Supersymmetric Clockwork Axion Model and Axino Dark Matter

PHYSICAL REVIEW D 102, 015011 (2020)

Supersymmetric clockwork axion model and axino dark matter

† Kyu Jung Bae1,* and Sang Hui Im 2, 1Department of Physics, Kyungpook National University, Daegu 41566, Korea 2Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Daejeon 34126, Korea

(Received 22 April 2020; accepted 29 June 2020; published 16 July 2020)

Implications of supersymmetrizing the clockwork axions are studied. Supersymmetry ensures that the saxions and axinos have the same pattern of the coupling hierarchy as the clockwork axions. If we assume supersymmetry breaking is universal over the clockwork sites, the coupling structure is preserved, while the mass orderings of the saxions and axinos can differ depending on the supersymmetry breaking scale. While the massive saxions and axions quickly decay, the lightest axino can be stable and, thus, a dark matter candidate. The relic abundance of the axino dark matter from thermal production is mostly determined by decays of the heavier axinos in the normal mass ordering. This exponentially enhances the thermal yield compared to the conventional axino scenarios. Some cosmological issues are discussed.

DOI: 10.1103/PhysRevD.102.015011

I. INTRODUCTION The clockwork theory presents a plausible mechanism to One of the strongest beliefs in particle physics is that build hierarchical mass spectra and interactions from a there exist extended sectors of new physics beyond the series of multiple nonhierarchical ones. An early form of standard model (SM). In theoretical aspects, it is invoked to the clockwork structure was studied to achieve a trans- resolve fine-tuning problems residing in the SM. In Planckian field excursion from two sub-Planckian fields in practical aspects, the SM does not contain physics for a natural inflation [12]. In further studies, it was shown that essential phenomena such as neutrino oscillation, matter- a number of axions with similar decay constants can – antimatter asymmetry, and dark matter (DM). A widely produce an exponentially large effective scale [13 15].It accepted notion of extensions of the SM is to introduce has been argued that the same mechanism is applicable for “dark” sectors which communicate with the SM via feeble more general systems with various spins, scales, and interactions, leading to rational explanations to those couplings [16]. In particular, the clockwork mechanism ≳109 phenomena. is able to construct an intermediate-scale ( GeV) A prominent fine-tuning problem in the SM is the strong axion decay constant from dynamics near the electroweak CP problem. It can be solved by introducing a sponta- scale [17]. ð1ÞNþ1 neously broken Peccei-Quinn symmetry [1] which involves In the case of the clockwork axion, a global U symmetry spontaneously breaks at scale f and conse- the QCD axion [2,3]. The axion couples to the gluon field þ 1 strength and dynamically relaxes the QCD θ term to zero. quently results in (N ) Goldstone bosons. The global Astrophysical observations constrain axion-gauge boson symmetry is explicitly but softly broken by N mass terms couplings (including the axion-gluon coupling) [4–7] so with clockwork structure. This specific structure leaves unbroken Uð1Þ and a corresponding massless degree of that the axion couplings are required to be suppressed by an þ 1 intermediate-scale dynamics. While such a large scale can freedom. If the SM sector couples to one end of (N ) be induced by exotic heavy quarks [8,9] or tiny coupling axions (clockwork gears), interactions of the massless with Higgs doublets [10,11], the origin of the hierarchical mode are exponentially suppressed compared to those f structure of new physics still remains unanswered. from the tangible symmetry breaking scale . Therefore, one can identify the massless degree with the QCD axion, and it provides a neat explanation why the axion decay constant is much larger than the electroweak scale. In this *[email protected] case, the massless degree becomes a good candidate of dark † [email protected] matter as the usual QCD axion, while the massive degrees quickly decay into visible particles in that they have Published by the American Physical Society under the terms of nonsuppressed couplings with the visible sector. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Intriguing phenomena in the dark sector (here axion the author(s) and the published article’s title, journal citation, sector) arise if one considers a supersymmetric model of the and DOI. Funded by SCOAP3. clockwork axion. Supersymmetry (SUSY) itself is also an

2470-0010=2020=102(1)=015011(10) 015011-1 Published by the American Physical Society KYU JUNG BAE and SANG HUI IM PHYS. REV. D 102, 015011 (2020) elegant solution to the gauge hierarchy problem, which is Let us consider N þ 1 pNGBs originating from a broken another fine-tuning problem in the SM. All pseudo-Nambu- global Uð1ÞNþ1 symmetry. Below the energy scale f, þ1 Goldstone bosons (pNGBs)1 corresponding to Uð1ÞN where all N þ 1 Uð1Þ symmetries are broken, Goldstone accompany their fermion partners, which we call axinos in fields are expressed by this context. The supersymmetry dictates the same clock- pffiffi work pattern to axinos and leads to clockwork fermions. iϕj=ð 2fÞ Uj ¼ fe : ð1Þ There are more interesting phenomena in the clockwork axinos. The R parity, if it is preserved, prevents the heavy The Lagrangian is given by axinos from decaying into only the SM particles. For example, if all the SUSY partners in the SM sector are XN XN−1 heavy and only the axinos are R-parity-odd particles near or L ¼ 2 ∂ ∂μ þ 2 2 ð † q þ Þþ f μUj Uj m f Uj Ujþ1 H:c: below the electroweak scale, axinos can decay only into j¼0 j¼0 another axinos with axions. It leads inter-dark-sector 1 XN transitions, which make all the axino states produced from ¼ ∂ ϕ ∂μϕ − ðϕ Þ ð Þ 2 μ j j V j ; 2 thermal bath contribute to dark matter number density. j¼0 In this paper, we consider a simple model of the supersymmetric clockwork axion, which consists of where the ellipsis denotes higher-order terms. The potential (N þ 1) chiral superfields containing axions, axinos, and of ϕ fields are given up to the quadratic order by also saxions (scalar partners of axions). In the SUSY preserving limit, all three components have the same XN−1 pffiffi 2 2 −iðϕj−qϕjþ1Þ= 2f clockwork structure for masses and couplings. Once the VðϕjÞ¼−m f e þ H:c: SUSY is broken, all three components receive SUSY j¼0 breaking masses, and, thus, masses of saxions and axinos 1 XN−1 deviate from the axion masses, while the couplings remain ¼ 2 ðϕ − ϕ Þ2 þ 2 m j q jþ1 the same clockwork structure. In a mass spectrum in which j¼0 the axinos are much lighter than the saxions and axions 1 XN (except the zero mode axion), the axinos are dominantly ¼ 2 M ϕ ϕ þ ð Þ 2 m CWij i j ; 3 produced via the gluon scattering mediated by gluinos. The i;j¼0 heavy axinos eventually decay into the lightest axino, M which is the dark matter in this model. Furthermore, due where a matrix CW which we call here the clockwork to the clockwork structure, the axino DM number density is matrix is given by determined by much more enhanced strengths than its 0 1 actual interactions with the SM sector but is independent of 1 −q 0 0 details of the clockwork gears (clockwork parameter and B C B − 1 þ 2 − 0 C number of gears). B q q q C B 2 C This paper is organized as follows. In Sec. II, we briefly B 0 −q 1 þ q 0 C M ¼ B C review a clockwork axion model to show essential elements CW B . . . . . C: B . . . . . C of the theory. In Sec. III, we consider a SUSYextension and B . . . . . C B C the mass spectrum for axions, saxions, and axinos. In @ 1 þ q2 −q A Sec. IV, we present a complete list of processes for axino 00 0 − 2 production and the axino abundance in a simple spectrum. qq In Sec. V, we discuss some cosmological issues related to ð4Þ the model. In Sec. VI, we conclude this paper. The matrix is real and symmetric and, thus, is diagonalized II. REVIEW OF CLOCKWORK AXION by an orthogonal matrix O. Hence, the mass eigenstate aj In this section, we briefly review a clockwork axion satisfies the relation model to elucidate essential features of the clockwork theory. In the next section, we will supersymmetrize the ϕj ¼ Ojkak ð5Þ clockwork axion and see what appears in the model. We follow a simple formulation shown in Refs. [15,16], but the with mass eigenvalues given by basic structure is the same as another formulations in OTM O ¼ ðλ … λ Þ ð Þ Refs. [13,14,17]. CW diag 0; ; k : 6

1The zero mode also becomes a pseudo-Nambu-Goldstone The eigenvalues and mixing matrix components are given, boson once one introduces the interaction with the QCD. respectively, by

015011-2 SUPERSYMMETRIC CLOCKWORK AXION MODEL AND AXINO … PHYS. REV. D 102, 015011 (2020) π α 2 k where em is the fine structure constant and Caγγ is a constant λ0 ¼ 0; λk ¼ q þ 1 − 2q cos ; ð7Þ N þ 1 determined by CaYY and chiral symmetry breaking effect ≃−1 92 (e.g., Caγγ . for Kim-Shifman-Vainshtein-Zakharov N 0 jkπ ðj þ 1Þkπ (KSVZ) model [18]). These states decay before the big bang O 0 ¼ ; O ¼ N q sin − sin ; ¼ 10 ¼ 1 j qj jk k N þ 1 N þ 1 nucleosynthesis for f TeV and m GeV. In most cases, therefore, the massive states do not make significant ¼ 0 … ; ¼ 1 … ð Þ for j ; ;N k ; ;N; 8 impacts on the evolution of the universe. where III. A SUPERSYMMETRIC EXTENSION sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 − 1 2 In this section, we consider a SUSY extension of the N ¼ q N ¼ ð Þ 0 2 −2N; k : 9 clockwork axion model. q − q ðN þ 1Þλk A. A model The axion masses are thus given by m2 ¼ m2λ . One can aj j Similar to a simple construction in Ref. [15], one can see that one degree remains massless, and it corresponds to consider a Kähler potential and a superpotential the Uð1Þ not broken by mass terms in Eq. (2). Suppose that the Nth field couples to the SM sector via XN topological terms, i.e., ¼ ð † þ † þ † Þ ð Þ K Xj Xj Yj Yj Zj Zj ; 13 j¼0 g2 g2C ϕ L ¼ s b ˜ bμν þ 1 aYY ˜ μν N ð Þ 2 GμνG 2 BμνB ; 10 XN 32π 16π f 2 W ¼ κZjðXjYj − v Þ ð3Þ ð1Þ j¼0 where gs and g1 are SU c and U Y gauge coupling b ˜ b ˜ XN−1 constants, Gμν, Bμν, Gμν, and Bμν are corresponding gauge 1 þ ð q þ 0 q Þ ð Þ q−1 mXjYjþ1 m YjXjþ1 ; 14 field strengths and their duals, respectively, and CaYY is a v j¼0 model-dependent constant of order unity. After clockworking, the above terms lead to interactions between all axions respectively, where charge assignment of Zj, Xj, and Yj and the SM gauge bosons: ð1Þ ð0 þ1 −1Þ under U j is ; ; . The first term reflects the spontaneous breaking of Uð1Þ global symmetry near v, g2 g2C L ¼ s b ˜ bμν þ 1 aYY ˜ μν 2 GμνG 2 BμνB while the second term corresponds to a small explicit 32π 16π breaking effect for m, m0 ≪ v. We consider a generic case 0 1 N XN π for m ≠ m leading to hXji ≠ hYji, which is important for 0 k k × a0 − ð−1Þ N q sin a : ð11Þ f qN k N þ 1 k inter-dark-sector couplings in Eq. (42). The fields are k¼1 stabilized at One can easily see that the coupling of the zero mode axion þ 1pffiffiffiffiffiffiffiffiffi h i¼−q 0 h i¼ h i¼ ð Þ is exponentially suppressed compared to that from the Zj κ mm ; Xj x; Yj y; 15 actual symmetry breaking scale f while the others are scaled by only 1=N3=2 for large N.Forq ¼ 2 and N ¼ 20, where2 the exponential factor is around 106, so one can achieve a 1=½2ðq−1Þ good QCD axion even from f ¼ 1 TeV. 2 m xy ¼ v ;x¼ v: ð16Þ If the zero mode is the QCD axion, it finally becomes m0 massive by the chiral symmetry breaking in the strong sector of the SM, but the mass is still tiny. As is well known, Below the spontaneous Uð1Þ symmetry breaking scale, this the QCD axion has very long lifetime, so it could be a dark theory can be described by chiral superfields containing matter component. On the other hand, massive states are pNGBs: rather strongly coupled to the SM sector. One can obtain 1 pffiffiffi decay widths of the massive modes to the photon pair as Φ ¼ pffiffiffi ðσ þ iϕ Þþ 2θψ þ θ2F ; ð17Þ j 2 j j j j 2 2 3 C γγα kπ m Γ ¼ a em N 2 2 2 ak a →γγ 3 q sin 2 k 256π k N þ 1 f 2 Here we can take a field basis where all parameters are taken 20 3 10 TeV 2 m 3 to be real and positive except κ. In this basis, the supersymmetric ∼ ð10−7 sÞ−1 ; ð12Þ effective action for the axion supermultiplets does not involve any N f GeV complex parameter as we will see below.

015011-3 KYU JUNG BAE and SANG HUI IM PHYS. REV. D 102, 015011 (2020) where σj and ψ j are scalar and fermion partners, respec- B. SUSY breaking effects and mass spectrum ϕ tively, of j. One can write Once the SUSY is broken, the mass spectrum for each component alters. The pNGBs and scalar partners would Φ −Φ ¼ j=v0 ¼ j=v0 ð Þ Xj xe ;Yj ye ; 18 receive mass contributions from SUSY breaking in the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi superpotential as 2 2 where v0 ¼ x þ y . The effective Kähler potential and Z superpotential become 2 2 L ¼ dθ ð1 þ msθ ÞW þ H:c: XN Φ þΦ† Φ þΦ† → ¼ − j j 2 ¼ 2 j j þξ j j ð Þ V mΦ ms v0 Keff v0 cosh sinh ; 19 −1 j¼0 v0 v0 XN pffiffi ϕ − ϕ ðσ − σ Þ 2 j q jþ1 j q jþ1 = v0 pffiffiffi × e cos þ δs 2v0 XN−1 j¼0 Φ − qΦ þ1 ¼ 2 j j ð Þ pffiffi ϕ − ϕ Weff mΦv0 cosh ; 20 −ðσ − σ Þ 2 j q jþ1 j q jþ1 = v0 pffiffiffi j¼0 v0 þ e cos − δs ; ð25Þ 2v0 2 2 2 respectively, where ξ ¼ðx − y Þ=v0 and where δs is the complex phase of ms. For simplicity, we will pffiffiffiffiffiffiffiffiffi 2 focus on parameter space where vacuum field configuration 0 v hΦ − mΦ ≡ 2 mm : ð21Þ is close to the supersymmetric minimum point j v0 qΦjþ1i¼0. Near the point, the above potential becomes approximately In the Kähler potential, we have omitted Z†Z since it is irrelevant in the low-energy dynamics. The above super- XN−1 σ − σ 2 j pffiffiffiq jþ1 potential shows that the supersymmetric minimum is Vσ ≃−2mΦjmsjv0 cos δs cosh ; ð26Þ 2v0 achieved for hΦj − qΦjþ1i¼0 and the supersymmetric j¼0 mass term indeed has the clockwork structure proportional XN−1 ϕ − ϕ to an overall mass scale mΦ. One can obtain superfields in 2 j q jþ1 Vϕ ≃−2mΦjm jv cos δ cos pffiffiffi ð27Þ the eigenbasis with mixing matrix in Eq. (8): s 0 s 2 j¼0 v0 Φ ¼ O ð Þ i ijAj: 22 along the scalar and pNGB directions, respectively. It contributes to squared masses with the clockwork structure Hence, one supermultiplet remains massless after clock- for the pNGBs and their scalar partners. The mass scale for working. this contribution is determined by Similarly to the clockwork axion model, one can introduce couplings of the Nth superfield to the SM gauge 2 m ≡ mΦjm j cos δ : ð28Þ fields as sb s s Z If SUSY breaking effects also arise in the Kähler potential g2 C L ¼ − s aGG 2θΦ WbαWb þ in Eq. (19), scalars and fermions acquire additional masses 2 d N α H:c: 32π v0 which are diagonal in the basis of chiral superfields. We 2 Z K K write mσ and mψ , respectively, for the scalars and fermions. − g1 CaYY 2θΦ WαW þ ð Þ 2 d N α H:c:; 23 ’ 16π v0 We further assume these terms are the same for all j s, and, thus, the mass matrices from this contribution are propor- where Wb is the gluon superfield, W is the hypercharge tional to the identity matrix. While it is expected to have K K K K mσ ∼ mψ mσ ≫ mψ superfield, and CaGG and CaYY are model-dependent in generic cases, it is possible to have 3 coefficients of the order of unity. After clockworking, in some cases. the zero mode superfield has exponentially suppressed Mass spectra for the pNGBs, scalars, and fermions are interactions as summarized, respectively, as Z 2 M2 ¼ 2 M2 þ 2 M ð Þ g C ϕ mΦ m CW; 29 L ¼ − s aGG 2θ WbαWb þ CW sb 2 d A0 α H:c: 32π f0 2 Z − g1 CaYY 2θ WαW þ ð Þ 2 d A0 α H:c:; 24 16π f0 3We refer readers to Refs. [19–22] for a general discussion for the mass generation and Refs. [23,24] for explicit models with N K K where f0 ¼ q v0. mσ ≫ mψ .

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M2 ¼ 2 M2 − 2 M þð KÞ2I ð Þ σ mΦ CW msb CW mσ ; 30 axinos is inverted. The ordering may be even not mono- K tonic if jmψ j

ma˜ N in a later discussion. The ðN þ 1Þ × ðN þ 1Þ identity matrix is denoted by I.We Some comments are in order about conditions to get the – emphasize that all the mass matrices are diagonalized by clockwork mixing pattern in Eqs. (32) (34), which is the same mixing matrix in Eq. (8). Hence, we write mass crucial for exponential coupling hierarchy. In the limit of 0 → 0 ð1ÞNþ1 eigenstates m; m , the global U symmetry is preserved, and, thus, there exist N þ 1 chiral superfields, Φj, corre- þ 1 ¼ 2 ϕj ¼ Ojkak; ð32Þ sponding to N flat directions, XjYj v . Once m and m0 are turned on, the global Uð1ÞNþ1 symmetry is broken ð1Þ ð1Þ σ ¼ O s ; ð33Þ down to U . The remaining U symmetry leaves one j jk k flat direction, while the others become massive. It can be explicitly seen by the fact that the superpotential does not ψ ¼ O ˜ ð Þ j jkak; 34 change under with mass eigenvalues −j Φj → Φj þ q α ð39Þ 2 2 2 2 ma ¼ mΦλ þ m λk; ð35Þ k k sb with a constant α. This ensures the superfield correspond-

2 2 2 2 K 2 ing to the remaining flat direction to have exponentially m ¼ mΦλ − m λ þðmσ Þ ; ð36Þ sk k sb k small couplings. The SUSY breaking in the superpotential (25) also respects it, so the flat direction remains. On the ¼ λ þ K ð Þ ma˜ k mΦ k mψ ; 37 other hand, the SUSY breaking in the Kähler potential develops masses of the scalars and fermions, while the and call these states axions, saxions, and axinos, respec- masses do not respect the above symmetry. This means tively. While the zero mode axion a0 is massless in that the that, except the axion, the saxion and axino may not get mass term is determined only by λ0, both s0 and a˜ 0 become small couplings if the SUSY breaking effect in the Kähler massive due to the SUSY breaking effect in the Kähler potential is significant. More quantitatively, those SUSY 2 2 ð KÞ potential. While mΦ is always positive by definition, msb > breaking contributions for their mass matrices mσ ij and − 2 ð − 1Þ2 ð KÞ mΦ q is required not to destabilize axion directions. mψ ij have to be sufficiently small compared to mΦ or msb δ 2 ≃ Once this condition is satisfied, the mass difference mak or closely proportional to the identity matrix as in Eqs. (30) 2 − 2 makþ1 mak is given by and (31) in order to preserve the clockwork coupling hierarchy. The hierarchy would be spoiled if departure from being proportional to the identity matrix is of the order 2 2 ðk þ 1Þπ δma > 2qmΦ λkþ1 1 − cos of mΦ or m . This argument is valid even when the k N þ 1 sb supersymmetric parameters κ, v, m, and m0 in (14) and the kπ − λ 1 − cos : ð38Þ SUSY breaking parameter ms in (25) are dependent on sites k N þ 1 j. Such dependency makes a difference only on mass eigenvalues in Eqs. (35)–(37) without qualitatively chang-

Since λkþ1 > λk and the cosine is monotonically decreas- ing our results. δ 2 Let us finally make a remark for a benchmark spectrum. ing, mak is always positive. Thus, the ordering of axion mass eigenvalues is the same as that in Eq. (6), although If we want to identify the zero mode axion a0 as a QCD mass differences alter. On the other hand, the ordering of axion with an intermediate scale decay constant, v0 can be ð1Þ ≲ 20 eigenvalues can be different for the saxions and axinos. If as low as O TeV for N . Effective descriptions in 0 ≪ 2 ≫ 2 j j δ ≫ λ Eqs. (19), (20), and (25) are valid only for m, m ,ms v0. msb mΦ (i.e., ms cos s mΦ), the k-dependent part becomes negative so as to destabilize the supersymmetric Hence, all states are expected to be near or below the K 2 weak scale. vacuum. Yet, if ðmσ Þ is large enough, the supersymmetric vacuum can be maintained. In this case, the largest 2 2 C. Interactions eigenvalue is ms0 , while the smallest one is msN . The mass ordering of the saxions is inverted when being compared to The axions have the same interactions as in the case of that of the axions. The same thing happens for the axinos. If the non-SUSY model in Eq. (11). The saxions also have K mψ < 0, a˜ 0 may not be the lightest mode. In the case similar interactions from the SUSY coupling term in K K jmψ j >mΦλN with negative mψ , the mass ordering of the Eq. (23). The saxion-gauge boson interactions are given by

015011-5 KYU JUNG BAE and SANG HUI IM PHYS. REV. D 102, 015011 (2020) 2 2 gsCaGG b bμν g1CaYY μν The axino production consists of the following channels: L ¼ GμνG þ BμνB sax 32π2 16π2 (i) gluino-mediated process, (ii) saxion- or axion-mediated process, and (iii) production from saxion or axino decay. In 1 N XN π 0 k k particular, we will consider a relatively low reheat temper- ×pffiffiffi s0 − ð−1Þ N qsin s : ð40Þ 2 N k þ 1 k v0 q k¼1 N ature TR below the SUSY breaking scale so that axino production is mainly from the SM thermal bath. The reason We neglect axion-gluino, saxion-gluino, and saxion-squark is that the thermal yield of the lightest axino can easily interactions derived from Eq. (23), since they are irrelevant saturate the DM abundance enhanced by a certain power of in the later discussion. The axino interactions are derived in the clockwork factor qN compared to the conventional the same way: scenarios as we will see. 1 N XN π 0 ¯ k k ¯ L ¼ pffiffiffi a˜ 0 − ð−1Þ N qsin a˜ A. Gluino-mediated process axn N k þ 1 k 2v0 q ¼1 N k From the interactions with gauge bosons in Eq. (41), g2C g2C axinos can be produced from the thermal plasma. If the s aGG b σμνγ5 ˜b þ 1 aYY σμνγ5 ˜ ð Þ × 32π2 Gμν g 16π2 Bμν B ; 41 temperature is larger than masses of the SUSY particles in the SM sector, the single-axino production is the dominant μν i μ ν where σ ≡ 2 ½γ ; γ . The gluino and bino are denoted by g˜ process which includes the other SUSY particles in either and B˜ , respectively. It is noteworthy that we use Majorana the initial or final state. This scenario has been intensively – spinors for axinos and gauginos in Eq. (41) and the later studied both for the KSVZ-type model [25 28] and for the discussion. Dine-Fischler-Srednicki-Zhitnitsky (DFSZ)-type model In addition, the Kähler potential in Eq. (19) generates [29–31]. If the temperature is smaller than masses of the qubic (and also higher-order) interactions between the SUSY particles in the SM sector but still larger than the ≪ ≪ ∼ axions, saxions, and axinos: axino mass, e.g., ma˜ T mg˜ mq˜ , the single-axino production is Boltzmann suppressed. Instead, the axino pair ξ XN Φ þ Φ† 3 production becomes more important [32]. By integrating ⊃ 2 j j K 3! v0 out the gluino field in Eq. (41), one can obtain an effective ¼0 v0 j Lagrangian for the axino pair production, i.e., gg → a˜ na˜ m: ξ XN → L ¼ pffiffiffi O O O α2C2 nml 2 jn jm jl L ¼ − s aGG O O v0 j gga˜ a˜ 2 2 Nn Nm 1024π v0mg˜ μ μ × ½snð∂μamÞð∂ alÞþsnð∂μsmÞð∂ slÞ ¯ μ ν ρ σ b b × a˜ n½γ ; γ ½γ ; γ a˜ mGμνGρσ: ð43Þ ¯ μ ¯ 5 μ þ isna˜ mγ ∂μa˜ l − ð∂μanÞa˜ mγ γ a˜ l: ð42Þ The squared amplitude for this process is given by From this Lagrangian, one can easily read off all trilinear α4 4 interactions which mediates inter-dark-sector transitions. g˜ 2 sC 2 3 2 jM j ¼ aGG jO O j s ð1 þ θÞ ; ð Þ ¼ 0 ’ nm 16π4 4 2 Nn Nm cos 44 Here we assume Fj for all j s. v0mg˜

IV. THERMAL PRODUCTION OF AXINOS where s is the square of the center of mass energy and θ is the angle between the incoming gluon and outgoing axino. In this section, we discuss thermal production of axinos Here we have summed over all possible degrees of freedom in the early Universe. Since the whole dark sector (i.e., for both the initial and final states. axion supermultiplets) communicates with the SM sector via the interactions in Eq. (23) and clockworking, all the axions, saxions, and axinos are produced from thermal B. Saxion- or axion-mediated process plasma after the primordial inflation. In a SUSY extension, Another channel for the axino pair production is realized the axinos are odd, while the saxions and axions are even by the saxion- or axion-mediated processes. The inter- under the R parity if it is preserved. Therefore, the lightest actions in Eqs. (40) and (42) lead to a scattering process → ð Þ → ˜ ˜ axino can be a dark matter candidate if it is the lightest R- gg sl or al anam, and its squared amplitude is parity-odd particle. The saxions and axions except a0, given by however, would normally disappear by decaying into another light species such as gluons and photons. In this ξ2α2 2 X 1 2 jMs=aj2 ¼ sCaGG O O O O respect, axino production is more prominent than the nm 2π2 4 l;j Nl jl jn jm − 2 v0 s ml others for dark matter physics. We focus on how axinos 2 3 × ðm˜ þ m˜ Þ s ; ð45Þ are produced. an am

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4 where m is a mass of s or a .Ifs ≫ m2, the squared α2C2 m l l l l R ∼ s aGG s=a ð52Þ amplitude is further simplified, so one can find 8π2ξ2 2ð þ Þ2 mg˜ ma˜ n ma˜ m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 s=a 2 ξ αsC 2 2 for s ≪ m . Thus, for m ≫ m˜ ðm˜ þ m˜ Þ, the jM j ≃ aGG jO O j ð þ Þ ð Þ s=a s=a g an am nm 2 4 Nn Nm ma˜ n ma˜ m s; 46 2π v0 gluino-mediated process dominates over the saxion- or axion-mediated process if the reheat temperature TR is where we have used an identity smaller than mg˜ . In this respect, we consider a simple particle mass spectrum with m˜ ≪ m ≪ m˜ and X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi an s=a g O O O O ¼ O O ð Þ ≫ ð þ Þ Nl jl jn jm Nn Nm: 47 ms=a mg˜ ma˜ n ma˜ m . In this spectrum, moreover, l;j the branching fraction of s=a → a˜ na˜ m is highly suppressed by small axino masses compared to saxion and axion ≪ 2 If s ml , the squared amplitude is approximately given by masses. Because of the supersymmetry, saxions, axions and axinos are produced with a similar amount in the large TR ξ2α2 2 limit, so the amount of axinos from saxion and axion s=a 2 s CaGG 2 2 3 jMnm j ≃ jO O j ðm˜ þ m˜ Þ s ; ð48Þ 2π2v4m4 Nn Nm an am decays is negligible in this case. Hence, axinos are 0 s=a predominantly produced in pairs via the gluino-dominated process. We call this a “secluded” spectrum. where we have assumed ml ∼ ms=a for all l; i.e., all masses are of the same order. In this argument, we have also E. Thermal yield of axinos neglected the zero mode axion contribution, since its coupling is exponentially suppressed. One can obtain the thermal-averaged axino production cross section from the squared amplitude. For a˜ na˜ m pair production, the thermal-averaged cross section is given by C. Production from saxion and axion decay Because of the interactions in Eq. (42), saxions and 6α4C4 T4 axions can decay into axino pairs. One can easily find their hσvi ≃ s aGG jO O j2Δ ; ð53Þ nm π5½ζð3Þ2v4m2 Nn Nm nm partial decay widths: 0 g˜

2 where T is the plasma temperature and ζ is the zeta ξ ml 2 Γðs =a → a˜ a˜ Þ¼ ðm˜ þ m˜ Þ ˜ ≡ l l n m 2 an am function. The yield of the an state, Ya˜ n na˜ n =s (na˜ n , 16πv0 number density of a˜ n; s, entropy density) is then given by X 2

× O O O Δ ; ð49Þ pffiffiffiffiffi jl jm jn nm 3 10 243α4 4 5 j sCaGGMPTR 2 Y ˜ ≃ jO j ; ð54Þ an ½ ð Þ3=2 16π12 4 2 Nn g TR v0mg˜ where Δnm ¼ 1 (1=2) for n ≠ m (n ¼ m). Meanwhile, saxions and axions can also decay into gluon pairs with where gðTRÞ is the effective degrees of freedom at TR and the partial decay widths MP is the reduced Planck mass. Here we have used an identity α2C2 m3 Γð → Þ¼ s aGG l jO j2 ð Þ sl=al gg 3 2 Nl : 50 X 64π v0 2 jONmj ¼ 1: ð55Þ m þ ≪ For ma˜ n ma˜ m ml, saxions and axions decay dominantly into gluons. It is noteworthy that we have included the correction from the continuous reheating process [33]. D. Secluded spectrum In the secluded spectrum, the heavier axinos eventually decay into the lightest axino, so the final yield of axino dark Comparing the gluino-mediated and saxion- or axion- matter is determined by the sum of all the axino yields: mediated processes, the relative ratio between squared amplitudes is given by pffiffiffiffiffi X 3 10 243α4 4 5 DM sCaGGMPTR Y ¼ Y ˜ ≃ ; ð56Þ g˜ 2 2 2 2 a˜ an 3=2 12 4 2 jM j α C s ½gðT Þ 16π v0m˜ R ≡ nm ∼ s aGG ð51Þ n R g jMs=aj2 8π2ξ2 2ð þ Þ2 nm mg˜ ma˜ n ma˜ m where we have used the identity in Eq. (55). The axino DM ≫ 2 for s ms=a or abundance is thus given by

015011-7 KYU JUNG BAE and SANG HUI IM PHYS. REV. D 102, 015011 (2020) 2 5 DM ma˜ due to the suppressed interactions between the axion and Ω˜ h ≃ 2.8 × 10 × Y a a˜ MeV clockwork gears. Yet relativistic axions produced from the clockwork gear domain wall contribute to dark radiation at C 4 TeV 4 10 TeV 2 ≃ 0.13 × aGG the recombination epoch as 1 v m˜ 0 g 2 2 5 vω mΦ v0 TR ma˜ Δ ≃ 0 1 × ; ð57Þ Neff . 6 40 10 1 10 TeV 10 GeV GeV keV ð Þ −4=3 −2 gS Ta0 Ta0 where we have used α ≃ 0.1 and m˜ denotes the lightest × ; ð59Þ s a 20 0.2 GeV axino mass. In the normal hierarchy, a˜ 0 is the lightest axino state and, where T is the temperature at which the axion a0 gets a thus, dark matter. Its interaction to the SM sector is highly a0 mass and vω ≤ 1 parametrizes the spectrum of small-scale suppressed by 1=qN, so most of the DM axinos are perturbations on the domain wall. For T , we use the value produced via decays of the heavier axinos which have a0 ∼1 3=2 of the QCD axion as the normalization. Observations of the interactions being mildly scaled by =N . In other Δ ≲ 0 1 comic microwave background require Neff . [37]. words, the clockwork mechanism realizes largely enhanced 2 Thus, it sets an upper bound on the quantity mΦv for a axino production in spite of the feebly interacting nature of 0 given T . In fact, this quantity corresponds to the domain DM species. Compared to the conventional nonclockwork a0 scenarios of the same axino coupling to the SM, the DM wall tension. On the other hand, the violent annihilation of 4 4N the clockwork domain walls gives rise to gravitational abundance is enhanced by the factor ðf0=v0Þ ¼ q . waves of frequencies of the order of the Hubble parameter. It turns out that we have a similar observational constraint V. COSMOLOGICAL ISSUES on the domain wall tension [36]. Using the estimation of A. Heavy axino decays Ref. [36], to be consistent with pulsar timing observations [38–42], our model parameters need to satisfy As discussed in Sec. IV, most of the DM axinos are produced via decays of the heavier axinos. In the secluded 2 mΦ v0 spectrum, an axino can decay into a lighter axino plus the 6 ˜ → ˜ þ 10 TeV 10 GeV zero mode axion, i.e., an am a0, n>m, due to the interaction in Eq. (42). The decay width is given by ϵ −2=11 Ω95 h2 4=33 ≲ 0.1 gw gw 0.7 2.3 × 10−10 1 ξ2 XN 2 −4=11 ð Þ 1=66 28=11 Γða˜ → a˜ þ a0Þ¼ O 0O O N g Ta0 Ta0 n m 16π v2 j jn jm × ; ð60Þ 0 j¼0 10 20 0.2 GeV 2 3 m˜ × m3 1 − am : ð58Þ where ϵ ≃ 0.7 0.4 is an efficiency parameter of the a˜ n 2 gw m˜ 95 an Ω gravitational wave emission [43] and gw is the current ν ≃ 3 10−8 While the DM axino yield is independent of the decay 95% confidence upper limit at 1 yr × Hz [40]. path, the phase space distribution of the DM axinos is These considerations imply that a small axino coupling ∼1 highly dependent on the decay path, lifetimes, and mass ( =v0) requires correspondingly light axions to be com- differences. Depending on the model parameters N, q, mΦ, patible with the observational data. In our benchmark K and mψ , the resulting phase space distribution can deviate parameter choice for the secluded spectrum and thermal from the conventional thermal distribution. Hence, it may yield of axinos, those constraints are safely satisfied. impact on the structure formation [34]. VI. CONCLUSIONS B. Axion string-wall network In this paper, we have studied implications of super- Since the clockwork axions and saxions have short symmetrizing the clockwork axion model. By supersym- lifetimes, their cosmological population from initial mis- metry, the superpartner axinos have the same clockwork alignment quickly decays without leaving substantial pattern with respect to the coupling hierarchy. The coupling impacts. However, a network of axion strings and domain hierarchy is not spoiled by SUSY breaking if the SUSY walls formed by the global Uð1ÞNþ1 symmetry breaking breaking is universal over the clockwork sites. Even for can sizably contribute to the dark radiation [35] and yield nonuniversal SUSY breaking, the coupling hierarchy is observable gravitational waves [36]. In Ref. [35],itis approximately maintained when the SUSY breaking scale argued that the axion DM production from collapse of the is sufficiently smaller than the clockwork mass scale. In the string-wall network of the clockwork gears is negligible universal SUSY breaking case, we find that the clockwork

015011-8 SUPERSYMMETRIC CLOCKWORK AXION MODEL AND AXINO … PHYS. REV. D 102, 015011 (2020) axino mass spectrum can be inverted in ordering when the formation [34]. Finally, the string-wall network from the SUSY breaking mass is larger than the clockwork mass superpartner clockwork axions has interesting cosmologi- scale. The same happens to the saxion sector. In this work, cal consequences on dark radiation and gravitational we have focused on the normal ordering, because it may waves, imposing an upper bound on the axino coupling have interesting consequences for axino dark matter. Under for a given clockwork mass scale. It may be interesting to the assumption that axinos are mainly produced from the examine further cosmological and collider consequences SM thermal bath, we find that the thermal yield of the for supersymmetric clockwork models. lightest axino is exponentially enhanced compared to the nonclockwork axino case with the same coupling to ACKNOWLEDGMENTS the SM. This is because the lightest axino production is dominated by the decay of heavy axinos which interact We thank Jeff Kost and Chang Sub Shin for with the SM thermal bath with exponentially larger helpful discussions. The work of K. J. B. was supported coupling. Thus, the relevant parameter space for axino by the National Research Foundation of Korea (NRF) dark matter is significantly different from the conventional grant funded by the Korean government (NRF- nonclockwork axino scenarios. It generally requires a lower 2020R1C1C1012452). S. H. I. acknowledges support from reheating temperature than the conventional scenarios for Basic Science Research Program through the National the same mass of axino dark matter. Furthermore, we Research Foundation of Korea (NRF) funded by the expect that the phase space distribution of the axino dark Ministry of Education (2019R1I1A1A01060680). S. H. I. matter is highly dependent on the detailed clockwork was also supported by IBS under the project code IBS- structure, which may have implications for the structure R018-D1.

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