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Some enhances matter deca exponentially scenarios. by dark This determined a mostly ordering. is thus mass production and thermal from stable matter saxion saxi be massive coupl the the can the of While ov axino orderings of scale. universal mass breaking pattern is the supersymmetry breaking the same while preserved, supersymmetry the is assume have structure we axinos If and axions. saxions the that .I atclr h lcwr mechanism clockwork the particular, In ]. uesmercCokokAinMdladAioDr Matter Dark Axino and Model Axion Clockwork Supersymmetric mlctoso uesmerzn h lcwr xosar axions clockwork the supersymmetrizing of Implications .INTRODUCTION I. 4 – 7 12 ota h xo opig r re- are couplings axion the that so ] 1 .I ute tde,i a shown was it studies, further In ]. eateto hsc,KugokNtoa nvriy Dae University, National Kyungpook Physics, of Department ag ffciesae[ scale effective large 2 , 3 nttt o ai cec IS,Deen 42,Korea 34126, Daejeon, (IBS), Science Basic for Institute .Teainculsto couples axion The ]. 2 etrfrTertclPyiso h Universe, the of Physics Theoretical for Center y ugBae Jung Kyu 13 1 – which ] 10 15 , .It ]. 11 1, ], ∗ n agHiIm Hui Sang and e hc saohrfietnn rbe nteS.All SM. prob- the hierarchy in (pNGBs) problem gauge bosons fine-tuning the pseudo-Nambu-Goldstone another itself to is solution (SUSY) which model elegant lem Supersymmetry an supersymmetric also a axion. is clockwork considers the one of if arise sector) sector. they visible that the in with particles couplings massive visible non-suppressed the have into while decay axion QCD candidate quickly usual In good degrees the a as scale. matter becomes dark electroweak degree of massless decay the the axion than case, the larger this why much explanation is neat axion constant QCD a the provides with it degree and massless the those identify scale can to breaking one massless compared symmetry tangible the suppressed the from of exponentially are interactions mode gears), (clockwork axions ihcokoksrcue hsseicsrcueleaves structure specific This unbroken structure. clockwork with reo.I h Msco ope ooeedo ( of end one to couples sector SM the If freedom. ymtyi xlctybtsfl rknby broken softly but explicitly is symmetry ( in results quently ymtysotnosybek tscale at breaks spontaneously symmetry sal ocntuta nemdaesae( scale intermediate an construct to able is n ntecokokaio.The axinos. clockwork the to phenom- interesting leads in more and ena are dic- axinos There to supersymmetry fermions. pattern The clockwork clockwork context. same the this tates in axinos call we to ing nteS etraehayadol h xnsare axinos the only and heavy partners SUSY are the sector all SM if the example, For in particles. only into SM decaying the from axinos heavy the prevents served, xnscndcyol noaohraio ihaxions. with axinos another into leads scale, only electroweak It the decay below can or axinos near particles odd parity xo ea osatfo yaisna h electroweak [ the scale near dynamics from constant decay axion 1 neoeitoue h neato ihteQCD. the with boson interaction pseudo-Nambu-Goldstone the a introduces becomes one once also mode zero The nrgigpeoeai h aksco hr axion (here sector dark the in phenomena Intriguing ntecs ftecokokain global a axion, clockwork the of case the In 17 U n n xnscndffrdpnigon depending differ can axinos and ons dcmae otecnetoa axino conventional the to compared ld n xosqikydcy h lightest the decay, quickly axions and s (1) inter-dark-sector ]. so h eve xnsi h normal the in axinos heavier the of ys U rtecokoksts h coupling the sites, clockwork the er tde.Sprymtyensures Supersymmetry studied. e N 1 n orsodn asesdge of degree massless corresponding a and (1) +1 bnac fteaiodark axino the of abundance c n irrh steclockwork the as hierarchy ing 2, copn hi emo ates which partners, fermion their accompany † N u456 Korea 41566, gu )Glsoebsn.Teglobal The bosons. Goldstone 1) + rniin,wihmk l the all make which transitions, R prt,i ti pre- is it if -parity, CTPU-PTC-20-08 N f 1 f & Therefore, . correspond- asterms mass n conse- and 10 U 9 (1) N GeV) 1) + N R +1 - 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. Af- A. A model ter clockworking, the above terms lead to interactions between all axions and the SM gauge bosons: Similar to a simple construction in Ref. [15], one can consider a K¨ahler potential and a superpotential
N 2 2 g g C K = X†Xj + Y †Yj + Z†Zj , (13) = s Gb G˜bµν + 1 aY Y B B˜µν j j j L 32π2 µν 16π2 µν j=0 X N N 1 0 kπ 2 a ( 1)k q sin a . (11) W = κZj XjYj v NN 0 k k − ×f q − − N N +1 ! j=0 k=1 X X N 1 1 − q q + mX Y + m′Y X , (14) vq 1 j j+1 j j+1 − j=0 One can easily see that the coupling of the zero mode X axion is exponentially suppressed compared to that from where charge assignment of Zj , Xj , and Yj under U(1)j the actual symmetry breaking scale f while the others is (0, +1, 1). The first term reflects the spontaneous − are scaled by only 1/N 3/2 for large N. For q = 2 and breaking of U(1) global symmetry near v while the second N = 20, the exponential factor is around 106, so one can term corresponds to a small explicit breaking effect for achieve a good QCD axion even from f = 1 TeV. m,m′ v. We consider a generic case for m = m′ ≪ 6 leading to Xj = Yj which is important for inter-dark- If the zero mode is the QCD axion, it finally becomes sector couplingsh i 6 inh Eq.i (42). The fields are stabilized at massive by the chiral symmetry breaking in the strong sector of the SM, but the mass is still tiny. As is well q +1 Z = √mm , X = x, Y = y (15) known, the QCD axion has very long lifetime, so it could h j i − κ ′ h j i h j i be a dark matter component. On the other hand, massive 2 states are rather strongly coupled to the SM sector. One where can obtain decay widths of the massive modes to the 1 2 m 2(q−1) photon pair as xy = v , x = v. (16) m′ Below the spontaneous U(1) symmetry breaking scale, this theory can be described by chiral superfields con- C2 α2 3 aγγ em 2 2 2 kπ mak taining pNGBs, Γak γγ = k q sin → 256π3 N N +1 f 2 1 3 2 2 3 Φj = (σj + iφj )+ √2θψj + θ Fj , (17) 7 1 20 10 TeV m (10− s)− √2 ∼ N f GeV (12) where σj and ψj are scalar and fermion partners of φj . One can write
Φj /v0 Φj /v0 Xj = x e , Yj = y e− , (18) where α is the fine structure constant and C is em aγγ where v = x2 + y2. The effective K¨ahler potential a constant determined by C and chiral symmetry 0 aY Y and superpotential become breaking effect (e.g., Caγγ 1.92 for Kim-Shifman- p ≃ − N Vainshtein-Zakharov (KSVZ) model [18]). These states † 2 Φj +Φj decay before the big bang nucleosynthesis (BBN) for Keff = v0 cosh v0 f = 10 TeV and m = 1 GeV. In most cases, therefore, j=0 " ! X the massive states do not make significant impacts on the Φj +Φ† evolution of the universe. +ξ sinh j , (19) v0 !# N 1 − Φ qΦ W = m v2 cosh j − j+1 , (20) eff Φ 0 v j=0 0 X
III. A SUPERSYMMETRIC EXTENSION 2 Here we can take a field basis where all parameters are taken to be real and positive except κ. In this basis, the supersymmetric In this section, we consider a SUSY extension of the effective action for the axion supermultiplets does not involve clockwork axion model. any complex parameter as we will see below. 4
2 2 2 where ξ = (x y )/v0 and Φj qΦj+1 = 0. Near the point, the above potential − becomesh − approximatelyi v 2 m 2√mm . (21) N 1 Φ ′ − ≡ v0 2 σj qσj+1 Vσ 2mΦ ms v cos δs cosh − ,(26) ≃− | | 0 √2v j=0 0 In the K¨ahler potential, we have omitted Z†Z since it X N 1 is irrelevant in the low energy dynamics. The above su- − 2 φj qφj+1 perpotential shows that the supersymmetric minimum is Vφ 2mΦ ms v cos δs cos − (27) ≃− | | 0 √ j=0 2v0 achieved for Φj qΦj+1 = 0 and the supersymmetric X mass term indeedh − has thei clockwork structure propor- along the scalar and pNGB directions, respectively. It tional to an overall mass scale m . One can obtain su- Φ contributes to squared masses with the clockwork struc- perfields in the eigenbasis with mixing matrix in Eq. (8): ture for the pNGBs and their scalar partners. The mass scale for this contribution is determined by Φi = Oij Aj . (22) 2 Hence one supermultiplet remains massless after clock- msb mΦ ms cos δs. (28) working. ≡ | | Similarly to the clockwork axion model, one can intro- If SUSY breaking effects also arise in the K¨ahler poten- duce couplings of the N-th superfield to the SM gauge tial in Eq. (19), scalars and fermions acquire additional fields as masses which are diagonal in the basis of chiral super- K K g2 C fields. We write mσ and mψ , repectively, for the scalars s aGG 2 bα b and fermions. We further assume these terms are the = 2 d θΦN α + h.c. L −32π v0 W W same for all j’s, and thus the mass matrices from this con- 2 Z g1 CaY Y 2 α tribution are proportional to the identity matrix. While 2 d θΦN α + h.c., (23) K K −16π v0 W W it is expected to have mσ mψ in generic cases, it is Z K K∼ 3 where b is the gluon superfield, is the hypercharge possible to have mσ mψ in some cases. W W Mass spectra for the≫ pNGBs, scalars and fermions are superfield, and CaGG and CaY Y are model-dependent co- efficients of the order of unity. After clockworking, the summarized as zero mode superfield has exponentially suppressed inter- M2 2 M2 2 M φ = mΦ CW + msb CW, (29) actions as M2 2 M2 2 M K 2 I 2 σ = mΦ CW msb CW + mσ , (30) gs CaGG 2 bα b − K = 2 d θA0 α + h.c. M M I L −32π f0 W W ψ = mΦ CW + mψ . (31) 2 Z g1 CaY Y 2 α The (N +1) (N +1) identity matrix is denoted by I. We d θA0 α + h.c., (24) −16π2 f W W emphasize that× all the mass matrices are diagonalized by 0 Z the same mixing matrix in Eq. (8). Hence we write mass where f = qN v . 0 0 eigenstates
φj = Ojkak, (32) B. SUSY breaking effects and mass spectrum σj = Ojksk, (33) ψ = O a˜ , (34) Once the SUSY is broken, the mass spectrum for each j jk k component alters. The pNGBs and scalar partners would with mass eigenvalues receive mass contributions from SUSY breaking in the 2 2 2 2 superpotential as mak = mΦλk + msbλk, (35) m2 = m2 λ2 m2 λ + (mK )2, (36) 2 2 sk Φ k sb k σ = dθ (1 + msθ )W + h.c. − L m = m λ + mK (37) Z a˜k Φ k ψ V = m m v2 → − Φ| s| 0 and call these states axions, saxions and axinos, respec- N 1 − tively. While the zero mode axion, a is massless in that (σj qσj+1)/√2v0 φj qφj+1 0 e − cos − + δs × √2v the mass term is determined only by λ0, both s0 and j=0 0 X a˜0 become massive due to the SUSY breaking effect in 2 (σj qσj )/√2v φj qφj+1 the K¨ahler potential. While m is always positive by +e− − +1 0 cos − δ , Φ √ − s 2v0 (25)
3 where δs is the complex phase of ms. For simplicity, we We refer readers to Ref. [19–22] for general discussion for the mK ≫ will focus on parameter space where vacuum field con- mass generation and Ref. [23, 24] for explicit models with σ mK . figuration is close to the supersymmetric minimum point ψ 5 definition, m2 > m2 (q 1)2 is required not to desta- argument is valid even when the supersymmetric parame- sb − Φ − bilize axion directions. Once this condition is satisfied, ters κ,v,m,m′ in (14) and the SUSY breaking parameter 2 2 2 the mass difference δmak mak+1 mak is given by ms in (25) are dependent on sites j. Such dependency ≃ − makes a difference only on mass eigenvalues in Eqs. (35) (k + 1)π - (37) without qualitatively changing our results. δm2 > 2qm2 λ 1 cos ak Φ k+1 − N +1 Let us finally make a remark for a benchmark spec- kπ trum. If we want to identify the zero mode axion a0 as λk 1 cos . (38) QCD axion with an intermediate scale decay constant, − − N +1 v0 can be as low as O(1) TeV for N . 20. Effective de- Since λk+1 > λk and the cosine is monotonically decreas- scriptions in Eqs. (19), (20), and (25) are valid only for ing, δm2 is always positive. Thus the ordering of axion m,m′,ms v0. Hence all states are expected to be near ak ≪ mass eigenvalues is the same as that in Eq. (6), although or below the weak scale. mass differences alter. On the other hand, the ordering of eigenvalues can be different for the saxions and axinos. 2 2 C. Interactions If msb mΦ (i.e. ms cos δs mΦ), the λk-dependent part becomes≫ negative| so| as to≫ destabilize the supersym- K 2 metric vacuum. Yet if (mσ ) is large enough, the super- The axions have the same interactions as in the case symmetric vacuum can be maintained. In this case, the of the non-SUSY model in Eq. (11). The saxions also 2 2 have similar interactions from the SUSY coupling term in largest eigenvalue is ms0 while the smallest one is msN . The mass ordering of the saxions is inverted when being Eq. (23). The saxion-gauge boson interactions are given compared to that of the axions. The same thing happens by K for the axinos. If mψ < 0,a ˜0 may not be the lightest 2 2 gs CaGG b bµν g1CaY Y µν K K sax = G G + Bµν B mode. In the case mψ >mΦλN with negative mψ , the L 32π2 µν 16π2 mass ordering of the| axinos| is inverted. The ordering may K N be even not monotonic if mψ < mΦλN . Nevertheless, 1 0 k kπ | | N s0 ( 1) kq sin sk . (40) we consider the ‘normal’ hierarchy, i.e., m2 <
IV. THERMAL PRODUCTION OF AXINOS axino. Here we have summed over all possible degrees of freedom for both the initial and final states. In this section, we discuss thermal production of axinos in the early Universe. Since the whole dark sector (i.e., axion supermultiplets) communicates with the SM sector B. Saxion/axion-mediated process via the interactions in Eq. (23) and clockworking, all the axions, saxions and axinos are produced from thermal Another channel for the axino pair production is re- plasma after the primordial inflation. In a SUSY exten- alized by the saxion- or axion-mediated processes. The sion, the axinos are odd while the saxions and axions are interactions in Eqs. (40) and (42) lead to a scattering pro- even under the R-parity if it is preserved. Therefore the cess gg (s∗ or a∗) a˜na˜m, and its squared amplitude → l l → lightest axino can be a dark matter candidate if it is the is given by lightest R-parity odd particle. The saxions and axions 2 2 ξ2α2C2 1 except a0, however, would normally disappear by decay- s/a = s aGG O O O O ing into another light species such as gluons and photons. Mnm 2π2v4 Nl jl jn jm s m2 0 l,j l In this respect, axino production is more prominent than X − 2 3 the others for dark matter physics. We focus on how (ma˜n + m a˜m ) s , (45) × axinos are produced. 2 where ml is a mass of sl or al. If s ml , the squared The axino production consists of the following chan- amplitude is further simplified, so one≫ can find nels: 1) gluino-mediated process, 2) saxion/axion- 2 2 2 2 mediated process, and 3) production from saxion/axino s/a ξ αsCaGG O O 2 nm 2 4 Nn Nm decay. In particular, we will consider a relatively low M ≃ 2π v0 | | 2 reheat temperature TR below the SUSY breaking scale (ma˜n + ma˜m ) s, (46) so that axino production is mainly from the SM thermal × where we have used an identity bath. The reason is that the thermal yield of the lightest axino can easily saturate the DM abundance enhanced ONlOjlOjnOjm = ONnONm. (47) by a certain power of the clockwork factor qN compared Xl,j to the conventional scenarios as we will see. 2 If s ml , the squared amplitude is approximately given by ≪ 2 2 2 2 A. Gluino-mediated process s/a ξ αsCaGG O O 2 nm 2 4 4 Nn Nm M ≃2π v0m | | s/a From the interactions with gauge bosons in Eq. (41), (m + m )2s3 (48) axinos can be produced from the thermal plasma. If the × a˜n a˜m where we have assumed m m for all l, i.e., all temperature is larger than masses of the SUSY particles l ∼ s/a in the SM sector, the single-axino production is the dom- masses are of the same order. In this argument, we have inant process which includes the other SUSY particles in also neglected the zero mode axion contribution since its either the initial or final state. This scenario has been in- coupling is exponentially suppressed. tensively studied both for the KSVZ-type model [25–28] and for the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ)- type model [29–31]. If the temperature is smaller than C. Production from saxion/axion decay masses of the SUSY particles in the SM sector but still larger than the axino mass, e.g., ma˜ T mg˜ mq˜, Because of the interactions in Eq. (42), saxions and the single-axino production is Boltzmann-suppressed.≪ ≪ ∼ In- axions can decay into axino pairs. One can easily find stead, the axino pair production becomes more impor- their partial decay widths: tant [32]. By integrating out the gluino field in Eq. (41), ξ2m Γ(s /a a˜ a˜ )= l (m + m )2 one can obtain an effective Lagrangian for the axino pair l l → n m 16πv2 a˜n a˜m production, i.e., gg a˜ a˜ : 0 → n m 2 2 2 αsCaGG O O O = O O jl jm jn ∆nm, (49) gga˜a˜ 2 2 Nn Nm × L −1024π v0 mg˜ j µ ν ρ σ b b X a˜¯n[γ ,γ ][γ ,γ ]˜amG G . (43) µν ρσ where ∆nm = 1 (1/2) for n = m (n = m). Meanwhile, × 6 The squared amplitude for this process is given by saxions and axions can also decay into gluon pairs with the partial decay widths α4C4 g˜ 2 = s aGG O O 2 s3(1 + cos θ)2, (44) α2C2 m3 nm 4 4 2 Nn Nm Γ(s /a gg)= s aGG l O 2. (50) |M | 16π v0mg˜ | | l l 3 2 Nl → 64π v0 | | where s is the square of the center of mass energy and For ma˜n + ma˜m ml, saxions and axions decay domi- θ is the angle between the incoming gluon and outgoing nantly into gluons.≪ 7
D. Secluded spectrum dark matter is determined by the sum of all the axino yields: Comparing the gluino-mediated and saxion/axion- DM mediated processes, the relative ratio between squared Ya˜ = Ya˜n n amplitudes is given by X 4 4 5 g˜ 2 2 2 2 3√10 243αsCaGGMP TR nm αsCaGG s , (56) R |M | (51) ≃ [g(T )]3/2 16π12v4m2 s/a 2 2 2 2 2 R ! 0 g˜ ≡ nm ∼ 8π ξ mg˜(ma˜n + ma˜m ) |M | for s m2 , or where we have used the identity in Eq. (55). The axino ≫ s/a DM abundance is thus given by 2 2 4 α C ms/a R s aGG (52) 2 5 DM ma˜ 2 2 2 2 Ωa˜h 2.8 10 Ya˜ ∼ 8π ξ mg˜(ma˜n + ma˜m ) ≃ × × MeV 4 4 2 2 CaGG TeV 10 TeV for s m . Thus, for ms/a mg˜(ma˜n + ma˜m ), 0.13 ≪ s/a ≫ the gluino-mediated process dominates over the ≃ × 1 v0 mg˜ p saxion/axion-mediated process if the reheat temperature T 5 m R a˜ , (57) TR is smaller than mg˜. In this respect, we consider a × 40 GeV 10 keV simple particle mass spectrum with m m m a˜n ≪ s/a ≪ g˜ and m m (m + m ). In this spectrum, where we have used αs 0.1 and ma˜ denotes the lightest s/a ≫ g˜ a˜n a˜m ≃ moreover, the branching fraction of s/a a˜na˜m is axino mass. p → highly suppressed by small axino masses compared to In the normal hierarchy,a ˜0 is the lightest axino state saxion and axion masses. Because of the supersymmetry, and thus dark matter. Its interaction to the SM sector saxions, axions and axinos are produced with the similar is highly suppressed by 1/qN , so most of the DM axi- amount in the large TR limit, so the amount of axinos nos are produced via decays of the heavier axinos which from saxion and axion decays is negligible in this case. have interactions being mildly scaled by 1/N 3/2. In ∼ Hence axinos are predominantly produced in pairs via other words, the clockwork mechanism realizes largely the gluino-dominated process. We call this a ‘secluded’ enhanced axino production in spite of the feebly interact- spectrum. ing nature of DM species. Compared to the conventional non-clockwork scenarios of the same axino coupling to the SM, the DM abundance is enhanced by the factor 4 4N E. Thermal yield of axinos (f0/v0) = q .
One can obtain the thermal-averaged axino produc- tion cross section from the squared amplitude. For a V. COSMOLOGICAL ISSUES a˜na˜m pair production, the thermal-averaged cross sec- tion is given by A. Heavy axino decays 6α4C4 T 4 σv s aGG O O 2 ∆ , (53) h inm ≃ π5[ζ(3)]2v4m2 | Nn Nm| nm As discussed in Sec. IV, most of the DM axinos are pro- 0 g˜ duced via decays of the heavier axinos. In the secluded where T is the plasma temperature and ζ is the zeta func- spectrum, an axino can decay into a lighter axino plus tion. The yield ofa ˜ state, Y n /s (n : number the zero mode axion, i.e.,a ˜n a˜m + a0, n>m due to n a˜n ≡ a˜n a˜n → density ofa ˜n, s: entropy density) is then given by the interaction in Eq. (42). The decay width is given by
2 √ 4 4 5 N 3 10 243αsCaGGMP TR O 2 1 ξ2 Ya˜n Nn , (54) O O O 3/2 16π12v4m2 Γ(˜an a˜m + a0)= j0 jn jm ≃ [g(TR)] ! 0 g˜ | | → 16π v2 0 j=0 X where g(TR) is the effective degrees of freedom at TR and 2 3 3 ma˜m MP is the reduced Planck mass. Here we have used an m 1 , (58) × a˜n − m2 identity a˜n
2 While the DM axino yield is independent of the decay O = 1 (55) | Nm| path, the phase space distribution of the DM axinos m X is highly dependent on the decay path, lifetimes and It is noteworthy that we have included the correction mass differences. Depending on the model parameters K from the continuous reheating process [33]. N, q, mΦ and mψ , the resulting phase space distribution In the secluded spectrum, the heavier axinos eventually can deviate from the conventional thermal distribution. decay into the lightest axino, so the final yield of axino Hence, it may impact on the structure formation [34]. 8
B. Axion string-wall network VI. CONCLUSIONS
Since the clockwork axions and saxions have short In this paper, we have studied implications of super- lifetimes, their cosmological population from initial mis- symmetrizing the clockwork axion model. By supersym- alignment quickly decays without leaving substantial im- metry, the superpartner axinos have the same clockwork pacts. However a network of axion strings and domain pattern with respect to the coupling hierarchy. The N+1 walls formed by the global U(1) symmetry break- coupling hierarchy is not spoiled by SUSY breaking if ing can sizably contribute to the dark radiation [35] and the SUSY breaking is universal over the clockwork sites. yield observable gravitational waves [36]. In Ref. [35] it Even for non-universal SUSY breaking, the coupling hi- is argued that the axion DM production from collapse of erarchy is approximately maintained when the SUSY the string-wall network of the clockwork gears is negligi- breaking scale is sufficiently smaller than the clockwork ble due to the suppressed interactions between the axion mass scale. In the universal SUSY breaking case, we and clockwork gears. Yet relativistic axions produced find that the clockwork axino mass spectrum can be in- from the clockwork gear domain wall contribute to dark verted in ordering when SUSY breaking mass is larger radiation at the recombination epoch as than the clockwork mass scale. The same happens to the saxion sector. In this work we have focused on the v 2 m v 2 ω Φ 0 normal ordering because it may have interesting conse- ∆Neff 0.1 6 ≃ 1 10 TeV 10 GeV quences for axino dark matter. Under the assumption 4/3 2 (59) g S(Ta ) − Ta − that axinos are mainly produced from the SM thermal ∗ 0 0 × 20 0.2 GeV bath, we find that the thermal yield of the lightest axino is exponentially enhanced compared to the non-clockwork where Ta0 is the temperature at which the axion a0 gets axino case with the same coupling to the SM. This is be- a mass, and vω 1 parametrizes the spectrum of small- cause the lightest axino production is dominated by the ≤ scale perturbations on the domain wall. For Ta0 , we decay of heavy axinos which interact with the SM ther- use the value of QCD axion as the normalization. Ob- mal bath with exponentially larger coupling. Thus the servations of the comic microwave background require relevant parameter space for axino dark matter is sig- ∆Neff . 0.1 [37]. Thus it sets an upper bound on the nificantly different from the conventional non-clockwork 2 quantity mΦv0 for a given Ta0 . In fact, this quantity cor- axino scenarios. It generally requires a lower reheating responds to the domain wall tension. On the other hand, temperature than the conventional scenarios for the same the violent annihilation of the clockwork domain walls mass of axino dark matter. Furthermore, we expect that gives rise to gravitational waves of frequencies of the or- the phase space distribution of the axino dark matter is der of the Hubble parameter. It turns out that we have a highly dependent on the detailed clockwork structure, similar observational constraint on the domain wall ten- which may have implications for the structure forma- sion [36]. Using the estimation of [36], to be consistent tion [34]. Finally the string-wall network from the su- with pulsar timing observations [38–42], our model pa- perpartner clockwork axions has interesting cosmological rameters need to satisfy consequences on dark radiation and gravitational waves, imposing an upper bound on the axino coupling for a m v 2 Φ 0 given clockwork mass scale. It may be interesting to ex- 6 10 TeV 10 GeV amine further cosmological and collider consequences for 4/33 2/11 95 2 supersymmetric clockwork models. ǫgw − Ωgwh . 0.1 10 0.7 2.3 10− ! × 4/11 1/66 28/11 N − g (Ta ) Ta ∗ 0 0 × 10 20 0.2 GeV ACKNOWLEDGMENTS (60) We thank Jeff Kost and Chang Sub Shin for helpful where ǫgw 0.7 0.4 is an efficiency parameter of the discussions. The work of KJB was supported by the Na- ≃ ± 95 gravitational wave emission [43] ,andΩgw is the current tional Research Foundation of Korea (NRF) grant funded 8 95% confidence upper limit at ν1yr 3 10− Hz [40]. by the Korean government (NRF-2020R1C1C1012452). These considerations imply that a small≃ × axino coupling SHI acknowledges support from Basic Science Research ( 1/v0) requires correspondingly light axions to be com- Program through the National Research Foundation of patible∼ with the observational data. In our benchmark Korea (NRF) funded by the Ministry of Education parameter choice for the secluded spectrum and thermal (2019R1I1A1A01060680). SHI was also supported by yield of axinos, those constraints are safely satisfied. IBS under the project code, IBS-R018-D1. 9
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