PPNUTP-19-A13
Freeze-in Axion-like Dark Matter
Sang Hui Im, Kwang Sik Jeong
Department of Physics, Pusan National University, Busan 46241, Korea
E-mail: [email protected] , [email protected] Abstract:
We present an interesting Higgs portal model where an axion-like particle(ALP) couples to the Standard Model sector only via the Higgs field. The ALP becomesstable due to CP invariance and turns out to be a natural candidate for freeze-in darkmatter because its properties are controlled by the perturbative ALP shift symmetry. Theportal coupling can be generated non-perturbatively by a hidden confining gauge sector, orradiatively by new leptons charged under the ALP shift symmetry. Such UV completionsgenerally involve a CP violating phase, which makes the ALP unstable and decay throughmixing with the Higgs boson, but can be sufficiently suppressed in a natural way by invokingadditional symmetries. a r X i v : . [ h e p - ph ] O c t ontents The existence of dark matter in the universe is a convincing evidence for physics beyond theStandard Model (SM) of particle physics. A weakly interacting massive particle (WIMP)has been considered as an attractive candidate for cold dark matter because it appearsoften and naturally if one extends the SM to address its puzzles such as the smallness ofthe electroweak scale [1, 2]. However, the negative results so far both in direct and indirectdetections of a WIMP dark matter lead to consider alternative dark matter scenarios.On the other hand, the LHC experiments have given no clear signals for new physicsbeyond the SM, and showed only small deviations of Higgs couplings from their SM values.This may indicate that a dark matter particle couples very weakly to the SM particles, forwhich its relic density generally depends on the history of the early universe. An axion-likeparticle (ALP) is an appealing candidate for such feebly interacting dark matter since theassociated shift symmetry controls its mass and couplings.If added to extend the SM, an ALP is usually considered to have an anomalous couplingto SM gauge bosons, and is subject to various experimental constraints depending on itsmass and decay constant. For instance, anomalously coupled to gluons, the QCD axionsolves the strong CP problem [3, 4], and it can also make up a significant fraction of darkmatter as produced by the misalignment mechanism or from topological defects. Anotherpossibility, which has recently attracted growing interest, is that an ALP couples to theSM sector via a Higgs portal. Such an ALP can play a crucial role in electroweak phasetransition, possibly providing an explanation why the electroweak scale is small [5–7], orhow the baryon asymmetry of the universe is produced [8–12].In this paper we explore if the dark matter of the universe can be explained by an ALPwhich couples to the SM sector only via a Higgs portal. The ALP is stabilized at a CP– 1 –onserving minimum and becomes stable if it has no other non-derivative interactions. Theportal coupling should then be tiny to make the ALP not thermalized with SM particlessince otherwise it would overclose the universe in most of the parameter space satisfyingthe experimental constraints on dark matter scattering with nuclei. In such a case, ALPdark matter can still be generated out of equilibrium via the freeze-in mechanism [13–16].It should be noted that the ALP shift symmetry can naturally suppress the portal couplingso that Higgs properties are rarely modified and the freeze-in occurs to produce the correctdark matter abundance.The portal coupling can be generated non-perturbatively from a hidden confining gaugesector, or radiatively by new leptons which have shift-symmetric interactions with the Higgsfield. In UV completions of the Higgs portal, one generally encounters a CP violating phasethat makes the ALP unstable and decay to SM particles through mixing with the Higgsboson. Such harmful coupling can be sufficiently suppressed in a natural way by invokingsupersymmetry. The ALP is never in thermal equilibrium as long as extra SM-chargedparticles responsible for a portal coupling are heavier than the reheating temperature ofinflation, or if ALP interactions are weak enough.This paper is organized as follows. In section 2 we introduce a novel model for freeze-indark matter where an ALP interacts with the SM only via the Higgs field, and examinethe parameter space where the ALP constitute all the dark matter of the universe. Section3 is devoted to discussions on a UV completion of the portal coupling and its cosmologicalaspects. The final section is for conclusions.
Having properties controlled by the associated shift symmetry, an ALP coupled to the SMvia a Higgs portal becomes a natural candidate for feebly interacting dark matter. In thissection we construct a portal model where the production of ALP dark matter takes placethrough freeze-in.
Coupling to the SM sector via a Higgs portal, an ALP can cosmologically relax the Higgsboson mass to the weak scale [5–7], or can drive a strong first-order phase transition requiredto implement electroweak baryogenesis [9, 10]. In this paper we explore the possibility thatthe dark matter of the universe is explained by an ALP with a Higgs portal − M cos (cid:18) φf (cid:19) | H | , (2.1)but without any anomalous coupling to gauge bosons or derivative couplings below a cutoffscale Λ. Here H is the Higgs doublet field, and f is the energy scale at which there occursa transition from linear to non-linear phase associated with the ALP shift symmetry. UVcompletions of the model will be presented in section 3.The ALP has a non-derivative coupling to the gauge-invariant Higgs squared operator,and thus its potential is radiatively generated from Higgs loops but without introducing a– 2 –ew CP violating phase. The scalar potential is thus written V = λ | H | + µ | H | − M cos (cid:18) φf (cid:19) | H | − π M Λ cos (cid:18) φf (cid:19) , (2.2)assuming the absence of other effects breaking the ALP shift symmetry explicitly. Here thelast term arises from a closed Higgs loop, and for simplicity we have absorbed a coefficientof order unity into the effective cutoff scale Λ. Note that f should be larger than Λ for theconsistency of the effective theory.It is clear that the ALP is stabilized at a CP preserving minimum φ = 0 and becomesstable due to Z symmetry, φ → − φ . The ALP can therefore contribute to the dark matterof the universe. Around the minimum φ = 0, the potential is expanded as V = λ | H | + ( µ − M ) | H | + 12 λ hφ | H | φ + 12 m φ φ + · · · , (2.3)where the ellipsis includes higher dimensional operators, and the ALP mass and couplingsare given by λ hφ = (cid:18) Mf (cid:19) ,m φ = (cid:112) πv/ Λ) × (cid:112) λ hφ π Λ , (2.4)with v (cid:39)
174 GeV being the Higgs vacuum expectation value. Note that Higgs mixing withthe ALP is subject to various experimental constraints [17, 18]. In our model, however,the neutral Higgs boson h does not mix with φ , implying that the Higgs couplings are notmodified. The LHC data on the 125 GeV Higgs boson constrain only the coupling λ hφ ifa Higgs invisible decay to ALPs is open.If produced by thermal freeze-out, ALP dark matter overcloses the universe in mostof the parameter space allowed by the experimental limit on spin-independent interactionswith nuclei and LHC searches for Higgs invisible decays [19]. To prevent the ALP fromthermalizing with the SM plasma, one needs to make it feebly interact with the SM sectorwith [20] λ hφ (cid:46) − . (2.5)In our scenario, this can be achieved in a natural way because all the ALP couplings arecontrolled by the scale f . The ALP never thermalizes, for instance, for M around the weakscale if f is above 10 GeV.
Even in the case that dark matter was never in thermal equilibrium with the SM plasma,it can be produced from decays or scatterings of thermal particles and make up a sizablefraction of the observed dark matter density [15, 16]. We note that the Higgs portal withan ALP provides a natural framework for such freeze-in dark matter. The ALP comoving– 3 –bundance freezes to a constant value as the Higgs boson number density is Boltzmann-suppressed to stop producing ALPs or the temperature of the universe drops below theALP mass.The time evolution of the ALP number density is described by a Boltzmann equation.In the Higgs portal scenario, the ALP couples to the SM sector only via the interactions λ hφ h φ + λ hφ v √ hφ (2.6)with a tiny portal coupling λ hφ = ( M/f ) (cid:46) − to avoid overclosure of the universe. ThenALP dark matter does not reach thermal equilibrium, but is produced via the processes h → φφ and hh → φφ depending on m φ , m h and λ hφ . The dominant freeze-in process is viadecays of Higgs bosons in thermal bath if kinematically allowed. Under the assumption thatthe initial abundance is negligible and the number of relativistic degrees of freedom doesnot change during ALP production, the approximate solution for the ALP relic abundanceis found to be [15] Ω φ h (cid:39) . × g ∗ s √ g ∗ m φ Γ( h → φφ ) m h (cid:12)(cid:12)(cid:12)(cid:12) T (cid:39) m h (cid:39) . × (cid:115) − m φ m h m φ v m h λ hφ g ∗ s √ g ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T (cid:39) m h , (2.7)for 2 m φ < m h with Γ( h → ii ) being the Higgs decay width of the indicated mode, while itis given by Ω φ h (cid:39) . × λ hφ g ∗ s √ g ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T (cid:39) m φ , (2.8)for 2 m φ > m h . Here g ∗ s and g ∗ are the effective numbers of degrees of freedom related tothe entropy and energy densities, respectively.Using the approximate solution, one can examine in which parameter space the freeze-in ALP accounts for the observed dark matter density. For the cutoff scale Λ above TeV,the ALP mass is given by m φ (cid:39) (cid:112) λ hφ Λ / π with λ hφ = ( M/f ) , and it is found that thecorrect dark matter density is obtained when the ALP has λ hφ (cid:39) − (cid:18) g ∗ s √ g ∗ (cid:19) / (cid:18) Λ10 GeV (cid:19) − / ,m φ (cid:39) (cid:18) g ∗ s √ g ∗ (cid:19) / (cid:18) Λ10 GeV (cid:19) / , (2.9)if 2 m φ < m h , and λ hφ (cid:39) × − (cid:18) g ∗ s √ g ∗ (cid:19) / ,m φ (cid:39)
380 GeV (cid:18) g ∗ s √ g ∗ (cid:19) / (cid:18) Λ10 GeV (cid:19) , (2.10)– 4 –therwise. The above shows that there are two viable regions of parameter space dependingon whether h → φφ is allowed or not:(a) f ∼ GeV (cid:18) M GeV (cid:19) (cid:18)
Λ10 GeV (cid:19) / and Λ (cid:46) × GeV , (b) f ∼ × GeV (cid:18) M GeV (cid:19) and Λ (cid:38) × GeV , (2.11)where we have taken g ∗ s √ g ∗ = 10 . In the region (b), where Higgs decay to ALPs iskinematically forbidden, the consistency condition f > Λ indicates that M is above TeV.In order to avoid large fine-tuning in electroweak symmetry breaking, one may constrain M to be not hierarchically larger than the weak scale. We also note that Ω φ h = 0 . m φ (cid:38)
100 keV for Λ above the weak scale, for which the ALP dark matter issufficiently cold to form structures of the universe. If light, the ALP can contribute to darkradiation during big-bang nucleosynthesis (BBN) but only with ∆ N eff below order 10 − for m φ above 100 keV.We close this subsection by discussing how much ALP coherent oscillations can con-tribute to the dark matter abundance. The ALP field starts to oscillate about the minimumat T = T osc when the expansion rate of the universe becomes comparable to its mass. Usingthe fact that ALP field oscillation behaves like cold dark matter, one can estimate the relicabundance to beΩ φ h (cid:39) . × − (cid:18) g ∗ ( T osc )100 (cid:19) − / (cid:16) m φ (cid:17) (cid:18) f GeV (cid:19) (cid:18) T osc GeV (cid:19) − θ , (2.12)where θ ini = φ ini /f is the initial misalignment angle, and T osc cannot exceed f because theALP appears at energy scales below f . The contribution from coherent oscillations is thusnegligible if T osc (cid:29) × GeV (cid:16) m φ (cid:17) (cid:18) f GeV (cid:19) , (2.13)taking g ∗ = 100. Combined with T osc < f , the above relation shows that coherent oscilla-tions can constitute a sizable fraction of the observed dark matter density if m φ is aroundor above 200GeV × (10 GeV /f ), i.e. in some part of the parameter space for freeze-in if θ ini is of order unity. Note that if the ALP mass does not depend on temperature, T osc is givenby T osc (cid:39) × GeV( m φ / MeV) / , and thus is high enough to suppress the contributionfrom coherent oscillations in most of the parameter space for freeze-in. On the other hand,if non-perturbatively generated by some hidden confining gauge interaction, the ALP massis turned on and grows only at temperatures around the confinement scale, implying that T osc is determined by the confinement scale, depending on how much the hidden sector iscolder than the SM plasma. If high energy dynamics generating the Higgs portal coupling involves a CP violating phase,there generally appears an effective potential of the form∆ V = − µ φ cos (cid:18) φf + α (cid:19) , (2.14)– 5 –hich shifts the minimum to a CP violating point, and consequently the ALP mixes withthe Higgs boson and couples to other SM particles. Let us consider the case with µ φ (cid:28) M Λ / π , for which ∆ V gives only a negligible contribution to the ALP mass. In suchcase, the mixing angle between φ and h is estimated by θ mix (cid:39) π µ φ ( m h − m φ )Λ vf sin α, (2.15)and it should be small enough in order for the ALP to live longer than the age of theuniverse, i.e. to ensure τ > × s where τ is the ALP lifetime. For m φ in the rangefor freeze-in, the mixing is more severely constrained by gamma ray observations becauseit makes the ALP directly decay into photons and possibly into other SM particles thatsubsequently produce photons via inverse Compton scattering [21–25]. The current exper-imental bound on τ ranges from about 10 s to 10 s depending on the ALP mass. Herewe shall conservatively take a lower bound on the ALP lifetime to be τ > s for theALP in the MeV to GeV range, and τ > s if above GeV.For m φ < m µ with m µ being the muon mass, the ALP dominantly decays intoelectrons via mixing with the Higgs boson, and its lifetime should be long enough to evadethe constraints from galactic photon spectra. The longevity bound on θ mix requires µ φ sin αM Λ (cid:46) − (cid:16) m φ
10 MeV (cid:17) − / (cid:16) τ s (cid:17) − / , (2.16)where we have used the relation f (cid:39) M Λ / (4 πm φ ), which holds for Λ above TeV. If theALP is heavy and decays into other SM particles, the constraint on the mixing gets severer.For instance, the longevity bound reads µ φ sin αM Λ (cid:46) − (cid:16) m φ GeV (cid:17) − / (cid:16) τ s (cid:17) − / , (2.17)if m φ (cid:38)
140 GeV, for which the ALP dominantly decays to W bosons. In this section we discuss how to UV complete the Higgs portal model. The UV completionshould be such that CP violation causing mixing between the ALP and the Higgs bosonif any is sufficiently suppressed, and the ALP is not thermalized with SM particles. Thiscan be achieved with help of perturbative ALP shift symmetry and supersymmetry.
The portal coupling (2.1) can be induced nonperturbatively if ALP shift symmetry isanomalously broken by hidden QCD confining at Λ hid . Let us introduce vector-like leptondoublets L + L c and singlets N + N c which are charged under the hidden QCD andhave interactions preserving the ALP shift symmetry. One can perform appropriate fieldredefinitions to write their interactions without loss of generality as yHLN c + y (cid:48) H † L c N + m L LL c + µ N e iα N N c , (3.1)– 6 –n which the Yukawa couplings and mass parameters are all real and positive. For the casewith µ N < Λ hid < m L , the singlet leptons have effective interactions yy (cid:48) m L | H | N N c + (cid:18) µ N e iα + yy (cid:48) π m L ln (cid:18) Λ ∗ m h (cid:19)(cid:19) N N c , (3.2)after integrating out heavy lepton doublets. Here Λ ∗ is the cutoff scale of the UV model,and we have included radiative contributions to the lepton singlet mass arsing from loopsof the Higgs and lepton doublets. If the ALP has an anomalous coupling to the hiddenQCD, the scalar potential is written V = λ | H | + µ | H | + ∆ V eff , (3.3)at energy scales below Λ hid , where ∆ V eff is the effective potential obtained by integratingout the heavy meson field from N + N c condensation∆ V eff = − M cos (cid:18) φf (cid:19) | H | − π M Λ cos (cid:18) φf (cid:19) − µ φ cos (cid:18) φf + α (cid:19) , (3.4)in which the involved parameters are determined by the couplings of high energy theoryas follows M = yy (cid:48) Λ m L , Λ = m L ln (cid:18) Λ ∗ m h (cid:19) ,µ φ = µ N Λ . (3.5)The effective potential assumes that the lepton doublets are heavier than the confinementscale while the lepton singlets are lighter, implying that Λ hid lies in the range yy (cid:48) π m L ln (cid:18) Λ ∗ m h (cid:19) < Λ hid < m L , (3.6)for the lepton singlets with a mass dominated by the radiative contribution. We note thatthe portal coupling is controllable in the sense that it vanishes in the limit that the gaugecoupling of hidden QCD goes to zero.The CP violating term in the effective potential should be highly suppressed because itmakes the ALP unstable and decay into SM particles via mixing with the Higgs boson. Oneway would be to impose that the hidden QCD sector preserves CP invariance, for which α = 0. Another way is to invoke supersymmetry to suppress the mass parameter µ N . Thiscan be achieved if the leptons couple to a singlet scalar X to acquire their masses afterspontaneous U(1) X breakdown in such a way that m L arises from superpotential while µ N is from K¨ahler potential. Here U(1) X symmetry, which can be either global or gauged, isassumed to be spontaneously broken at a scale lower than f . If global, we further assumethat some hidden confining dynamics makes the phase component of X much heavier thanthe ALP. To promote U(1) X to a gauge symmetry, one should include additional fermionsto satisfy anomaly cancellation. – 7 –s an explicit example with global U(1) X , let us consider a simple model where theoperators responsible for lepton masses are given by K (cid:51) X ∗ M P l
N N c + h . c .,W (cid:51) XLL c + H d LN c + H u L c N, (3.7)omitting dimensionless coupling constants, in the framework of flavor and CP conservingmediation of supersymmetry breaking, such as gauge, anomaly, and moduli mediation,so that soft supersymmetry breaking terms do not contain new sources of flavor and CPviolation. Here M P l denotes the reduced Planck mass, and H u and H d are the conventionalHiggs doublet superfields which are singlets under U(1) X and ALP shift transformations.The involved superfields carry charges given by H u H d L L c N N c X SU(2) L U(1) Y +1 / − / / − / N ) U(1) X − / − / / / N ) for a hidden confining gauge group. From the above interactions, which pre-serve both U(1) X and ALP shift symmetries, it is found that spontaneous U(1) X breakinginduces lepton masses according to µ N ∼ m SUSY M P l m L , (3.8)where m SUSY is the supersymmetry breaking scale. For m L < Λ ∗ = m SUSY , one obtainsthe effective theory given by eq. (3.1) at energy scales below m SUSY . A large hierarchy A Dirac mass for the singlet leptons N + N c is radiatively generated from the loops formed by theHiggs scalars and doublet leptons, and also from the loops of their superpartners. Those would make theALP unstable if the soft supersymmetry breaking parameters associated with H u H d and XLL c introducea new CP violating phase, i.e. a CP phase that cannot be rotated away by field redefinitions. Note that there are gravity mediation effects from non-renormalizable Planck-suppressed operators,which generally induce CP violations and thus should be highly suppressed to satisfy the longevity condi-tions. Using the relations (3.5) and (3.6), one finds that the condition (2.16) requires (cid:15)m / m SUSY (cid:46) − ( yy (cid:48) ) (cid:16) m φ (cid:17) − / (cid:16) τ s (cid:17) − / , for m φ < m µ , barring an alignment of the associated CP violating phases. Here m / is the gravitinomass, and (cid:15) represents the degree of sequestering between the visible sector and the supersymmetry breakingsector. The above indicates that the ALP becomes a proper dark matter candidate if (cid:15) (cid:28) yy (cid:48) (cid:28) π ) m SUSY M Pl < m / m SUSY (cid:28) , for the messenger scale higher than 16 π m SUSY as required to give non-tachyonic masses to the messengerscalars. – 8 –etween µ N and m L can be generated naturally because the lepton singlet mass additionallyrequires supersymmetry breaking. For the ALP to be stable enough and constitute all ofthe dark matter, the mixing between h and φ should be tiny, putting an upper bound onthe supersymmetry breaking scale roughly as m SUSY (cid:46)
10 TeV yy (cid:48) sin α (cid:16) m φ
10 MeV (cid:17) − / (cid:16) τ s (cid:17) − / , (3.9)for m φ < m µ , where we have used the relations (2.16) and (3.5) for the confinement scalelying in the range (3.6). Having a heavier mass, the ALP can decay to other SM particles,and the constraint becomes severer. For instance, for the case that the ALP dominantlydecays to W bosons, the longevity condition (2.17) requires m SUSY (cid:46) − TeV yy (cid:48) sin α (cid:16) m φ GeV (cid:17) − / (cid:16) τ s (cid:17) − / , (3.10)showing that m SUSY above TeV requires small Yukawa couplings and/or small CP phaseleading to yy (cid:48) sin α (cid:46) − for m φ around TeV.Since SM gauge charged particles participate in generating the effective portal coupling,it is important to examine if the ALP is thermalized due to the interactions with them. Ifthe reheating temperature of inflation T reh is higher than m L , the hidden lepton doubletsare in thermal bath. Because they are charged also under hidden QCD, the ALP would bethermalized via its anomalous coupling to hidden QCD unless T reh < T dec , (3.11)where the ALP decoupling temperature is roughly given by the larger of the confinementscale Λ hid and 10 GeV( f / GeV) . On the other hand, if T reh is lower than m L , hiddensector plasma can be colder than the SM plasma, and the ALP is never thermalized aslong as T dec is higher than the hidden sector temperature after inflation.Another issue to be considered is the cosmological effect of the N N c meson η , whichis integrated out to give the effective potential (3.4). The heavy meson has a mass of theorder of Λ hid , and the relevant interaction is κηφh, (3.12)where the coupling constant is given by κ ∼ M vf Λ hid . (3.13)where we have used that the mixing angle between the meson and Higgs is roughly givenby θ mix ( m h − m φ ) f / Λ , and it is tiny in the region of parameter space of our interest.The model possesses an approximate Z symmetry, η → − η and φ → − φ , which is brokenonly slightly by nonzero α and thus can ensure that the ALP lives sufficiently long. Onemay however wonder if the meson is a long-lived particle causing cosmological problems.It would destroy light elements synthesized by BBN if decays during or after BBN, ormay alter the freeze-in production of ALP dark matter via the κ interaction. These aresimply avoided if the reheating temperature is lower than Λ hid so that the meson is neverin thermal equilibrium. – 9 – .2 Radiative Higgs Portal Another interesting way to generate a Higgs portal interaction is to consider a slight break-ing of ALP shift symmetry that makes the ALP radiatively couple to the Higgs squaredoperator [26]. A simple model is obtained by adding vector-like lepton doublets L + L c anda lepton singlet N with interactions respecting the ALP shift symmetry∆ L = m L LL c + ye i φf HLN + y (cid:48) H † L c N + 12 µ s e i ( φf + α ) N N + h . c ., (3.14)where we have taken a field basis such that all the parameters are real and positive, whichis always possible without loss of generality. Note that the first three interactions areresponsible for an effective portal coupling, and in their presence the fourth term cannotbe forbidden by the ALP shift symmetry. The interactions depending on φ can arise, forinstance, in a low energy effective theory below f if ALP shift symmetry is linearly realizedand spontaneously broken at a scale f .Now we introduce a small mass term for the lepton singlet to break slightly the ALPshift symmetry ∆ L sb = 12 µ sb N N + h . c ., (3.15)where µ sb is real and positive. Then, an effective potential is radiatively generated∆ V eff = − M cos (cid:18) φf (cid:19) | H | − π M Λ cos (cid:18) φf (cid:19) − µ φ cos (cid:18) φf + α (cid:19) , (3.16)with couplings given by M = yy (cid:48) π µ sb m L ln (cid:18) Λ m L (cid:19) ,µ φ = 116 π µ sb µ s Λ . (3.17)The potential term involving the CP phase α is sensitive to the cutoff scale Λ as radiativelygenerated, differently from that in the case of a non-perturbatively generated portal. For α (cid:54) = 0, the ALP behaves like dark matter only when µ s is sufficiently small to suppressALP-Higgs mixing.The singlet mass term µ sb explicitly breaks the ALP shift symmetry at the perturbativelevel in the effective theory. As a result, there arises an ALP coupling to the lepton doublets yy (cid:48) π µ sb ln (cid:18) Λ m h (cid:19) e i φf LL c , (3.18)at the loop level, and thus the ALP interacts with electroweak gauge bosons via leptondoublet loops. Such couplings make the ALP decay to gauge bosons, but can be suppressedif the shift symmetric mass of lepton doublets is sufficiently large. In the parameter spaces(a) and (b) for freeze-in, the longevity condition requires(a) m L (cid:38) × GeV (cid:18) M GeV (cid:19) / (cid:16) m φ
10 MeV (cid:17) / (cid:16) τ s (cid:17) / , – 10 –b) m L (cid:38) × GeV (cid:18) M GeV (cid:19) / (cid:16) m φ GeV (cid:17) / (cid:16) τ s (cid:17) / , (3.19)respectively, where we have used the relations (2.9) and (2.10).To render the ALP stable further against α (cid:54) = 0, we embed the interactions for a portalcoupling in a supersymmetric model where the leptons acquire masses from the vacuumexpectation value of X after spontaneous breaking of U(1) X . Let us assume that ALPshift symmetry is linearly realized by a superfield Φ, which implies Φ = f √ e iφ/f + · · · afterspontaneous U(1) Φ breaking at f . We assign nonzero U(1) X charges only to X and theleptons, and nonzero U(1) Φ charges only to Φ and the leptons while making the leptonbilinear LL c neutral so that the ALP does not have an anomalous coupling to SM gaugebosons. For instance, one can take the charge assignment H u H d L L c N X
ΦSU(2) L U(1) Y +1 / − / / − / X − / − / / Φ − / / − / K (cid:51) X ∗ M P l Φ N N + h . c .,W (cid:51) XLL c + Φ H d LN + H u L c N, (3.20)where we have omitted dimensionless coupling constants, and the cutoff scale of Φ-dependentnon-renormalizable operators, which is higher than f . The lepton doublets become massiveafter U(1) X breaking, while the lepton singlet mass parameter µ s arises after supersym-metry and U(1) Φ are broken further. It thus follows µ s = y m SUSY M P l m L , (3.21)with m L < Λ = m SUSY , which indicates that µ s can be highly suppressed. On the otherhand, as the origin of the shift-symmetry breaking coupling µ sb , one can consider a holo-morphic operator N N in the superpotential or X ∗ N N in K¨ahler potential generated bynon-perturbative effects breaking U(1) Φ such as stringy instantons or field theoretic gauginocondensation.Combining the relations (2.16) and (2.17) with the couplings (3.17), one finds that thelongevity condition is not sensitive to the supersymmetry breaking scale and is translatedinto M (cid:46) y (cid:48) ln( m SUSY /m L )sin α (cid:16) m φ (cid:17) − / (cid:16) τ s (cid:17) − / , (3.22)for m φ < m µ with the ALP mass roughly given by m φ ∼ × ( m SUSY / TeV) / , whileit leads to M (cid:46) × − GeV y (cid:48) ln( m SUSY /m L )sin α (cid:16) m φ GeV (cid:17) − / (cid:16) τ s (cid:17) − / , (3.23)– 11 –ith m φ ∼
380 GeV × ( m SUSY / GeV) if the ALP decays mainly to W bosons. In theabove, the relation between the ALP mass and supersymmetry breaking scale assumes thatthe ALP makes up all of the dark matter of the universe. As discussed below eq. (2.11),the consistency condition requires M above TeV if the ALP is heavier than m h /
2. Thelongevity bound (3.23) would thus indicate that it is hard to realize a scenario of radiativeHiggs portal if the ALP is heavy.Finally we discuss the condition for the ALP to remain unthermalized. The interactionsrelevant to thermalization of the ALP read µ s f φN N + y f φHLN + y (cid:48) H † L c N, (3.24)where the second operator induces an effective Yukawa interaction but with a tiny couplinggiven by yv/f after electroweak symmetry breaking. Thus, if the reheating temperatureis lower than the lepton doublet mass, the ALP never enters thermal equilibrium with theSM plasma. In the opposite case with T reh > m L , the Yukawa coupling µ s /f should besmaller than about 10 − to avoid thermalization of the ALP. An ALP coupled to the SM sector via a Higgs portal has recently been noticed to providean explanation for the puzzles in the SM such as the smallness of the electroweak scale andthe origin of baryon asymmetry. In this paper we have explored if it can solve the darkmatter problem. The ALP is stabilized at a CP conserving vacuum and becomes stable ifit has no other non-derivative interactions. The perturbative shift symmetry then makesthe ALP a natural and appealing candidate for freeze-in dark matter.To UV complete the portal coupling, one can rely upon non-perturbative effects from ahidden confining gauge group or radiative corrections from new leptons charged under theshift symmetry. UV models generally involve a CP violating interaction, which makes theALP decay into SM particles through mixing with the Higgs boson. We found that suchmixing is sufficiently suppressed in a natural way if embedded into a supersymmetric theory.To avoid overclosure of the universe, ALP dark matter should be never in equilibrium withSM particles, constraining the properties of particles involved in generating an effectiveportal coupling.
Acknowledgments
We thank Kyu Jung Bae for helpful discussions. This work was supported by the NationalResearch Foundation of Korea (NRF) grant funded by the Korea government (MSIP)(NRF-2018R1C1B6006061). SHI also acknowledges support from Basic Science ResearchProgram through the National Research Foundation of Korea (NRF) funded by the Min-istry of Education (2019R1I1A1A01060680).– 12 – eferences [1] G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, M. Pierre, S. Profumo, and F. S.Queiroz, “The waning of the WIMP? A review of models, searches, and constraints,”
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