov Chain Monte Carlo analysiswith the theoretical expectation that the proton S critical temperature is higher than the neutron P critical temperature [58, 65, 66], they found that the critical temperature for proton singlet pairing mustbe larger than × K in most of the neutron star core, while the critical temperature for neutrontriplet pairing must be less than × K in the entire core. The authors of [72] found similar results,comparing the results of their neutron star cooling simulations with cooling data from a wide variety ofisolated neutron stars.Finally, although we will not discuss the neutron star crust in this review, it is expected that freeneutrons in the inner crust of the neutron star pair in the S channel [49, 58]. The phenomenologicalimplications of this pairing are discussed in [65].
2. ALP emissivity
We will, in the coming paragraphs, develop the calculation of the axion emissivity from a neutronstar core. The emissivity measures the amount of energy lost via axion radiation per time, per volume.Integrating the emissivity over the core gives the axion luminosity, which can be used to calculate theexpected cooling rate of a neutron star due to axion emission [53, 71, 73–75]. Later in this section, wewill also be interested in the energy spectrum of the emitted axions, for the purpose of understandingthe spectrum of the photons into which the axions have a probability of converting in the neutron starmagnetosphere.In this section of the review, we focus on axion emission from the core of magnetars, although at theend of this section we briefly discuss more conventional neutron stars. Magnetars are believed to beyounger neutron stars (see Fig. 9 of [76]) with relatively high core temperatures and very strong magneticfields. While the surface temperature of a magnetar can be inferred from fitting the soft X-ray emissionto a blackbody spectrum [77], modeling is required to deduce the core temperature. For standard neutronstars, this relationship is relatively well understood [70, 78], but many magnetars have an anomalouslyhigh surface temperature, which would seemingly require their core temperature to be well above K.Such a hot magnetar core is not believed to be sustainable over the lifetime of the magnetar, due to thelarge neutrino emissivity that would be generated [79, 80]. As a result, various mechanisms of surfaceheating due to the strong magnetic field have been proposed [81]. Ordinarily, it is expected that themagnetar core is isothermal, due to its high thermal conductivity [82]. However, if it is the case that thehigh surface temperature is generated by magnetic heating in the interior of the star, then the core wouldno longer be isothermal [83].Following [84], we will assume that the magnetar core temperature lies in the range K to a few times K. In this case, the magnetar core temperature is expected to be larger than or comparable to thecritical temperature of neutron triplet superfluidity, in which case it is common to consider superfluidityonly in the proton singlet channel [84, 85], neglecting superfluidity of the neutrons. The inclusion ofneutron superfluidity is briefly discussed at the end of this section.Following the lead of [86], the matrix elements for the three bremsstrahlung processes N + N (cid:48) → N + N (cid:48) + a are usually computed modeling the strong interaction of the two nucleons by assuming thatthe two nucleons exchange a pion. This one-pion exchange (OPE) approximation [87, 88] is used for manydifferent types of nucleon bremsstrahlung processes including those that emit a neutrino-antineutrino pair[59, 86], a CP-even scalar [52, 89–92], a saxion [93], a Kaluza-Klein graviton [94, 95], or a dark photon[96, 97].In the case of axion production, to which we now focus, the matrix element for n + n → n + n + a and p + p → p + p + a are [27] S (cid:88) spins |M| = C π m n /m π ) f G an (cid:20) k ( k + m π ) + l ( l + m π ) + k l − k · l ) ( k + m π )( l + m π ) (cid:21) , (2)assuming that the axion-neutron and axion-proton couplings have the same strength. In (2), the symmetryfactor is S = 1 / to account for the identical particles in both initial and final states. The matrix elementfor n + p → n + p + a is S (cid:88) spins |M| = C π m n /m π ) f G an (cid:20) k ( k + m π ) + 4 l ( l + m π ) − k l − k · l ) ( k + m π )( l + m π ) (cid:21) , (3)where the symmetry factor is simply S = 1 due to the absence of identical particles in the initial orfinal states. Although the matrix element (3) was first computed in [27], a minus sign error was recentlycorrected in [50]. In both (2) and (3), the factor f ≈ is the pion-nucleon coupling constant [86] andits definition introduces the extra neutron mass factor m n . As such, this factor is the vacuum neutronmass, not the effective neutron mass stemming from strong interactions in the magnetar core. The three-vectors k = p − p and l = p − p represent the nucleon momentum transfer. Finally, since the OPEapproximation is known to overestimate the strength of the nuclear interaction, we introduce in (2) and(3) a factor C π = 1 / .The axion emissivity for any one of the nucleon bremsstrahlung processes is given by the phase spaceintegral Q = (cid:90) d p (2 π ) d p (2 π ) d p (2 π ) d p (2 π ) d ω (2 π ) S (cid:80) |M| E ∗ E ∗ E ∗ E ∗ ω ω (4) × (2 π ) δ ( p + p − p − p − p a ) f f (1 − f ) (1 − f ) . where ω is the energy of the axion and f i is the Fermi-Dirac factor of the appropriate nucleon. The factorsof energy in the denominator relate to the normalization of the nucleon wavefunctions. In magnetars,which contain strongly degenerate nuclear matter due to the fact that their temperatures lie well below1 MeV, the emissivity can be calculated in the “Fermi surface approximation”, described in [84] (see also[42, 108, 112]), which assumes that nucleons near their Fermi surface provide the dominant contribution The OPE potential is known to reproduce some, but not all, features of the nucleon-nucleon interaction [98, 99]. In[100], the authors pointed out that the OPE approximation drastically underestimates the nucleon-nucleon scatteringcross-section at low center-of-mass energy, and overestimates it by an increasingly large factor as the collision energy rises(see Fig. 3 in their paper). To get an estimate of the discrepancy C π between the OPE approximation and QCD, theauthors of [101] calculated the rate of n + n → n + n + a in the soft radiation approximation (SRA) where, as the axionenergy goes to zero (technically, as the axion energy drops far below the center-of-mass energy of the nucleon scattering),the bremsstrahlung rate is directly related to the on-shell nucleon-nucleon scattering amplitude. The calculation ismodel-independent, but neglects many-body effects. This approach shows that the OPE approximation overestimatesthe emissivity by about a factor of 4 near nuclear saturation density, and so it has become common to choose C π = 1 / [71, 84], although slightly different values are also used [52, 73]. The SRA was also used to improve the OPE estimate ofthe emissivity of neutrinos [102, 103], Kaluza-Klein gravitons and dilatons [104], and dark photons [100].The above divergence is regulated by two effects. For massive emitted particles, the energy of the emitted particlemust be larger than its mass. For both massive and massless emitted particles, the divergence is regulated by the finitedecay width of the nucleon, the so-called Landau-Pomeranchuk-Migdal (LPM) effect [105, 106], that kicks in only whenthe energy of the emitted particle is less than nucleon decay width. Since the nucleon decay width is small for lowtemperatures, the LPM effect becomes important only when the temperature rises above 5-10 MeV [103]. When the nuclear interaction is described with a relativistic mean field theory, the nucleon energy denominators aregiven by E ∗ = (cid:112) p + m ∗ , where m ∗ is the Dirac effective mass of the nucleon [41]. The nucleon wavefunctions anda discussion of calculating spin-summed matrix elements in a relativistic mean field theory are given in [107], although[108–110] also contain useful information. In other treatments of the nucleons, the extreme nonrelativistic limit is taken,and the nucleon energy E ∗ = m ≈ MeV is chosen (see [97] and Appendix C of [111]). Q nn = 312835 π C π ( m n /m π ) f G an p F n F ( c ) T (5) Q Spp = 312835 π C π ( m n /m π ) f G an p F p F ( d ) T R pp ( n B , T ) (6) Q Snp = 1242835 π C π ( m n /m π ) f G an p F p G ( c, d ) T R np ( n B , T ) . (7)The derivations of these expressions are given in [84], as are the definitions of the functions F and G ,which depend on density through c = m π / (2 p F n ) and d = m π / (2 p F p ) . The functions R np and R pp arisefrom the gap in the proton energy spectrum, and Boltzmann suppress the corresponding bremsstrahlungrate by some function of exp ( − ∆ /T ) , where ∆( n B , T ) is the superfluid gap in the proton energy spec-trum. Processes involving multiple “gapped” particles will experience additional suppression. When thetemperature rises above the critical temperature for proton pairing, the gap goes to zero and R → .The suppression factors R are defined in [84] and are further discussed in [67].In nuclear matter below the critical temperature for proton pairing, Cooper-paired protons and un-paired protons coexist and at finite temperature, Cooper pairs are constantly breaking and reforming[68, 113]. When a Cooper pair forms, at least of energy is liberated and can be carried by an emittedaxion. This process happens slowly far below the critical temperature, but the rate increases rapidlyas the critical temperature is approached from below, since the fraction of protons in excited, unpairedstates is higher. Above the critical temperature, Cooper pairs cannot form. The emissivity of axions dueto S proton Cooper pair formation is given by [114] Q CP = G an π ν (0) v F p ∆ T (cid:90) ∞ dx x (1 + e x/ ) θ ( x − /T ) (cid:112) x − /T , (8)where ν (0) = m Lp p F p / ( π ) is the density of states at the proton Fermi surface [115], the proton Fermivelocity is v F p = ∂E p /∂p | p = p Fp = p F p /m Lp , and m Lp is the Landau effective mass of the proton [116, 117].The authors of [84] examine a selection of magnetars from the McGill magnetar catalog in an effort toconstrain couplings of the axion to nucleons and photons. As no mass measurements for the magnetarsare available, the authors of [84] assume that each magnetar is a . M (cid:12) neutron star with nuclear matterdescribed by the IUF equation of state [118]. Because the core temperatures are assumed to lie above K, neutron superfluidity is neglected, while S proton pairing is included. The zero-temperatureproton gap ∆( T = 0 , n B ) is given by the parametrization (from [119]) of the calculation done by CCDK[120], and is consistent with the constraints from neutron star cooling mentioned above. Further details,including the relationship between the zero-temperature gap and the critical temperature as well as plotsof the critical temperature and superfluid gaps throughout the . M (cid:12) magnetar core are given in [84].In the rest of this section, we will consider axion emission from this . M (cid:12) magnetar.In the left panel of Fig. 1, we plot the luminosity of axions emitted from the . M (cid:12) magnetar discussedabove, split into the individual contributions of each of the three bremsstrahlung processes and the S proton Cooper pair formation process. When the temperature of the magnetar core is well lessthan the critical temperature (which is always above K in the CCDK model considered here), theBoltzmann suppression exp ( − ∆ /T ) for any process involving a proton is very strong, and so the neutronbremsstrahlung process dominates the axion production. For core temperatures well less than K, n n → n n an p → n p ap p → p p ap p → {pp} a A x i o n l u m i n o s i t y L a [ e r g s / s ] Temperature [K]10 K 10 K5×10 K10 K NN' → NN'app → {pp}a d N / d ω −6 ω [keV]0 500 1000 1500 2000 FIG. 1. Axion emission from the characteristic . M (cid:12) magnetar with CCDK S proton pairing discussed inthe text. In both panels, the axion-nucleon coupling constant has been chosen to be at the upper limit set bySN1987a G an = 7 . × − GeV − [23]. Left panel: Axion luminosity from the magnetar, split into contributionsfrom three different nucleon bremsstrahlung processes N + N (cid:48) → N + N (cid:48) + a and from the formation of protonCooper pairs p + p → { pp } + a . At low temperatures, the protons are superfluid and processes involving protonsare strongly suppressed. As the core temperature approaches the critical temperature in much of the magnetar,the Cooper pair formation process becomes more prominent. When the core rises above T ≈ × K, none ofthe star is superfluid (we ignore the crust, which might have a high critical temperature for S neutron pairing)and the bremsstrahlung processes operate as they do in ungapped nuclear matter. Right panel: Spectrum ofaxions emitted from the magnetar due to the three nucleon bremsstrahlung processes (combined) and Cooperpair formation. The low-energy axion spectrum comes from the bremsstrahlung processes, but the Cooper pairformation process produces most of the high-energy axions, as long as the protons in most of the star are superfluid. most protons near the Fermi surface are already Cooper paired, so the rate of axion production fromthe Cooper pair formation process is small. As the core temperature approaches K, the fractionof protons near the Fermi surface that are unpaired increases and therefore the Cooper pair formationprocess becomes an important source of axion production and competes with n + n → n + n + a . At thesame time, while the protons are still superfluid, the Boltzmann suppression of n + p → n + p + a is notvery large at these higher temperatures, and so it too becomes an important source of axions.As the core temperature rises above K, the protons in the center of the magnetar are no longersuperfluid and once the core passes T ≈ × K, protons in most of the neutron star are no longerCooper-paired. At this point, the rate of Cooper pair formation dramatically falls off and the threenucleon bremsstrahlung processes produce the vast majority of the axions. For core temperatures aboveabout × K, none of the core remains superfluid. In ungapped nuclear matter, all three nucleonbremsstrahlung processes are relevant, though n + p → n + p + a produces the largest contribution to theaxion emissivity. Profiles of the axion emissivity throughout the . M (cid:12) magnetar, for different values ofthe core temperature, can be seen in Fig. 2 of [84].2
3. ALP spectrum
It is important to understand the energy spectrum of axions emitted from neutron star cores, becausethose axions, with some probability, convert into photons with the same energy (see Sect. II C). The axionspectrum and resultant photon spectrum will not be identical, as the conversion probability depends onthe axion energy. In the right panel of Fig. 1 we plot dN a / dω , the number of axions produced withenergies between ω and ω + dω per time in the core of the . M (cid:12) magnetar considered above, separatingthe emission into the contribution from the three bremsstrahlung processes and the contribution fromCooper pair formation. The quantity dN a / dω is related to the differential luminosity of axions, whichcan be written as an integral of the axion emissivity over the core of the magnetar dN a dω = 1 ω dL a dω = 1 ω (cid:90) d r dQ a dω = 4 πω (cid:90) R crust dr r dQ a dω . (9)Expressions for the differential emissivity dQ a / dω coming from the integrated expressions Eqs. 5, 6, 7,and 8 are given in [84].In degenerate nuclear matter, the nucleon bremsstrahlung process produces axions with an energyspectrum that peaks at around ω = 2 T [28]. In contrast, the Cooper pair formation process producesaxions only with energies greater than or equal to twice the superfluid gap, which is the binding energy ofa Cooper pair. In a section of the neutron star where the proton gap is ∆ , the spectrum of axions producedfrom Cooper pair formation is at its maximum very close to ω = 2∆ . When studying the spectrum ofaxions produced by Cooper pair formation in the star as a whole, it is clear that the spectrum will beginat ω = 2∆ min , where ∆ min is the minimum (but nonzero) value of the gap found in the magnetar.As can be seen in the right panel of Fig. 1, when the core has T = 10 K, the bremsstrahlung processesproduce axions with a spectrum peaked at 17 keV. At this temperature, the superfluid gap is about 160 keVat the center of the magnetar, and rises to 1 MeV near the crust (see Fig. 1 of [84]). Therefore, the axionsproduced by Cooper pair formation in the center of the star will have energies greater than 320 keV, andthose produced from the outer regions of the star will have energies greater than 2 MeV. Therefore, at thiscore temperature, all axions produced with energies below about 320 keV are from bremsstrahlung, whilethose produced with energies greater than 320 keV are produced about evenly from bremsstrahlung andfrom Cooper pair formation. This trend holds for core temperatures of × K and K, however at K the superfluid gap at the center of the magnetar is slightly smaller than at the lower temperaturesdisplayed here, and thus the axion spectrum coming from Cooper pair formation starts at about 200keV. At these temperatures, the high-energy axion spectrum comes largely from Cooper pair formation.When the core temperature is × K, only a tiny part of the star is superfluid, so bremsstrahlungdominates the axion production. However, the Cooper pair formation can produce axions of arbitrarilylow energy in this case, because the superfluid gap rises continuously from zero as, moving outward fromthe magnetar center, the protons transition from ungapped to gapped at about 8 km from the center ofthe magnetar.Since the authors of [84] studied the hard X-ray spectrum coming from axion-photon conversion, onlyaxions emitted with energies between 20-150 keV needed consideration. Thus, the authors neglected axionproduction from the formation of S proton Cooper pairs, because the axions emitted in this channelhave energies above 200 keV (except in the case where the core temperature is × K, but in thatcase the channel is subdominant to axion production from bremsstrahlung since protons are no longersuperfluid in much of the magnetar).The above analysis of the total energy loss from axion emission and the spectra of the emitted axionsfollowed the investigation of axion emission from magnetars in [84], which considered only proton singletsuperfluidity due to the expected high core temperature of the magnetars. If the magnetar core is colderthan assumed in [84], then neutron superfluidity should be considered, which would strongly suppress theaxion emission from bremsstrahlung processes. In addition, the breaking of P neutron Cooper pairs3could also produce axions. In an analysis of a group neutron stars called the Magnificent Seven, which(for the most part) have core temperatures less than the magnetars considered in [84], the authors of[121] discussed the effects of including neutron superfluidity, although the main results of their studyassume the absence of both superfluid neutrons and protons. Neutron Cooper pairing and its effect onthe production of light particles is discussed further in [58, 59, 67]. C. Conversion of ALPs to photons near neutron stars
Having discussed the production of axions from neutron stars, we now go on to a discussion of thesubsequent fate of these axions as they enter the magnetosphere.ALP production in the neutron star core peaks at energies of a few keV to a few hundreds keV; theALPs subsequently escape and convert to photons in the magnetosphere. The emissivity has a parametricdependence on the core temperature that goes as ∼ T , and the conversion probability is enhanced bylarge magnetic fields. This makes magnetars, with high core temperatures and strong magnetic fields,ideal targets for probing ALPs. Other systems – such as magnetic white dwarfs [54] and the MagnificentSeven [121, 122] – have also been studied recently in the same framework. Quiescent emission in thehard X-ray and soft gamma-ray band from neutron stars will be our data of interest. There is a separateprogram of cold ambient axions from the halo converting to photons in the magnetosphere, but theputative signal in that case will lie in the radio band [123]. In our opinion, radio signals are complicatedby plasma effects which must be carefully treated; such effects are negligible at the higher energies wewill review.The inter-conversion of ALPs and photons in the dipolar magnetic fields near neutron stars has onlybeen fully understood recently, and semi-analytic solutions for the probability of conversion obtained[84, 124–126]. We now summarize these results.
1. ALP-photon conversion in dipolar magnetic fields
The evolution equation for the ALP-photon system propagating in the magnetic field of a neutron starmagnetosphere in terms of the dimensionless distance x = r/r is given by [15] i ddx aE (cid:107) E ⊥ = ωr + ∆ a r ∆ M r M r ωr + ∆ (cid:107) r
00 0 ωr + ∆ ⊥ r aE (cid:107) E ⊥ , (10)where ∆ a = − m a ω , ∆ (cid:107) = ( n (cid:107) − ω, ∆ ⊥ = ( n ⊥ − ω, ∆ M = 12 g aγγ B sin θ. (11)The ALP field has been represented by a ( x ) while the parallel and perpendicular electric fields are denotedby E (cid:107) ( x ) and E ⊥ ( x ) , respectively. The energy is denoted by ω , the neutron star radius by r , and theangle between the direction of propagation and the magnetic field by θ . The magnetic field B is given bya dipole with strength B at the surface r .In the strong magnetic fields near neutron stars, the photon refractive indices n (cid:107) and n ⊥ can departsignificantly from their standard vacuum values of unity. The refractive indices can be derived from thephoton polarization tensor and perturbative expansions are possible in various limits of the magneticfield B and photon energy ω . For ALPs with energies in the hard X-ray and soft gamma-ray range ω ∼ O (100 − keV that convert near the “radius of conversion” [that we define below in Eq. (15)],4the appropriate limits are eBm e ≡ BB c (cid:28) (cid:18) BB c (cid:19) ω sin θm e (cid:28) , (12)and the corresponding indices of refraction are given by (cid:26) n (cid:107) n ⊥ (cid:27) = 1 + α π (cid:18) BB c (cid:19) sin θ (cid:26) (cid:27) + O (cid:0) ( eB ) (cid:1) . (13)Here, B c denotes the critical magnetic field: B c = m e /e = 4 . × G, where e = √ πα is the chargegiven in terms of the fine structure constant α ∼ / .The mixing angle ϕ mix between the ALP field and the perpendicular photon field can be extractedfrom Eq. (10) and is given by tan(2 ϕ mix ) = 2∆ M ( x )∆ (cid:107) ( x ) − ∆ a . (14)The mixing angle (14) at the magnetar surface is negligible, which led the classic paper by Raffeltand Stodolsky [15] to dismiss this particular avenue of probing ALPs. However, it is clear that the r -dependence of ∆ M and ∆ (cid:107) lead to an eventual increase of the mixing angle away from the magnetar:indeed, ∆ (cid:107) ∼ /r , while ∆ M ∼ /r . Thus the denominator decreases faster than the numerator, andthe conversion probability peaks near the radius of conversion where the off-diagonal ∆ M contributionbecomes the same order as the diagonal ∆ (cid:107) term. This is given by r a → γ = r (cid:18) α π (cid:19) / (cid:18) ωm a B B c | sin θ | (cid:19) / . (15)For large conversion radius where r a → γ (cid:29) r which is typically the case, the probability of conversiontakes a simple form in both the small and large | ∆ a r a → γ | regimes, P a → γ = (cid:18) ∆ M r r a → γ (cid:19) × π | ∆ a r a → γ | e ara → γ | ∆ a r a → γ | (cid:38) . Γ ( ) | ∆ a r a → γ | | ∆ a r a → γ | (cid:46) . . (16)These expressions constitute semi-analytic solutions to the ALP-photon propagation system and can beused to obtain the final spectrum of photons coming from axion conversions.
2. X-rays and gamma-ray signatures of ALPs: current constraints
From Eq. (9), the total energy of photon emission from ALP conversion in the magnetosphere isobtained by multiplying the ALP emission spectrum by the ALP to photon conversion probability [125]: L a → γ = (cid:90) ∞ dω π (cid:90) π dθ · ω · dN a dω · P a → γ ( ω, θ ) , (17)where P a → γ can be obtained from Eq. (16).5 FIG. 2. Left panel: The maximal conversion probability is shown as a function of B /B c where B is the surfacemagnetic field and B c the critical QED field strength. Right panel: The mixing angle is shown as a function ofthe dimensionless distance r/r from the magnetar surface. The benchmark values are chosen to be ω = 80 keV, g aγγ = 10 − GeV − and B = 1 . × G corresponding to the magnetar 4U 0142+61.
The photon spectrum per area observed at the earth is then obtained by dividing dL a → γ /dω by πD with D the distance between the magnetar and the earth. The experimental data of the spectrum isusually expressed on the νF ν plane, which is related to the above definition by the following relation: νF ν ( ω ) = ω πD ω dL a → γ dω . (18)This is the master equation that can be used to place constraints on G an × g aγγ .The most promising targets for this direction are magnetars, with their high core temperatures andstrong magnetic fields. Magnetars are a group of neutron stars with dipole magnetic fields of strengthsup to – G (see [77, 127] for recent reviews). Apart from short X-ray bursts and giant flares, theyexhibit persistent emission in the ∼ . – keV band, with luminosity L ∼ – erg s − . Thispersistent emission is of interest for our purposes. It consists of a soft quasi-thermal component up toaround 10 keV, and a very flat hard X-ray tail extending to beyond 200 keV. While the soft componentis due to thermal emission from the surface, the hard X-rays are generally attributed to resonant inverseCompton scattering of thermal photons.Data from INTEGRAL [128–135],
Suzaku [136–139],
RXTE [140],
Swift [141],
XMM-Newton [142],ASCA and
NuSTAR [143–145] has revealed the hard X-ray component for around nine magnetars. More-over, data from
INTEGRAL and COMPTEL can be used to place upper limits on the soft gamma-rayemission from a subset of these magnetars. Experimental data in both bands can be compared to thetheoretical spectrum coming from ALP-photon conversion, and constraints can be placed on the productof ALP couplings G an × g aγγ by demanding that the theoretical spectrum obtained from Eq. (17) notexceed the observed spectrum in any energy bin. It is important to note that ALP-photon conversionis not being advanced as the underlying model of magnetar emission; in particular, no excess is beingclaimed over astrophysical “background”. Rather, upper limits of G an × g aγγ are being obtained by6 - - - - - - - - - FIG. 3. The
CL limits on G an × g aγγ obtained by comparing Eq. (18) against data. The blue linesare the limits obtained using the X-ray band (10 −
160 keV) from 8 magnetars with the S proton superfluidityincluded [84]. The magenta lines use the soft gamma-ray band ( keV − × K (dashed) and K (solid). The limit corresponding to the dot-dashed line and denoted by Magnificent Seven is taken from [121],and the gray shaded region is the exclusion derived from both CAST and SN 1987a from [121]. directly comparing with observational data.The constraints can be obtained by comparing the theoretical spectrum against data from magnetarsin the hard X-ray (10-160 keV) and soft gamma-ray (300-1000 keV) bands. This analysis was performedin [84, 126]. The results are displayed in Fig. 3, with the constraints coming from the analysis of the hardX-ray (soft gamma-ray) spectrum being depicted by the blue (magenta) curves. For both analyses, twocore temperatures of the magnetars are used: × K (dashed) and K (solid). It should be notedthat [84, 126] performed the analysis for several magnetars. What is being depicted in Fig. 3 are the bestcase scenarios: magnetar 1E 1547.0 − F ermi -GBM, AMEGO, and future telescopes may yield stronger constraints on the axion-photon coupling andshould be a component of the fundamental physics case for such experiments.7 . . . . . − − − − −
10 ¯ I k / ¯ I ⊥ ¯ I a / ¯ I ⊥ Q / ¯ I ⊥ Q/ ¯ I ⊥ for ω = 1 keV − − − − − . . . . . . . . . − . − . − . I k / ¯ I ⊥ ¯ I a / ¯ I ⊥ Q / ¯ I ⊥ Q/ ¯ I ⊥ for ω = 100 keV − . . FIG. 4. O-mode polarization in X-rays from ALP-photon conversion: The normalized Stokes parameter Q/ ¯ I ⊥ atinfinity is shown for the benchmark point m a = 10 − eV, g/e = 5 × − keV − , r = 10 km, B = 20 × Gand θ = π/ in the plane ( ¯ I (cid:107) / ¯ I ⊥ , ¯ I a / ¯ I ⊥ ) . The ALP energies are soft X-rays ω = 1 keV (left panel) and hardX-rays ω = 100 keV (right panel). Astrophysical models predict mostly X-mode polarization and one has Q/ ¯ I ⊥ ∼ , ¯ I (cid:107) / ¯ I ⊥ ∼ , and ¯ I a / ¯ I ⊥ ∼ in the absence of ALPs. In the presence of ALPs, Q/ ¯ I ⊥ may lie anywhere on theplane, depending in the averaged intensity ¯ I a of the ALP field. Figures reproduced from [124].
3. Polarized X-rays and gamma-rays from magnetars
One of the most crucial signatures of ALP-induced emission versus astrophysical background is thedifference in polarization. In the strong magnetic fields near magnetars, photons are polarized in twonormal modes, the ordinary (O) and extraordinary (X) ones, which correspond to the electric fieldbeing respectively parallel and perpendicular to the plane containing the external magnetic field and thedirection of propagation. For photon energies lower than the electron cyclotron energy ∼ . B/ G) keV, astrophysical models predict that the X-mode opacity is suppressed with respect to the O-mode.Thus, the thermal radiation is almost completely polarized in the X-mode [146, 147]. Remarkably,resonant inverse Compton upscattering, which is the putative astrophysical model underlying the hardemission, also predicts a strongly polarized X-mode emission [148]. One therefore expects astrophysicalemission to be polarized along the X-mode, in all energy ranges from the soft to the hard spectrum.Since the magnetic field is dipolar, the net polarization has to be averaged over emissions from regionswith different orientations of the field direction. The effect is generally to obscure the overall polarizationpattern. However, the QED vacuum birefringence effect can lock the polarization vector to the magneticfield direction up to the polarization radius r P L , where the polarization vector stops tracking the magneticfield, which is more uniform [149]. The adiabatic tracking of the polarization vector continues up to thepolarization radius r P L . This leads to astrophysical predictions of polarization fractions as high as 40%to 80% at the detector [146]. Therefore, not only is the astrophysical emission X-mode, the birefringenceeffect in the magnetosphere near magnetars “locks in” this mode as far as detection is concerned. This“locking in” is especially efficient in magnetars with their strong magnetic fields. The radius of polarizationis given by [146] r P L = (cid:16) α νc (cid:17) / (cid:18) B B c r sin β (cid:19) / ∼ . (cid:16) ω keV (cid:17) / (cid:18) B G (cid:19) / (cid:16) r km (cid:17) / km , (19)where β is the angle between the dipole axis and the line of sight.8The unique observational signature of ALP-photon conversion is the change in the predicted polariza-tion pattern. ALPs add to the astrophysical picture described above by producing O-mode photons, forwhich the electric field is parallel to the plane containing the external magnetic field and the direction ofpropagation. The radius of conversion, where the probability of conversion becomes significant, is typi-cally larger than the polarization radius, implying an overall O-mode superposed on the X-mode comingpurely from astrophysics [124]. The ratio is given by r P L r a → γ ∼ . (cid:18) keV ω (cid:19) / (cid:18) B G (cid:19) / (cid:16) m a − keV (cid:17) / (cid:16) r km (cid:17) / . (20)We note that r P L (cid:28) r a → γ for typical ALP benchmark points. The effect of ALPs on the observedpolarization pattern will thus be to add an O-mode intensity to the purely astrophysical X-mode intensity.The relative strength of the O-mode component depends on the intensity of ALPs produced in the coreand the probability of conversion. This can be quantified by considering X-ray emission produced bothby astrophysical processes and by ALP-photon conversion, in an uncorrelated fashion, and in differentrelative proportions. The uncorrelated production of photons and ALPs allows one to average over theinitial phase difference ∆ φ between the fields E (cid:107) ( x ) and a ( x ) (the sum phases decouples). It is convenientto introduce the following quantities, all obtained by averaging over the phase difference ∆ φ : the sum( I = ¯ I ⊥ + ¯ I (cid:107) ) and difference ( Q = ¯ I ⊥ − ¯ I (cid:107) ) of the phase-averaged photon intensities in the parallel andperpendicular planes, and the phase averaged ALP intensity ¯ I a . These intensities can be normalized bythe total amplitude of the E (cid:107) ( x ) and a ( x ) fields.The resulting Stokes parameter Q is shown in Fig. II C 2, for a photon energy of 1 keV and 100 keV.The astrophysical prediction is at the top left corner, where Q/ ¯ I ⊥ = 1 , ( ¯ I (cid:107) / ¯ I ⊥ = 0 , and ¯ I a / ¯ I ⊥ ) = 0 .With ALPs, on the other hand, Q/ ¯ I ⊥ can take values anywhere on the planes, depending on the ALPintensity emitted. We can thus see a significant departure from the astrophysical prediction.There are several missions that may be able to explore these features. The Imaging X-ray PolarimetryExplorer (IXPE), Small Explorer (SMEX) and the Enhanced X-Ray Timing and Polarimetry Mission(eXTP) missions will launch in the next few years, and look for signals in the − keV range [149].The next generation of X-ray polarimeters and increasingly sophisticated modeling of the astrophysics ofmagnetars provide an opportunity to investigate ALPs using polarization. D. Photon-to-ALP conversion in galactic and extra-galactic magnetic fields
We finally circle back to the topic of axion conversions in large-scale magnetic fields. Photons producedby distant sources will convert to ALPs in large-scale magnetic fields as they propagate towards the earth.This is a long-standing topic of interest in ALP searches with a vast amount of associated literature thatmerits its own dedicated review. We only mention some broad ideas and a sliver of the recent literature.
X-ray Sources:
A number of studies have concentrated on X-ray point sources located within orbehind galaxy clusters. The intracluster magnetic field serves as the medium in which ALP-photoninterconversion occurs, as X-rays travel from the source to the earth. The result is that there are spectraldistortions of the X-ray source, which can be leveraged to place constraints on the ALP-photon coupling.As we have indicated before, a major challenge is the limited knowledge that we have about the mor-phology of the magnetic field along our line of sight to the source. Typically, a domain-like configurationis assumed, with the field being assumed to be constant in each such domain. The expression for theconversion probability in the constant magnetic field within a given domain is standard and was given inthe classic paper by Raffelt and Stodolsky [15]. We follow the notation of, for example, [150]: P γ → a = 4 ϑ sin (∆ osc L/ , (21)9where L is the domain length, ϑ ≈ tan(2 ϑ ) = ∆ gaγγ ∆ γ − ∆ a and ∆ osc = ∆ γ − ∆ a . Here the expressions forthe various quantities are ∆ γ = − ω ω p , ∆ a = − m a ω p , ∆ g aγγ = Bg aγγ , and ω p = (cid:113) παn e m e is the plasmafrequency. The intracluster magnetic field is expected to have a central strength of B ∼ − µ G andtypical coherence lengths of L ∼ − kpc.The simplicity of the conversion probability in a given domain belies the complexity of obtaining thetotal probability and finally the signatures of ALP-photon interconversion. This involves several steps,which we describe in some detail following, for example, Ref. [33]; similar methods are employed for otherX-ray and gamma-ray sources. ( i ) A radial dependence for the magnetic field is assumed, with a benchmark central strength B inthe range stated above; B ( r ) ∝ B [ n e ( r )] η , with η expected to be between 0.5 and 1. For example, forthe Perseus AGN NGC1275 B ( r ) ∝ [ n e ( r )] . [33]. The electron density n e ( r ) is given as a function ofradial distance and has functional form n e ( r ) ∝ n [1 + ( r/r c ) ] β , where n is the central electron density, r c is the radius of the core, and β is a constant. A double β dependence is appropriate for cool-coreclusters, for example in the specific case of Perseus β = − . , n , = 3 . × − , r c, = 80 kpc and β = − . , n , = 4 . × − , r c, = 280 kpc. For more details, we refer to [33]. ( ii ) The total propagation distance is divided into a large number of domains ( ∼ ), with domainlengths L drawn as a random parameter in the range 1-10 kpc. Within each domain, the magnetic field isassumed to be constant at a value given by the radial dependence described above, with r being taken asthe distance from the center of the cluster to the near end of the domain. The direction of the magneticfield is taken from a flat distribution between [0 , π ] . ( iii ) A simulation is then run for a range of ALP couplings g aγγ and masses m a ; for each choice of ( g aγγ , m a ) there are ∼ choices for domain lengths, and ∼ choices of the magnetic field in eachdomain.The above steps can be used to obtain the final survival probability. The observed spectrum is the sourcespectrum modulated by the survival probability. These quasi-sinusoidal modulations can be analyzedusing global goodness-of-fits or machine learning techniques. A variety of X-ray sources have been studiedin this framework [33, 151–153]. Gamma-ray Sources:
In parallel with distant X-ray sources, there is a substantial amount ofliterature on studying gamma-ray sources within the same framework of photon-ALP conversion. Thesources studied include flat spectrum radio quasars and BL Lacs, supernova remnants, galactic pulsars,etc. [38, 40, 154, 155]. The intervening magnetic field is that of the host galaxy or cluster [156–160],extragalactic space between galaxy clusters [38, 161, 162], the field within the blazar jet [40], etc. As inthe previous case, a domain-like modeling of the large-scale magnetic field is employed, with each domainlength being equal to the field’s coherence length, and the field being assumed to be constant in a givendomain. The ALP-photon transfer function is computed by performing simulations of various magneticfield configurations and computing the mean and variance of the gamma-ray flux [163, 164].
ALPs from Supernovae:
The galactic-scale scenarios described above study photon-to-ALP con-versions. One could also study the opposite scenario of ALP-to-photon conversion within the sameframework. ALPs produced from supernovae will convert to photons in the magnetic field of the MilkyWay on their way to the earth. The result will be, for example, a gamma-ray flash during a supernova[36, 165, 166] or a diffuse supernova flux with energies ∼ O (50)
MeV [35]. Using the absence of such agamma-ray flash in observations of SN1987a or using data from the Fermi satellite to constrain a diffusesupernova flux leads to constraints on ALP couplings.The properties of the interstellar medium of the Milky Way affect the ALP-to-photon conversionprobability in such scenarios, and hence the final constraints on ALP parameter space. A recent modelingof the galactic magnetic field which has been used in the ALP literature is the one by Jansson and Farrar[167]. The field in this model is composed of a sum of coherent, random, and striated components. On0the other hand, the electron density of the interstellar medium, which controls the photon mass, can bemodeled using the thin and thick disc components of the NE2001 model. For more details, we refer to[39].
ALPs from other Astrophysical Sources:
It is known that CMB photons can convert to axionson their way to the earth [168, 169]. This causes the CMB spectrum to deviate from being a perfectblack body, ultimately putting constraints on g aγγ B IGM . Assuming some large value for B IGM , such as B IGM ∼ nG , the constraint on ALPs lighter than − eV is g aγγ (cid:46) − GeV − .In a recent study [170], different cosmic distance measurements were combined to constrain the axion-photon coupling. The constraints included the luminosity distance of type Ia supernovae and angulardiameter distance of galaxy clusters. The idea is that due to the photon-to-axion conversion, the apparentmagnitude of the luminosity is decreased compared to the would-be value if there had been no axionconversion. Any distance measurement that relies on the apparent magnitude is affected. The constraintsrely on either intergalactic magnetic field or the intracluster magnetic field (ICM), depending on wherethe conversion happens. Therefore this provides a constraint that scales differently with the intergalacticmagnetic field alone as in [168, 169]. It was shown that for the axion masses in the range m a (cid:46) − eV ,a constraint of the photon-axion coupling can be put in the range of g aγγ (cid:46) × − GeV − if optimisticICM model is adopted, or × − (nG /B ) GeV − if ICM conversion is neglected altogether. III. NEUTRON STAR MERGERS
In this section, we review the inspiral and postmerger stages of a binary neutron star merger, and thendiscuss what each stage of the merger can tell us about the physics of axions.
A. Dynamics of a neutron star merger
As a binary neutron star system evolves in time, the individual stars emit gravitational radiation asthey orbit each other. The gravitational waves carry energy away from the system (see [171, 172] forthe relevant calculations), causing the distance between the two stars to decrease. The orbit is graduallycircularized if it had an initial eccentricity. The frequency of the emitted gravitational waves is twicethe orbital frequency, and thus as the two stars move closer together, the gravitational wave frequencyincreases. After tens or hundreds of millions of years [172, 173], the two stars get sufficiently close to eachother that they begin to tidally deform, which takes additional energy away from their orbit, bringingthe system closer to the eventual merger. During most of this “inspiral” phase of the neutron star merger,the internal condition of one neutron star is unaffected by the other. The nuclear matter within the starsis cold due to millions of years of neutrino cooling [70], and it remains cold until the stars collide [174].On August 17, 2017, the LIGO-Virgo collaboration observed the gravitational wave signal from theinspiral of the binary neutron star merger, GW170817 [175]. The signal was detected when the frequencyof the emitted gravitational waves from the inspiral rose above 24 Hz, and continued for about 100seconds until the two stars collided. No postmerger gravitational wave signal was detected. To date, onlygravitational waves from the inspiral phase of a neutron star merger have been measured [177, 178].From the early stages of the inspiral, little about the neutron stars can be learned, because they aresufficiently separated to be indistinguishable from point masses. Once the stars are close enough totidally deform each other, the deformation alters the phase of the gravitational wave signal, from which The LIGO-Virgo collaboration has also measured the gravitational wave signal from many binary black hole mergers, thefirst of which was GW150914 [176]. FIG. 5. Temperatures and densities attained during a neutron star merger, according to a numerical simulation.The simulation does not include axions. Each point corresponds to a fluid element, colored by its distance fromthe center of the merger. While two milliseconds after the merger (left panel) there is little correlation between thetemperature and density of fluid elements in the merger, after ten milliseconds (right panel) the nuclear matterhas settled down to a configuration with warm, low density matter on the outskirts of the merger, hot medium-density matter in the outer core, and cooler, high density matter in the core. Figure courtesy of M. Hanauskeand the Rezzolla group, see also [200]. information about the nuclear equation of state can be inferred [179, 180]. In Sect. III B we will reviewthe information that can be learned about axions from the neutron star inspiral.From the beginning of tidal deformation onward, numerical simulations are necessary to understandthe dynamics of the neutron star merger. Numerical simulations evolve Einstein’s equations to determinethe spacetime metric throughout the merger. The nuclear matter in the neutron stars is treated as afluid and evolved according to the equations of relativistic hydrodynamics. The fluid has an equation ofstate derived from nuclear theory [46]. As their mean free path in hot and dense matter is macroscopic[107, 112], neutrinos are treated separately from the nuclear matter, either with a kinetic formalism or asa separate fluid with a loss term [181, 182]. Merger simulations are reviewed in [172, 183–185] and willbe featured in Sect. III C 2.Snapshots of the configuration of the merger at various times in its evolution, obtained from numericalsimulations, can be found in [186–192]. A few orbits after the stars begin to tidally deform, they collidewith a relatively large impact parameter, which generates a shear interface that may lead to a Kelvin-Helmholtz instability [184, 193]. When the two neutron stars merge, unless there is prompt collapse toa black hole, a differentially rotating mass of hot nuclear matter emerges. Right at the time of merger,the nuclear matter at the collision interface heats to several tens of MeV. The differential rotation [194],but also perhaps the thermal pressure, keep the star from collapsing for a period of time, ranging froma few milliseconds to as much as a few seconds [196, 197]. If the mass of the spinning remnant is belowthe maximum mass of a stable neutron star [41, 199], the remnant will survive indefinitely.In Fig. 5 we show results from a merger simulation conducted by the Frankfurt group, where theyhave plotted the temperature and density of fluid elements in the merger at two snapshots in time. We The influence of the thermal pressure on the lifetime of the remnant is complicated - see the discussion in [195]. It is believed that the remnant created in GW170817 lasted for approximately one second before collapsing to a blackhole [198]. . n , but the right panel shows that the maximum density reached in the remnant is almost twice this.The trends seen in this simulations are seen in many other simulations as well [201–203], although thespecifics depend on the initial masses of the two stars and the nuclear equation of state. After the collisionof the two stars, very little – if any – of the nuclear matter is superfluid, since the temperature of the vastmajority of the remnant is above 1 MeV. The thermodynamic conditions of the nuclear matter in themerger have a large impact on the role of axions in the merger remnant, which we discuss in Sect. III C. B. Ultralight axions in the neutron star inspiral
It is unlikely that the inspiral of a binary neutron star system can tell us much more about “particle-like”axions (axions with a Compton wavelength λ Compton = 1 /m a much smaller than the size of a neutronstar) than we already know from isolated neutron stars (Sect. II), since for most of the inspiral the internalconditions of one star are not modified by the presence of its binary companion. However, axions withvery small masses have long Compton wavelengths, and when the Compton wavelength reaches the sizeof a neutron star or larger (corresponding to axion masses of less than − eV) the dynamics of theinspiral could potentially be impacted by the existence of ultralight axions.The authors of [205] found that ultralight axions can be sourced by high-baryon density objects likeneutron stars, endowing those objects with a scalar charge. They studied a “tuned” version of the QCDaxion and showed that this type of axion can mediate a force [208] F = − Q Q πr e − m a r ˆ r, (22)between two neutron stars with scalar charge. Here, m a is the axion mass and the scalar charge is Q = ± π f a R , where f a is the axion decay constant and R is the radius of the neutron star. Theforce between the two stars can be either attractive or repulsive. When the separation between the twoneutron stars drops to about /m a , deviations from the standard general-relativistic inspiral dynamicswill become evident and could be seen in the gravitational wave signal. In addition, once the frequencyof the inspiral rises above m a , scalar Larmor radiation can occur, dissipating orbital energy from themerging system.The authors of [208] provided a forecast for how well Advanced LIGO would be able to constrain suchan ultralight axion from a measurement of the gravitational wave signal from the neutron star inspiral.They also speculated that by measuring the scalar charge of the neutron star (from the deviation of theinspiral dynamics from their expected general-relativistic behavior), one could determine the compactness M/R of the star, which constrains the nuclear equation of state [209].The work in [210, 211] sets upper limits on the axion decay constant by comparing the measured decayof the period of several binary neutron star and neutron star - white dwarf systems to the expected resultfrom the energy loss due to gravitational radiation, allowing them to constrain the amount of scalarLarmor radiation that could have been produced by the compact objects. An exception would be the f-mode oscillations that are induced in binaries with eccentric orbits [179, 204]. As part of a detailed extension of the QCD axion to finite density, the authors of [206] note that it is unlikely that thestandard QCD axion will be able to significantly alter the inspiral dynamics of a merger. Tuning of the axion potential (asdone in [205]) is likely required for the axion to modify the inspiral dynamics. In [205], the authors provide an examplescenario that gives rise to the tuned axion potential. See also the very recent work [207]. C. Axions in merger remnants
Following in the path of studying the role of “particle-like” axions in supernovae [28, 212–217] andisolated neutron stars [53, 71, 73, 84, 122], recent progress has been made in understanding the role ofaxions in neutron star merger remnants, which will be discussed in this section. The study of ultralightaxions in merger remnants [218] is less developed, and we will not discuss it here.
1. Axion mean free path in hot, dense matter
The potential role of axions in a neutron star merger after inspiral is determined by their mean free pathin the hot, dense conditions present in the merger remnant. If their mean free path is small comparedto the size of the remnant, they would form a thermally equilibrated Bose gas and would be able tocontribute to transport processes inside the star, for example, thermal conductivity or shear viscosity(see Sect. III C 3). If the axion mean free path is comparable to or larger than the size of the remnant,then the axions would free-stream through the system, taking some energy with them. This would serveto cool down the merger remnant [108] (see Sect. III C 2). The mean free path of axions is determinedby how often they are absorbed by the inverse bremsstrahlung process N + N (cid:48) + a → N + N (cid:48) where N and N (cid:48) are either neutrons or protons.The mean free path of axions in nuclear matter is determined from the individual mean free paths dueto each possible absorption process λ − a = λ − nn + λ − np + λ − pp , (23)and is a function of the baryon density and temperature, but also of the axion energy. The mean freepath of a given absorption process, for example, n + n + a → n + n , is given by the phase space integral λ − nn = (cid:90) d p (2 π ) d p (2 π ) d p (2 π ) d p (2 π ) S (cid:80) spins |M| E ∗ E ∗ E ∗ E ∗ ω (2 π ) δ ( p + p − p − p + p a ) f f (1 − f )(1 − f ) . (24)The factors in this equation have the same meaning as in Eq. (4). Using the Fermi surface approximation,the mean free path of an axion with energy ω has been calculated in degenerate nuclear matter (forexample, in cold neutron stars), and has the simple form λ − nn = 118 π f G an ( m n /m π ) p F n F ( c ) ω + 4 π T − e − ω/T . (25)Here, c = m π / (2 p F n ) and the derivation and definition of F ( c ) – which is a function of the baryon density– is given in [108] (the original calculation was done in [92], however). The mean free path of the axionin nondegenerate nuclear matter (seen, for example, in a supernova) is calculated in [219, 220] and inmatter of arbitrary degeneracy (although assuming non-relativistic nucleons) in [50]. These calculationsare reviewed and extended to the case of arbitrary degeneracy and fully relativistic nucleons in [108],where the full phase space integration is performed. In Fig. 6, we show the results of a calculation of theaxion mean free path (from [108]) in the absorption process n + n + a → n + n .Fig. 6 shows that even for the strongest allowed axion-nucleon coupling, in the thermodynamic con-ditions likely encountered in neutron star mergers, the axion mean free path is quite long and axiontrapping in mergers is unlikely. This mean free path calculation considers only n + n + a → n + n . The inclusion of p + p + a → p + p and n + p + a → n + p will shrink the mean free path by a factor of a few. However, the calculation in Fig. 6 also does not include the factorof (approximately) four reduction in the square of the matrix element due to improvements to the one-pion exchangeinteraction (Sect. II B 2). This leads to a lengthening of the mean free path by a factor of 4. Thus, the results shown inthis figure are likely close to the correct answer. T e m p e r a t u r e ( M e V ) Baryon number density (units of n ) k m k m k m k m k m Axion MFP (G = G
SN1987A , ω = 3T)
FIG. 6. Axion mean free path (for an axion with energy ω = 3 T ) due to absorption via n + n + a → n + n . Theaxion-neutron coupling constant is chosen to be at the upper bound set by SN1987a [23], so this plot indicatesthe minimum mean free path an axion with ω = 3 T could have at each density and temperature. The dashedcontours correspond to the axion MFP calculated in strongly degenerate nuclear matter [Eq. (25)], while the solidcontours are the result of the full integration over phase space [Eq. (24)]. Figures reproduced from [108].
2. Axion cooling of merger remnants
Since axions likely free-stream from the nuclear matter in a neutron star merger, the temperature of afluid element decreases due to the energy loss from axion emission according to dTdt = − Qc V , (26)where Q is the axion emissivity and c V is the specific heat of the nuclear matter at constant volume(or baryon density). Since the vast majority of the matter in neutron star mergers has a temperaturewell above an MeV, we can ignore nucleon superfluidity and thus axions are produced only by the threenucleon bremsstrahlung processes N + N (cid:48) → N + N (cid:48) + a . The total emissivity is the sum of the emissivityfrom each axion production process Q = Q nn + Q pp + Q np . (27)The emissivities are calculated in the degenerate limit in the context of magnetars (Sect. II B 2) (one canconsider strongly degenerate nuclear matter that is not superfluid by just setting the reduction factors R np and R pp to one). The emissivity of axions due to bremsstrahlung processes has also been calculatedfor nonrelativistic and nondegenerate nucleons in [221–223] and extended to nonrelativistic nucleons witharbitrary degeneracy in [50]. The full phase space integration was performed in [108] to obtain the axionemissivity with relativistic nucleons of arbitrary degeneracy. As in the mean free path calculation, theemissivity calculation in [108] chose C π = 1 and neglected axion production processes involving protons.The specific heat is dominated by the particle species with the largest number of low energy excitations,which in dense matter is the neutron. The specific heat of a degenerate Fermi liquid of neutrons is [224] c V = 13 m Ln p F n T, (28)5 Radiative cooling time (1n ) s m s . m s m s m s SN1987a
Axions not free-streaming (a) G a n ( G e V - ) −10 −9 −8 Temperature (MeV)
20 40 60 80 100
Radiative cooling time (7n ) s m s . m s m s m s SN1987a
Axions notfree-streaming G a n ( G e V - ) −10 −9 −8 Temperature (MeV)
20 40 60 80 100 (b)
FIG. 7. Radiative cooling time due to axion emission from the process n + n → n + n + a at densities of n (leftpanel) and n (right panel). All couplings stronger than the dotted blue line are disallowed by the observation ofSN1987a. As long as the axion-neutron coupling is not too far below the bound set by SN1987a, the lower densityregions of the star can cool substantially in tens of milliseconds, while the highest density regions of the mergercould possibly cool in under 10 milliseconds (although it is not expected that the densest regions of the mergerreach temperatures of more than about 10 MeV - see Fig. 5). The emissivity used in this calculation only tookinto account axion production from n + n → n + n + a , neglecting the other two bremsstrahlung processes. Also, C π = 1 was chosen, not the more realistic value of around / . Including the other two bremsstrahlung processesand choosing C π = 1 / will not significantly change the results shown here. Figures reproduced from [108]. where m Ln is the Landau effective mass of the neutron. The specific heat does not change significantlyeven when the neutrons become nondegenerate, which occurs in the regions of the merger which have lowdensity but high temperature [108].The time that it takes for a fluid element to cool to half of its current temperature by radiating axionsis calculated by solving the differential equation in (26). The cooling time is calculated in [108], wherethe nuclear matter is described by the NL ρ equation of state [225] and the emissivity is calculated byperforming the full phase space integration [Eq. (4)]. The results are plotted in Fig. 7 at saturationdensity n (left panel) and at n (right panel). Along the y-axis is the strength of the axion-neutroncoupling. Couplings above the blue dotted line are ruled out by SN1987a [23]. For very large (and thus,ruled out) values of the coupling, axions would be trapped inside the merger and cooling would actuallyoccur on the diffusive timescales of several seconds [226], not the radiative timescales shown in the redhatched region of Fig. 7. As the coupling becomes weaker, the axion mean free path grows and axionscan escape the merger, cooling it. The calculations shown in Fig. 7 indicate that, as long as the axion-neutron coupling is not too much smaller than the SN1987a bound, at low densities fluid elements cancool significantly in timescales of tens of milliseconds and at high densities, significant cooling can takeplace on timescales under ten milliseconds. Simulations (conducted without axion cooling) show that thehottest matter in the merger exists at around − n and can reach temperatures of at least 50 MeV(see Fig. 5). The calculations shown in Fig. 7 predict axion emission could cool this region of the mergerto 25 MeV in approximately 30 milliseconds, well within the lifetime of many merger remnants.The effects of axion cooling were included in a neutron star merger simulation by the authors of [227].6
10 0 10 x y
10 0 10 x y l o g () l o g ( T M e V )
10 0 10 x y
10 0 10 x y l o g () l o g ( T M e V ) FIG. 8. Density profile (left panel) and temperature profile (right panel) from the neutron star merger simulationconducted in [227]. Snapshots are all from 15 ms after merger. The top row neglects axion cooling, and thebottom row incorporates axion cooling assuming (for illustration) an axion-nucleon coupling approximately 100times the SN1987a bound [23], significantly overestimating the effect that axions could have on the cooling. Thesimulation neglects neutrino cooling. Both neutron stars in the merger initially had masses of . M (cid:12) . Figuresadapted from [227]. They included an energy-loss term in their hydrodynamic equations that was set by the axion emissivitiescalculated by the authors of [27]. This method of incorporating cooling due to particle emission is similarto an early way of treating neutrino cooling in supernovae simulations [228, 229] and, later, compact starmergers [230]. In Fig. 8 we display results of the simulations including cooling from axion emission. Theleft column plots are colored by density and the right column plots are colored by temperature. The toprow corresponds to a simulation without axion cooling, while the bottom row corresponds to a simulationwhere axion cooling is included, with a value of the axion-nucleon coupling constant approximately 100times the value allowed by SN1987a. While this coupling is unrealistically large, and thus overestimatesthe effects of cooling, it was chosen so that the effects of axion cooling can be easily seen by eye. It iseasy to see from the right column of Fig. 8 that the axions indeed cool the remnant significantly in 15milliseconds (the time between the collision and the snapshots in Fig. 8) compared to the case where nocooling mechanism exists – note that the simulations discussed here neglect cooling of the remnant due7to neutrinos so as to focus on the effects of axion cooling. In addition, the energy loss due to axionssphericalizes the remnant, eliminating the bar shape that was present when axion cooling was neglected.The density of the central region is observed to be slightly larger when the axion-neutron coupling isstronger. In general, the authors of [227] observed that with increased axion-neutron coupling, the lifetimeof the remnant decreases, as is expected because the remnant is sustained partially by thermal pressure,as discussed in Sect. III A. However, the lifetime is only shortened by a couple milliseconds between thetop and bottom rows of Fig. 8, indicating that even strong axion emission is unlikely to significantlyinduce gravitational collapse. This is consistent with the expectation that differential rotation is thedominant influence on the lifetime of the remnant [194]. The difference in the gravitational wave signalpredicted by the simulation with and without axions was small.The conclusion of the work including cooling from axion emission in neutron star mergers [108, 227] isthat when the axion-neutron coupling is not too much smaller than the SN1987a bound, axion emissioncan lead to substantial cooling of the merger remnant. However, this cooling only has minor changeson the dynamics of the merger at the level of the simulations conducted in [227], raising the possibilitythat, at present, it may be difficult to use the postmerger phase to constrain axion physics any furtherthan has been done through other astrophysical environments. This is especially true given the currentuncertainties in, for example, the nuclear equation of state, which has a large impact on the post-mergerdynamics.However, if additional complexities are considered, axions again have the chance to substantially influ-ence the dynamics of neutron star mergers. As merger simulations develop, there is interest in includinga possible phase transition to quark matter [188, 200, 231–238] – or another exotic phase – at highdensity or high temperature. In this case, cooling on merger timescales that is due to axion emissioncould trigger an unexpected phase transition in the nuclear matter, which would almost certainly haveobservable consequences. In addition, efforts are starting to be made to include transport processes inmerger simulations [111]. Many transport processes are strongly temperature dependent, and thuscooling from axion emission would impact transport during the merger.
3. Transport from trapped dark sector particles
Thermal transport is due to particles with long mean free paths. If the mean free path is long, but stillshorter than the system size, the particle will contribute to evening out thermal gradients in the system.In a neutron star merger, neutrinos – if they are trapped – dominate the thermal conductivity becausetheir mean free path is always longer than that of the neutron, proton, or electron. The authors in [243]calculated that a temperature gradient over a distance of 100 meters could be eliminated by neutrino-driven conduction in tens of milliseconds. In a region of the star where neutrinos are not trapped, itwould take months for electron-driven conduction to eliminate the same temperature gradient.Dark sector particles, if they are trapped inside the merger remnant, could provide a more expedientpath towards thermal equilibration. While the axion is not likely to be trapped in a merger, as anillustration Ref. [108] calculated the thermal equilibration timescale due to axion emission and absorptionvia nucleon bremsstrahlung processes in a merger remnant. The timescale is highly dependent on theaxion-neutron coupling. It would be useful to consider the possibility of thermal transport due to otherdark sector particles, including CP-even scalars [89, 249], dark photons [96, 97], or other light dark mattercandidates. It may also be possible for dark sector particles to contribute to shear or bulk viscosity insidethe merger remnant. For example, pions [44, 239], hyperons [240, 241], or quarkyonic matter [242]. For example, bulk viscosity [241, 243–246], shear viscosity [243, 247, 248], and thermal conductivity [243, 247, 248]. IV. LABORATORY-PRODUCED ALP SEARCHES
If an axion or ALP exists, it could be produced in the laboratory, without relying on any extraterrestrialsources, and leave experimental signatures within a detector. Unlike the cases in Sects. II and III,environmental parameters associated with production of axions or ALPs are more under control, andtherefore, the relevant searches can set more conservative and model-independent constraints on modelsof axion or ALP. In particular, in regards to the PVLAS anomaly [29] various models or mechanisms toavoid astrophysical and/or cosmological bounds have been proposed. They typically introduce featuresin the hidden sector that suppress or turn off Primakoff production in stellar environments but restorethem elsewhere: for example, choices of couplings to facilitate trapping [250], phase transitions [251, 252],and chameleon-like screening effects [253] (see [254] for a more detailed discussions on these and otherstudies). More recently, the EDGES [31] and Xenon1T [32] anomalies have stimulated similar efforts inconstructing viable ALP models, e.g., [255] and [256], and the importance of laboratory-based searchesreceives attention again as robust consistency checks of those models.In the following subsections, we elaborate the ALP searches at beam-dump and reactor neutrino exper-iments and review related studies, followed by a survey on the recent developments in the collider ALPprobes. Light-shining-through-wall (LSW) experiments, e.g., ALPS I/II [257, 258], CROWS [259], andOSQAR [260], and polarization experiments, e.g., PVLAS [29], are other important classes of laboratory-produced ALP probes, for which comprehensive reviews can be found in [24, 25].
A. Beam-dump and reactor neutrino experiments
The laboratory-produced ALP searches are extensively performed in beam-dump type experiments(i.e., fixed target experiments) including E137 [297], E141 [298], reanalyses [299] of CHARM [300] andNuCal [301] ALP searches, NOMAD [302], and NA64 [280], and reactor experiments [303–311] for the pastdecades, and an increasing number of studies have pointed out that the next-generation beam neutrinoexperiments [277, 299, 312–319] and high-power reactor neutrino experiments [285, 293] can explore awider range of ALP parameter space. We provide summaries tabulating key specifications of existingand future beam-dump type (neutrino) experiments in Table I and reactor (neutrino) experiments inTable II. The first four experiments in Table I utilize charged pions and their decay product muons thatare stopped inside the target in order to produce an isotropic neutrino flux. By contrast, in the otherproton-beam experiments, a large fraction of the charged pions and muons escape from the target anddecay outside, resulting in a neutrino flux orienting in the forward direction. While proton or electronbeam-dump experiments and their physics opportunities have been extensively investigated, photon-beamexperiments are receiving attention as a complementary avenue of exploring ALP parameter space, e.g.,PrimEx [322] and GlueX [323]. In addition, proposed muon beam experiments, e.g., NA64 µ [324] andM [325], would provide opportunities for the ALP search.The calculation of the expected signal rate in these experiments essentially comprises of three parts:i) production of ALPs at a source point (IV A 1), ii) transportation of ALPs from the source point to thedetector of interest (IV A 2), and iii) detection of ALPs at the detector (IV A 3). Here the productiontakes into account only the portion of the ALP flux directed toward the detector. Given a source particle(say, i th particle) which would be converted to an ALP, one can understand that its associated signalrate n i is a product of the probabilities corresponding to the three parts [319]: n i = P prod × P tran × P det , (29) See also [320] proposing new physics searches at reactor neutrino experiments, using the ALP-dark photon-photon cou-pling. See also [321] for an extensive list of accelerator-based experiments including the decommissioned ones. Experiment Beam E beam POT/EOT Target Detector Mass Distance Angle[GeV] [yr − ] [m]CCM [261–263] p . × W LAr 7 t 20 90 ◦ COHERENT [264–266] p . × Hg CsI[Na] 14.6 kg 19.3 ◦ LAr 24 kg (0.61 t) 28.4 ◦ JSNS [263, 267, 268] p . × Hg Gd-LS 17 t 24 ◦ MiniBooNE [269] p ∼ × ) Be Mineral oil 450 t 541 On-axisMicroBooNE [270, 271] p . × Be LArTPC 89 t 470 On-axisSBND [270] p . × Be LArTPC 112 t 110 On-axisICARUS [270] p . × Be LArTPC 476 t 600 On-axisT2K [272] p . × Graphite Water ∼ . t 280 . ◦ Gas TPC kLWater + PS 2.2 tNO ν A [273] p . × Graphite PVC-LS 125 t 1,000 0.84 ◦ DUNE [274, 275] p . × Graphite LArTPC 67.2 t 574 MovableGArTPC 1.8 tSHiP [276] p . × TZM Pb-ECC 9.6 t ∼ On-axisECAL/HCAL – ∼ LDMX [277] e − W ECAL/HCAL – O (1) On-axisBDX [278, 279] e − ∼ Al ECAL – 20 On-axisNA64 [280] e −
100 ( . × ) PRS/ECAL HCAL – O (1) On-axis
TABLE I. Key specifications of existing and future beam-dump type (neutrino) experiments. In the first fourexperiments, charged pions and their decay product muons created in the target are stopped inside the target,whereas in the other proton-beam experiments, a large fraction of them decay outside the target, resulting ina neutrino flux in the forward region. The POT of MiniBooNE corresponds to the data collection in-between2002 and 2019, the EOT of LDMX is for Phase 2, and the EOT of NA64 corresponds to the data collection in-between 2016 and 2018. The mass value of the COHERENRT-LAr detector in parentheses is for a future upgrade.The (underlined) values in the Mass column are fiducal (active) masses. [POT/EOT: protons/electrons on target,TZM: titanium-zirconium doped molybdenium, LAr/GAr: liquid/gaseous argon, Gd-LS: gadolinium-loaded liquidscintillator, PVC-LS: PVC cells filled with liquid scintillators, TPC: time projection chamber, Pb-ECC: emulsioncloud chamber with lead plates, PRS: pre-shower detector, ECAL/HCAL: electromagnetic/hadronic calorimeter,PS: plastic scintillators] where P prod , P tran , and P det are the probabilities associated with i), ii), and iii) above, respectively. Herewe omitted the number of source particles in the right-hand side of Eq. (29), since it is unity. So, n i can be interchangeably used with P i , the signal detection probability with respect to the given sourceparticle. If N src such source particles are available for a given exposure, the total number of ALP signalevents detected at the detector of interest (denoted by N sig ) is given by N sig = N src (cid:88) i n i = N src (cid:104) n i (cid:105) , (30)where the second equality implies that equivalently, one may evaluate the average of n i , (cid:104) n i (cid:105) , for asufficiently large sub-set out of N src and multiply it by N src . This approach essentially allows us tofactorize the rate calculation into ALP physics and source physics.
1. Production of ALP
Depending on the underlying ALP model details, ALPs can be produced in various ways inside the tar-get of beam-dump type experiments or the reactor core of reactor neutrino experiments. Among possible0
Experiment Thermal power [GW] Detector Mass Distance [m]CONNIE [281, 282] 3.95 Skipper CCD 52 g 30CONUS [283] 3.9 Ge 3.76 kg 17.1MINER [284, 285] 0.001 Ge + Si 4 kg 1 – 2.25NEON [286] 2.82 NaI[Tl] ∼ / / kg (Ph1/2/3) 24 ν -cleus [287] 4 CaWO + Al O + ν GeN [288] ∼ Ge 1.6 – 10 kg 10 – 12.5RED-100 [289, 290] ∼ DP-Xe ∼ kg 19Ricochet [291] 8.54 Ge + Zn 10 kg 355/469SBC-CE ν NS [292, 293] 0.68 LAr[Xe] 10 kg 3/10SoLid [294] 40 – 100 PVT + LiF:ZnS(Ag) 1.6 t 5.5 – 12TEXONO [295] 2.9 Ge 1.06 kg 28vIOLETA [296] 2 Skipper CCD 1 – 10 kg 8 – 12
TABLE II. Key specifications of existing and future reactor (neutrino) experiments. [CCD: charge couple device,DP-Xe: dual-phase xenon, Ph1/2/3: phase1/phase2/phase3, N/M/F: near/medium/far] source particles, the photon is particularly interesting as it is one of the most copiously produced particlespecies in the target or the reactor core. Photons are typically produced by decays of mesons including π , η , etc. and bremsstrahlung of the incoming beam particle and secondary charged particles inside thetarget in the beam-dump experiments, and produced by nuclear transitions and neutron captures insidethe reactor core. While standard event generators, e.g., PYTHIA [326, 327],
HERWIG [328], and
SHERPA [329],can describe the meson decays and beam-induced bremsstrahlung, photon production by secondary par-ticles such as ionized electrons, nuclear transitions, and neutron captures require a dedicated detectorsimulation, e.g.,
GEANT4 [330] and
FLUKA [331]. One may estimate the photon fluxes induced by the firsttwo mechanisms (semi-)analytically. For example, production of pions through energetic proton beams ontarget is empirically parametrized by the Sanford-Wang description [332] and the BMPT model [333], andtheir decay photons can be used as an injection photon flux [316]. For another example, the equivalentphoton approximation, also known as Fermi-Weizsäker-Williams method [334–336], provides a convenientframework to estimate bremsstrahlung photons from an energetic charged particle. See e.g., [299] for amore systematic discussion in the context of ALP searches in the beam-dump experiments.Once a photon emerges, it can be converted to an ALP via the Primakoff process, i.e., γ + A → a + A ,with A denoting the atomic system of interest in the target or the core material, if the ALP-photoncoupling g aγγ in (1) is non-zero. The differential cross-section in the angle of ALP with respect to theincoming photon direction θ a [337] is dσ Primprod d cos θ a = 14 g aγγ αZ F ( t ) | (cid:126)p a | sin θ a t , (31)where α , Z , and (cid:126)p a are the electromagnetic fine structure constant, the atomic number of the targetor reactor core material, and the ALP three-momentum, respectively. Here t denotes the square of thefour-momentum transfer: t = ( p γ − p a ) = m a − E γ ( E a − | (cid:126)p a | cos θ a ) . (32)Note that the typical momentum transfer to the target nucleus is much smaller than E γ , so that E γ ≈ E a valid under the collinear limit can provide a good approximation and simplify calculational procedures.Finally, F ( t ) describes a form factor as a function of t . Depending on the coherency length scale, anatomic or nuclear form factor has to be taken into account. In the beam-dump experiments where thecoherency length is usually at the nuclear scale, F ( t ) will be a nuclear form factor such as the Helmparametrization [338].1If the ALP-electron coupling g aee is non-vanishing, ALPs can be produced via s -channel and t -channelCompton-like scattering processes on electrons inside an target atom, i.e., γ + e − → a + e − . Its differentialcross-section has the form of dσ Compprod dx = g aee αZx s − m e )(1 − x ) (cid:20) x − m a s ( s − m e ) + 2 m a ( s − m e ) (cid:18) m e − x + m a x (cid:19)(cid:21) , (33)where s is a Mandelstam variable given by s = ( p γ + p e ) = m e + 2 m e E γ . (34)Here x is the fractional light-cone momentum whose value lies in-between 0 and 1. In the laboratoryframe, one may perform a change of variables, using the relation x = 1 − E a E γ + m a m e E γ . (35)The fiducial (total) production cross-section σ fidprod ( σ totprod ) can be obtained by integrating (31) and/or(33) over the phase space in which the produced ALPs are directed toward the detector of interest (overthe entire phase space). However, the Primakoff and/or Compton-like processes do not always arise, butthey basically compete with standard interactions such as pair production, photoelectric absorption, etcthat the photons usually get through. Denoting the total cross-section of the standard interactions by σ SM , we therefore write the probability of ALP production P prod as follows: P prod = σ fidprod σ SM + σ totprod ≈ σ fidprod σ SM . (36)where the approximation is usually valid due to σ SM (cid:29) σ totprod in most of the well-motivated ALP parameterspace. Note that σ SM is generally a function of photon energy E γ for which the measurement data isavailable in e.g., [339].While we have focused on P prod of photon-initiated Primakoff or Compton-like ALP production, thisapproach is straightforwardly applicable to other production mechanisms. For example, if a photonoriginates from neutron captures or nuclear deexcitations and an ALP is produced via its coupling tonucleons, P prod is given by the branching ratio of ALP emissions in the transitions [311]. For anotherexample, if ALP is mixed with pseudoscalar mesons such as π , η ( (cid:48) ) , etc in the presence of the ALP-gluoncoupling [318], the source particle and the probability can be replaced by the mesons and the associatedmixing angle squared.
2. Transportation of ALP
Once an ALP is produced in the target or the reactor core, it should neither decay before reaching thedetector of interest nor interact with target material. The usual decay law defines the former probability P decaytran : P decaytran = exp (cid:18) − d ¯ (cid:96) lab a (cid:19) , (37)where d is the distance between the source point and the detector. Here ¯ (cid:96) lab a stands for the laboratory-frame mean decay length of ALP which is a function of the total decay width of ALP, Γ tot a , and the boost2factor of ALP, γ a , ¯ (cid:96) lab a = (cid:112) γ a − tot a . (38)The partial particle widths of the diphoton and the e + e − -pair decay modes are, respectively, Γ a → γ = g aγγ m a π , (39) Γ a → e + e − = g aee m a π (cid:115) − m e m a . (40)The latter probability that no ALP interactions arise in the target (say, P inttran ) can be evaluated ina similar fashion. Assuming that the ALP interactions are dominated by the ALP scattering and thedistance between the source point and the target end is denoted by D , we can express P inttran as P inttran = exp( − ρ T σ totscat D ) , (41)where ρ T is the number density of target particles in the beam target with respect to σ totscat . Here σ totscat describes the total scattering cross-section of ALP for which the related formulation is discussed in thenext subsection. Therefore, the transportation probability is P tran = P decaytran × P inttran . (42)However, since D is much smaller than the mean free path of ALPs defined by / ( ρ T σ totscat ) in most of theexperiments, P inttran becomes approximately unity, and in turn, P tran ≈ P decaytran .
3. Detection of ALP
There are broadly three channels to detect ALPs at a detector: through their decay, through theirscattering with detector material, and through their conversion to photons. Most of the beam-dump typeexperiments have performed ALP searches in the first channel [277, 280, 297–301, 312–315, 317, 318].Since the ALP should decay to visible particles before escaping from the detector fiducial volume, inorder to leave an experimental signature, the probability of ALP detection in this decay channel, P decaydet ,is again described by the decay law and has the form of P decaydet = 1 − exp (cid:18) − L det ¯ (cid:96) lab a (cid:19) , (43)where L det is the cord length along which the ALP would sweep in the detector fiducial volume. Sincethe decay within the detector is crucial, this channel is suited for ALPs with a relatively sizable decaywidth. If m a is too small, the associated decay length of ALP is too large to have decay signals.In Fig. 9, we display existing (color-shaded regions) and future expected (dashed lines) limits of g aγγ in m a for laboratory-produced ALPs. The laboratory-based experiments contributing to the existinglimits are listed in the figure caption, and we include astrophysical and cosmological limits (gray-shadedregions) compiled in [340] for reference purposes. Future prospects of beam-dump experiments in thedecay channel are shown in the right panel of Fig. 9: DUNE-GAr (one year exposure) [317], LDMX-vis. and LDMX-inv. ( EOTs of an 8 GeV beam) [314], NA62 ( POTs) [315], NA64 ( × EOTs) [341], SeaQuest (Phase I) [313], and SHiP ( × POTs) [312]. We further include expectations3
FIG. 9. Existing (color-shaded regions) and expected (dashed lines) limits of laboratory-produced ALPs in the ( m a , g aγγ ) parameter plane. For reference purposes, we show astrophysical and cosmological limits (gray-shadedregions) compiled in [340]. Left panel: Limits for m a < MeV to which the scattering and the conversionchannels in beam-dump and reactor neutrino experiments are relevant. The current constraints include e + + e − → γ + inv . [344–348], Υ → γ + inv . [349, 350], LSW-type experiments [351], NOMAD [302], and beam-dumpexperiments [297]. Future expected limits include a PASSAT interpretation [316] of NOMAD [302], PASSATimplementations at the BDF facility with the CAST or BabyIAXO magnets with × POTs [316], a PASSATimplementation at the DUNE MPD with a 7-year exposure [319], PASSAT implementations at DUNE withthe CAST or BabyIAXO magnets with a 7-year exposure [319], and reactor searches at CONNIE, CONUS,and MINER [285]. Right panel: Limits for m a > MeV to which the decay channels in beam-dump andreactor neutrino experiments are relevant. The current constraints include e + + e − → γ + inv . [344–348], Belle-II [352], CHARM [300], E137 [297], E141 [298], LEP [353], LHC (Pb) [354, 355], NA64 [280], NuCal [301], andPrimEx [342]. Future prospects shown here include Belle II in the γ and the γ + inv . modes with 20 fb − and g aγZ = 0 [356], DUNE-GAr with one year exposure [317], FASER [343], GlueX with 1 pb − [342], LDMX-vis. andLDMX-inv. with EOTs of an 8 GeV beam [314], LHC (Pb) with 10 nb − [357], NA62 with POTs [315],NA64 with × EOTs [341], PrimEx with all runs [342], SeaQuest in Phase I [313], and SHiP with × POTs [312]. of photon-beam experiments, e.g., GlueX (Pb target with 1 pb − ) [342] and PrimEx (all runs) [342], anda forward-physics experiment, FASER [343]. The expected limits of the reactor neutrino experiments inthe decay channel are shown in the left panel of Fig. 9, including CONNIE, CONUS, and MINER [285].Since the energy of photons in the reactor core is limited, the reactor experiments are not sensitive tothe region of m a (cid:38) MeV. However, the close proximity to their detector enables them to explore theregions of smaller m a and g aγγ than what typical beam-dump experiment would reach.Similarly, we exhibit existing (color-shaded regions) and future expected (dashed lines) limits on ( m a , g aee ) in Fig. 10, for ALPs purely coupled to electrons. Future prospects include LDMX-vis. andLDMX-inv. ( EOTs of an 8 GeV beam) [314], but the reactor searches are sensitive to the ALP signalsof the decay channel only within a narrow mass range since most of the energy values of the photons cre-ated in their reactor core are not big enough to overcome the production threshold s ≥ ( m a + m e ) [285].Light ALPs which do not decay inside a detector may leave an experimental signature via its scatteringoff detector material. This possibility has been recently pointed out in [285], and ALP detection prospectsin the scattering channel have been investigated in reactor neutrino experiments [285, 293] and neutrinobeam experiments [317]. For example, in the presence of the ALP-photon interaction, an ALP can inducea photon via an inverse Primakoff process, i.e., a + A → γ + A . Depending on the detector capability,4 FIG. 10. Existing (color-shaded regions) and expected (dashed lines) limits of laboratory-produced ALPs in the ( m a , g aee ) parameter plane. The current constraints from the laboratory-produced ALP searches include beam-dump experiments performed at SLAC [358] and limits derived from the dark photon search of BaBar [359, 360]under the assumption of the approximate universality of the ALP-lepton couplings. For reference purposes, weshow the astrophysical limits (gray-shaded regions) set by EDELWEISS-III. [361]. Future prospects shown hereinclude LDMX-vis. and LDMX-inv. with EOTs of an 8 GeV beam [314] and reactor searches at CONNIE,CONUS, MINE, and ν -cleus [285]. one may observe a photon and potentially a nucleus recoil. The differential scattering cross-section inthe outgoing photon angle θ γ is almost the same as in (31): dσ Inv Primdet d cos θ γ = 12 g aγγ αZ F ( t ) | (cid:126)p a | sin θ γ t , (44)where Z and F are associated with the detector material and where (cid:126)p a is the three-momentum of theincoming ALP. The alteration that the prefactor 1/4 becomes 1/2 is because the initial spin states includea spin-0 ALP rather than a spin-1 photon. For another example, the ALP could scatter off electronsthrough the (inverse) Compton-like process, i.e., a + e − → γ + e − , in the presence of a non-zero g aee , andthe resulting final state involves a recoiling electron and a photon. The differential cross-section for thisscattering process in the solid angle of the outgoing photon [362, 363] is given by dσ Inv Compdet d Ω γ = g aee αZE γ πm e | (cid:126)p a | (cid:34) m e E γ (2 m e E a + m a ) − m e E γ m e E a + m a − m a | (cid:126)p a | m e E γ (2 m e E a + m a ) sin θ γ (cid:35) , (45)where again Z is the atomic number of the detector material.As for the production cross-section, the fiducial scattering cross-section σ fiddet can be obtained by integrat-ing (44) and/or (45) over the phase space consistent with detector thresholds, cuts, etc. In well-motivatedregions of ALP parameter space, the total scattering cross-section is sufficiently small that the scatteringprobability for a given ALP P scatdet is P scatdet = n T σ fiddet L det , (46)where n T is the number density of target particles in the detector under consideration and where L det isthe length through which the ALP would travel in the detector fiducial volume.5The expected sensitivity reaches of reactor neutrino experiments in this channel are shown in theleft panel of Fig. 9: the horizontal pieces of the sensitivity curves of CONNIE, CONUS, and MINERexperiments [285]. The scattering channel allows for the exploration of the ALP parameter space to whichthe decay channel would be insensitive, providing complementary information in the ALP search. Similaranalyses also appear in [293] by taking sets of benchmark ALP production and detection parameters,not specifying concrete experiments. When it comes to the beam-dump type neutrino experiments, theauthors of [317] provide a preliminary discussion on the experimental reach of ALP parameter space inthe scattering channel, taking the DUNE LArTPC near detector and assuming the ALP-photon coupling.Similarly, the expected sensitivity reaches of reactor neutrino experiments in the ( m a , g aee ) plane areshown in Fig. 10: the horizontal through rising pieces of the sensitivity curves of CONNIE. CONUS,MINER, and ν -cleus [285]. As in the g aγγ case, the scattering channel provides a complementary probe.Finally, if a detector carries a magnetic field region, it can be sensitive to models of ALP interactingwith photons through the ALP-to-photon conversion. For an ALP traveling a distance L B in a magneticfield B , the conversion probability P convdet [24] is P convdet = (cid:18) g aγγ BL B (cid:19) (cid:18) qL B (cid:19) sin (cid:18) qL B (cid:19) , (47)where the product of the second and third factors parametrizes the form factor describing the decoherencyof the conversion. In vacuum and in the relativistic limit, q is given by q = 2 (cid:115)(cid:18) m a E a (cid:19) + (cid:18) g aγγ B (cid:19) . (48)In the limit of m a , g aγγ → , this q becomes small and the conversion probability is simplified to P convdet ≈ (cid:18) g aγγ BL B (cid:19) . (49)This hybrid type experiment of beam-dump and helioscope approaches was first proposed in [316],dubbed Particle Accelerator helioScopes for Slim Axion-like-particle deTection or PASSAT as shorthand;the ALP decay is replaced by the ALP conversion from the perspective of conventional beam-dump ALPsearches, while the sun is substituted by a target from the perspective of a traditional helioscope. Asis implied by the decoherency form factor, the ALP-to-photon conversion can be maximally effective forlight ALPs. Therefore, this channel enables the exploration of ALP parameter space to which the ALPsearches in the decay channel are not sensitive, and complements the scattering channel [316].Expected sensitivity reaches under the PASSAT framework are shown in the left panel of Fig. 9. Theauthors in [316] interpreted the ALP search result in NOMAD [302] by a LSW-like regeneration method(ALP production in the magnetic field of the beam focusing horn and ALP detection in the magneticfield of the detector) in terms of PASSAT (see the line for PASSAT-NOMAD). They further proposed arealization of PASSAT by recycling the magnets of the CAST or the proposed BabyIAXO experiments(after decommissioned) and placing them at the proposed beam-dump facility [364] at CERN (see thelines for PASSAT-BDF-CAST and PASSAT-BDF-BabyIAXO). Similar proposals were made for DUNEor DUNE-like neutrino facilities [319]. Since the multi-purpose detector (MPD) of DUNE [274] carries amagnetic field region, the idea of PASSAT is readily applicable (see the line for PASSAT-DUNE-MPD).Alternatively, one could reuse the magnets of CAST or BabyIAXO and place them near the dumparea, reducing the distance between the target (i.e., the source point) and the magnetic field region toimprove the signal sensitivity (see the lines for PASSAT-DUNE-CAST and PASSAT-DUNE-BabyIAXO).As mentioned above, this conversion channel is better suited for the regimes of small m a where thedecoherency form factor gets negligible hence the conversion probability becomes maximized, and Eq. (49)6suggests that the conversion probability is essentially governed by B and L B . Therefore, large-scalestronger magnets enable the investigation of ALP parameter space that the decay and scattering channelswould never access. B. Collider searches
Collider probes have formed an important branch of the ALP search efforts as they are sensitive toALP signals directly and indirectly [365], and have mostly constrained models of ALP interacting withthe SM photon. Due to their relatively large center-of-mass energy, colliders have provided particularopportunities in the search for MeV-to-TeV mass-range ALPs.For example, the authors of [366] interpreted LEP measurement data available in [367–371] using aprocess of ALP interacting with photons, e + + e − → γ ∗ → γ + a , a → γ + γ , and filled the missing m a gapbetween ∼ MeV and ∼
10 GeV . In this mass regime, the produced ALP would be so significantly boostedthat its decay products, two photons would be merged hence appear single-photon-like. Therefore, thenull signal observation in the diphoton channel can set the limits in the ( m a , g aγγ ) parameter space.LEP data in the mono-photon channel (i.e., e + + e − → γ + inv . ) [344–348] and radiative decays ofUpsilon mesons (i.e., Υ(1 S ) → γ + inv . ) [349, 350] have set the limits up to m a ∼ MeV, as no excesseswere observed. Proton colliders have also provided the bounds: limits were derived from CDF data in thetriphoton channel [372, 373] and pp collision data of ATLAS and CMS in channels involving ≥ photonsin the final state [374–377], covering up to the ∼ TeV scale. Beyond pp collisions, it was pointed out thatlight-by-light (LBL) scattering in heavy ion collisions enables the investigation of unexplored regions ofparameter space below m a ∼ GeV [357], and ATLAS and CMS Collaborations have reported theirsearch results [354, 355]. Recently, Belle-II searched for an ALP process, e + + e − → γ + a , a → γ + γ using data corresponding to 445 pb − , and set new limits as no evidence was found [352].Future sensitivity reaches of existing collider experiments in the ( m a , g aγγ ) plane have been investigatedin various parts of the literature. Example studies include Belle II in the γ and the γ + inv . modes with20 fb − and 50 ab − [356] and LHC Pb-Pb collisions with 10 nb − [357], as also shown in the rightpanel of Fig. 9. For the cases where the produced ALPs are long-lived, the two photons from theALP decay would be appreciably displaced so that existing searches may not be sensitive enough to theassociated signature and a more dedicated search would be needed [373]. If ALPs decay even outside thedetector because they are very light and/or very weakly coupled, their decay signature could be observedby a future surface-based detector specifically designed for the purpose of searching for very long-livedparticles, e.g., MATHUSLA [380]. Future lepton colliders, e.g., FCC- ee [381], ILC [382], CEPC [383], andCLIC [384, 385], hadron colliders, e.g., HL-LHC [386], HE-LHC [387], FCC- hh [387], and SPPC [383],and electron-hadron colliders, e.g., LHeC [388] and FCC- he [388], can provide unprecedented sensitivitiesto ALP signals especially in the high mass regime, and we provide a summary of key parameters of future(proposed) energy-frontier colliders in Table III for reference purposes. Related investigations of ALPphenomenology have been performed, including [340, 366, 373, 389–395].For models of ALPs interacting with photons, a limited number of search channels are available atcolliders. By contrast, for ALPs with couplings to other SM particles like gluons, massive gauge bosons,Higgs, and leptons, a number of search channels become available and richer phenomenology is expected.For example, for ALPs coupling to the hypercharge boson, the ALP- Z -photon coupling is non-zero so thatALPs can be produced by the decay of on-shell Z gauge bosons. Studies of ALPs using on-shell decays h → a + a , h → Z + a and Z → γ + a have been conducted by [340, 366, 398, 399]. As an example of thesestudies, we briefly discuss the work of [340]. Figure 14 of [340] contains the projected discovery contours Recent studies on the displaced vertex signature of ALPs interacting with gluons appear in [378, 379], using the dedicateddisplaced vertex trigger at the HL-LHC. Collider Particles collided Center-of-mass energy [TeV] Integrated L [ab − ]HL-LHC [386] pp
14 3HE-LHC [387] pp
27 10SPPC [383, 396] pp
75, 100 3FCC- hh [387] pp
100 30FCC- ee [381] e + e − e + e − e + e − e + e − e − p E e = 0 . , E p = 6 . ) 1FCC- he [388, 397] e − p E e = 0 . , E p = 20 ), 12 ( E e = 0 . , E p = 50 ) 3, 3TABLE III. Key parameters of future (proposed) energy-frontier colliders. on the plane of ( m a , g aγγ ) in the channel Z → a + γ with Br( a → γ + γ ) = 1 , and assuming 3,000 fb − of pp collision data at the LHC. The authors calculate the relevant cross-sections and determine the signalsignificance by requiring a minimum signal yield of 100 events, and find that g aγγ ∼ × − GeV − canbe probed for m a ∼ − GeV. Far more conservative results are obtained once one incorporates γ and γ backgrounds at the LHC, which are certainly not negligible, with important contributions arisingdue to genuine γ production, “fake γ ” backgrounds due to neutral pion decays and large bremsstrahlungfrom electrons [400]. The situation is even worse if one considers realistic experimental uncertainty.Conservatively, the best systematic uncertainties on genuine tri-gamma backgrounds at CMS and ATLASare probably about 15%. Realistic background modeling and uncertainty estimation were accounted forby the authors of [400], who presented a feasibility study for the detection of ALPs produced throughVBF processes and decaying via a → γ + γ . For 3,000 fb − of data, the discovery reach was found to be g aγγ ∼ × − GeV − , for m a from 10 MeV to 100 GeV.There are several other studies along these lines: mono-gauge-boson (including mono-photon) signa-tures were proposed in the search for ALP signals by [373, 398]; ALPs interacting with electroweak gaugebosons have been studied by for example [401]; it has been suggested that non-resonant ALP-mediateddiboson production could be a promising channel due to the derivative nature of ALP couplings [402];and it has been claimed that a triboson search would be useful in the search for ALPs giving rise toboosted four photons which appear diphoton-like [403].Since the LHC features a large luminosity of gluons, it can provide particular opportunities for ALPscoupled to gluons [404]. For example, stronger limits between m a ∼ GeV and ∼ GeV can be derivedin the diphoton channel together with the ALP-photon coupling [405]; new search channels are availablesuch as ALP production in association with a gluon-induced jet [406] and ALP production in associationwith a jet and a photon [407]; and gluon-fusion-induced ALP production in the pp , Pb-Pb collisions inthe next LHC runs [408]. ALP production via light-by-light (LBL) scattering has also been investigated:LBL in pp collisions with proton tagging [409]; LBL in heavy-ion collisions [410, 411]; and SuperChic ,a simulation package for LBL in pp , p -heavy ion, heavy ion-heavy ion collisions [412]. Exotic decays of Z or Higgs also can be good channels to look for ALP signals [360, 376, 413]. Finally, the scenario thatALP couples to sterile neutrinos and its collider signature were investigated [414]. V. BOSE-EINSTEIN CONDENSATES
It is known that ALP could form Bose-Einstein Condensate (BEC) if they constitute a fraction of darkmatter (DM). In this section, we discuss the BEC properties and their implications on both the galacticscale and stellar scales. In what follows, we assume the scalar to be as light as sub-eV, while the reasons8for it will be elaborated shortly.
A. Theoretical estimate
Due to its bosonic nature, ultra-light bosonic dark matter can exhibit collective behaviors at themacroscopic level that are not obvious at the Lagrangian level. It has been observed and well understoodin condensed matter physics that for bosons there exists a phase, BEC phase, once the ensemble is cooledto below the critical temperature. In the case of ultra-light dark matter, one can estimate the mass rangeone requires for it to be in the BEC phase in a fashion similar to the estimate of the critical temperatureof BEC [415–418]. We start with a potential of the following type, partly motivated by the axion cosinepotential L = 12 ∂ µ φ∂ µ φ − (cid:18) m φ + λ m f φ + ... (cid:19) . (50)By requiring the de Broglie wavelength to be longer than the inter-spacing between dark matter particles,one has πmv (cid:38) (cid:16) mρ (cid:17) / , which gives m (cid:46) (cid:18) − v (cid:19) / (cid:18) ρρ DM (cid:19) / , (51)where the average of the dark matter density is ρ DM = ρ c Ω DM ≈ . × − GeV / cm , with ρ c beingthe critical density of the universe. Assuming negligible self-interaction, numerically solving the equationof motion in the non-relativistic regime leads to a BEC mass [419] M BEC ≈ . × M (cid:12) (cid:16) m − eV (cid:17) − (cid:18) φ (0) M P l,r (cid:19) , (52)where M P l,r is the reduced Planck mass, φ (0) the value of the wave function at r = 0 . One can seethat m ∼ − eV (10 − eV) corresponds to galactic (stellar) scale BEC structures. The mass profileis parametrized by two variables, the scalar mass m and the central density related to φ (0) . Whilethere is no simple analytical expression for the mass profile, a few approximations exist that are in goodagreement with the numerical solutions, such as [420–424].
1. Cosmological evolution
It is pointed out that ultra-light dark matter relieves the core-cusp problem [425, 426]. This can beeasily seen through the equations of motion of the inhomogeneous perturbation of the field. In Newtoniangauge, we denote the metric perturbation following the notation of [427] ds = − a ( τ )(1 + 2Ψ) dτ + a ( τ )(1 − dx , (53)where a is the scale factor. Writing φ ( t, x ) = φ ( t ) + δφ ( t, x ) , minimizing the action order by order givesthe equations of motion for both the homogeneous background and the fluctuation. In the momentumspace they are [428, 429] φ (cid:48)(cid:48) + 2 H φ (cid:48) + V ,φ a = 0 ,δφ (cid:48)(cid:48) + 2 H δφ (cid:48) + ( k + V ,φφ a ) δφ = − V ,φ a Ψ + (Ψ (cid:48) + 3Φ (cid:48) ) φ (cid:48) , (54)9where () (cid:48) is the derivative with respect to the conformal time τ , H = a (cid:48) /a , and V ,φ = dV /dφ, V ,φφ = d V /dφ , and k is the comoving momentum. In the non-relativistic limit, the equation of motion for theperturbation reduces to δφ (cid:48)(cid:48) + 2 H δφ (cid:48) + c s k a δφ = 4 πG ¯ ρ δφ, (55)where c s is the sound speed of the φ fluid. In the limit k (cid:28) m , it is given by [417, 430] c s ≈ k m a + λ ¯ ρ φ m f , (56)where ¯ ρ φ ≈ m φ . Comparing the pressure term with the gravity term gives an estimate of the Jeansscale. There are a few competing forces in question. When λ is negligible, i.e., λφ (cid:28) m φ , gravitycompetes with the quantum pressure, which gives k J = (16 πG N m ¯ ρ φ a ) / ≈ (cid:16) m − eV (cid:17) / (cid:18) ¯ ρ φ, . × − GeV / cm (cid:19) / (cid:18) aa eq (cid:19) / Mpc − , (57)where a eq is the matter-radiation equality, ρ φ, the φ energy density today. When λ > and the self-interaction term is non-negligible, gravity needs to compete directly with λφ term, which gives a Jeansscale k J = (16 πG N /λ ) / mf a = 2 . λ − / (cid:16) m − eV (cid:17) (cid:18) f GeV (cid:19) (cid:18) aa eq (cid:19) Mpc − . (58)When λ < and non-negligible, the quantum pressure competes with both gravity and the attractiveself-interaction. This leads to k J = a (cid:34) πG N m ¯ ρ φ + (cid:18) λ ¯ ρ φ f (cid:19) (cid:35) / − λ ¯ ρ φ f , (59)which leads to possible gravitational collapse assisted by the attractive self-interaction at even smallerscales. As a special case, in [428, 429], it is pointed out that if φ starts to roll with a specific initialcondition, such as from the hilltop part of a cosine potential, ( V ,φφ + k /a ) term can be negative alltogether for certain k modes, without the need of gravity. This allows a quick growth of these k modesbefore matter domination.Alternatively, one can understand the deviation of ultralight dark matter from cold dark matter (CDM)as whether the density contrast of a given k mode (in comoving frame), δρ φ ( k ) /ρ φ , grows the same as δρ CDM ( k ) /ρ CDM . Roughly speaking, modes that enter the horizon after m ∼ H ( z ) grows the sameas CDM. Those that enter the horizon before this do not immediately grow as δ ∝ a during matterdomination, because the background ρ φ is still frozen, hence the deviation from CDM. This leads to thesame estimate as comparing the pressure and gravity.
2. Stable self-gravitating structures
Spherically symmetric self-gravitating BEC structure is verified to be stable against radial perturba-tion [431, 432]. There are many studies of the stability of stellar scale BEC’s [421, 422, 431, 433–437] and0 P ) R2001751501251007550250 ( m M P / f ) H attractive 9 fM P / m fM P / m fM P / m P ) R60504030201001020 ( m M P / f ) H repulsive 9 fM P / m fM P / m fM P / m FIG. 11. We show the Hamiltonian in the case of attractive self-interaction (left) and repulsive self-interaction(right). In the left panel, from N = 9 fM P /m to N = 10 fM P /m , the local maximum moves to the right andthe local minimum moves to the left. At N = 11 fM P /m , the local minimum is lost, so there is no stable darkstar beyond this number of particles. In the right panel, there is always a minimum (corresponding to a stablesolution) at any N . galactic scale BEC’s [438] to name a few. Instead of using approximations of high precision [422, 439],here we briefly outline the estimate using a simple ansatz, which is shown to agree with the numericalsolution reasonably well: φ ( r ) ≈ (cid:114) NmπR e − r/R , (60)where N is the total number of particles, and R is the radius where φ starts to drop exponentially. Inother words, R parametrizes the characteristic size of the BEC object. The energy of such a system canbe broken into kinetic, self-interaction, and gravitational energies, H = H kin + H int + H grav , (61)where H kin = − (cid:90) d x φ ( r ) ∇ φ ( r ) = N mR ,H int = λm f (cid:90) d x φ ( r ) = λN πf R ,H grav = − G N m (cid:90) ∞ (cid:18)(cid:90) r φ ( r ) πr (cid:48) dr (cid:48) (cid:19) r φ ( r ) πr dr = − G N m N R , (62)and the full Hamiltonian is H = N mR ∓ | λ | N πf R − G N m N R , (63)where the upper (lower) sign in the second term corresponds to the attractive (repulsive) case. A fewbenchmarks of H ( N, R ) are plotted out with arbitrarily rescaled units in Fig. 11. From this one caneasily see that for λ > the system is stablized while for λ < the system can be de-stabilized when1 λφ term becomes large compared to the quantum pressure from the k φ term. When λ is negligible, agiven m leads to a class of solutions that are related to each other by a M ∼ /R scaling. This scalingbreaks down when general relativity effect kicks in at large density as shown in [437].It is noted that there are a few variations beyond the simplest spherical setup. There are studies onself-gravitating BEC structures without the assumption of a spherical symmetry. The spherical symmetrycan either be broken by a non-spherical source, such as baryons disk [440], dark disk [441], or due torotational excitations [422, 442–444], to give a few examples.Beyond the single scalar assumption, there are studies of the BEC with multiple species. To name a few,assuming non-interacting ultralight scalars, [445] proposes a solitonic origin of the Nuclear Star Clusterin the Milky Way. Assuming non-zero self-interactions, [446] analytically studies the mass-radius scalingrelation in the context of multiple scalars; [432] studies the stability and scaling behavior numericallyand provides some analytical interpretation. Besides the self-interaction, [432] also studies the casewhere non-gravitational interaction is present between the two species. It is observed that a repulsiveinteraction (+ φ φ ) between the two scalars can stabilize the BEC structure even if each has attractive( − φ ) self-interactions. B. Simulations of the ultra-light dark matter
There have been a few simulations showing evidence of BEC structures forming on the galactic scalesfrom gravitational relaxation [420, 447–450]. The simulations are performed with the scalar mass to be m ∼ − eV . It is observed that a BEC core, i.e., soliton, forming at the center of simulated galaxies,which matches to the Navarro-Frenk-White (NFW) dark matter halo profile [451, 452] at large radius.While the soliton core profiles are consistent with each other, the NFW tail and the transition betweenthe BEC core and NFW tail are slightly different. For a direct comparison of a few simulated profiles,see [419].While the relation between the BEC core and the halo remains an open question, in particular, theauthors of [447] observe a relation between the two, which can be phrased as M BEC ≈ (cid:18) | E h | M h (cid:19) / M P l m , (64)where E h is the virial energy of the halo, M h the virial mass. Utilizing the scaling of the BEC core,without loss of any information, one can express this empirical relation as [419] E BEC M BEC ≈ E h M h . (65)In [453], the interplay between BEC formation and baryonic physics is studied. It is observed that theBEC formation is largely unaffected by the baryonic feedback, while the BEC imprints the distributionof gas and stars with cored structure. In [454], the role of attractive self-interaction in BEC formation isverified in simulations. C. Experimental probes
From bullet cluster, potential self-interaction is constrained to be σ/m (cid:46) / g if φ makes up alldark matter. This translates to a constraint on the self-interaction [417] λ (cid:18) mf (cid:19) (cid:46) − (cid:16) m eV (cid:17) / . (66)2If ultralight dark matter takes upon a significant fraction of the total dark matter density, from Eq. (57)one can see that structures will be suppressed at a scale that is below the Jeans scale. In the range of m < − eV , [455] shows that there is suppression in the linear regime of the matter power spectrum,starting as small as k ∼ .
03 Mpc − , which leads to significant change of CMB anisotropy compared toCDM. This constrains the fraction of ultralight dark matter of the total dark matter to be below ∼ in the range of − eV to − eV . In [456], a study of weak gravitational lensing for the CMB athigh (cid:96) shows future experiment CMB-S4 can probe ultralight dark matter with mass up to − eV .At higher mass, nonlinear perturbations or simulations are needed to distinguish between CDM scenarioand ultralight dark matter scenario. The work in [457–459] shows that for m = 10 − eV , the halo massfunction is affected at scale as large as M (cid:12) , and sharply cut off at − M (cid:12) .Probes of structure suppression at this scale include Lyman- α (excluding − eV (cid:46) m (cid:46) . × − eV ) [426, 460–464], halo mass function from stellar stream [465, 466] and strong lensing [466–476]( m (cid:38) . × − eV ), from Milky Way satellite counting [477–482], and galaxy UV luminosity function[483–487]. Alternatively, if one takes the empirical soliton-halo relation from simulation [420, 447], thesoliton-halo profile can be constrained by rotation curve data [419, 440] and stellar orbits around Sgr A*(to exclude × − eV (cid:46) m (cid:46) × − eV ) and M87* (to exclude m (cid:46) × − eV ) [488].On the stellar scale, m ∼ − eV , possible BEC structures can exist in the form of exotic compactobjects ∼ M (cid:12) named boson stars [421–424, 431, 433–437, 439, 442, 444, 489–492]. That includes gravita-tional wave from boson star mergers [437, 493–498], boson stars in an extreme mass ratio inspiral system[499], and boson star decay products [443, 498]. VI. CONCLUSIONS
We bring our review to a conclusion by discussing, in turn, the current state and future prospects ofeach of the topics we have covered.High density astrophysical environments have been used quite successfully to constrain axions sincethe 1980s. While powerful, at present many of these constraints are subject to uncertainties coming fromastrophysics and nuclear theory. The production rate of axions in the core of a neutron star is verysensitive to the types of pairing in nuclear matter and, although not the focus of our review, the presenceof exotic phases deep in the neutron star. The effects of axions on neutron star mergers - either in theinspiral or postmerger (see Sec. III A) - appear to be subtle and will likely require (at the very least) amuch better understanding of the nuclear equation of state, the behavior of the neutrinos, and the roleof transport in the remnant in order to measure. Fortunately, there are a myriad of recent observationsthat have been able to constrain some of these uncertainties, and more observations and experimentsare on the horizon. Much progress has been made in constraining the size of the superfluid gaps, forexample, by studying the cooling of neutron stars (as discussed in Sect. II B 2). Some constraints on theparticle content of dense matter (including the presence of some exotic phases) have been found throughstudying the thermal relaxation of neutron stars [47, 500, 501]. We have learned about the stiffness ofthe nuclear equation of state at different densities as a result of our observation of several M (cid:12) neutronstars [502–504] and through a recent measurement of the neutron star radius by the NICER experiment[505–507]. The NICER experiment also found evidence that the magnetic field near the surface ofPSR J0030+0451 deviates from the expected dipole behavior [508], which may have implications foraxion-photon conversion near the surface of the star. On the merger side, numerical simulations andgravitational and electromagnetic observations of neutron star mergers are used in concert to developan understanding of these complex events. Future runs of Advanced LIGO and Advanced Virgo shouldprovide us with many more merger events to study, and future gravitational wave detectors will enableus to see the postmerger gravitational wave signal which has so far been hidden.Magnetars, covered in Sec. II, are a promising tool to constrain axions because of their relativelyhigh core temperature combined with a high magnetic field strength, creating the possibility of a unique3photon signal coming from the existence of axions. However, studying magnetars comes with its own un-certainties. The surface temperature of magnetars is anomalously high, seemingly violating the core-crusttemperature relationship that exists in normal neutron stars, leading to the belief that the magnetic fieldis somehow involved in heating the neutron star crust. A consequence of our ignorance about the crustalheating mechanism is that it is difficult to be precisely know the temperature of the magnetar core, uponwhich the axion luminosity strongly depends. Beyond the core temperature, the extremely high magneticfields present in magnetars can lead to other axion production processes, for example, the transitionof electrons or protons between Landau levels [509–511], the scattering of electrons from magnetic fluxtubes, among other processes discussed in [78]. These processes have yet to be included in efforts toconstrain axions using magnetars (or neutron star mergers, where it is possible that hydrodynamic andmagnetohydrodynamic instabilities produce extremely strong magnetic fields [190, 512]). As theoreticalmodels of magnetars develop (see [513]) and as more magnetars are discovered, magnetars will play anincreasingly important role in constraining axions.There have been recent developments in expectations of the fundamental properties of axions in densematter. In particular, it was recently found that a finite baryon density environment causes the axionmass to decrease and the axion coupling to neutrons to be enhanced by up to one order of magnitude[206]. Even more significant changes can occur for axions in a kaon-condensate or a color-flavor-lockedquark matter phase. How this finite-density modification affects the phenomenology of axions in neutronstars is yet to be determined.Beyond the usual couplings of ALPs with nucleons, electrons, and photons, the possibility of couplingsbetween ALPs and other standard model particles is beginning to be investigated. For example, theaxion-muon coupling can be constrained quite dramatically by SN1987a [213, 514]. Recent calculationsindicate that a significant thermal population of pions might exist in supernovae [239], reviving thepossibility of axion production from the process π − + p → n + a . This process was found to significantlyenhance axion production in supernovae and push the peak of the spectrum of emitted axions to higherenergies [214].We now turn to the second set of topics discussed in our review: laboratory-produced axion or ALPsearches. These searches are less reliant on astrophysical assumptions on which the previously discussedsearches are often based; hence they hold out the opportunity of setting the most conservative limits in theaxion parameter space. We have discussed this aspect of axion searches in Sect. IV, focusing on recentdevelopments in accelerator-based and reactor-based facilities. The first half was devoted to neutrinofacilities including beam-dump type (Table I) and reactor neutrino (Table II) experiments, where theaxions can be copiously produced in addtion to neutrions. There are three axion detection channels:through their decay, through their scattering on detector material, and through their conversion tophotons in the presence of a magnetic field. These three channels are suited for different regions of axionparameter space, so that they can provide information complementary to one another and allow for moreexhaustive exploration of axion parameter space in the ongoing and upcoming neutrino experiments. Bycontrast, ALP searches at energy-frontier colliders (Table III) were discussed in the other half of Sect. IV.Due to their large center-of-mass energy, they will allow for unprecedented opportunities to investigateaxion parameter space toward the higher end of the axion mass through various search channels.Finally, we turn to the last set of topics we have covered in this review: the topic of BECs. Towards thelighter end of the axion mass spectrum, collective behavior due to BEC formation could imply new probeson stellar and galactic scales. At axion masses around − eV and − eV , they can be potentiallyprobed by stellar dynamics and galactic dynamics, respectively. The former indicates great opportunitiesfrom gravitational wave astronomy including binary systems consisting of boson stars and observation ofits decay products. The latter motivates cosmological measurements of the power spectrum toward evensmaller scales. In addition, understanding the behavior of the BEC calls for dedicated simulations. Forexample, the origin of the relation between the BEC mass and the halo mass, observed empirically in afew simulations, is currently an open question. New simulations with different scalar masses are requiredto address this and distinguish competing explanations.4As a final remark, we emphasize that axion or ALP physics has served as a well-motivated frameworkin terms of beyond-Standard Model model building and related new signal searches. We hope that thisreview will provide guidance to beginning researchers, and be of use to experts as well.We end this review with a list of detection strategies that we did not discuss. Many of these topics arestandard and the latest results are covered in the excellent reviews [23–25]. ( i ) Helioscopes such as CAST [30, 515] and IAXO [516] constrain the ALP-photon coupling g aγγ bysearching for photons obtained from conversion of ALPs emitted by the sun. The sensitivity of thissearch method to the coupling goes as ∼ g aγγ , with a contribution of g aγγ coming from Primakoffproduction of ALPs in the sun, and a contribution of g aγγ coming from the subsequent conversionof the ALPs to photons in the magnetic field of the apparatus. ( ii ) Cavity haloscope experiments that probe cold ALP dark matter: ADMX [517], HAYSTAC [518],etc. These searches can constrain g aγγ × √ ρ , where ρ is the local ALP dark matter density,for some ALP masses between − eV and ∼ O (few times 10 − ) eV. There are also other ALPsearches that exploit the fact that ALP cold dark matter behaves as a classical oscillating field in thecurrent universe: experiments include those using wire arrays (ORPHEUS [519]) , dielectric plates(MADMAX [520]), NMR (CASPEr [521]), LC circuits (ABRACADABRA [522, 523]), birefringentcavities (ADBC [524], DANCE [525]) and interferometry [526, 527]. ( iii ) Light-shining-through-walls (LSW) experiments including ALPS I/II [257, 258], CROWS [259], andOSQAR [260] which constrain m a < × − eV and g aγγ < . × − GeV − . VII. ACKNOWLEDGEMENTS
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