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Title QCD Axion Dark Matter with a Small Decay Constant.
Permalink https://escholarship.org/uc/item/61f6p95k
Journal Physical review letters, 120(21)
ISSN 0031-9007
Authors Co, Raymond T Hall, Lawrence J Harigaya, Keisuke
Publication Date 2018-05-01
DOI 10.1103/physrevlett.120.211602
Peer reviewed
eScholarship.org Powered by the California Digital Library University of California PHYSICAL REVIEW LETTERS 120, 211602 (2018)
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QCD Axion Dark Matter with a Small Decay Constant
Raymond T. Co,1,2,3 Lawrence J. Hall,2,3 and Keisuke Harigaya2,3 1Leinweber Center for Theoretical 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 (Received 20 December 2017; published 23 May 2018)
The QCD axion is a good dark matter candidate. The observed dark matter abundance can arise from 11 misalignment or defect mechanisms, which generically require an axion decay constant fa ∼ Oð10 Þ GeV (or higher). We introduce a new cosmological origin for axion dark matter, parametric resonance from 8 11 oscillations of the Peccei-Quinn symmetry breaking field, that requires fa ∼ ð10 –10 Þ GeV. The axions may be warm enough to give deviations from cold dark matter in large scale structure.
DOI: 10.1103/PhysRevLett.120.211602
Introduction.—The absence of CP violation from QCD is unstable and decays into axions [10], yielding a dark is a long-standing problem in particle physics [1] and is matter density [11,12] elegantly solved by the Peccei-Quinn (PQ) mechanism 1.19 [2,3] involving a spontaneously broken anomalous sym- 2 fa Ω h j − ≃ 0.04–0.3 : ð2Þ metry. The scheme predicts the existence of a light boson a string DW 1011 GeV [4,5], the axion, which is constrained by data to be extremely light and, hence, is stable on cosmological time These mechanisms most naturally lead to axion dark scales, and a dark matter (DM) candidate. 11 12 matter for fa ∼ ð10 –10 Þ GeV, and the misalignment What cosmological production mechanism yields axions θ mechanism could also yield dark matter for larger fa as mis that account for dark matter? Axions produced from the is reduced. However, axions are not expected to be dark thermal bath have too low an abundance to be dark matter, matter for smaller fa. (The special cases for axion dark and are too hot. Two production mechanisms have been matter with lower fa are as follows. References [13–15] considered; both are IR mechanisms with axions produced study the anharmonicity effects of the axion cosine potential near the QCD phase transition. While they depend on the θ π when mis is tuned to approach , requiring a small inflation PQ symmetry breaking scale, fa, they are insensitive to scale to avoid quantum fluctuations. Reference [16] ana- details of the UV axion theory and dynamics of the PQ lyzes axion production in nonstandard cosmological eras phase transition. (i) Initially, the axion is nearly massless with kinetic energy domination even at temperatures below a θ and takes a generic field value misaligned by angle mis GeV. References [11,17–19] consider topological defects from the vacuum value. Around the QCD phase transition, for a domain wall number larger than unity with an explicit the axion obtains a mass, and the energy density in the PQ breaking, where a fine tuning is required to solve the resulting oscillations account for the dark matter abundance strong CP problem). – [6 8] In this Letter, we introduce a mechanism for UV production of dark matter axions from the early evolution f 1.19 Ω h2j ≃ 0.01θ2 a : ð1Þ of the PQ symmetry breaking field S in a relatively flat a mis mis 1011 GeV potential. We assume S has a large initial value, Si ≫ fa, e.g., from a negative Hubble induced mass during inflation (ii) When PQ symmetry is broken after inflation, cosmic [20]. After S begins oscillating, parametric resonance strings are produced [9]. Domain walls form between these [21,22] creates a huge number of axions. Assuming no strings around the QCD phase transition and, if the domain subsequent entropy production, wall (DW) number is unity, the string-domain wall network 2 1 9 10 2 10 Ω 2 ≃ 0 1 Si TeV GeV ð Þ ah . 16 ; 3 10 GeV mS;i fa Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. where m is the initial S mass. Unlike (1) and (2), the abun- Further distribution of this work must maintain attribution to S;i the author(s) and the published article’s title, journal citation, dance grows at low fa and sufficient dark matter can be obtai- 3 8 11 and DOI. Funded by SCOAP . ned for 10 GeV 0031-9007=18=120(21)=211602(7) 211602-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 120, 211602 (2018) efforts are ongoing for axions with small f , including IAXO p ðTÞ m a a ≃ 0.5 × S ; ð8Þ [23,24] and TASTE [25] for solar axions, Orpheus [26] and 1=3 ðΓ Þ1=2 s MPl MADMAX [27] for halo axions, and ARIADNE [28,29] for axion mediated CP-violating forces. See Refs. [30–32] for assuming that axions are produced with momenta other proposals. Helioscopes will explore the lower end of ∼m ðS Þ ≡ m . Axions are colder for a larger Γ since this range, and it is worth noting that our mechanism allows S S they receive more redshifting after production. Given the values of f even below 108 GeV, which are also consistent a recent constraint on the warmness of dark matter (see, e.g., with the supernova bound [33–37] if there is a mild [38,39]), we require that the axion velocity is smaller than cancellation in the axion-nucleon coupling. 10−3 at a temperature of 1 eV, In our mechanism, axions are initially produced with momenta of the order mS;i. After subsequent redshifting, ð Þ 109 the axion velocity at a temperature of 1 eV is pa T −6 GeV ≲ 6 × 10 ; ð9Þ T T¼1 eV fa 1 j ∼ 10−4 mS;i 2 fa ð Þ va eV × 6 9 : 4 10 GeV 10 GeV placing a lower bound on Γ . Let us assume that saxion to axion conversion occurs Γ ∼ 3 2 For a sufficiently large mS;i, the axion is warm. via a perturbative decay rate of mS =Sþ , where — Axion production by oscillating PQ field. The axion Sþ ¼ maxðS ;faÞ. The number density and momenta of mass is [4] the axions are given by 9 10 GeV 2 3 ¼ 6 ð Þ n S Sþ p Sþ ma meV : 5 a ∼ ; a ∼ : ð10Þ fa s 7=2 3=2 s1=3 1=2 1=2 mS MPl mS MPl The axion number density na0 that explains the observed ρ ≃ 0 4 dark matter abundance, DM=s . eV, is One can verify that the bound from coldness, (9), and the abundance requirement, (6), are incompatible with each other for any ðS ;m Þ, for all f larger than the exper- na0 fa S a Y ≡ ≃ 70 ; ð6Þ 8 a0 s 109 GeV imental lower bound of 10 GeV. Parametric resonance.—In fact, the production rate is where s is the entropy density of the Universe. The required typically much larger than the perturbative decay rate. The 8 Ya is much larger than thermal for all fa ≥ 10 GeV, so saxion couples to the axion through the potential of the PQ axions must be produced nonthermally. breaking field, so that the axion mass oscillates. Axion The radial direction of the PQ symmetry breaking field, modes with nonzero momenta, which we call fluctuations, which we call the saxion, S, (following terminology of grow rapidly by parametric resonance [21,22] from initial supersymmetric theories) is taken to have a large initial seeds set by quantum fluctuations. The axion momentum is field value. As the Hubble scale H becomes smaller than typically of the order mS. The energy of the fluctuations the saxion mass, the saxion beginspffiffiffiffiffiffiffiffiffiffiffiffi coherent zero-mode grows exponentially and, at some point, typically soon after 00 oscillations begin, becomes comparable to that of the saxion oscillations. The mass mSðSÞ ≡ V ðSÞ is generally S dependent and differs from the vacuum value. For a wide oscillation. At this stage, the back reaction on the saxion is range of potentials VðSÞ, the saxion yield is conserved with non-negligible and parametric resonance ceases. Because of the initial value efficient scattering between the oscillating saxion field and the axion fluctuations, the entire zero-mode energy of the 2 saxion is converted into comparable saxion and axion nS Si Y ≡ ∼ Y ∼ ; ð7Þ fluctuations, YS ∼ Ya ∼ YS , given in Eq. (7). S s Si 1=2 3=2 i mS;i MPl The evolution of the fluctuations after this stage is model dependent, and a model-by-model lattice simulation is ≃ in a radiation-dominated universe, where MPl needed to rigorously follow the dynamics. In this Letter, 2.41018 GeV. A result for the matter-dominated case can we consider the case where the comoving momentum and be straightforwardly derived. Suppose that this saxion yield number density of the fluctuations are approximately is converted into axions, e.g., by decay. For fa ≪ conserved, so that the axion yield is comparable to that Si 211602-2 PHYSICAL REVIEW LETTERS 120, 211602 (2018) 1 3 3 1 ð Þ 2 0 04 2 2 2 pa T ∼ mS;i ð Þ ≃ . Si ≃ 0 05 Si fa ð Þ 1=3 ; 11 Ya 1=2 . ; 14 s MPl λ MPl MPl mS;0 1 1 1 that easily satisfy the coldness constraint (9) for sufficiently p S 2 S 2 m 0 2 a ≃ λ1=2 i ≃ i S; ð Þ small saxion masses. 1=3 ; 15 s MPl MPl fa So far, we assumed that entropy is conserved after axion production, but this depends on the fate of the saxions. To giving constraints on the parameter space, avoid overclosure, the energy of the saxions must be Γ transferred to the thermal bath with some rate s and, 3 9 Si 10 GeV hence, thermalize at Tth. However, thermalization after m 0 ≃ 400 ; S; MeV 2 1017 matter domination generates entropy, leading to an axion × GeV fa 17 yield that is independent of Si Si ≲ 2 × 10 GeV: ð16Þ n T pffiffiffiffiffiffiffiffiffiffiffiffi ¼ a ∼ th ∼ Γ ð Þ The axions should not be thermalized. For small fa Ya ;Tth sMPl; 12 s mS;0 thermalization seems inevitable, but in our mechanism, it is suppressed by the large field value of S. The axions scatter with the thermal bath with a rate where mS;0 ≡ mSðfaÞ is the vacuum saxion mass. In the following, we apply the above mechanism to a few T3 1 1 simple models of PQ symmetry breaking, giving results Γ ≃ 10−5 ; ≡ : ð Þ a 2 2 2j 2j 17 with and without matter domination by saxions. Seff Seff P Production of axions via parametric resonance is dis- – cussed in the literature. References [40 42] consider During the zero-mode saxion oscillation, Seff may be much production of QCD axions by oscillations of the PQ smaller than the amplitude of the oscillation as P coherently symmetry breaking field, with axions identified as dark gets small during the oscillation [45]. However, Seff is still radiation. Reference [43] investigates production of axion- large enough to prevent thermalization. After the saxion like particle dark matter via derivative couplings with a oscillation energy is converted into fluctuations, Seff is of scalar condensation. However, these papers do not consider the order of the amplitude of the fluctuation and decreases production of QCD axion dark matter from parametric in proportion to the temperature, until it becomes of the ¼ resonance. order fa, after which Seff fa. The ratio of the scattering — Quartic potential. We consider a potential of the PQ rate to the Hubble scale is maximized when Seff first symmetry breaking field P, reaches fa. Requiring this ratio to be smaller than unity gives f2 2 V ¼ λ2 jPj2 − a : ð13Þ 5 2 Si 2 m 0 ≲ 1 GeV ; ð18Þ S; 2 × 1017 GeV pffiffiffi m ≃ 3λS S ≫ f The saxion masspffiffiffi is field dependent, S for a after removing f using the constraint Eq. (16) for the dark ¼ 2λ a and mS;0 fa. Thepffiffiffi saxion begins to oscillate when the matter abundance. λ 3 ≡ Hubble scale is Si= Hosc. The upper panel in Fig. 1 summarizes the constraints on After the saxion oscillation energy is transferred to axion mS;0 and Si. On each contour, dark matter is explained by and saxion fluctuations, the fluctuation momentum, ini- axions produced from parametric resonance for the corre- λ tially of the order S, slowly grows via number-reducing sponding fa. In the pink region, the axions are too warm to scatterings as k ∝ R1=7 [44] when the fluctuation amplitude be dark matter; the y axis on the right labels the axion is larger than fa. Here, R is the scale factor of the Universe. velocity va at T ¼ eV. In the yellow region, the resonantly The evolution of the momentum for a smaller amplitude is produced axions thermalize and are not dark matter. The not known, but we expect that the growth of the momentum supernova (SN) or dark radiation (DR) constraint leading to becomes ineffective since the interaction term λ2jPj4 is less the orange region is explained later. effective. The overall growth of the momentum is, at the Once the zero-mode saxion oscillation is transferred into most, Oð10Þ in the parameter space of interest, and we fluctuations of axions and saxions, their energy density ignore it. evolves as radiation. After Seff becomes of the order fa, the First, we discuss the case of rapid thermalization of the saxion behaves as matter, and may dominate the energy saxion fluctuations so that they never dominate. The axion density of the Universe. The axion abundance is then given yield and momentum are given by by Eq. (12), and the axion momentum is 211602-3 PHYSICAL REVIEW LETTERS 120, 211602 (2018) 2 2 2 2jPj Axion too warm –3 ¼ j j − 1 ð Þ 10 GeV 10 V m P ln 2 : 22 =10 fa fa 1017 9 GeV =10 –4 ev For simplicity, we approximate the dynamics of the saxion fa 10 a (GeV) v i as the oscillation by an almost constant quadratic term with 8 S GeV ≃ ≡ ∼ SN or DR =10 mS;i mS;0 mS mP, m. fa –5 In these models, the saxion starts oscillating when the Axion thermalized 10 ≃ 3 Hubble scale is Hosc mS= . We parametrize the time of 1016 parametric resonance as t ≡ Np=mS. After the zero-mode 18 GeV saxion oscillation energy is transferred to axion fluctua- 10 8 excluded for =10 tions, the axion yield is of the order of the original saxion in f a f =109 GeV SN or DR a Eq. (7). The axion momentum is 10 GeV 1 (GeV) 17 2 i 10 p 1 2 m 10 fa = a = S S ≃ 0 7 ð Þ 9 GeV 1 3 . Np : 23 Axion underabundant s = M fa =10 Pl Axion too warm Axion thermalized We have checked with a lattice simulation using 1016 LATTICEEASY [49] that, in the one field model of –3 –2 –1 10 10 10 1 10 Eq. (22), the number density is conserved, the axion mS,0 (GeV) momentum does not grow due to number-reducing scatter- ¼ Oð100Þ FIG. 1. Constraints on the quartic theory in the parameter space ings, and Np . We discuss this issue in detail in a of the saxion initial amplitude and vacuum mass, assuming separate publication [50]. The results below also apply to saxion oscillations begin during a radiation-dominated era. The the two-field theory if the comoving number density and upper (lower) panel assumes an era of saxion domination does not momentum do not vary much. (does) exist. The solid lines of the two panels coincide; the The constraints on the quadratic theory derived in a observed dark matter abundance arises on (above) these lines for similar manner as for the quartic potential are shown in the upper (lower) panel. Fig. 2. In comparison with the quartic theory, a large vacuum mass of the saxion is allowed since the mass is 1 2 1 independent of the saxion field value. p n 3 m 0 n 3 a ≃ 36λ2 a ¼ 18 S; a : ð19Þ The additional feature seen in the axion thermalization 1=3 s f2 s s a constraint arises from the following observations. If the With this dilution, the dark matter abundance requires the saxion transfers its energy to the thermal bath before S saxion mass to be smaller than in Eq. (16). The constraint 1017 10–2 from axion thermalization is derived in a similar manner to Np=100 Axion too warm Eq. (18), and is given by 10–3 SN 3 1016 11 GeV –4 ) 10 fa Si or =10 m 0 ≲ 200 : ð Þ fa S; GeV 9 20 DR ev 10 10 GeV GeV M GeV –5 Pl a ( 10 v i =10 fa S 15 10 9 GeV 10–6 The constraints on parameters with a saxion domination era = 10 are shown in the lower panel of Fig. 1. The region to the fa 8 GeV Axion thermalized 10–7 =10 right of each color line is excluded for the corresponding fa 1014 fa. In regions between the orange boundary and the colored 1018 Np=100 lines, the dark matter abundance is explained with Tth given by Eq. (12). GeV 1017 8 ) Quadratic potential.—In supersymmetric QCD axion 10 SN = 11 models, the saxion potential may be approximately quad- a GeV 16 f (GeV 10 or fa =10 ratic. One example is a two-field model with superpotential i 10 GeV Axion S DR f =10 and soft supersymmetry breaking terms a 9 GeV underabundant 15 =10 10 fa 9 GeV too warm f = 10 ¼ ð ¯ − 2Þ ¼ 2 j j2 þ 2 j ¯ j2 ð Þ d for a thermalized W X PP fa ;Vsoft mP P mP¯ P : 21 exclude 1014 10–3 10–2 10–1 1 10 102 103 104 105 Another example is a one-field model with a potential given mS,0 (GeV) by soft supersymmetry breaking and radiative corrections [46–48], FIG. 2. The same as Fig. 1 but for the quadratic theory. 211602-4 PHYSICAL REVIEW LETTERS 120, 211602 (2018) reaches the minimum, Seff suddenly drops to fa, enhancing fa of interest, from saxion cooling of supernovae Γa. In the upper panel, the yellow region is derived by [33–37,56]. For small mH, on the other hand, saxions assuming that saxions are destroyed at the temperature of decouple from the thermal bath while relativistic, and decay saxion domination. If saxions are destroyed at higher temper- dominantly to a pair of axions at temperature atures, the constraint above the kink becomes stronger. For 3 8 m 0 2 10 GeV large Si, axion thermalization occurs after saxion domination ≃ 1 S; ð Þ Tdec MeV : 27 rather than during the radiation-dominated era, resulting in the 10 MeV fa kinks of the dotted lines in the lower panel. 15 The resulting axions are observed as dark radiation of the For S smaller than 6 × 10 GeV, saxions begin oscil- i Universe, with an energy density, normalized to one lating in the thermal potential generated by the free energy generation of neutrinos, given by of quarks and gluons [51]. (In principle, this can be avoided by giving a large mass to gluons.) This not only changes the 120ζð3Þ ð1 Þ4=3 Δ ¼ mS;0 g MeV saxion evolution but may also lead to formation of clumpy Neff 4 1=3 7π T g ðT Þg ðT Þ objects such as Q balls [52] and I balls or oscillons dec D dec 1 1 – 10 2 80 10 75 3 [53 55], which may drastically change axion production. ≃ 0 3 MeV fa . . 8 ð Þ ð Þ ; We discuss this in a future work [50]. mS;0 10 GeV g TD g Tdec The fate of the saxion.—Remnant saxions must be ð28Þ removed without causing cosmological problems. If saxions, with yield (6), do not decay or scatter with the where g ðTÞ is the number of degrees of freedom of the thermal bath, they dominate the energy density below thermal bath at temperature T, and TD is the saxion temperature Tdom, decoupling temperature. The experimental upper bound Δ 0 6 10 is Neff < . [57].FormS;0 < MeV, there is too much Tdom ≃ 100 fa ð Þ 9 : 24 dark radiation (small mH) or too rapid supernova cooling mS;0 10 GeV (large mH), as shown by the orange region in Figs. 1 and 2. Discussion.—In a wide class of theories, a large initial In this case, Eqs. (6) and (12) imply Tth >mS;0. Without saxion domination, the saxion must be thermalized at a value of the PQ breaking field leads to cosmological production of axions via parametric resonance. The result- temperature above Tdom, larger than mS;0. In both cases, ing dark matter abundance scales as 1=fa and dominates Tth >mS;0 is required, and hence, scattering with the thermal bath is important for destruction. One may naively at low fa. Figures 1 and 2 show that, in theories with expect that scattering between axions and the thermal bath quartic and quadratic potentials, there are large regions is also effective, thermalizing the axion. of parameter space allowing axion dark matter with ∼ ð108–1011Þ This is, however, not always true. Consider a coupling fa GeV. The parts of these regions close “ ” between P and the Higgs field H, V ¼ κjPj2jHj2. This to the too warm boundaries have large scale structure that 2 j j2 deviates from that of cold dark matter. The required flatness induces a saxion-Higgs coupling, mHS H =fa, but not an 2 of the potential leads to a mass hierarchy, mS;0 ≪ fa, that axion-Higgs coupling. Here, mH is a contribution to the Higgs mass squared from PQ symmetry breaking. The can be understood with supersymmetry. saxion scattering rate with the thermal bath, before the Parametric resonance randomizes the axion direction so that axions are also produced from the misalignment electroweak phase transition, and for T>mS,is[45] θ ¼ Oð1Þ mechanism with mis , overproducing axions for 1 4 2 f ≳ 1012 GeV. In the quartic theory, PQ symmetry is Γ ≃ mH S ð Þ a s;H π 2 : 25 restored, leading to domain wall formation [58,59]. This faT fa might also occur in the one-field quadratic model. The With this interaction, the saxion is thermalized below a domain wall number should be unity so that domain walls temperature Tth, are unstable [60]. The unstable string domain wall network ≳ 1012 4 9 2 overproduces axions if fa GeV. A numerical sim- 3 10 3 ≃ 1 mH GeV ð Þ ulation [61] suggests that domain walls are not produced in Tth TeV ; 26 200 GeV fa the two-field quadratic model. which may be large enough. The authors thank Kyohei Mukaida and Yue Zhao for Thermalized saxions interact with standard model fer- discussions. This work was supported in part by the mions via mixing with the Higgs field. For sufficiently Director, Office of Science, Office of High Energy and large mH, saxions are still in thermal equilibrium while Nuclear Physics, of the U.S. Department of Energy under nonrelativistic and, hence, disappear without leaving any Contract No. DE-AC02-05CH11231 and by the National imprint by decaying into standard model fermions. 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