CTPU-PTC-20-08 Supersymmetric Clockwork Axion Model and Axino Dark Matter 1, 2, Kyu Jung Bae ∗ and Sang Hui Im † 1Department of Physics, Kyungpook National University, Daegu 41566, Korea 2Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Daejeon, 34126, Korea 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. I. INTRODUCTION is able to construct an intermediate scale (& 109 GeV) axion decay constant from dynamics near the electroweak One of the strongest beliefs in particle physics is that scale [17]. N+1 there exist extended sectors of new physics beyond the In the case of the clockwork axion, a global U(1) standard model (SM). In theoretical aspects, it is invoked symmetry spontaneously breaks at scale f and conse- to resolve fine-tuning problems residing in the SM. In quently results in (N + 1) Goldstone bosons. The global practical aspects, the SM does not contain physics for symmetry is explicitly but softly broken by N mass terms essential phenomena such as neutrino oscillation, matter- with clockwork structure. This specific structure leaves antimatter asymmetry, and dark matter (DM). A widely unbroken U(1) and a corresponding massless degree of accepted notion of extensions of the SM is to introduce freedom. If the SM sector couples to one end of (N + 1) ‘dark’ sectors which communicate with the SM via fee- axions (clockwork gears), interactions of the massless ble interactions leading to rational explanations to those mode are exponentially suppressed compared to those phenomena. from the tangible symmetry breaking scale f. Therefore, A prominent fine-tuning problem in the SM is the one can identify the massless degree with the QCD axion strong CP problem. It can be solved by introducing a and it provides a neat explanation why the axion decay spontaneously broken Peccei-Quinn symmetry [1] which constant is much larger than the electroweak scale. In involves the QCD axion [2, 3]. The axion couples to this case, the massless degree becomes a good candidate the gluon field strength and dynamically relaxes the of dark matter as the usual QCD axion while the massive QCD θ-term to zero. Astrophysical observations con- degrees quickly decay into visible particles in that they strain axion-gauge boson couplings (including the axion- have non-suppressed couplings with the visible sector. gluon coupling) [4–7] so that the axion couplings are re- Intriguing phenomena in the dark sector (here axion quired to be suppressed by an intermediate scale dynam- sector) arise if one considers a supersymmetric model ics. While such large scale can be induced by exotic heavy of the clockwork axion. Supersymmetry (SUSY) itself arXiv:2004.05354v2 [hep-ph] 20 Jul 2020 quarks [8, 9] or tiny coupling with Higgs doublets [10, 11], is also an elegant solution to the gauge hierarchy prob- the origin of the hierarchical structure of new physics still lem which is another fine-tuning problem in the SM. All 1 remains unanswered. pseudo-Nambu-Goldstone bosons (pNGBs) correspond- N+1 The clockwork theory presents a plausible mechanism ing to U(1) accompany their fermion partners, which to build hierarchical mass spectra and interactions from we call axinos in this context. The supersymmetry dic- a series of multiple non-hierarchical ones. An early form tates the same clockwork pattern to axinos and leads to of the clockwork structure was studied to achieve a trans- clockwork fermions. There are more interesting phenom- Planckian field excursion from two sub-Planckian fields in ena in the clockwork axinos. The R-parity, if it is pre- a natural inflation [12]. In further studies, it was shown served, prevents the heavy axinos from decaying into only that a number of axions with similar decay constants can the SM particles. For example, if all the SUSY partners produce an exponentially large effective scale [13–15]. It in the SM sector are heavy and only the axinos are R- has been argued that the same mechanism is applicable parity odd particles near or below the electroweak scale, for more general systems with various spins, scales and axinos can decay only into another axinos with axions. couplings [16]. In particular, the clockwork mechanism It leads inter-dark-sector transitions, which make all the ∗ [email protected] 1 The zero mode also becomes a pseudo-Nambu-Goldstone boson † [email protected] once one introduces the interaction with the QCD. 2 axino states produced from thermal bath contribute to by dark matter number density. N 1 In this paper, we consider a simple model of the super- − 2 2 i(φj qφj )/√2f V (φ )= m f e− − +1 + h.c. symmetric clockwork axion, which consists of (N +1) chi- j − j=0 ral superfields containing axions, axinos and also saxions X N 1 (scalar partners of axions). In the SUSY preserving limit, 1 − = m2 (φ qφ )2 + , all three components have the same clockwork structure 2 j − j+1 ··· j=0 for masses and couplings. Once the SUSY is broken, X N all three components receive SUSY breaking masses and 1 thus masses of saxions and axinos deviate from the axion = m2 M φ φ + , (3) 2 CWij i j ··· masses, while the couplings remain the same clockwork i,j=0 X structure. In a mass spectrum in which the axinos are M much lighter than the saxions and axions (except the where a matrix CW which we call here the clockwork zero mode axion), the axinos are domaninatly produced matrix is given by via the gluon scattering mediated by gluinos. The heavy 1 q 0 0 axinos eventually decay into the lightest axino which is − 2 ··· the dark matter in this model. Furthermore, due to the q 1+ q q 0 − − 2 ··· clockwork structure, the axino DM number density is de- 0 q 1+ q 0 MCW = . −. ··· . (4) termined by much more enhanced strengths than its ac- . .. . tual interactions with the SM sector but is independent 2 1+ q q of details of the clockwork gears (clockwork parameter −2 0 0 0 q q and number of gears). ··· − This paper is organized as follows. In Sec. II, we briefly The matrix is real and symmetric, and thus is diagonal- review a clockwork axion model to show essential ele- ized by an orthogonal matrix O. Hence the mass eigen- ments of the theory. In Sec. III, we consider a SUSY states aj satisfies the relation extension and the mass spectrum for axions, saxions and axinos. In Sec. IV, we present a complete list of processes φj = Ojkak (5) for axino production and the axino abundance in a sim- ple spectrum. In Sec. V, we discuss some cosmological with mass eigenvalues given by issues related to the model. In Sec. VI, we conclude this paper. OT M O = diag(λ , , λ ). (6) CW 0 ··· k The eigenvalues and mixing matrix components are given II. REVIEW OF CLOCKWORK AXION by kπ In this section, we briefly review a clockwork axion λ =0, λ = q2 +1 2q cos , (7) 0 k − N +1 model to elucidate essential features of the clockwork theory. In the next section, we will supersymmetrize the 0 jkπ (j + 1)kπ O = Nj , O = q sin sin ,(8) clockwork axion and see what appears in the model. We j0 q jk Nk N +1 − N +1 follow a simple formulation shown in Refs. [15, 16], but for j =0, ,N; k =1, ,N, the basic structure is the same as another formulations ··· ··· in Refs. [13, 14, 17]. where Let us consider N + 1 pNGBs originating from a bro- ken global U(1)N+1 symmetry. Below the energy scale f q2 1 2 where all N +1 U(1) symmetries are broken, Goldstone 0 = 2 − 2N , k = . (9) N sq q− N s(N + 1)λk fields are expressed by − 2 2 √ The axion masses are thus given by m = m λ . One U = feiφj /( 2f). (1) aj j j can see that one degree remains massless and it corre- The Lagrangian is given by sponds to the U(1) not broken by mass terms in Eq. (2). Suppose that the N-th field couples to the SM sector N N 1 2 µ 2 2 − q via topological terms, i.e., =f ∂ U ∂ U + m f U †U + h.c. + L µ j j j j+1 ··· j=0 j=0 2 2 X X gs b bµν g1CaY Y µν φN N = G G˜ + B B˜ , (10) 1 L 32π2 µν 16π2 µν f = ∂ φ ∂µφ V (φ ), (2) 2 µ j j − j j=0 X where gs and g1 are SU(3)c and U(1)Y gauge coupling b ˜b ˜ where the ellipsis denotes higher order terms. The po- constants, Gµν , Bµν , Gµν and Bµν are corresponding tential of φ fields are given up to the quadratic order gauge field strengths and their duals, respectively, and 3 CaY Y is a model-dependent constant of order unity.