PHYSICAL REVIEW D 99, 015003 (2019)
Supersymmetric R-parity violating Dine-Fischler-Srednicki-Zhitnitsky axion model
† ‡ Stefano Colucci,1,* Herbi K. Dreiner,1, and Lorenzo Ubaldi2,3, 1Physikalisches Institut der Universität Bonn, Bethe Center for Theoretical Physics, Nußallee 12, 53115 Bonn, Germany 2SISSA International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy 3INFN - Sezione di Trieste, Via Bonomea 265, 34136 Trieste, Italy
(Received 31 August 2018; published 2 January 2019)
We propose a complete R-parity violating supersymmetric model with baryon triality that contains a Dine-Fischler-Srednicki-Zhitnitsky axion chiral superfield. We parametrize supersymmetry breaking with soft terms and determine under which conditions the model is cosmologically viable. As expected we always find a region of parameter space in which the axion is a cold dark matter candidate. The mass of the axino, the fermionic partner of the axion, is controlled by a Yukawa coupling. When heavy [OðTeVÞ], the axino decays early and poses no cosmological problems. When light [OðkeVÞ], it can be long lived and a warm dark matter candidate. We concentrate on the latter case and study in detail the decay modes of the axino. We find that constraints from astrophysical x rays and gamma rays on the decay into photon and neutrino can set new bounds on some trilinear supersymmetric R-parity violating Yukawa couplings. In some corners of the parameter space the decays of a relic axino can also explain a putative 3.5 keV line.
DOI: 10.1103/PhysRevD.99.015003
I. INTRODUCTION and provides a dark matter (DM) candidate in the form of a weakly interacting massive particle (WIMP). This is the Supersymmetry (SUSY) is the best candidate to most popular and most searched for DM candidate. explain the hierarchy problem of the Standard Model However, to date no neutralino dark matter has been found (SM) [1,2], and thus a leading candidate for the discovery [7–9]. It is thus prudent to investigate other DM candidates, of new physics effects at the LHC. Most supersymmetric also within alternative supersymmetric models. models which have been searched for at the LHC make R-parity does not forbid some dimension-five operators the assumption of conserved R-parity [3]. A very restric- dangerous for proton decay [10]. This issue can be resolved tive and widely considered version with universal boun- by imposing a Z6 discrete symmetry, known as proton dary conditions at the unification scale, the constrained hexality (P6), which leads to the same renormalizable minimal supersymmetric Standard Model (CMSSM) [4] superpotential as R-parity [11,12], but is incompatible with is now in considerable tension with collider data [5]. a grand unified theory symmetry. Alternatively, one can The R-parity violating (RpV) CMSSM is still very much impose a Z3 symmetry known as baryon-triality (B3) allowed [6]. [11,13–18]. The latter allows for lepton number violating R-parity is a discrete multiplicative symmetry invoked in operators in the superpotential thanks to which the neu- the CMSSM to forbid baryon and lepton number violating trinos acquire Majorana masses, without introducing a new operators which together lead to rapid proton decay. The very high energy Majorana mass scale [15–17,19–24]. This bonus of imposing R-parity is that the lightest super- 1 is a virtue of B3 models. A possible handicap is that the symmetric particle (LSP), typically the neutralino, is stable LSP is unstable and is not a dark matter candidate. This can be naturally resolved when taking into account the strong *[email protected] CP problem, as we discuss in the following. † [email protected] Besides the hierarchy problem, every complete model ‡ [email protected] should also address the strong CP problem [30], which Published by the American Physical Society under the terms of 1 the Creative Commons Attribution 4.0 International license. Other than P6 and B3, one can also consider R-symmetries to Further distribution of this work must maintain attribution to restrict the renormalizable Lagrangian resulting in R-parity the author(s) and the published article’s title, journal citation, conservation or violation [25–29]. For our purposes this is and DOI. Funded by SCOAP3. equivalent.
2470-0010=2019=99(1)=015003(20) 015003-1 Published by the American Physical Society COLUCCI, DREINER, and UBALDI PHYS. REV. D 99, 015003 (2019) plagues the SM, as well as its supersymmetric generaliza- supersymmetric Standard Model (MSSM) plus three super- tions. In its simplest forms this necessitates introducing the fields, which are necessary to construct a self-consistent pseudoscalar axion field [31,32]. In the supersymmetric axion sector as discussed in Ref. [48]. The particle content, versions, the axion is part of a chiral supermultiplet and is the quantum numbers with respect to the SM gauge sector 1 2 3 2 1 accompanied by another scalar, the saxion, and a spin- = SUð Þc × SUð ÞL × Uð ÞY, and the charges under the 1 fermion, the axino. global Peccei-Quinn (PQ) symmetry [49], Uð ÞPQ, are In this paper we propose to study a B3 model with the summarized in Table I. The renormalizable superpotential inclusion of an axion supermultiplet of the Dine-Fischler- is given by Srednicki-Zhitnitsky (DFSZ) type [33,34]. The axion contributes to the DM energy density in the form of cold
DM [35–38]. Depending on the value of its decay constant, W ¼ WB3 þ WI þ WPQ; ð2:1Þ fa, it constitutes all or a fraction of the DM. The gravitino is heavy [OðTeVÞ], decays early in the history of the 1 universe, and does not pose cosmological problems. The ˆ ˆ ¯ˆ ˆ ˆ ¯ˆ ˆ ˆ ¯ˆ λ ˆ ˆ ¯ˆ WB3 ¼ YuQHuU þ YdQHdD þ YeLHdE þ L L E axino mass is proportional to λχvχ [see Eq. (2.18)], where 2 ˆ vχ is the vacuum expectation value (VEV) of a scalar field þ λ0Lˆ Qˆ D;¯ ð2:2Þ of the order of the soft masses [OðTeVÞ], and λχ is a dimensionless Yukawa coupling in the superpotential. We consider two cases: (i) a heavy axino (∼TeV, which ˆ ˆ ˆ ˆ ˆ ˆ WI ¼ c1AHuHd þ c2A L Hu; ð2:3Þ requires λχ ∼ 1), and (ii) a light one (∼keV, requiring λχ ≪ 1). In case (i) the axino decays early and is not a DM candidate. In the more interesting case (ii), it is the LSP, its 1 λ χˆ ˆ ¯ˆ − 2 lifetime is longer than the age of the universe, and it WPQ ¼ χ A A 4 fa : ð2:4Þ contributes as warm DM [39]. We study in detail its three decay channels: into an axion and a neutrino, into three neutrinos, and into a neutrino and a photon, the last one Here and in the following we suppress generation, as subject to constraints from x- and gamma-ray data. well as isospin and color indices. Superfields are denoted In 2014 a potential anomaly was observed in x-ray data by a hat superscript. We have imposed the discrete Z3 coming from various galaxy clusters and the Andromeda symmetry [11] known as baryon triality, B3. This leaves galaxy [40,41]. There have been several papers discussing 1 ˆ ˆ ¯ˆ 0 ˆ ˆ ¯ˆ ˆ ˆ ˆ only the RpV operators λL L E, λ L Q D,andc2A L H . this in terms of an axino in R-parity violating supersym- 2 u μ metry [42–44], although we were able to show that this We have generalized the usual bilinear operators HuHd κ explanation is not probable [45]. and iLiHu to obtain PQ invariance. The PQ charges The outline of the paper is as follows. In Sec. II,we satisfy introduce the model and describe its parameters and mass spectrum. In Sec. III we compute in detail the axino decay QA þ QH þ QH ¼ 0;QA þ QH þ QL ¼ 0; ð2:5Þ rates and branching fractions. In Sec. IV we consider u d u cosmological and astrophysical constraints on the model, with a focus on the more interesting case of a light axino. TABLE I. Charge assignments for the chiral superfields. Note We conclude with a discussion of our results and point out in our R-parity violating supersymmetric models QH ¼ QL. possible future directions of investigation. d 3 2 1 1 In Appendix A we include some general comments on Superfield SUð ÞC SUð ÞL Uð ÞY Uð ÞPQ the axino mass, we discuss its dependence on the SUSY ˆ Q 321=6 QQ breaking scale, and we point out that DFSZ models ¯ˆ 3¯ −2=3 Q ¯ accommodate more easily a light axino, as opposed to U 1 U ¯ˆ 3¯ 1 3 ¯ models ala` Kim-Shifman-Vainshtein-Zakharov (KSVZ) D 1 = QD ˆ [46,47]. In Appendix B we furthermore give details on the L 12−1=2 QL derivation of the mixings of the axino mass eigenstates. ¯ˆ ¯ E 111QE Hˆ 12−1=2 Q II. THE RpV DFSZ SUPERSYMMETRIC d Hd Hˆ 121=2 Q AXION MODEL u Hu ˆ A 110QA A. The Lagrangian ¯ˆ − A 110QA The building blocks of the minimal supersymmetric χˆ 1100 DSFZ axion model are the particle content of the minimal
015003-2 SUPERSYMMETRIC R-PARITY VIOLATING DINE- … PHYS. REV. D 99, 015003 (2019) from the terms in Eq. (2.3), and similar relations for the Below the PQ breaking scale, we generate an effective μ μ ˆ ˆ κ other fields that are readily obtained from the terms in the -term, effHuHd, and effective bilinear RpV -terms, κ ˆ ˆ rest of the superpotential. eff;iLiHu, with The PQ symmetry is spontaneously broken at the μ c1ffiffiffi scale fa, due to the scalar potential resulting from WPQ, eff ¼ p vA; ˆ 2 ˆ ¯ ¯ and the scalar parts of A and A get VEVs vA and vA, c2 κ ¼ pffiffiffi;i v : ð2:9Þ respectively, eff;i 2 A
1 1 For successful spontaneous breaking of the electroweak ffiffiffi ϕ σ ¯ ffiffiffi ϕ σ μ A ¼ p ð A þ i A þ vAÞ; A ¼ p ð A¯ þ i A¯ þ vA¯ Þ: (EW) symmetry, eff should be of the order of the EW scale, 2 2 κ while eff is constrained by neutrino physics, as discussed 9 ð2:6Þ below, and should be ≤OðMeVÞ. Since fa > 10 GeV from astrophysical bounds on axions [50], the couplings −6 c1 and c2 must be very small, roughly c1 < 10 and The VEVs must fulfill −11 c2 < 10 . They are, nonetheless, radiatively stable. W R 1 2 We have assumed that PQ respects an -symmetry, vA · vA¯ ¼ 2 fa; ð2:7Þ under which the field χˆ has charge 2. This forbids the superpotential terms χˆ2 and χˆ3. See also Ref. [25]. and we denote their ratio as The soft-supersymmetric terms of this model consist of scalar squared masses, gaugino masses, and the counter- v ¯ tan2 β0 ≡ A : ð2:8Þ parts of the superpotential couplings. The full soft vA Lagrangian reads
˜ ˜ ˜ ˜ ˜ † 2 ˜ 2 2 2 2 2 ¯ 2 2 2 2 2 −L ¼ðM1B B þM2W W þM3g˜ g˜ þH:c:Þþf m˜ f þ mχjχj þ m jAj þ m ¯ jAj þ m jH j þ m jH j soft f A A Hu u Hd d 2 ˜ † † ˜ ˜ ˜ ˜ ˜ þ m ðl H þ HulÞþðBμH · H þ H:c:ÞþðT u˜ qH˜ þ T d qH˜ þ T e˜ l H þ Tλl l e˜ lHu u d u u u d d e d ˜ ˜ ˜ ¯ 0 l ˜ l χ χ þ Tλ q d þTc1AHuHd þ Tc2A Hu þ Tλχ AA þ LV þ H:c:Þ; ð2:10Þ
˜ ˜ ˜ ˜ aμν μνρσ a with f ∈ fe;˜ l; d; u;˜ q˜g. We assume that SUSY breaking adjoint gauge group index, with X ≡ ϵ Xρσ, and the violates the R-symmetry of the axion sector; hence we coefficients are given in terms of the PQ charges χ ¯ χ include the soft terms Tλχ AA and LV . Note that, even if set to zero at some scale, these terms would be generated CaBB ¼ 3ð3QQ þ QLÞ − 7QA; ð2:12Þ at the two-loop level because of the small coupling to the MSSM sector, where no R-symmetry is present. C ¼ −3ð3Q þ Q ÞþQ : ð2:13Þ At low energy the Lagrangian contains interactions of aWW Q L A the axion with the gauge fields. These are best understood by performing a field rotation2 that leads to a basis in which In the supersymmetric limit the couplings of the axino to all the matter fields are invariant under a PQ transformation gauginos and gauge fields are related to those in Eq. (2.11). [52]. We have We are, however, interested in the case of broken SUSY. In general, such axino couplings can be calculated explic- a itly, by computing triangle loop diagrams, once the full L 2 ˜ μν 2 a ˜ aμν aVV ¼ 2 ðg1CaBBBμνB þ g2CaWWWμνW Lagrangian is specified, as it is in our model. Later in the 32π f a paper we use the full Lagrangian to compute the one-loop 2 α ˜ αμν þ g3GμνG Þ: ð2:11Þ decay of the axino into a photon and a neutrino, including the couplings. Here a is the axion field which in our model is given by 1 3 ≃ pffiffi σ − σ ¯ B. The mass spectrum a 2 ð A AÞ to a good approximation ; cf. Eq. (2.6). a a 1 2 Bμν, Wμν, Gμν are the SM field strength tensors, here a is an We now turn our attention to the spin- = neutral fermion mass spectrum of this model. The 10 × 10 mass ˜ 0 ˜ 0 ˜ 0 ˜ ¯˜ 2 matrix in the basis λ ˜ ; W ; H ; H ; ν ; A; A; χ˜ , which is See, for example, Appendix C of Ref. [51] for details. ð B u d L;i Þ 3 a generalization of the MSSM neutralino mass matrix, The axion also has a small admixture of Hu and Hd which we neglect here. reads
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2 3 g1v g1v M1 0 u − d 000 000 6 2 2 7 6 0 − g2vu g2vd 000 0007 6 M2 2 2 7 6 7 g1v g2v c2 1v c2 2v c2 3v c1v 6 u − u 0 − cp1vffiffiA p; ffiffi A p; ffiffi A p; ffiffi A − pffiffid 007 6 2 2 2 2 2 2 2 7 6 7 g1v g2v c1v 6 − d d − cp1vffiffiA 0 000− pffiffiu 007 6 2 2 2 2 7 6 7 c2 1v c2 1v 6 00p; ffiffi A 0 000p; ffiffi u 007 6 2 2 7 6 7 M 0 6 c2;2ffiffivA c2;2ffiffivu 7 χ ¼ 6 00p 0 000p 007: ð2:14Þ 6 2 2 7 6 c2;3vA c2;3vu 7 6 00pffiffi 0 000pffiffi 007 6 2 2 7 6 c1v c1v c2 1v c2 2v c2 3v vχ λχ v ¯ λχ 7 6 00− pffiffid − pffiffiu p; ffiffi u p; ffiffi u p; ffiffi u 0 pffiffi pA ffiffi 7 6 2 2 2 2 2 2 2 7 6 λ λ 7 6 vχ ffiffiχ vA ffiffiχ 7 6 0 0 0 0 000p 0 p 7 4 2 2 5 v ¯ λχ v λχ 0 0 0 0 000pA ffiffi pA ffiffi 0 2 2
f Mχ0 was obtained by first implementing our model with ¯ paffiffi β0 1 in the limit vA ¼ vA ¼ 2, corresponding to tan ¼ , and the computer program SARAH [53–55], and then setting the c1;2 → 0. One finds for the axinos [48] sneutrino VEVs to zero, which corresponds to choosing a specific basis [56,57]. This is possible, since the superfields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 2 ˆ ˆ − pffiffiffi λχvχ; pffiffiffi ðλχvχ λχ vχ þ 4f Þ: ð2:17Þ Hd and Li have the same quantum numbers in RpV models. 2 2 2 a The choice of basis is scale dependent, in the sense that sneutrino VEVs are generated by renormalization group The lightest of these three states corresponds to the (RG) running, even if set to zero at a given scale. Hence, fermionic component of the linear combination of super- 1 one needs to specify at what scale the basis is chosen. For pffiffi ¯ − fields 2 ðA AÞ. It is interpreted as the axino with a later convenience, when we compute the axino decays, we 4 choose this scale to be the axino mass. Neglecting for a mass [48] moment the axino sector, which couples very weakly to the 1 7 7 ffiffiffi rest, and Takagi diagonalizing the upper-left × block ma˜ ≃−p λχvχ: ð2:18Þ [58], one finds [20,59] two massless neutrinos and a 2 massive one with For the more general case of tan β0 ≠ 1, we cannot give κ2 analytical expressions. Instead, we find at the lowest order 2 2 2 eff mν ≈ m β ξ ≈ m β ; : Zcos sin Zcos μ2 ð2 15Þ in vχ=fa eff ffiffiffi p 2 2 2tan β0 vχ where ≃− λ O λ ma˜ 4 0 χvχ þ χvχ 2 : ð2:19Þ 1 þ tan β fa μ β vu ξ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieff tan ¼ and cos ¼ : We also give the power of the next term in the expansion, vd μ2 κ2 κ2 κ2 eff þ eff;1 þ eff;2 þ eff;3 where fa enters. This is the main contribution to the axino ð2:16Þ mass, and we neglect small corrections proportional to c1 and c2. We expect vχ to be at the soft SUSY breaking ≈ 0 1 β ≈ 1 μ ≈ 1 scale, OðTeVÞ. The Yukawa coupling λχ then is our main Requiring mν . eV, with tan and eff TeV, κ parameter to control m˜ . λχ is radiatively stable: if we set it one finds that eff;i is of order MeV and correspondingly a smaller for larger tan β. The two neutrinos which are small at tree level, it remains small. See also Ref. [60] for a massless at tree level acquire a small mass at one loop quantitative treatment of Yukawa renormalization group [19,23,24], which we have not included in the 10 × 10 equations in supersymmetric models. The most interesting mass matrix above. case we consider later is that of a light axino, with O λ In addition to the neutrinos, there are four eigenstates, ma˜ ¼ ð keVÞ, which means χ must be very small. λ ˜ 0 ˜ 0 ˜ 0 mainly built from the MSSM neutralinos ð B˜ ; W ; Hu; HdÞ, with masses between 100 and 1000 GeV, and three axino 4Negative fermionic masses can be eliminated by a chiral eigenstates with masses that can be calculated analytically rotation.
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In order to determine the interactions of the axino, in easily be diagonalized. We find a massless axion and a particular the potential decay modes, we would like to ∼ 2 2 saxion with a mass squared mA or mA¯ , the parameters in estimate the mixing between the axino and the neutrinos. O the soft Lagrangian [mA, mA¯ ¼ ðTeVÞ; cf. Eq. (2.10)]. To do this one first diagonalizes the upper 7 × 7 block, They are, respectively, the CP-odd and the CP-even mass 1 ¯ neglecting the axinos which are very weakly coupled. The eigenstates, corresponding to the combination pffiffi ðA − AÞ. massive neutrino eigenstate will be mostly a combination of 2 The result is not appreciably altered once we turn on the the gauge eigenstates νL;i, with small components of bino, small couplings c1 and c2. The other four real scalar wino, and Higgsinos. Next include the lower 3 × 3 axino degrees of freedom in this sector are heavy, with masses of block. The off-diagonal entries between the neutrino and pffiffiffi order λχf . The masses of the scalars in the MSSM sector 2 ≃ κ a axino blocks are c2vu= effvu=fa. Now rotate the are to a very good approximation those already well studied 3 3 × axino block to the mass eigenstates. The axino is a in the RpV SUSY literature [19,60]. linear combination of A and A¯ , and the other two heavier fermions are a combination of A, A¯ , and χ. The important III. LIGHT AXINO DECAY MODES point is that the entries in the 3 × 3 Takagi diagonalization axino matrix are of order one. This implies that the pieces in In this section we consider an axino mass lower than κ the off-diagonal blocks remain of order eff vu=fa after twice the electron mass, in which case the axino has only rotating the axino block, and we obtain the form of the matrix three decay modes: 2 3 (1) into a neutrino and an axion (via a dimension-five κ eff v operator), a˜ → ν a; mν O i þ 6 fa 7 (2) into three neutrinos (tree-level), a˜ → ν þ ν þ ν ; M˜ ν ¼ 4 5: ð2:20Þ i j k a; κ ˜ → ν γ eff v (3) into a neutrino and a photon (one-loop), a i þ . O ma˜ fa We discuss them in turn. From this we can read off the axino-neutrino mixing A. Decay into neutrino and axion κ eff v The dominant operator responsible for the decay a˜ → xa;˜ ν ≈ : ð2:21Þ ma˜ fa a þ ν is dimension five:
For theqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi purpose of this estimate and those in Sec. IV,wetake 1 μ ð∂ aÞψ¯ ˜ γμγ5ψ ˜ : ð3:1Þ ≡ 2 2 ∼ ∼ f a a v vu þ vd vu vd to be the weak scale. a The axino-Higgsino mixing can be estimated by observ- μ This is in the basis of the mass matrix in Eq. (2.20), before ing that the Higgsino mass is of order eff, and the off- we diagonalize to the final mass eigenstate. Here a is the diagonal element between axino and Higgsino is of order μ axion, ψ a˜ is the four-component spinor denoting the axino c1v ¼ eff v. Thus, the mixing is fa mass eigenstate, and the neutrino arises due to mixing with the axino, once we diagonalize. A simple way to under- ≈ c1v v xa;˜ H˜ μ ¼ : ð2:22Þ stand the origin of this operator is to consider the kinetic eff fa μ iγ5a=fa term ψ¯ a˜ γμ∂ ψ a˜ . After the chiral rotation ψ a˜ → e ψ a˜ , ∼ μ ∼ O we obtain the above operator when we expand the Considering M1 eff MSUSY ¼ ðTeVÞ, we can esti- v mate the bino-Higgsino mixing as g1 . Multiplying this exponential. The diagram for the decay is shown in MSUSY Fig. 1(a), where we include the axino-neutrino mixing times x ˜ we obtain the axino-bino mixing a;˜ H [see Eq. (2.21)] in the final state. The partial decay width into this channel is v2 x˜ ˜ ≈ g1 : ð2:23Þ a;B 2 3 2 2 MSUSYfa x˜ ν m 1 v κ Γ a; a˜ ≈ u eff a˜→νa ¼ 16π 2 16π 4 ma˜ ð3:2Þ fa fa The Higgsino-wino mixing is enhanced by cot θW, with θW the Weinberg angle, compared to Higgsino-bino. Hence we 2 11 4 κ 10 GeV m˜ write the axino-wino mixing as ¼ 6 × 10−54 GeV eff a : ð3:3Þ MeV fa keV θ xa;˜ W˜ ¼ cot Wxa;˜ B˜ : ð2:24Þ
Next we briefly consider the scalar sector. It is instructive B. Decay into three neutrinos to start by considering the limit c1;2 → 0, in which there is This tree-level decay proceeds via the axino mixing with no mixing between the MSSM sector and the axion sector, the neutrino and the exchange of a Z boson between the and the axion block of the scalar squared mass matrix can fermionic currents, as shown in Fig. 1(b). Assuming for
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as well as the corresponding decay width, without includ- ing interference terms. If not otherwise specified, we make use of the formulas provided in Ref. [63] to compute the loop integrals. We do not include diagrams with sneutrino VEV insertions (see e.g., Ref. [64]), since, as previously mentioned, we work in the basis, defined at the axino mass scale, where such VEVs are zero. The first diagrams we consider are those depicted in (a) (b) Figs. 2(a) and 2(b). The corresponding decay width has FIG. 1. Left: Feynman diagram for the axino decay into an been computed in the analogous case of a sterile neutrino axion and a neutrino, as described by the five-dimensional decaying into a neutrino and a photon [65] (see also operator of Eq. (3.1). Right: One of the Feynman diagrams Ref. [66] for a review), describing the decay of the axino into three neutrinos. In our 9α 2 9α 2 κ2 2 convention the ⊗ on a fermion line stands for a mixing between GF 2 5 GF effvu 3 Γν ¼ x˜ νm˜ ≈ m˜ fermionic eigenstates. ðWlÞ 1024π4 a; a 1024π4 f2 a a κ 2 1011 2 3 3 10−58 eff GeV ma˜ simplicity that the neutrino admixture of the axino is ¼ × GeV : MeV fa keV measured by x˜ ν for all neutrino flavors, the decay width a; 3:8 into all possible flavors of final neutrinos reads [61] ð Þ Here, the first subscript on Γ indicates the state the 1 2 Γ GF 2 5 incoming axino mixes with, while the subscripts in paren- a˜→3ν ¼ 3 xa;˜ νma˜ ð3:4Þ 3π 128 theses denote the particles running in the loop. We use this κ 2 1011 2 3 notation for the rest of the section. Let us anticipate that in −56 eff GeV ma˜ Γ ¼ 3×10 GeV : ð3:5Þ most of the parameter space we consider, νðWlÞ gives the MeV fa keV dominant contribution. It is therefore useful to compare Γ the other partial widths to νðWlÞ. Incidentally, a decay Γa˜→3ν and Γa˜→νa give the main contribution to the total −58 width of Γa˜ ¼ 3 × 10 GeV corresponds to a lifetime decay width in most of the parameter space. We find that 23 10 12 τ ˜ ¼ 2 × 10 s and the lifetime of the universe is about for 10 GeV 015003-6 SUPERSYMMETRIC R-PARITY VIOLATING DINE- … PHYS. REV. D 99, 015003 (2019) (a) (b) (c) (d) (e) (f) FIG. 2. Diagrams with no dependence on the RpV trilinear couplings λð0Þ. The cross on a fermion line indicates a mass insertion needed for the chirality flip, while the ⊗ on a scalar or fermion line is a (dimensionless) mixing. We write the coupling constants in green at the vertices. μ2 μ2 2 3 2 2 1 eff − eff κ μ ð þ ln 2 2 Þ cluttering the equations, we use λ to denote either. For the ma˜ eff 2 2 eff mτ˜ mτ˜ Γ ˜ ˜ ≈ xτ˜ Yτ 2 : ð3:11Þ ðH lÞ 11 5 2 LR 4 μ 4 diagrams of Figs. 3(a) and 3(b) we find 2 π fa mτ˜ 1 − eff ð m2 Þ τ˜ μeff mτ 3 2 2 2 Here, xτ˜ ≈ 2 tan β is the mixing between the left and m˜ Q m m LR m 2 a f 2 2 2 f 2 f τ˜ Γ ˜ ˜ ≈ N g λ x ˜ ln ; ð3:13Þ μ BðffÞ c 211π5 1 a;˜ B m4 m2 right staus, and eff is the Higgsino mass. Note that the f˜ f˜ μ expression is finite in the limit eff ¼ mτ˜. We find the ratio μ2 μ2 2 and for the diagrams of Figs. 3(c) and 3(d) eff eff Γ 2 4 2 1 2 − 2 ˜ ˜ 1 Yτ μ mτ ð þ ln m m Þ ðH lÞ eff 2β τ˜ τ˜ ¼ 2 2 8 tan μ2 4 3 2 2 2 Γν 18παG v mτ˜ eff ðWlÞ F u ð1 − 2 Þ 2 ma˜ Qf 2 2 2 mf 2 mf mτ˜ Γ ˜ ˜ ≈ λ WðffÞ Nc 11 5 g2 xa;˜ W˜ 4 ln 2 : ð3:14Þ 4 −8 2 π m˜ m˜ μ mτ˜ f f ≈ 10−12tan2β eff : ð3:12Þ TeV 2 TeV 4 4 The latter is a factor of cot θW ¼ðg2=g1Þ ¼ 12.57 larger This is also highly suppressed, and we neglect it in the than the former, following. Next, we examine the diagrams of Fig. 3, which involve Γ 4 θ Γ at least one trilinear RpV coupling, λ or λ0. To avoid W˜ ðff˜Þ ¼ cot W B˜ ðff˜Þ: ð3:15Þ 015003-7 COLUCCI, DREINER, and UBALDI PHYS. REV. D 99, 015003 (2019) (a) (b) (c) (d) (e) (f) (g) (h) FIG. 3. Diagrams with an explicit dependence on the RpV trilinear couplings λð0Þ where the axino mixes with the bino or the wino. The notation is the same as in Fig. 2. 2 4 2 2 2 2 2 In these formulas, NC is the number of colors of the Γ ˜ ˜ N g2Q λ v m m ˜ WðffÞ c f f 2 f fermions/sfermions running in the loop. f (f) denotes Γ ¼ 18πα 2 2 κ2 4 ln 2 νðWlÞ GF MSUSY eff m˜ m˜ either a down-type quark (squark), in the case of a λ0 f f coupling, or a lepton (slepton), in the case of a λ coupling. 2 λ 2 Q 2 2 ≈ 20 Nc f MeV θ −2 Qf is the electric charge of f in units of e, g1 ¼ e= cos W 3 10 1=3 κ θ 2 eff the hypercharge gauge coupling, g2 ¼ e= cos W the SUð ÞL m2 2 f ˜ 2 gauge coupling, xa;˜ B the axino-bino mixing from Eq. (2.23), 2 2 4 ln m TeV mf TeV f˜ and x˜ ˜ the axino-wino mixing from Eq. (2.24). The largest × ; ð3:16Þ a;W M m m˜ 120 contribution to Eq. (3.13) and Eq. (3.14) comes from a SUSY b f Γ Γ ˜ loop of bottom-sbottom. Comparing W˜ ðff˜Þ to Eq. (3.8), where mb is the mass of the bottom quark. We see that W˜ ðffÞ, 1 we find even considering mf˜ > TeV as motivated by recent 015003-8 SUPERSYMMETRIC R-PARITY VIOLATING DINE- … PHYS. REV. D 99, 015003 (2019) (a) (b) (c) (d) FIG. 4. Diagrams with an explicit dependence on the RpV trilinear couplings λð0Þ where the axino mixes with the Higgsino or the neutrino. analyses [6,67] which disfavor lighter squarks and sleptons, Here, f (f˜) can be either a down-type quark (squark), in the Γ λ0 λ can give a contribution larger than νðWlÞ. We comment case of a coupling, or a lepton (slepton), in the case of a further on this contribution below. Diagrams in Figs. 3(e), coupling. Y is either the Yd or the Ye Yukawa, and the 3(f), 3(g),and3(h) differ from those of Figs. 3(a), 3(b), 3(c), axino-Higgsino mixing is given in Eq. (2.22). As in the and 3(d) in that there is no mass insertion in the propagator previous case, the main contribution to these diagrams is of the fermion, but there is a mixing in the sfermion line in the loop. Again the contribution from diagrams where the axino mixes with the wino is larger. We estimate the decay width corresponding to Figs. 3(g) and 3(h) to be m3Q2 2 m2 Γ˜ ≈ 2 a˜ f 2λ2 2 2 ma˜ 2 f W˜ ðff˜Þ Nc 11 5 g2 x˜ ˜ x˜ 4 ln 2 ; ð3:17Þ 2 π a;W fLR m m f˜ f˜ μeff mf where x˜ ≈ 2 tan β is the mixing between left and right fLR m f˜ sfermions. We have ˜ Γ ˜ ˜ 2 μ2 WðffÞ ≈ ma˜ eff 2β ≪ 1 2 2 tan ; ð3:18Þ Γ ˜ ˜ m m WðffÞ f˜ f˜ ≪ Γ˜ due to ma˜ mf˜ , and the same suppression for B˜ ðff˜Þ.As ˜ ˜ FIG. 5. We show the branching ratios of the axino decay a result, both Γ ˜ ˜ and Γ ˜ ˜ are negligible compared to WðffÞ BðffÞ channels into axion and neutrino, three neutrinos, photon and Γ Γ W˜ ðff˜Þ and B˜ ðff˜Þ. neutrino, as a function of the axino mass. The total width is For the diagrams of Figs. 4(a) and 4(b) we find (note Γa˜→aν þ Γa˜→3ν þ Γa˜→γν, where in Γa˜→γν we take into account Γ Γ Γ there is no mass insertion in the fermion line in the loop) only the three dominant contributions, W˜ ðff˜Þ þ B˜ ðff˜Þ þ νðWlÞ, as discussed in the main text. We consider a loop of bottom- 3 2 2 Γ Γ 2 sbottom for the evaluation of W˜ ðff˜Þ and B˜ ðff˜Þ, with 2 ma˜ Qf 2 2 2 ma˜ 2 mf Γ ˜ ˜ ≈ N Y λ x : : 1 λ 10−2 HðffÞ c 11 5 a;˜ H˜ 4 ln 2 ð3 19Þ mb˜ ¼ TeV, and an RpV trilinear coupling ¼ . Continu- 2 π m˜ m˜ 10 12 f f ous (dashed) lines are for fa ¼ 10 GeV (fa ¼ 10 GeV). 015003-9 COLUCCI, DREINER, and UBALDI PHYS. REV. D 99, 015003 (2019) TABLE II. Upper bounds on the magnitude of R-parity the main contributions being again from bottom-sbottom violating couplings at the 2σ confidence level, taken from Ref. [6]. and tau-stau loops, and The constraints arise from indirect decays. The concrete proc- esses are described in detail in Ref. [68]. Γ 2 2 4 2 2 νðff˜Þ NcQf λ mf Γ ¼ 18πα 2 4 log 2 ijk λ M λ0 M νðWlÞ GF m˜ m˜ ijkð WÞ ijkð W Þ f f qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 2 m˜ λ Q 133 0 0060 mτ˜ 0 0014 b −8 Nc f . × 100 GeV . × 100 GeV ≈ 10 qffiffiffiffiffiffiffiffiffiffiffiffiffi 3 10−2 1=3 mτ˜ 233 0 070 R m˜ . × 100 GeV 0.15 × b m2 100 GeV 2 f 4 ln m2 333 0.45 (1.04) TeV f˜ × 120 : ð3:22Þ mf˜ Γ ˜ ˜ Γ Γ Γ from the bottom-sbottom loop. HðffÞ is suppressed com- From our estimates we see that W˜ ðff˜Þ, B˜ ðff˜Þ, and νðWlÞ Γ 2 2 give the dominant contributions, the other diagrams being pared to B˜ ðff˜Þ by a factor of ma˜ =mf. We have always negligible. Thus the axion decay width into photon 2 2 2 Γ ˜ ˜ λ2 2 2 HðffÞ NcQf Y ma˜ 2 mf and neutrino is given to a good approximation by ¼ 2 2 4 ln 2 Γ ˜ Γ ˜ Γ Γν 18παG κ m m W˜ ðffÞ þ B˜ ðffÞ þ νðWlÞ. In Fig. 5 we plot the branching ðWlÞ F eff f˜ f˜ ratios corresponding to the three decay channels of our 2 λ 2 Q 2 2 ≈ 10−11 Nc f Y light axino. 3 10−2 1=3 Y Γ b It is worth examining W˜ ðff˜Þ in more detail. First, we m2 2 f reintroduce the generation indices in the superpotential 2 2 4 ln m2 MeV ma˜ TeV f˜ terms × κ 10 120 : eff keV m˜ f 1 ˆ ˆ λ Lˆ Lˆ E¯ λ0 Lˆ Qˆ D¯ : : ð3:20Þ 2 ijk i j k þ ijk i j k ð3 23Þ Γ The last two diagrams to evaluate are those of Figs. 4(c) As the main contributions to W˜ ðff˜Þ are from tau-stau and and 4(d). The two RpV couplings appearing in each of bottom-sbottom loops, the trilinear couplings of interest are these diagrams have no restrictions on their flavor index λ λ0 those shown in Table II. The first index of ijk and ijk structure, contrary to those of Figs. 3(a) to 4(b), which are refers, in our case, to the neutrino in the lepton doublet. We diagonal in the singlet-doublet mixings. For the sake of our see in Fig. 6 that when we consider the decay a˜ → ν2γ, with discussion we also do not distinguish between the two 0 λ233 and λ saturating the upper bounds in Table II, the potentially different λ here, as we consider only one RpV 233 diagrams giving Γ ˜ ˜ provide the dominant contribution. coupling to be different from zero at a time. We find WðffÞ We can use this fact to set new bounds on these trilinear 3 2 2 2 couplings by considering constraints from x rays and 2 ma˜ Qf 4 2 ma˜ 2 mf Γν ˜ ≈ N λ x˜ ν ln ; ð3:21Þ gamma rays [69,70] on the decays a˜ → ν2 3γ. We proceed ðffÞ c 211π5 a; m4 m2 ; f˜ f˜ as follows. First, we assume that the entire dark matter is 105 4 233 10 104 233 1000 1000 100 100 10 10 133 1 1 0.1 133 1000 2000 3000 4000 5000 1000 2000 3000 4000 5000 Γ W˜ ðff˜Þ 1 κ 1 FIG. 6. We show the ratio Γ from Eq. (3.16) as a function of the sfermion mass. We have set MSUSY ¼ TeV and eff ¼ MeV. νðWlÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi λ λ 0 006 mτ˜ The left plot corresponds to the tau-stau loop, with mf ¼ mτ. The solid line is obtained by taking ¼ 133 ¼ . 100 GeV, while for λ λ 0 07 mτ˜ the dashed line ¼ 233 ¼ .qffiffiffiffiffiffiffiffiffiffiffiffiffi100 GeV. The right plot corresponds to the bottom-sbottomqffiffiffiffiffiffiffiffiffiffiffiffiffi loop, with mf ¼ mb. The solid line is obtained m˜ m˜ λ0 λ0 0 0014 b λ λ0 0 15 b by taking ¼ 133 ¼ . 100 GeV, while for the dashed line ¼ 233 ¼ . 100 GeV. The values chosen for the trilinear couplings saturate the up-to-date bounds of Table II. 015003-10 SUPERSYMMETRIC R-PARITY VIOLATING DINE- … PHYS. REV. D 99, 015003 (2019) FIG. 7. We show regions excluded by x- and gamma-ray constraints [69,70]. Here we use Eq. (3.24) and assume the axino to 10 λ0 contribute to the totality of dark matter. The dark (light) gray region is excluded when ma˜ ¼ keV (100 keV). In i33, the index i can ˜ → ν γ Γ be either 2 or 3, in which case the dominant contribution to the decay a i is from W˜ ðff˜Þ. Ω 2 Ω constituted by the axino, a˜ h ¼ DM. This fixes fa for With these values, the axino constitutes the entire Γ ≈ any given value of ma˜ . Then the excluded region corre- dark matter and has a partial decay width a˜→νγ Γ ≈ 6 10−53 sponds to W˜ ðff˜Þ × GeV, needed to explain the putative line. The benchmark complies with the bounds we men- 1 Γ ˜ ˜ > Γ ; : tioned above. We emphasize, however, that evidence that WðffÞ bound ¼ τ ð3 24Þ bound the 3.5 keV line is due to decaying DM is not conclusive [71–79]. τ where bound is given, for instance, in Fig. 9 of Ref. [70] as a function of the mass of the decaying particle, the axino in our IV. COSMOLOGICAL AND ASTROPHYSICAL case. In Fig. 7 we see that, for ma˜ ¼ 100 keV, we can set 0 0 −3 CONSTRAINTS bounds on the trilinears λ233, λ233, λ333 of order 10 .These are to be compared with the bounds in Table II.However,itis A. The gravitino important to keep in mind that these new bounds, contrary to We assume a supergravity completion of our model, with pffiffiffiffi those in Table II, rely on the presence of the axino and on the SUSY broken at a high scale ( F ∼ 3 × 1010 GeV) and the assumption that it is the whole dark matter. breaking effects mediated via Planck suppressed operators, In Fig. 10 we show the x- and gamma-ray constraints on so that the soft scale is M ∼ F ¼ OðTeVÞ. The SUSY MPl the plane fa vs ma. The excluded regions in purple gravitino is heavy, with m3 2 ∼ M , decays before big correspond to = SUSY bang nucleosynthesis, and poses no cosmological problems. Ω 1 a˜ Γ Γ Ω a˜→νγ > bound ¼ τ : ð3:25Þ DM bound B. The saxion Γ Γ Γ In the context of supersymmetric axion models one Here, a˜→νγ ¼ max½ ˜ ˜ ; νðWlÞ , and we multiply by WðffÞ usually has to worry about the saxion: if it is too light, it can Ω˜ =Ω to account for the region of parameter space a DM pose serious issues (see e.g., Ref. [80]). We briefly review where the axino is underabundant. We discuss the axino why. The saxion is a pseudomodulus, meaning that its relic abundance in Sec. IV. potential is quite flat. During inflation we expect it to sit at a distance of order fa from the minimum of its zero- D. Comment on the 3.5 keV line temperature potential. After inflation, when the Hubble This model can fit the 3.5 keV line observed in galaxy parameter H becomes comparable to the saxion mass, ms, clusters [40–42,44]. A possible benchmark point, for the saxion starts to oscillate about its minimum. From the 2 instance, is the following: condition HðToscÞ ∼ m , with HðTÞ ∼ T , we find the s s MPl 10 temperature at which the oscillations start, m˜ ¼ 7 keV;f¼ 3.5 × 10 GeV; a a pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 λ0 0 07 osc ∼ mb˜ ¼ TeV; 233 ¼ . : ð3:26Þ Ts msMPl: ð4:1Þ 015003-11 COLUCCI, DREINER, and UBALDI PHYS. REV. D 99, 015003 (2019) 1.19 Λ At that time the energy density in the oscillating field is Ω 2 ≈ 0 2ξ¯ fa QCD ρ ∼ 2 2 ρ ∼ osc 4 a;strh . r 12 : ð4:6Þ s ms fa. The energy density of radiation is r ðTs Þ , 10 GeV 400 MeV 2 ρs fa so that ðρ Þ ∼ 2 . The energy density of the oscillating r osc MPl 3 These strings are topological field configurations contain- saxion scales as T , behaving like matter, which implies ing loops in space where the axion field wraps the domain that in the absence of decays it would come to dominate the ½0; 2πfa nontrivially [85,86]. In Eq. (4.6), ξ is the length energy density of the universe at a temperature parameter which represents the average number of strings in a horizon volume. It is determined by numerical f2 f2 pffiffiffiffiffiffiffiffiffiffiffiffiffi dom ∼ a osc ∼ a simulations [87], ξ ≃ 1.0. The parameter r¯ is related to Ts 2 Ts 2 msMPl: ð4:2Þ MPl MPl the average energy of the axions emitted in string decays. Its value has been the subject of a long debate. Two It is important to determine whether it decays before or after scenarios have been put forth: one predicts r¯ ≈ 1, and the domination. To proceed with this simple estimate we other r¯ ≈ 70. See Ref. [85] and references therein for details 3 1 ms and Ref. [86] for recent considerations on the issue. parametrize the decay rate as Γs ∼ 16π 2 , neglecting the fa 2 While the misalignment mechanism produces axions masses of the decay products. The factor of fa at the ∼ Λ when the temperature is T QCD, the strings form at a denominator is due to the fact that the saxion couplings to much higher temperature T ∼ fa. If the PQ phase transition matter are suppressed by fa. Then the decay temperature is happened before the end of inflation, or equivalently if the obtained from HðTdecÞ ∼ Γ : s s reheating temperature (TRH) were lower than fa, then the strings would be inflated away and we would no longer 3=2 1=2 ms M have a contribution corresponding to Eq. (4.6). In this Tdec ∼ Pl : ð4:3Þ s scenario the axion abundance is from Eq. (4.5), and θ1 can 10fa take any value between −π and π. In the scenario with the PQ phase transition post inflation (TRH >fa), one has to From these estimates we find 2 2 2 average θ1 over many QCD horizons, hθ1i¼π =3, and the Tdec m 1012 GeV 3 contribution from string decays can be comparable to that s ∼ 104 s : ð4:4Þ from misalignment or larger, depending on the value of r¯. dom 1 Ts TeV fa In our model the saxion mass is of order TeV, so it decays D. A heavy axino λ ∼ 1 ∼ ∼ way before it comes to dominate the energy density of the For χ ,wehavema˜ vχ MSUSY, an axino mass of universe.5 Note the temperature when it decays is of order order TeV. Such an axino has many open decay channels GeV, safely above big bang nucleosynthesis. into MSSM particles, with a total width that can be 3 1 ma˜ estimated as Γa˜ ∼ 8π 2 . It decays safely before big bang fa C. Axion dark matter nucleosynthesis and does not pose cosmological issues. In this model the axion is a dark matter candidate. We do not review here the calculations of its relic abundance, but E. A light axino refer the reader to Refs. [83,84]. There are two possible The axino, if light, can also be a dark matter candidate mechanisms to produce axion cold DM, which depend on – whether the PQ symmetry is spontaneously broken before [39,88 93]. It was pointed out in Ref. [39] that for TRH >fa or after the end of inflation. In the former case, axion DM is a stable axino should not exceed the mass of 1 or 2 keV; solely produced via the misalignment mechanism, which otherwise, it would result in overabundant DM. More results in the abundance recently Bae et al. [84,94] showed that the axino production rate is suppressed if MΦ ≪ fa, where MΦ is the mass of the heaviest PQ-charged and gauge-charged matter supermul- f 1.19 Λ Ω h2 0 18θ2 a QCD : : tiplet in the model. This suppression is significant in DFSZ a;mis ¼ . 1 1012 400 ð4 5Þ GeV MeV models, where MΦ corresponds to the Higgsino mass. The consequence of such a suppression is that the upper bound of Here θ1 is the initial misalignment angle. If the PQ 2 keVon the axino mass quoted in Ref. [39] is relaxed. In our symmetry is broken after the end of inflation, on top of model the axino is not stable. However, if its mass is below this contribution there is another one coming from non- the electron mass, the only decay modes are the following: relativistic axions emitted by global strings: (1) into a neutrino and an axion (tree-level, with dimension-five operator), a˜ → νi þ a; 5 ˜ → ν ν ν A saxion that comes to dominate the energy density can have (2) into three neutrinos (tree-level), a i þ j þ k; other interesting implications; see e.g., Refs. [81,82]. (3) into a neutrino and a photon (one-loop), a˜ → νi þ γ. 015003-12 SUPERSYMMETRIC R-PARITY VIOLATING DINE- … PHYS. REV. D 99, 015003 (2019) FIG. 8. We show, in the plane fa vs the axino mass ma˜ , the regions where the combination of axion and the axino dark matter exceeds Ω Ω 2 Ω 2 0 12 the observed abundance, ð a þ a˜ Þh > DMh ¼ . [96]. In the grey region the main contribution is from the axion, and in the red from the axino [Eq. (4.7)]. Along the black solid line the axion alone results in the correct abundance, while along the black dashed line Ω the axino alone accounts for DM. In the left panel the reheat temperature, TRH, is below the PQ phase transition scale, fa, and the only contribution to axion dark matter is from the misalignment mechanism, Eq. (4.5). Here we take the initial misalignment angle θ1 ¼ 1. Ω In the white region, the combination of axion plus axino results in a total abundance below DM. In the right panel TRH is above fa, and we also have a contribution to axion dark matter from the decay of axionic strings, Eq. (4.6). We show the excluded region for two different values of the parameter r¯. Note that for r¯ ¼ 70, the axion plus axino are overabundant on the entire plane. Furthermore, the lifetime of the axino is longer than the age of the universe in the entire region plotted. The resulting lifetime, as we showed in Sec. III, is longer than the axion abundance can be different [95], but we will not the age of the universe, thus making the axino a dark matter consider such scenarios in this work. We learn that, for 12 candidate, with relic abundance (see Sec. 3.1 and fa ¼ 10 GeV, the axino can have a mass as large as a few Appendix C in Ref. [84] for a derivation of this result) MeV before it becomes overabundant. The important feature 12 2 of Eq. (4.7) is that the abundance is inversely proportional to m˜ 10 GeV Ω 2 ≃ 2 1 10−5 a f2. It is easy to understand why. The axino never reaches a˜ h . × 1 : ð4:7Þ a keV fa thermal equilibrium in the early universe because its inter- 104 actions are very weak. It is produced in scattering processes This does not depend on TRH, as long as TRH > GeV (listed e.g., in Table 1 of Ref. [84]) with a cross section [84,94]. For lower TRH and different cosmological histories 2 2 FIG. 9. Scatter plot of the allowed parameter space in the plane Ωah against Ωa˜ h . In the left panel (TRH 015003-13 COLUCCI, DREINER, and UBALDI PHYS. REV. D 99, 015003 (2019) FIG. 10. We add to the right plot of Fig. 8 the constraints from x rays and gamma rays [see Eq. (3.25)], shown in light purple. When these regions overlap with the red region, the parameter space is excluded by the overabundance of the axino. On the left we set the trilinear coupling to λ ¼ 10−1, and on the right λ ¼ 10−2. The lighter (darker) purple region corresponds to a mass of the sfermion, running in the loop, of 1 (2) TeV. 2 proportional to 1=fa. The larger the cross section the more a corner of parameter space where a 7 keV axino could fit 2 the axino is produced, hence the 1=fa dependence in the 3.5 keV line observed in galaxy clusters. Eq. (4.7).InFig.8 we show the allowed region of parameter This model, like any axion model (whether supersym- space, in the plane fa vs ma˜ , for axion and axino dark matter. metric or not), is very difficult to test at colliders. The main 2 2 reason is that the coupling to the Higgs portal, c1 in our In Fig. 9 we show scatter plots in the plane Ωah vs Ωa˜ h .Itis specific case, must be very small for the axion to be apparent from this figure that when TRH 015003-14 SUPERSYMMETRIC R-PARITY VIOLATING DINE- … PHYS. REV. D 99, 015003 (2019) use of SARAH. We are grateful to Manuel Drees for carefully corrections tend to raise it to GeV values. Thus a KSVZ reading the manuscript and for his comments. L. U. thanks axino prefers to be heavier than a few keV. Francesco D’Eramo for discussions and acknowledges In DFSZ models the field AðxÞ couples to two Higgs support from the Progetti di Rilevante Interesse Nazionale doublets, Hu and Hd, which are also charged under the “ 1 (PRIN) project Search for the Fundamental Laws and Uð ÞPQ, as well as the other SM fields are. Thus the Constituents” (2015P5SBHT_002). H. K. D. expresses spe- supersymmetric version of the DFSZ axion model is more cial thanks to the Mainz Institute for Theoretical Physics economical than the KSVZ counterpart, as SUSY already (MITP) for its hospitality and support during part of this work. requires at least two Higgs doublets. The SUSY coupling of ˆ ˆ ˆ axions to Higgses can bewritten as [39] WDFSZ ¼ c1AHuHd. APPENDIX A: THE UNBEARABLE The field A has to get a large VEV hAi ∼ fa,with LIGHTNESS OF THE AXION 9 fa > 10 GeV; otherwise, the corresponding axion would The axino mass has been extensively discussed in the be excluded by supernova constraints [50]. In turn, this literature [39,97–100]. It is commonly held to be highly implies a tiny coupling c1. Note that the corresponding model dependent, taking on almost any value [89,92]. coupling in nonsupersymmetric DFSZ models also has to be In this Appendix we point out that a keV mass axino is very small for the axion to be invisible. In the SUSY context it ˆ ˆ ˆ more easily embedded in the DFSZ class of models rather has been proposed that the operator c1AHuHd could be than in the KSVZ models [46,47]. We then discuss how the g ˆ 2 ˆ ˆ replaced [103] by a nonrenormalizable one A HuHd. mass is related to the PQ and SUSY breaking energy scales. MPl With the latter operator one easily obtains a μ-term at the TeV scale, while with the former we are forced to take very small 1. KSVZ vs DFSZ values for the coupling c1. We do not address the μ-problem, Axion models fall into two broad categories denoted as but we point out that Ref. [104] showed that the operator we KSVZ and DFSZ. Both contain a complex scalar field, consider can be derived consistently within a string theory 1 AðxÞ, charged under a global Uð ÞPQ, PQ symmetry [49]. framework. The fact that c1 is tiny helps with the axino mass: 6 The axion field, aðxÞ, is identified with the phase of AðxÞ. once we set it to keV values at tree level we are guaranteed In the KSVZ model, AðxÞ couples to exotic heavy quark that radiative corrections, proportional to c1,willbenegli- fields, QðxÞ, that are charged under color and under the gible as opposed to the KSVZ case we discussed above. 1 1 Uð ÞPQ [and possibly Uð ÞEM], while the rest of the SM ¯ fields do not carry any PQ charge. The coupling fQAQQ 2. Low-scale vs high-scale SUSY breaking generates, via a triangle diagram, the interaction term ˜ μν In a previous paper [48], we indicated a simple way to aGμνG which is crucial in solving the strong CP problem. understand that in models where the SUSY breaking scale, ˜ μν Here GμνðxÞ is the gluon field strength tensor and G ðxÞ its MSB, is lower than the PQ scale, fa, the axino mass is of M2 dual. In the supersymmetric version, the KSVZ model is O SUSY order ð Þ, with MSUSY the scale of the soft super- ˆ ¯ˆ ˆ fa defined by the superpotential WKSVZ ¼ fQA Q Q, where symmetry breaking terms, while in models with MSB >fa we use hats to denote chiral superfields and Q and Q¯ are two the axino mass is typically of order MSUSY. The former distinct superfields in the 3 and 3¯ representations of SU 3 , ð Þc models, with low MSB, are representative of global SUSY, respectively, but with equal PQ charge. Embedding the for which the best known framework is gauge mediation of model in supergravity results in a one-loop contribution to supersymmetry breaking [105]. The latter ones, with high the axino mass of order [39,101] MSB, usually fall in the scheme of gravity mediation of 2 supersymmetry breaking [106–108], in local supersym- m3=2fQ m˜ ∼ 10 GeV ; ðA1Þ metry, and the scale of the soft terms is that of the gravitino a 100 GeV ∼ mass, MSUSY m3=2. In light of these considerations one would think that a where m3 2 is the gravitino mass. Moreover, even if super- = light axino is more natural in the context of gauge gravity effects are neglected, there is an irreducible two-loop mediation. However, it turns out that this is difficult to contribution involving the gluino [102] accommodate. The problem is with the saxion. Consider a 2 mg˜ fQ model of minimal gauge mediation (see e.g., Ref. [105]), m˜ ∼ 0.3 GeV ; ðA2Þ a 1 TeV where the messengers do not carry any PQ charge and communicate with the visible sector only via gauge where mg˜ is the gluino mass. This implies that even if the interactions. The leading contribution to the mass of the axino mass is tuned to keV values at tree level, loop sfermions of the MSSM is generated at two loops. However, the saxion is a gauge singlet, and its mass can 6This is not rigorously exact in the DFSZ models, but true to a be generated only at three loops, as shown schematically in good approximation. Fig. 11. The coupling c1 that appears twice in the diagram 015003-15 COLUCCI, DREINER, and UBALDI PHYS. REV. D 99, 015003 (2019) gravitino. This represents an explicit exception to the generic argument that the axino should be heavier than the gravitino, put forward in Ref. [111]. APPENDIX B: AXINO-GAUGINO MIXING Most of the amplitudes involved in the radiative decay of the axino bear a strong dependence on its mixing with some FIG. 11. Three-loop contribution to the saxion mass in minimal of the other neutralino mass eigenstates. In this work we gauge mediation. At the top we have a loop of messenger fields, rely on well-motivated estimates for these quantities, but in Φ, drawn as a circle, and at the bottom a loop of Higgsinos, drawn order to obtain their precise values one should diagonalize as a rectangle. The wiggly lines are gauge bosons. The coupling the neutralino mass matrix Mχ0 . While an exact analytical c1 of the saxion to the Higgsinos is defined in Eq. (2.3). algorithm exists for the case of the MSSM [112], adding one more mass eigenstate, as for the case of the next-to- is the one we discussed in the previous subsection, and the minimal supersymmetric Standard Model (NMSSM), does same that appears in the superpotential of Eq. (2.1). The not allow for a closed-form solution. In this section we 2 2 c1 show how the approximate block-diagonalization pro- saxion squared mass, ms, is suppressed by a factor of ∼ 2 16π cedure of Ref. [113] for the NMSSM 5 × 5 neutralino compared to the squared masses of the MSSM sfermions, mass matrix can be used to obtain a numerical value for the m2, so we can estimate f˜ axino-gaugino mixing. The only requirement is a small mixing between the singlet and the doublet states, which in c1 m˜ m ∼ 0 1 f : our case corresponds to the mixing between any of the two s . keV 10−9 1 ðA3Þ TeV Higgs doublets and the axion field A. From Mχ0 in Eq. (2.14) we see that such mixing is indeed small, since This light saxion could pose serious cosmological problems c1v μ −7 pffiffiu;d ≈ eff ≲ 10 . Recalling that the axino mass eigenstate [109] as it would come to dominate the energy density of 2 fa consists of a linear combination of the fermionic compo- the universe for a long time before it decays. We reviewed ¯ some of the issues associated with saxion cosmology in nents of the fields A and A only, we define the following Sec. IV. One way out of this problem would be to make neutralino mass 5 × 5 submatrix: the saxion heavier. This could be achieved by coupling the 2 3 g1v g1v M1 0 u − d 0 axion superfield either to the messengers or to fields in the 6 2 2 7 6 0 − g2vu g2vd 0 7 hidden sector responsible for SUSY breaking. Apart from 6 M2 2 2 7 the extra field content that the procedure would bring into 6 g1v g2v c1v c1v 7 M 6 u − u 0 − pffiffiA − pffiffid 7 the model there is another issue that seems difficult to 5 ¼ 6 2 2 2 2 7; ðB1Þ 6 g1v g2v c1v 7 6 − d d − cp1vffiffiA 0 − pffiffiu 7 overcome: the same couplings needed to make the saxion 4 2 2 2 2 5 heavier would very likely produce a heavy axino [110]. c1v c1v 00− pffiffid − pffiffiu m˜ Perhaps there is a clever way to arrange for a heavy saxion 2 2 a and a light axino in gauge mediation, but in light of our considerations it seems that such a model would have to be where the axino mass ma˜ is the lightest eigenvalue which 3 3 quite complicated. We do not investigate this aspect further results from the diagonalization of the × lower-right M 0 in this work. Rather we choose a supergravity (SUGRA) block of the neutralino mass matrix χ in Eq. (2.14). model. We showed in Ref. [48] that in SUGRA the axino The diagonalization of M5 proceeds in two steps: first, the would typically have a mass comparable to the gravitino, 4 × 4 matrix V rotates only the MSSM upper-left 4 × 4 thus in the TeV range. However, we can adjust a single block into a diagonal form; next a second matrix block 5 5 parameter, λχ, to lower the axino mass down to the keV diagonalizes the full resulting × matrix, in a way that 4 4 M range. Doing so does not affect other masses, such as those the × MSSM submatrix 4 is still diagonal up to corrections of the second order in the singlet-doublet of saxion and gravitino for instance. Also, as we argued μ mixing parameter eff. The 5 × 5 unitary matrix U which above, since the axino we consider is of the DFSZ type, fa once we set its mass at tree level it will not be affected combines these two steps is appreciably by loop corrections. Therefore the SUGRA 1 − 1 Λ Λ T Λ model can be kept minimal, as opposed to a possible model 4 2 ðV ÞðV Þ ðV Þ V 0 U ¼ ; with gauge mediation, at the expense of some tuning, T 1 T 01 −ðVΛÞ 1 − 2 ðVΛÞ ðVΛÞ needed to lower the mass of the axino. In the model presented in this paper the axino is much lighter than the ðB2Þ 015003-16 SUPERSYMMETRIC R-PARITY VIOLATING DINE- … PHYS. REV. D 99, 015003 (2019) M 105 FIG. 12. Distribution of the mixings xa;˜ B˜ and xa;˜ H˜ obtained by numerical evaluation of the eigenvalues of 4.Werun points 2 4 10 12 2 randomly varying 10 GeV 3 v2 − 2 − 2 andP M⋆ ¼ 4 ½ðM1 ma˜ Þg2 þðM2 ma˜ Þg1 . The axino-gaugino mixings can then be read off from the off-diagonal entry 4 Λ ˜ ˜ ˜ 0 ˜ 0 j¼1 Vij j of U, with j ¼ B0, W0, Hu, Hd. In Fig. 12 we show two examples of how a more careful analysis can change significantly such quantities, and thus ultimately the phenomenology of the model. This possibly constitutes caveats to our β above reasoning. 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