Dark Photon Dark Matter Produced by Axion Oscillations

PHYSICAL REVIEW D 99, 075002 (2019)

Dark photon dark matter produced by axion oscillations

Raymond T. Co,1 Aaron Pierce,1 Zhengkang Zhang,1,2,3 and Yue Zhao1,4 1Leinweber Center for Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 2Department of Physics, University of California, Berkeley, California 94720, USA 3Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 4Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA

(Received 8 November 2018; published 3 April 2019)

Despite growing interest and extensive effort to search for ultralight dark matter in the form of a hypothetical dark photon, how it fits into a consistent cosmology is unclear. Several dark photon dark matter production mechanisms proposed previously are known to have limitations, at least in certain mass regimes of experimental interest. In this paper, we explore a novel mechanism, where a coherently oscillating axionlike field can efficiently transfer its energy density to a dark photon field via a tachyonic instability. The residual axion relic is subsequently depleted via couplings to the visible sector, leaving only the dark photon as dark matter. We ensure that the cosmologies of both the axion and dark photon are consistent with existing constraints. We find that the mechanism works for a broad range of dark photon masses, including those of interest for ongoing experiments and proposed detection techniques.

DOI: 10.1103/PhysRevD.99.075002

I. INTRODUCTION production of DPDM via this mechanism is difficult absent a nonminimal coupling of A0 to gravity [25], which The identity of dark matter remains unknown. One 0 introduces a quadratic divergence for m 0 [26]. An alter- candidate is a dark photon A , a novel gauge boson with A native approach relying on quantum fluctuations during an unknown mass and tiny coupling to the Standard Model −5 0 10 (SM). Recent years have seen a growing interest in inflation can realize DPDM with mA > eV [26],but experimental techniques for detecting light dark photon the Stückelberg mass and high scale inflation essential to dark matter (DPDM), and many ideas have been studied. the mechanism may be in tension with the weak gravity 0 0 3 These include microwave cavity experiments such as conjecture unless mA > . eV [27], and the viable mass ADMX [1], a dark matter radio [2], dish antennas [3–8], range may shrink further with improved CMB bounds on dielectric haloscopes [9], absorption in various targets the scale of inflation (see, however, Ref. [28]). [10–18], the use of dark matter detectors as helioscopes In light of the extensive experimental searches and [19], and the repurposing of gravitational wave detectors limitations of these existing mechanisms, it is important [20] as well as other accelerometers [21]. All these to explore alternative ideas of how DPDM may be realized techniques focus on DPDM with masses ≲keV, in which in a consistent cosmology. In this paper we study a novel case it must be produced nonthermally to avoid constraints possibility, in which energy stored in coherent oscillations ϕ 0 on warm dark matter [22,23]. It is thus important to of an axionlike field can be efficiently transferred to A understand precisely how this could occur. via a so-called tachyonic instability. Explosive particle Several mechanisms for nonthermal production of production via tachyonic instabilities was first studied DPDM have been studied in the literature. One possibility in the context of preheating [29,30] (for reviews, see is a misalignment mechanism [24] similar to that of axions: Refs. [31,32]) and has been discussed as a mechanism A0 is initially displaced from the minimum of its potential, to produce vector fields in the context of generating and the energy stored in its oscillations may act as dark primordial magnetic fields [33–35]. Axionlike fields are matter. It was later pointed out, however, that sufficient a well-motivated ingredient in many theories beyond the SM and numerous string constructions. While they may be dark matter themselves, their abundance could be easily suppressed in the presence of couplings to hidden sector Published by the American Physical Society under the terms of fields (such as a dark photon) [36,37]. Here we show that the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to even in this case, an axionlike field can be closely tied the author(s) and the published article’s title, journal citation, to the cosmological origin of dark matter in the form of a and DOI. Funded by SCOAP3. dark photon. In what follows, we specify the details of

2470-0010=2019=99(7)=075002(8) 075002-1 Published by the American Physical Society CO, PIERCE, ZHANG, and ZHAO PHYS. REV. D 99, 075002 (2019) our DPDM production mechanism and discuss possible when the produced A0 backreacts on the ϕ condensate to subsequent thermal histories. A key ingredient is the excite higher momentum modes so that ϕ is no longer depletion of the residual ϕ relic that remains after its described by a coherently oscillating field. Detailed numeri- partial conversion to A0. This is achieved via couplings to cal simulations are needed to accurately determine the final the visible sector. We consider two possibilities, where momentum distribution and yield. For the present study, 2 0 ∼ ∼ 0 ¼ ∼ ϕ the ϕ relic either comes to dominate the energy density of we take pA pϕ mϕ, nA nϕ mϕ i after backreac- the Universe, or remains subdominant to SM radiation, tion sets in, consistent with the lattice results in Ref. [37]. before being thermalized into the SM bath and eventually The uncertainty in these quantities should not affect our depleted. In each case, we find DPDM production con- conclusions. sistent with all cosmological constraints for a wide range Shortly after the era of tachyonic instability, ρϕ will scale 0 of ϕ and A masses. Our findings strengthen the case for as nonrelativistic matter and quickly dominate over ρA0. DPDM searches. In the absence of additional couplings, ϕ would either decay to A0 at a later time or survive until today. In the first ϕ II. TACHYONIC INSTABILITY case, DPDM might be dominantly produced from decay but tends to be too hot. In the second case, dark matter ϕ We consider an axionlike particle (referred to as the would be dominantly composed of axions if mϕ ≫ mA0 [36]. “ ” ϕ axion hereafter) with a homogeneous initial value i after In principle, these two species can coexist as comparable inflation and reheating. It starts coherent oscillations when components if mϕ ≃ mA0 [39]. This comes at the cost of the Hubble rate is comparable to its mass, 3H ∼ mϕ, around fine-tuning, because these masses with wildly different pffiffiffiffiffiffiffiffiffiffiffiffiffiffi origins can span many decades and have no a priori reason ≃ 0 3 ð Þ Tosc . mϕMPl; 1 to be close. Here we instead consider the possibility that the residual ϕ is depleted via couplings to the visible sector, assuming an SM radiation-dominated epoch, where 0 ¼ 2 4 1018 so that dark matter is composed of just the A produced from MPl . × GeV. Our DPDM production mecha- ϕ 0 the tachyonic instability. The remaining sections concentrate nism centers around a coupling of to a dark photon A , on understanding this ϕ depletion, which is largely inde- α 0 D 0 ˜ 0μν pendent of the A production mechanism discussed above. LϕA0A0 ¼ ϕFμνF ; ð2Þ 8πfD III. DEPLETION OF THE AXION RELIC with F0 being the dark photon field strength tensor and ˜ 0μν αβμν 0 ϕ ð1Þ F ¼ ϵ Fαβ=2. Here, we have introduced a fine struc- We consider a setup where couples to the U Y hypercharge gauge boson, ture constant αD and an axion decay constant fD for the dark sector. This interaction affects the dark photon α Y μν ˜ Lϕ ¼ ϕB Bμν; ð4Þ equation of motion, written in Fourier space as [38] BB 8πf   B 2 0 ∂ α 0 ∂ϕ A 2 2 DkA 0 induced by Peccei-Quinn (PQ) fermions ψ charged under þ m 0 þ k 0 A ¼ 0; ð3Þ B ∂η2 A A 2π ∂η ð1Þ α ≃ 10−2 fD U Y, where Y . We will treat fB and fD as independent parameters, keeping in mind that each can be η 0 0 where is the conformal time, mA and kA are the mass and separated from the PQ breaking scale f [40]. A0 PQ momentum of , and indicates helicity. We have An example of how hierarchical decay constants could A0 assumed negligible self-interactions; this implies impor- be realized is a clockwork theory [41–44] where PQ tant constraints [39] that we find may be always satisfied in fermions charged under different gauge groups reside on the parameter space of interest. ≲ different sites. As a result, it is possible to have fPQ Tachyonic instability refers to an efficient energy transfer ≪ α 0 fB;D fa with either ordering of fB and fD. Here fa is the 0 2 2 DkA ∂ϕ from ϕ to A , which occurs when m 0 þ k 0 < 0. A A 2πfD ∂η inverse axion coupling to a postulated dark QCD sector on This negative quantity can be regarded as a tachyonic the last site, which sets the range of axion field excursions. effective mass for A0 and leads to an exponentially growing Meanwhile, small explicit breaking of the clockwork solution to Eq. (3). This tachyonic condition can be satisfied symmetry can give additional contributions to mϕ without ∂ϕ ∂η ≠ 0 only after the axion starts rolling at Tosc, i.e., = .Ata spoiling the clockwork mechanism, in which case mϕ is given time, one of the A0 helicities exhibits exponential bounded only by fPQ. In sum, we have αD ∂ϕ growth, with a peak momentum kA0 ∼ 4π j ∂η j ≫ mA0 , fD ≲ ≪ ∼ ϕ ð Þ obtained from minimizing the tachyonic effective mass. mϕ

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We now discuss processes mediated by the ϕBB cou- because a fermion flips helicity after absorbing a (pseudo) pling of Eq. (4) that may reduce the ϕ abundance by scalar. Subsequently, if T falls below mψ before reaching ϕ B thermalizing into the SM bath. At high temperatures mϕ, ψ B decouples and the previous discussion around 2 T ≳ mϕ=α , the axion mass is smaller than the thermal 2 Y Eqs. (6) and (7) applies. Otherwise, if mψB T;2mψ Þ: ð10Þ estimate the rate as B B 8π 2 B fB m2 T Γ ≃ 10−3 ϕ ð α2 Þ ð Þ Equations (6), (7), (9), and (10) above, valid in different ϕB→B 2 mϕ= Y leptons, consistent with ψ 0 ρ 3 precision electroweak constraints, would allow B to decay nA nϕ rad Tth 0 ¼ ∼ ¼ ¼ ð Þ fast enough without having dangerous cosmological YA 4 : 12 s T s T mϕs mϕ impact; see, e.g., Ref. [50] and references therein. th th Tth ψ In this setup, B itself can play a role in axion thermal- Using the observed dark matter abundance ization. After DPDM production at Tosc, there is a period ρA0 =s ¼ mA0 YA0 ≃ 0.44 eV, we thus obtain when T>mψB , mϕ, and scattering processes involving the ϕψ ψ¯ mϕ B B coupling, ≃ 0 6 ≡ ð Þ Tth . eV TR; 13 mA0 mψ L ¼ B ϕψ¯ γ5ψ ð Þ ϕψ Bψ¯ B Bi B; 8 fB where we have introduced the notation TR to indicate that Tth acts as a reheat temperature in the MD case. are dominant. We estimate the rate from 0 If mψB is given in addition to mϕ and mA , we would then be able to solve for f from ΓðT Þ¼HðT Þ. To ensure 2 B R R mψ T −2 B consistency, we need to check that for the fB value Γϕ →ψ ψ¯ ≃ 10 α ðT>mψ ;mϕÞ: ð9Þ B B B Y 2 B Γð Þ¼ ð Þ Γð Þ ð Þ fB determined by TR H TR , T TR where different processes may be rel- Note that unlike the gauge boson case, axion dissipation evant; otherwise, the axion would have been thermalized via the ϕψBψ¯ B coupling cannot be resonantly enhanced, earlier. When performing this check, we maximize HðTÞ by

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−3 allowing matter domination to begin as early as possible, after TR. Requiring kA0 < 10 mA0 at T ¼ eV [22,23] and up to Tosc. This allows us to identify the maximal viable making use of Eq. (13), we obtain parameter space.    2 If a consistent solution for f exists for given mϕ meV B ϕ ≳ 4 × 1010 GeV : ð19Þ ðmϕ;m 0 ;mψ Þ, we then need to check if it satisfies the i A B GeV mA0 following constraints: First, since mϕ

pffiffiffi  1  2 2πf mϕ 2 m 0 ð Þ ϕ ≳ 10 · D ≳ 106 GeV A : ð20Þ fB >mϕ;fB > mψ B : 14 i 4π αD GeV meV ð1Þ Next, we require that U PQ not be restored by the thermal Finally, the isocurvature perturbation constraint from 0 ϕ mass of the PQ breaking field, S, generated by ψ . A HI 2 −11 B Planck [59], P ≃ P ≃ ðπϕ Þ ≲ 8.7 × 10 , where HI At minimum, this condition should be satisfied when iso iso i ϕ 2 2 24 þ 2 ∼ 2 2 is the Hubble rate during inflation, translates into starts to oscillate, i.e., yψ B Tosc= mS mψB Tosc= ð12 2 Þ − 2 0 2 ∼− 2 Oð1Þ fPQ fPQ < , where mS fPQ for an quartic 4 4 ϕi ≳ 3 × 10 HI > 10 mϕ: ð21Þ coupling. As a result,     3 1 1 Here we have used HI >mϕ, which is necessary for mϕ 4 mψ 2 ≳ ≳ 105 B ð Þ ϕ fB fPQ GeV : 15 ensuring that starts oscillating only after inflation. GeV 100 GeV Lastly, we note there can be a contribution to DPDM from inflationary quantum fluctuations [26], estimated as To satisfy astrophysical constraints [51–55], we require     1 2 Ωinf 1 0 2 7 A0 mA HI f > 10 GeV if mϕ < 10 MeV: ð16Þ ≃ ð Þ B Ω −6 14 ; 22 DM D 6 × 10 eV 10 GeV pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Meanwhile, an upper bound on fB may apply if thermal- ¼ ð1 0 ð ÞÞ ization is not sufficient to deplete the ϕ relic; i.e., if the where D max ; mA =H TR accounts for dilution ϕ thermal abundance of ϕ at T ¼ T is higher than the dark from entropy production when is thermalized. Requiring R Eq. (22) to be less than 1, we obtain matter abundance. This happens if TR >mϕ, in which case

ϕ acquires a yield 1=g upon thermalization. We require  1 13 meV 4 1 that this thermal abundance decay away fast enough so 2 mϕ < 3HI ≲ 8 × 10 GeV D : ð23Þ as not to dominate the energy density of the Universe again, mA0 or to inject energy into the SM bath at late times, which may be subject to constraints from big bang nucleosyn- The aforementioned constraints are summarized in the ð 0 Þ thesis (BBN) or the CMB [56–58]. This requires mϕ > mA ;mϕ plane in Fig. 1. The red region is excluded ϕ T (taken to be 3 MeV), and Γϕ→BB ψ ψ¯ >H at because TR ; GeV GeV min B 2 Td We scan over mψ B (restricted to be above 100 GeV to avoid possible tension with collider searches) to find the applicable if TR >mϕ: ð17Þ maximal parameter space allowed by Eqs. (14)–(17), which Further constraints on this scenario come from several excludes the gray regions. The irregular shape of these requirements on ϕi. First, TM >TR implies regions reflects the nature of this optimization procedure,

namely adjusting mψB to open the least constrained  1 1 mϕ 4 meV 2 thermalization channel, among a set of options that differs ϕ ≳ 2 × 1015 GeV : ð18Þ i from point to point in the ðm 0 ;mϕÞ plane. For example, for GeV mA0 A mA0 ≳ 0.6 eV, TR

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where we have introduced the notation ϕ0 to indicate that ϕi is fixed (given mϕ, mA0 ) in the RD case. On the other hand, the axion thermalization temperature Tth is not fixed, but is subject to the following constraints. First, Eq. (25) combined with Eq. (11) fixes TM, which sets a lower bound on Tth:

mϕ ≃ 0 6 ð Þ Tth >TM . eV : 26 mA0

Another lower bound comes from requiring that the ϕ relic not decay to A0 before thermalization because this would give a potentially large contribution to hot dark matter. Γ 0 0 ð Þ From ϕ→A A

 5 1 mϕ 4 m 0 2 T ≳ 0.3 keV A ; ð27Þ th GeV meV

where we have used αD=ð2πfDÞ ≳ 10=ϕ0. Finally, we FIG. 1. Parameter space of the dark photon and axion masses in require Tth >TBBN to avoid potentially dangerous energy ϕ ϕ the scenario where is thermalized after dominating the Uni- injection after BBN, and Tth TR, which again requires considering different (17)], with an additional constraint from freeze-in (FI) processes for different mψ . 0 B overproduction of A . Note that for mA0 ≳ Oð100 eVÞ, the The availability of the remaining parameter space required dark matter yield is less than the thermal equi- depends crucially on the PQ fermion mass. Interestingly, librium value, and hence any additional thermal production if we are to live in the region below the lower solid (dashed) may result in an overabundance. In the current setup, a curve (including the lowest-mA0 region our mechanism can 0 ¯ thermal abundance of A can freeze in via ψ Bψ B → accommodate), a concrete prediction would be the exist- ϕ → A0A0, which is most effective at the highest temper- ence of a fermion with mass below 1(10) TeV. If a heavy ature Tmax. The FI abundance is computed to be [after using lepton mixes with the SM as discussed above, it can decay Eq. (25)] to SM gauge bosons and fermions, a signature possibly         accessible at the LHC or future colliders. Alternately, FI 2 2 2 Ω 0 α ϕ0 m 0 mψ T discovering a fermion with a sub-TeV mass would exclude A ≃ D A B max ð Þ Ω 2π : 28 all parameter space above the upper solid curve. DM fD keV fB Tosc

At least, this ratio should be less than 1 for αDϕ0=ð2πfDÞ¼ V. RADIATION-DOMINATED UNIVERSE 10 ¼ and Tmax Tosc. Thus, We now consider the second possibility, where the axion   is thermalized before TM, so the Universe remains radiation mA0 f ≳ mψ : ð29Þ dominated (RD). In this case, the DPDM relic abundance B B 0.1 keV ϕ fixes i rather than Tth. This is because in the absence of 0 entropy production, the A yield is constant since Tosc: This constraint does not apply for the MD case, as the FI

ϕ2 abundance is sufficiently diluted by entropy production. nA0 nϕ mϕ i Y 0 ¼ ∼ ∼ : ð24Þ In addition, constraints from isocurvature perturbations A s s sðT Þ ϕ ¼ ϕ Tosc Tosc osc and inflationary production, Eqs. (21) [with i 0 defined in Eq. (25)] and (23) (with D ¼ 1) apply equally The observed dark matter abundance then determines here. One last constraint is from the coldness of DPDM, −3 0 10 0 ¼  1 1 kA < mA at T eV. The dark photon momentum is mϕ 4 meV 2 3 ∼ 3 ð Þ 15 estimated using the invariant ratio k 0 =s mϕ=s Tosc , and ϕi ≃ 2 × 10 GeV ≡ ϕ0; ð25Þ A GeV mA0 the bound reads

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investigation of the subject in light of the near-future experimental prospects of detecting such a dark matter candidate over a wide range of masses. While our mechanism does not rely on interactions of A0 with the SM, a kinetic mixing ϵ with the photon is generically expected to exist, whose value is crucial for the potential detectability of DPDM. Constraints on ϵ specific to our mechanism include requiring that the A0-photon mixing not disturb the momentum coherence necessary for the tachyonic instability, or thermalize the produced A0 at any time afterward. These constraints are, however, rather loose, because ϵ is suppressed at high temperatures by the large thermal mass of the SM photon. As a result, the only relevant constraints are those consid- ered in Refs. [19,25], from a resonant conversion to SM photons. Experimental searches for DPDM will probe deeper into the ðmA0 ; ϵÞ parameter space, where our mechanism may be realized. The most promising access to the PQ sector of our setup may be via a PQ fermion ψ B rather than the axion ϕ [which FIG. 2. Similar to Fig. 1, but for the scenario where ϕ is cannot be lighter than ∼20 MeV and has a photon coupling thermalized in a radiation-dominated Universe. that is already strongly constrained; see, e.g., Eq. (15)]. We have seen that the viable parameter space depends on   2 mψ , because it affects the thermalization history of ϕ. ≲ 2 106 mA0 ð Þ B mϕ × GeV : 30 Interestingly, the lowest-m 0 region allowed by our mecha- meV A nism (which goes beyond the mass range viable for inflationary production) is accompanied by mψ ≲ TeV. The result of all these constraints for this RD case is B Discovery of such a PQ fermion at the LHC or a future shown in Fig. 2, where we scan over both mψB and Tth to find the maximal allowed parameter space. The additional collider, combined with positive results from light DPDM searches, may therefore hint toward an intertwined cos- freedom of adjusting Tth opens up more parameter space compared to the MD case. mological history of the PQ and dark photon sectors.

VI. DISCUSSION ACKNOWLEDGMENTS 0 We have shown that dark photon dark matter A The authors thank K. Harigaya, V. Narayan, and B. Safdi produced by coherent oscillations of an axionlike field ϕ 0 for useful discussions. The work of R. C. was supported in can have a viable cosmology for a broad swath of A part by DOE Early Career Grant No. DE-SC0019225. The 10−8 ∼ masses, ranging from a few × eV up to GeV. Two work of A. P., Z. Z., and Y. Z. was supported in part by key ingredients are a tachyonic instability that leads to the DOE under Grant No. DE-SC0007859. Z. Z. was also 0 ϕ explosive production of A from , and a mechanism to supported by the Summer Leinweber Research Award, ϕ deplete the residual relic. For the latter, we have by NSF Grant No. PHY-1638509, and by DOE Contract ϕ considered a minimal possibility of coupling to the No. DE-AC02-05CH11231. Z. Z. thanks the CERN theory visible sector as a proof of principle and discussed two group for hospitality during the completion of this work. possible thermalization histories, mapping out regions in the ðmA0 ;mϕÞ plane consistent with all constraints (see Figs. 1 and 2). Our work shows that to explicitly realize the Note added.—Recently, we became aware of Refs. [39, often-quoted “nonthermal production” of ultralight dark 60,61], which also consider new production mechanisms photon dark matter can be nontrivial, and motivates further for light vector dark matter.

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