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PHYSICAL REVIEW D 102, 115030 (2020)

Cosmological tension of ultralight dark and its solutions

† Jeff A. Dror* and Jacob M. Leedom Theory Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA, and Berkeley Center for Theoretical , University of California, Berkeley, California 94720, USA

(Received 14 August 2020; accepted 24 November 2020; published 23 December 2020)

A number of proposed and ongoing experiments search for axion with a nearing the limit set by small scale structure [Oð10−21 eVÞ]. We consider the late of these models, showing that requiring the axion to have a matter-power spectrum that matches that of constrains the magnitude of the axion couplings to the visible sector. Comparing these limits to current and future experimental efforts, we find that many searches require with an abnormally large coupling to fields, independently of how the axion was populated in the early Universe. We survey mechanisms that can alleviate the bounds, namely, the introduction of large charges, various forms of kinetic mixing, a clockwork structure, and imposing a discrete symmetry. We provide an explicit model for each case and explore their phenomenology and viability to produce detectable ultralight axion dark matter.

DOI: 10.1103/PhysRevD.102.115030

I. INTRODUCTION Axion couplings to the Standard Model fields are Axions with well below the electroweak scale are intimately related to the shape of the axion potential. simple dark matter candidates [1–3] with novel experi- For a given axion decay constant, fa, axion couplings to ∼1 mental signatures [4–6], a potential solution to the apparent matter are suppressed by =fa while the ratio of the axion incompatibility of cold dark matter with small scale mass to its quartic coupling is typically at least on the order ∼ structure [7,8], and a common prediction of of fa. Such nonquadratic contributions to the axion [9,10]. Axions can arise as pseudo-Goldstone bosons of a potential play an important role in cosmology and the ’ spontaneously broken global symmetry or as zero modes of interplay between the axion s couplings and potential is the antisymmetric tensor fields after compactification of extra focus of this . – dimensions. In either case, their parametrically suppressed Measurements of the matter-power spectrum [20 22] mass can result in a length scale comparable to the size of find that the cosmic density of dark matter ∝ −3 dwarf , the so-called fuzzy dark matter regime. Dark as R , where R is the , until well before matter with a macroscopic Compton wavelength allows for recombination. Such a scaling is satisfied for an oscillating if (and only if) the scalar potential is solely novel detection opportunities and many on-going and 1 2 2 future experimental efforts search for ultralight axion relics composed of a mass term, 2 maa . This potential is typically with masses around this limit. Purely gravitational searches only a good approximation if the field amplitude is look for erasure of structure on small scales as a conse- sufficiently small and may not hold for ultralight axions quence of the axion’s sizable wavelength, limiting the axion in the early Universe. Deviation from a simple quadratic −21 1 term results in a perturbation spectrum that is no longer mass, ma ≳ Oð10 eVÞ [11–17]. If axion dark matter has sizable nongravitational coupling to the visible sector, scale invariant, constraining the axion potential. This in there are additional detection strategies dependent on the turn places a powerful bound on conventional ultralight of the coupling. axion dark matter candidates due to the relationship between fa and the axion-matter couplings. While this point is implicitly acknowledged in some of *[email protected] † the axion literature, its significance is not widely empha- [email protected] 1Searches for oscillations in the local stress-energy tensor sized, and its implications for current and future searches is detectable with timing arrays [5] set slightly weaker missing altogether. In this work, we provide the constraints bounds [18,19]. from the matter-power spectrum and thereby motivate a natural region of mass and coupling values wherein the Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. axion can constitute dark matter without any additional Further distribution of this work must maintain attribution to model building. Circumventing the cosmological bounds the author(s) and the published article’s title, journal citation, requires breaking the parametric relationships between the and DOI. Funded by SCOAP3. axion potential and couplings. There are few techniques

2470-0010=2020=102(11)=115030(11) 115030-1 Published by the American Physical Society JEFF A. DROR and JACOB M. LEEDOM PHYS. REV. D 102, 115030 (2020)

α that can successfully accomplish this task. We consider the Caγ ˜ CaN ¯ μ L ⊃ aFF þ ∂μaNγ γ5N; ð2Þ possibility of large charges, kinetic mixing, a clockwork 8πfa fa structure, and discrete symmetries in the context of ultralight dark matter. These models typically predict additional where here and throughout we suppress the Lorentz ˜ ˜ μν ˜ μν states in the spectrum and we survey their phenomenology. indices on the gauge interactions, FF ≡ FμνF and F ≡ The outline of the paper is as follows. In Sec. II we 1 μναβ ϵ Fαβ. The parameters C γ and C represent combi- review features of axion models, with particular emphasis 2 a aN nations of couplings in the UV theory and are Oð1Þ for on the coupling of axions to visible matter and the axion generic axions. Demanding that the theory be invariant potential. In Sec. III we study the impact of ultralight axion under axion discrete shift transformations requires the dark matter on the matter-power spectrum and derive the 3 coefficient C γ to be an integer and hence cannot represent associated bound. In Sec. IV we examine the axion a a large ratio of scales without additional model building. detection prospects of various experiments in light of the There may also be contributions to the above couplings bounds. In Sec. V we study the robustness of the constraints from the IR if the axion mixes with dark sector particles, by exploring ways to disrupt the relationship between similar to the QCD axion-meson mixing, but such con- axion-matter couplings and the axion potential. Finally, we tributions will be unimportant for our considerations. The conclude in Sec. VI. relationship between the axion potential and its coupling to matter is made manifest in (1) and (2). To make contact II. AXION MASS AND COUPLING with other studies, we define

The axion decay constant, fa, relates the terms of the C γα axion potential to its couplings with Standard Model fields. ≡ a ≡ CaN ð Þ gaγ 2π ;gaN : 3 The potential arises from nonperturbative contributions of fa fa gauge or string theories and explicitly breaks the continu- ous shift symmetry of the axion. We use the standard In principle it is possible that the particle searched for by parametrization of the potential, which is a simple cosine of dark matter experiments is not a true axion, in the sense that the form it is not shift symmetric, but a light pseudoscalar with a potential, a VðaÞ ≃ μ4 cos ; ð1Þ 1 f L ¼ 2 2 ð Þ a a 2 maa : 4 where μ is a scale associated with the explicit breaking of In this case the corresponding coefficients in front of the the global symmetry. If the potential arises from a terms in (2) do not correspond to any symmetry breaking composite sector (as in the case of the QCD axion), the scale, but are instead completely free parameters associated explicit breaking scale corresponds to the maximum scale with the scale of integrating out heavy fields. This would at which states must show up in the spectrum. The full prevent us from using the arguments of Sec. III to restrict axion potential is expected to be more complicated than the the dark matter parameter space. While such models seem simple cosine above, but we can consider (1) to be the first viable, they are highly fine-tuned and do not exhibit the term in a Fourier decomposition of the potential. An desirable features of axion models. One way to see the important feature of (1) is the existence of terms beyond tuning is to consider the additional terms in the effective the mass term and that the coefficients of these higher order theory that arise when integrating out the heavy fields that terms are not arbitrary. The size of the quartic determines lead to the couplings in (2). For example, in addition to the the point at which the quadratic approximation breaks aFF˜ term, the low energy theory of a simple pseudoscalar down and has significance for axion cosmology. will include terms such as a2FF, a3FF˜ , etc. These terms Axions may also couple to Standard Model fields, with will always be generated as they are no longer forbidden by the leading operators obeying the axion shift symmetry. In any symmetry, and lead to corrections to the scalar potential this work we focus on two types of operators for axion dark 2 which destabilize the light scalar. Thus any simple pseu- matter, the prospective and nucleon couplings, doscalar becomes unnatural and the motivation to consider such a particle as dark matter is rendered null. There- 2Other couplings of ultralight axions with matter are the gluon fore, we take the position that the target particles of ˜ μν α operator, aGμνG , the operator, ∂αae¯γ e, and the muon α experimental searches are indeed ultralight axions with a operator, ∂αaμγ¯ μ. The gluon operator requires tuning to be sizable around the fuzzy dark matter regime and has other constraints [23–25], the electron coupling can be probed using 3If there is additional axion coupling in the phase of the mass torsion pendulums [26], and the muon operator has other strong matrix of some new , Caγ only needs to sum to an integer constraints making it difficult to see experimentally [27,28]. with the coefficient of the coupling, see e.g., [29].

115030-2 COSMOLOGICAL TENSION OF ULTRALIGHT AXION DARK … PHYS. REV. D 102, 115030 (2020) full trigonometric potential, and we now examine the energy stored in the axion potential is ∼μ4 and, assuming a cosmological limitations of such dark matter candidates. simple cosmology from the start of oscillations to recom- bination, demanding that this be less than the dark matter ρ ð Þ ≲ μ4 III. AXION MATTER-POWER SPECTRUM at zc: DM zc . This restricts, μ4 ≲ 4ð Þ3 eV zc=zeq , or equivalently, A scalar field evolving in a purely quadratic potential has   a scale-invariant matter-power spectrum, matching that of 10−21 1=4 ≳ 1017 eV ð Þ ð Þ ΛCDM. However, if the potential contains higher order fa GeV misalignment : 7 m terms, the scalar equation of motion will possess nonlinear a terms which impact the growth of perturbations, with The authors of [32] also consider temperature-dependent positive (negative) contributions wiping out (enhancing) axion masses which relax the misalignment constraint. small scale structure. For an axion with field amplitude While both types of bounds in [32] are more stringent than a0ðzÞ at z the condition for the axion fluid to ð Þ ≪ 1 the matter-power spectrum bound, they are also less robust behave like cold dark matter is a0 z =fa . The cosmic since they rely on misalignment and on having a simple background is the most sensitive probe of the cosmology from z to recombination. matter-power spectrum, measuring deviations at a part per c The bound on fa can be translated into a bound on the thousand, and sets a bound around recombination on any coupling to and nucleons using the relations in (2) additional energy density fluctuations, δρ=ρ ≲ 10−3, cor- for given values of C γ and C . The results are displayed in ð Þ2 2 ≲ 10−3 a aN responding to, a0 zrec =fa . It is important to note Figs. 1 and 2 in black for several values of the coefficients. that this bound does not rely on the specific production Since generic axion models predict Caγ, CaN that are at most mechanism and must be satisfied for any light axion Oð1Þ, this is a powerful bound on the ultralight axion making up the entirety of dark matter. parameter space. Additional model building beyond the This constraint was studied quantitatively for misaligned minimal scenario is required to access regions with larger axions in a trigonometric potential in [30] (see also [31] for coupling. For comparison, we include the regions related discussions). The authors considered an axion with the potential in (1) and a field value frozen by Hubble friction until zc, the redshift at which the axion mass is comparable to the Hubble rate and oscillations begin. The matter-power spectrum then constrains the fraction of dark matter made up by axions as a function of zc. The authors of [30] find that in order for the axion to constitute all of 4 dark matter, zc must be ≳9 × 10 . This can be translated onto a constraint on fa by noting that the axion field amplitude is fixed today by the measured dark matter ρ ð Þ¼1 2 ð Þ2 energy density with DM z 2 maa0 z . Since the ampli- 3=2 tude redshifts as a0ðzÞ ∝ ð1 þ zÞ , requiring the axion to oscillate before it exceeds its field range, a0ðzcÞ ≲ fa, requires sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ρ ðz0Þ ≳ DM ð1 þ Þ3=2 ð Þ fa 2 zc ; 5 FIG. 1. Ultralight axion dark matter mass vs photon-coupling ma parameter space. Requiring the ultralight axion to exhibit a   matter-power spectrum consistent with ΛCDM sets the bound 10−20 eV ≳ 1 2 1013 ð Þ shown in black for several values of Caγ. The region below the or fa . × GeV : 6 ¼ 1 ma Caγ line permits natural axions without any additional model building (see text). The purple and dotted purple lines 2 2 The rough expressions motivated above, a0ðz Þ =f ≲ display the weakest and strongest bounds, respectively, arising rec a – 10−3, give a similar result. Note that while [30] assumed a from purely gravitational considerations [33 36]. The lack of misalignment mechanism, it is more general, and will apply axion-to-photon conversion of axions produced during super- nova-1987A [37] gives the bound in green. Additional bounds (approximately) to any axion dark matter production from current (solid) and proposed searches (dashed) are from mechanism as suggested by the rough estimate. active galactic nuclei [38] (red), protoplanetary disk polarimetry The constraint proposed in this work utilizes the matter- [39] (light blue), CMB birefringence [40] (brown), power spectrum and is distinct from the work of [32], which [41,42] (orange), optical rings [43] (dark blue), and heterodyne presented a bound assuming the misalignment mechanism. superconductors [44] (olive). The misalignment bounds for 2 4 The limit in [32] is derived by noting that the maximum Caγ ¼ 10 , 10 are displayed by the wavy contours (gray).

115030-3 JEFF A. DROR and JACOB M. LEEDOM PHYS. REV. D 102, 115030 (2020)

be determined by utilizing constraints on the subhalo mass function from gravitational lensing and stellar streams [16,17]. Recently the Lyman-α bound was reanalyzed −20 and strengthened to ma ≳ 2 × 10 eV [15]. Since the precise restriction on the axion mass varies between these studies, and so we include both the weakest and strongest bounds in the plots below. In addition, there are astro- physical bounds on axions that are independent of their energy density. Axions released during (SN) 1987A would have produced a flux of axions that could convert as they passed through the galactic magnetic fields [37,48,49] and the nonobservation of this conversion sets the strongest bounds on low mass axions coupled to photons. For axion-nucleon coupling, the strongest dark FIG. 2. Ultralight axion dark matter mass vs nucleon-coupling matter-independent bounds arise from forbidding excess parameter space. Requiring the ultralight axion to exhibit a cooling of SN-1987A [33] and [34–36]. There matter-power spectrum consistent with ΛCDM sets the bound are also the bounds arising from superradiance shown in black for several values of CaN. The region below the [50,51], but these are relevant for larger masses or smaller ¼ 1 CaN line permits natural axions without any additional model couplings than we consider. building (see text). The solid purple and dotted purple lines There are a large number of searches looking for axion display the weakest and strongest bounds, respectively, arising dark matter that rely on its relic abundance. Efforts to – from purely gravitational considerations [33 36]. Supernova- discover a photon coupling include looking for deviations 1987A and neutron cooling [33–36] bounds are shown in in the polarization spectrum of the cosmic microwave green and the bound from old comagnetometer data is shown in background [52] (with updated bounds in [40]), searching red [45]. Additional bounds for nucleon couplings are shown ’ from projections of the CASPEr-Zulf experiment [46,47] (dark for the axion s influence on the polarization of light blue), interferometry (brown) [26], atomic magnetometers from astrophysical sources [38,39,41,42], and terrestrial 4 (light blue) [26], and storage rings [27] (pink). The misalignment experiments [43,44,53]. Searches for an axion-nucleon 2 4 bounds for CaN ¼ 10 , 10 are displayed by the wavy contours coupling focused on the ultralight regime include axion- (gray). wind spin precision [54], using nuclear magnetic resonance [45–47,55,56], and using storage rings [27]. Several spin precession experimental setups are considered in [26].5 constrained assuming misalignment as wavy gray lines for The photon bounds are compiled in Fig. 1 and nucleon different values of C γ and C . a aN bounds in Fig. 2. The matter-power spectrum bound The constraints we derive here assumed the axion field derived in Sec. III is displayed in both figures. We use makes up dark matter until prior to recombination. An solid (dashed) lines to denote current (prospective) bounds. alternative scenario is the case where an axion is only We conclude that many experimental proposals in this produced at late times, such as through the decay of a ultralight regime are inconsistent with a generic axion dark heavier state. While an intriguing possibility, decay of matter and require C γ ≫ 1 or C ≫ 1. Reaching the large heavy states will produce relativistic axions which will in a aN couplings considered in various experiments is an issue of turn modify the of the Universe. Thus additional model building, and is the focus of the next evading the matter-power spectrum bound by tweaking section. cosmology at late times is a formidable task. V. ENHANCED AXION COUPLINGS IV. COMPARISON WITH EXPERIMENTS We have presented stringent bounds on axions arising We now consider the prospects of ultralight axion dark from the relationship between their field range, fa, and matter searches in light of the matter-power spectrum their coupling to photons or nucleons. However, there exist restriction derived above, starting with a summary of model-building techniques that can relax this relationship, α current experimental constraints. First, the Lyman- -flux which have often been discussed in the context of power spectra sets a bound on the axion mass, independent axion . These methods may also be applied to of the size of nonlinear terms in the potential. These mea- ultralight axion dark matter and have distinct low energy surements are sensitive to sharp features in the matter- power spectrum on small scales, which would be if 4We used the “realistic” projections of [43]. the axion has a mass comparable to the size of dwarf 5 ≳ We selected the most stringent bounds from [26,27,44] and galaxies, and set a bound on the axion mass of ma continued these bounds to lower ULA mass values than what the 10−21 eV [11–14]. A mass bound of similar magnitude can original works consider.

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ε α phenomenology as a consequence of the lightness of the F1 ˜ μν L ⊃− a1FμνF : ð10Þ axion and the requirement of matching the observed matter- F2 8πF1 power spectrum. In this section we review these mecha- nisms, provide explicit realizations of such models, and Taking a1 to be the axion dark matter candidate, we study their phenomenology. We focus on the photon conclude that kinetic mixing gives Caγ ¼ εF1=F2.Ifε is coupling, but similar models can be built for the nucleon held fixed and the decay constants have a large hierarchy coupling. (F1 ≫ F2), then a1 will have Caγ ≫ 1. While this appears to be a simple solution, it is not A. Large charges possible to have Caγ ≳ 1 within most field theories. This is One way to enhance the axion coupling to visible matter a consequence of axions arising as Goldstone bosons of an is to introduce fermions with large charges or a large number extended scalar sector and hence the axion kinetic mixing is of fermions (see, e.g., [57,58] for a discussion in the context not a free parameter but must be generated. There are two of inflation). This strategy is limited by the requirement of possible sources for ε: group flow (“IR”) perturbativity of and the presence of light and higher dimensional operators (“UV”) contributions. To fermions charged under electromagnetism. see the suppression from IR contributions, consider a To be explicit, consider a KSVZ-like model [59,60] theory of two axions with a , χ, Φ where a complex scalar (whose phase will be identified μ μ ∂ a1 ∂ a2 with the axion), has Yukawa couplings with a set of Weyl L ⊃ χγ¯ μγ5χ þ χγ¯ μγ5χ: ð11Þ fermions with an electromagnetic charge Qf. Integrating F1 F2 out the fermions leads to an axion-photon coupling, The induced kinetic mixing of the axion is quadratically αQ2N divergent and goes as L ⊃ f f ˜ ð Þ 8π aFF: 8 fa Λ2 ε ∼ ð Þ 2 ; 12 The presence of charged fermions renormalizes the ð4πÞ F1F2 electric charge as computed through corrections to the photon gauge kinetic term. Perturbativity requires that where Λ represents the cutoff scale. Since Λ ≲ F1;2 2α 4π ≲ 1 ¼ 2 NfQf = . Since Caγ NfQf, the perturbativity (otherwise the effective theory is inconsistent), the kinetic ε ≲ 4π constraint sets a bound C γ ≲ 4π=α. We conclude that mixing is bounded by F2= F1 and will result a ≲ 1 large charges can at most enhance the axion-photon in Caγ . coupling by Oð103Þ.6 Alternatively, it is possible to induce an axion kinetic mixing through higher dimensional operators (see, e.g., B. Kinetic mixing [61,64]). Taking a1 and a2 to be the phases of complex scalar fields Φ1 and Φ2, there can be an operator, Kinetic mixing of multiple axion fields can raise the axion coupling to visible matter by (potentially) allowing 1 ↔ ↔ L ⊃ Φ† ∂ Φ Φ† ∂ Φ ð Þ an axion with a large field range to inherent couplings of an 2 1 1 2 2; 13 2M axion with a smaller field range (see [57,58,61–67] for discussions in other contexts). As a simple example ↔ where Φ† ∂ Φ ≡ Φ†∂Φ − ð∂Φ†ÞΦ. Once the scalar fields consider two axions a1 and a2, where a1 obtains a potential take on their values, the axions get a mixing term while the lighter axion, a2 (which is massless here), couples ε ¼ 2 to photons: with F1F2=M . This is again suppressed since con- sistency of the effective theory requires M ≳ F1;2 and 1 1 μ μ cannot result in Caγ ≳ 1. L ⊃ ∂μa1∂ a1 þ ∂μa2∂ a2 þ ε∂μa1∂a2 2 2 While these examples show that kinetic mixing is not α 4 a1 ˜ μν typically sizable for field theory axions, it has been þ μ cos þ a2FμνF : ð9Þ F1 8πF2 suggested that certain string constructions allow for sizable mixing coefficients [57]. While we are not aware of a The kinetic term can be diagonalized by the shift concrete string construction where this is true, this may be a a2 → a2 − εa1, which induces an a1-photon coupling, way to achieve Caγ ≳ 1. Interestingly, for ultralight axion dark matter, kinetic 6 Here we have taken the Peccei-Quinn charges of the fermions mixing has additional phenomenological implications. In Oð1Þ to be . If one chooses larger Peccei-Quinn charges such that order for the Lagrangian in (9) to result in a photon the fermion mass only arises through higher dimensional oper- ators, then the photon coupling can be slightly amplified. coupling for a1 that is not suppressed by a ratio of axion However, requiring a hierarchy between fa and the cutoff masses, a2 must be lighter than a1. Since the a2 photon strongly constrains this possibility [57]. coupling is not suppressed by factors of ε, it may be more

115030-5 JEFF A. DROR and JACOB M. LEEDOM PHYS. REV. D 102, 115030 (2020)   detectable than a1 and drastically influence direct con- a1 a2 a ¼ −F¯ þ straints, such as from supernova axion cooling or con- F1 F2 version. This would need to be studied with care for a   a a particular realization of a value of ε. ¼ ¯ 1 − 2 ð Þ b F 1 ; 17 In addition to axion mixing, kinetic mixing of Abelian F2 F gauge fields can boost the axion-photon coupling, as so that the physical axion interactions are considered in [68]. In this case the coupling may be enhanced if the axion-photon coupling inherits the dark β 1 L ⊃− aGG˜ − aFF;˜ ð18Þ photon gauge coupling. To see this explicitly, we consider 8πF¯ 8πF¯ an axion coupled to a dark U(1) gauge field, A0, which ¯ ¼ ð 2 þ 2Þ1=2 kinetically mixes with the electromagnetism, and F F1F2= F1 F2 . The axion b remains charged and provides a mass for the dark . The α0 1 1 ϵ surviving axion, a, is neutral under the dark U(1) and is 0 ˜ 0 0 0 0 ¯ L ⊃ aF F − FF − F F − FF ; ð14Þ the dark matter candidate. Since F is smaller than F1 and 8πF 4 4 4 a F2, a is more strongly coupled to photons than either of the original axions [29]. Nevertheless, this does not result in where α0 is the dark gauge coupling. If A0 has a mass below Caγ ≳ 1. This is because the decay constant of the surviving the photon mass, then a basis rotation can be axion, F¯ , appears in the anomalous coupling to both the performed to diagonalize the kinetic terms through non-Abelian and electromagnetic gauge sectors and so the → − ϵ 0 A A A . This transformation leaves the canonical relationship between axion potential and matter approximately massless and gives the axion a coupling to coupling is maintained. We conclude that axion-vector photons as mixing cannot be used to evade the cosmological bounds on ultralight dark matter axions. ϵ2α0 L ⊃ aFF;˜ ð15Þ 8πFa C. Clockwork Clockwork models provide a means to disturb the 2 0 such that Caγ ¼ ϵ α =α. Direct constraints on dark photons canonical relationship between the axion potential and permit ϵ ∼ 1 (see [69–71] for the bounds on ultralight dark photon coupling by introducing a large number of axions, photons) while α0 can be ∼1. Taken together, gauge kinetic each interacting with both its own confining gauge sector 2 “ ” mixing permits an amplification factor C γ ∼ Oð10 Þ. and its neighbor. After a rotation to the axion mass a ’ So far we have considered the cases of axion-axion and basis, the lightest axion s potential can be exponentially vector-vector mixing. It is also possible for axions to mix suppressed without introducing an exponential number of with a vector if the axions transform under the gauge fields. This light axion can be understood as the Goldstone symmetry, as is the case for Stückelberg axions (see, e.g., boson of a global symmetry between scalar fields in a UV – [63,66] for discussions in the context of inflationary model completion (see, e.g., [57,58,73 76] for discussions in building, as well as [29,72]). As a simple model, we different contexts). consider the case of two Stückelberg axions that have As an explicit model, we consider a set of N axions, ai, ð Þ gauge interactions with a dark U(1) gauge field, A0, and with couplings to N SU ni gauge sectors with field nearly identical interactions with electromagnetism and a strengths Gi, and a photon coupling only for aN:   dark confining gauge sector: XN−1 α β L ⊃ s;iþ1 iai þ aiþ1 ˜ 8π Gðiþ1ÞGðiþ1Þ ¼1 Fi Fiþ1 1 0 2 1 0 2 i L ⊃ ð∂μa1 − q1F1AμÞ þ ð∂μa2 − q2F2AμÞ α α 2 2 s;1 ˜ ˜     þ a1G1G1 þ aNFF: ð19Þ βα α 8πF1 8πFN þ s a1 þ a2 ˜ þ a1 þ a2 ˜ ð Þ 8π GG 8π FF: 16 F1 F2 F1 F2 The βi factors are integers greater than or equal to unity and we have omitted a bare θ term. Upon confinement, the 7 Gauge invariance requires q2 ¼ −q1 ≡ −q. We perform a gauge sectors give rise to the potential for the axions, field redefinition,   XN−1 β ð Þ ≃ μ4 iai þ aiþ1 þ μ4 a1 ð Þ V ai iþ1 cos 1 cos ; 20 7 F F þ1 F1 This Lagrangian is invariant under the U(1) gauge trans- i¼1 i i → þ α → þ α → þ formation a1 a1 q1F1 , a2 a2 q2F2 , and Aμ Aμ μ ∂μα if q1 þ q2 ¼ 0. The Lagrangian we consider is a simplified where the i are the confinement scales and represent the version of the setups in [29,63,66], but the conclusions are maximum possible masses for the dark composite states. To unchanged in the more general scenarios. get an enhanced photon coupling, we require

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μ1 ≪ μ2; μ3; … ð21Þ Standard Model, it may be possible to observe these in terrestrial experiments. and we take the Fis to be comparable to each other. In this From the low energy perspective, the clockwork model case, up to Oðμ1=μiÞ corrections, integrating out the we have described appears to permit arbitrarily large Caγ heavy axions corresponds to iteratively introducing the values. However, there may be limitations on this enhance- substitution: ment factor if one attempts to embed the model into a string construction. In heterotic string models, the four-dimen- β iai þ aiþ1 ≃ 0 ∀ ¼ 1 2 … ð Þ sional gauge groups descend from the rank 16 gauge i ; ; : 22 ð32Þ Fi Fiþ1 groups E8 × E8 or SO . Demanding that the Standard Model’s rank 4 gauge group be present in the low energy This transformation produces the effective Lagrangian theory restricts the rank of the dark sector to be ≤ 12 and so N could be severely limited [77].8 We leave an extensive α L ≃ μ4 aQN þ ˜ ð Þ study of string compactification restrictions on clockwork 1 cos β 8π aNFF: 23 FN i i FN models to future work. Q β The aN potential is exponentially suppressed by i i D. Discrete symmetry while the photon coupling remains unchanged, resulting in an axion potential exponentially flatter than the naive Finally, axion couplings tovisible matter can be augmented estimate. Redefining the axion decay constant as in (1) by introducing multiple non-Abelian gauge sectors related by gives adiscretesymmetry[78]. When the axion potential contri- Y butions from the confinement of each gauge sector are summed together, one finds the potential may be exponen- Caγ ¼ βi; ð24Þ i tially suppressed compared to the naive expectation. As an example, we consider a theory with a single axion, thereby boosting the photon coupling relative to a a, that couples to N confining gauge sectors with field generic axion. strengths, GðnÞ, and impose a discrete symmetry under We now consider the phenomenological implications of which clockworked axions as dark matter. First, in addition to the light axion, there exist N − 1 axions with masses propor- a → a þ 2πFa=N tional to μi≥2 (the only bound on these is from unitarity, GðnÞ → Gðnþ1Þ: ð25Þ requiring μi ≲ Fi [58]). These would be populated in the early Universe if the new non-Abelian gauge groups confine after reheating (from their own misalignment The symmetry forces all the non-Abelian gauge sectors to mechanisms) or if they are thermalized. Assuming the share a common gauge coupling and fermion content. confinement scales μi≥2 are comparable, the energy density Including an axion-photon coupling, the Lagrangian con- of the heaviest axion would dominate. However, the photon sistent with the symmetry has couplings of the N − 1 heavy axions are suppressed by   products of βi relative to the coupling of the lightest axion, βα XN 2π α s a n ˜ ˜ and so they cannot be the target particles of the above L ⊃ þ GðnÞGðnÞ þ aFF: ð26Þ 8π F N 8πF experimental searches. Furthermore, if the lightest clock- n¼1 a a work axion is to be dark matter and the experimental target, the heavier axions must decay into Standard Model In contrast to clockwork, the integer β serves no essential particles (if the axions decayed into lighter axions, they purpose here and can be taken to be unity. would produce excess in conflict with Each of the N gauge sectors contribute to the axion Δ measurements of Neff). This mandates the need for potential after they confine. If we were to use the leading substantial couplings of the heavy axions to the contribution to the axion potential from (1) for each sector, Standard Model and may lead to observable effects in the total axion potential would vanish. Therefore we must terrestrial experiments. include corrections associated with higher modes in the In addition to the heavy axions, the clockwork model Fourier expansion of the potential, which depend on the predicts the existence of a non-Abelian gauge sector with light fermion content of the theory. For a sector with two composite states well below the electroweak scale and fermions with masses m1 and m2 below the composite μ scale, chiral perturbation theory yields the leading order masses below 1. Demanding thatQ FN

115030-7 JEFF A. DROR and JACOB M. LEEDOM PHYS. REV. D 102, 115030 (2020)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  XN−1 π ð Þ¼−μ4 1 − 2 a þ n ð Þ V a z sin 2 27 n¼0 Fa N

2 with z ¼ 4m1m2=ðm1 þ m2Þ . After the sum in (27) is carried out, one finds the axion mass is exponentially suppressed if there is a small hierarchy between the light quark masses. Taking m2 >m1 the axion mass dependence on N is, approximately,   2 2 m1 N= μ ma ∼ : ð28Þ m2 Fa

Canonically normalizing the decay constant, we get N=2 Caγ ∼ ðm2=m1Þ , breaking the relation between the axion mass and photon coupling for N ≫ 1. While discrete symmetries produce axions with FIG. 3. The coefficients of the axion potential arising from a ≫ 1 Caγ , they do not evade the bounds from the matter- discrete symmetry normalized to their expected values. We see power spectrum. This is a consequence of the axion the exponential drop in C4 and C2, however their ratio grows with potential from (27) giving unusually large higher order N. Thus discrete symmetries strengthen the matter-spectrum axion terms. Unlike clockwork, which keeps the axion bounds instead of weakening them (see text). potential of the form in (1) and simply extends the field range, discrete symmetries break this relationship entirely. VI. CONCLUSIONS To see this behavior, we expand (27) about one of its minima, giving the potential, In this work, we considered the experimental prospects of detecting ultralight axion dark matter through its μ4 μ4 couplings to the visible sector, focusing on photon and ð Þ¼C2 2 − C4 4 þ V a 2 2 a 4! 4 a ; nucleon interactions. We presented a stringent bound on Fa Fa axions by requiring that their matter-power spectrum match 1 1 Λ ¼ m2a2 − λa4 þ; ð29Þ that of CDM and concluded that generic axions are 2 a 4! constrained to have couplings significantly smaller than is often assumed. This bound makes use of the relationship ’ where the Ci s are constants that arise from the sum in (27). between axion-matter couplings and the axion potential and The coefficient C2 determines the exponential suppression is independent of the dark matter production mechanism. of the axion mass and C4 fulfills a similar role for the This discussion displays the tension between experimental quartic. It is convenient to recast the mass suppression projections and cosmological bounds and has not been factor into anp axion-photonffiffiffiffiffiffi coupling enhancement factor widely emphasized in previous literature. ≡ ¼ μ2 λ ¼ μ4 4 via fa Fap=ffiffiffiffiffiffiC2 such that ma =fa, C4 =C2fa, Given the up and coming experimental program, the and Caγ ¼ C2. need to understand the landscape of ultralight axion dark The key observation is that the dependence on N is matter models with detectable couplings is clear. As such, different for the two constants C2 and C4, as displayed in we studied various strategies to boost axion couplings that Fig. 3. For large N, C4 decreases more slowly than C2 with were introduced previously in the literature and applied increasing N. The approximate condition presented above them to ultralight axion dark matter. In particular, we for the axion to behave sufficiently like cold dark matter is considered models with large charges, diverse forms of kinetic mixing, a clockwork mechanism, and a discrete

4 4 4 2 symmetry. We examined the extent to which axion cou- λ λ C γ a0 ∼ eV ¼ C4 eV a ≲ 10−3 ð Þ 2 2 4 2 2 : 30 plings can be boosted in each mechanism, if at all, and maa eq ma C2 ma fa explored their distinct predictions and phenomenology. In brief, Oð102–103Þ coupling enhancements are possible by 4 2 2 2 The factor eV Caγ=mafa is restricted to be greater than introducing large charges or vector kinetic mixing. unity to get a large enhancement in the photon coupling. Significantly larger enhancements are possible with clock- From Fig. 3, we see that C4=C2 will also be greater than work models if one takes an agnostic view towards UV unity and the bound cannot be satisfied. We conclude that completions, but arbitrarily large amplifications may be this variety of model cannot be used to boost the axion- stymied in string embeddings. Inversely, axion-axion photon coupling for ultralight axion dark matter. kinetic mixing can only be effective if some string

115030-8 COSMOLOGICAL TENSION OF ULTRALIGHT AXION DARK … PHYS. REV. D 102, 115030 (2020) construction allows one to bypass the field theory argu- are desirable. Some examples include inflation (where most ments presented above. Finally, discrete symmetries and of these mechanisms first arose, see text for references), axion-photon kinetic mixing are ineffective in raising the looking for parametric resonance during axion minicluster axion coupling to visible matter. If a discovery of ultralight mergers [85,86], monodromy axions [87,88], vector dark axion dark matter is made by a search in the near future, it matter production [89,90], and addressing the H0 tension would be a clear sign of new dynamics with possible [91,92]. Lastly, while we focused primarily on the axion- implications for other low energy terrestrial experiments. photon and axion-nucleon couplings, similar bounds can be There are several phenomena not discussed above that constructed for axion-electron couplings and, potentially, may place further restrictions on ultralight axion models. ultralight -philic scalars [93] (whose potential First of all, if a symmetry is restored in the early Universe, likely also needs to arise from breaking of a shift symmetry topological defects such as domain walls and cosmic to be protected against quantum corrections from ). strings could form. Axion emission from cosmic strings We leave a study of such scalars to future work. would contribute to the energy density of axions present today and any stable domain walls may dominate the ACKNOWLEDGMENTS energy density and thereby drastically alter the cosmology. This may further constrain variants of axions models, such We thank Tristan Smith for clarification of the bounds as clockwork axions, whose UV completions could have derived in [30], Anson Hook, Lawrence Hall, and Prateek multiple restored symmetries. Additionally, if an axion Agrawal for helpful comments on the manuscript, and symmetry is restored, axions may form miniclusters [80] Nicholas Rodd, Gustavo Marques-Tavares, , which would contribute to dark matter small scale structure. and Matthew Moschella for useful discussions. This work These may be observed using probes such as microlensing was supported in part by the Director, Office of , [81], pulsar timing [82,83], and 21 cm cosmology [84].We Office of High Energy and Nuclear Physics, of the U.S. leave the consideration of these issues for future work. Department of Energy under Contract No. DE-AC02- We considered ultralight axion dark matter, but the 05CH11231 (J. D. and J. M. L.) and by the National mechanisms discussed here may be applied in other Science Foundation under Grant No. PHY-1316783 contexts where large axion couplings to the visible sector (J. M. L.).

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