A clockwork WIMP

Journal of High Energy Physics, Jul 2017

We embed a thermal dark matter (DM) candidate within the clockwork framework. This mechanism allows to stabilize the DM particle over cosmological time because it suppresses its decay into Standard Model (SM) particles. At the same time, pair annihilations are unsuppressed, so that the relic density is set by the usual freeze-out of the DM particle from the thermal bath. The slow decay of the DM candidate is induced by “clockwork” particles that can be quite light (rather than at some GUT or Planck scale) and could be searched for at current or future colliders. According to the scenario considered, the very same particles also mediate the annihilation process, thus providing a connection between DM annihilation and DM decay, and fixing the mass scale of the clockwork states, otherwise unconstrained, to be in the TeV range or lighter. We then show how this setup can minimally emerge from the deconstruction of an extra dimension in flat spacetime. Finally, we argue that the clockwork mechanism that we consider could induce Majorana neutrino masses, with a seesaw scale of order TeV or less and Yukawa couplings of order unity.

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A clockwork WIMP

Revised: June clockwork WIMP Thomas Hambye 0 1 2 Daniele Teresi 0 1 2 Michel H.G. Tytgat 0 1 2 0 ories in Higher Dimensions , Neutrino Physics 1 Boulevard du Triomphe , CP225, 1050 Brussels , Belgium 2 Service de Physique Theorique 3 Universite Libre de Bruxelles We embed a thermal dark matter (DM) candidate within the clockwork framework. This mechanism allows to stabilize the DM particle over cosmological time because it suppresses its decay into Standard Model (SM) particles. At the same time, pair annihilations are unsuppressed, so that the relic density is set by the usual freeze-out of the DM particle from the thermal bath. The slow decay of the DM candidate is induced by clockwork" particles that can be quite light (rather than at some GUT or Planck scale) and could be searched for at current or future colliders. According to the scenario considered, the very same particles also mediate the annihilation process, thus providing a connection between DM annihilation and DM decay, and xing the mass scale of the clockwork states, otherwise unconstrained, to be in the TeV range or lighter. We then show how this setup can minimally emerge from the deconstruction of an extra dimension in at spacetime. Finally, we argue that the clockwork mechanism that we consider could induce Majorana neutrino masses, with a seesaw scale of order TeV or less and Yukawa couplings of order unity. Beyond Standard Model; Cosmology of Theories beyond the SM; Field The- - A 1 Introduction 2 3 4 5 6 1 Dark matter phenomenology Clockwork WIMP from deconstruction Majorana neutrino masses Conclusions Introduction model, it could be absolutely stable or very long lived (see e.g. [1] for a general discussion of the stability issue). Absolute stability may be due, for instance, to a gauge symmetry, as for the electron in the SM. Alternatively, DM could be long-lived for accidental reasons, as the proton, whose stability is associated to a global continuous symmetry, such as the baryon number. The possibility that DM may be long lived is actually very interesting as its decay could be probed through indirect-detection searches. Constraints from uxes of cosmic rays impose that the lifetime of DM candidates in the WIMP mass range, which we will consider in the sequel, must be typically larger than 1026 sec. The most straightforward way to get such a large lifetime is to assume that the decay is induced by the exchange of very heavy particles, encapsulated at low energies by mass-suppressed e ective operators. Assuming couplings of order one, instability associated to a dimension5 interaction would require that the heavy degrees of freedom lie way above the Planck scale. In contrast, a dimension-6 operator only requires that these particles are around the GUT scale. This is appealing but also means that the nature of these very heavy degrees of freedom would be impossible to test directly. Alternatively, one could assume that the particles that trigger the decay of DM are much lighter, but then with much smaller couplings (something that we are usually reluctant to assume, unless protected from large quantum corrections by a symmetry). However this cannot be tested either. In the following we propose that, along a clockwork mechanism [2{7], a DM particle could be made accidentally stable but have a decay into SM particles induced by order unity interactions with particles that could be produced at colliders. Such a clockwork mechanism does not prevent the DM to undergo fast annihilations into SM particles (or { 1 { hidden-sector particles) so that its relic density is determined by the standard freeze-out mechanism, i.e. it is a WIMP. As we will see, we need to introduce a lot of new elds (and speci c couplings). One may right away wonder whether this is worth explaining the mere stability of dark matter. the presence of all these elds can be the 4-dimensional consequence of theories with extradimensions. We will show in particular that a fermion clockwork chain can arise in a simple way from the deconstruction of a single eld in at spacetime. The plan of the article is as follows. In the next section 2 we build the model, based on the clockwork mechanism and expose the basic properties of the DM candidate. The basic phenomenology is developed in section 3. In section 4 we discuss how this kind of setup can be minimally obtained from the deconstruction of an extra dimension, whereas in section 5 we comment on the possibility of inducing Majorana neutrino masses from a similar mechanism. We nally draw our conclusions in section 6. 2 Clockwork dark matter One may | but does not have to | think of the clockwork mechanism as arising from the deconstruction [8] of an extra spatial dimension. The resulting \theory space" consists of a series of elds with adjacent interactions. Here we start with N Dirac fermions i (Li; Ri) (with i = 1 : : : N ), similarly to [4], together with a single right-handed chiral fermion R0. We thus have the following set of (2N + 1) chiral elds R0; L1; R1; : : : ; LN ; RN : We refer to this set as the clockwork chain. Each chiral fermion is charged under a U( 1 ) chiral symmetry. The total symmetry group is thus U( 1 )R0 U( 1 )L1 U( 1 )R1 : : : U( 1 )LN U( 1 )RN ; and we identify the last abelian factor with the lepton-number symmetry of the SM: U( 1 )RN U( 1 )LSM : The latter choice makes natural a Yukawa coupling y of RN with the SM left-handed leptons, here denoted by LSM . We also introduce, as in ref. [4], two sets of N spurions Si and Ci with Si (1; 1) under U( 1 )Li+1 U( 1 )Ri and Ci (1; 1) under U( 1 )Li U( 1 )Ri , which break the symmetry (2.2) to U( 1 ). With these ingredients, the Lagrangian takes on the form L = LSM + Lkinetic N X i=1 ySSiLiRi 1 yC CiLiRi + h.c. (2.1) (2.2) (2.3) { 2 { 1 2 y LSM He RN + h.c. mN R0c R0 + h.c. : (2.4) For simplicity we take all spurions to be universal ySSi = m ; yC Ci = M q m We have also introduced a soft breaking of U( 1 )R0 which gives a Majorana mass mN to R0 (thus breaking the residual U( 1 ) symmetry). The fermionic For the phenomenological analysis in the next section we will focus on the regime q 1, for simplicity. Physically, we thus have a chain of N massive Dirac fermions, with relatively weaker nearest-neighbour couplings (that allow the fermions to hop from one site to the other). The Majorana particle R0 lies at one extremity of the chain, while the SM fermions live at the other extremity. The lightest state is (essentially) the Majorana particle R0, but it communicates to the SM through a chain of (relatively heavy, with respect to the hopping scale m) particles, which, as in ref. [4], we dub \clockwork gears" in the sequel. Going to the mass eigenbasis we get a band of N Dirac fermions ( Li Ri ), i = 1 : : : N , with masses m i = p i m (q 1)m (q + 1)m ; (the exact form of i can be found in [4]) and a light Majorana eld N which will be our DM candidate. The mixing matrices are L = U L L ; R = U R R ; with (in the limit N 1, q 1, for the general formulas see [4]) L m LiRi 1 q LiRi 2 mN R0c R0 + h.c. (2.6) f =N qg ; (N ! h; Z; lW ) mN y 8 2 q2N : { 3 { (2.7) (2.8) (2.9) (2.10) (2.11) By requiring that the lifetime is larger than the typical indirect-detection bound we get instead As an example, for mN 100 GeV and y 1, this can be satis ed by For mN < mW the decay processes are at 1-loop level, and the bound on q and N below is slightly weaker. By requiring that the decays (2.11) are slower than the age of the Universe AU ' 4 1017s, we get the bound: giving rise to the usual clockwork suppression by a factor qN . So far the Lagrangian of eq. (2.4) leads to a long lived DM candidate but does not provide any DM annihilation process, apart from N N ! LL and N N ! HHy, which are doubly suppressed by the clockwork chain and thus far too slow to account for the DM relic density. Therefore a clockwork WIMP DM scenario requires extra interaction(s) that are not suppressed by the clockwork chain. In the following we will consider two simple possibilities. Scenario 1. A rst one consists in simply promoting the spurions into dynamical scalar elds, acquiring a vev so that (again, taking them universal for simplicity) the m and M parameters above are now de ned as yShSii = m ; yC hCii = M q m S, the sums in eq. (2.19) are dominated by the i = 1 term for q For simplicity we consider here that all the quartic couplings are of the same order (and take them to have the same value C;S;CS along the clockwork chain). Depending on the mass spectrum of the model, the quartic couplings may or may not be necessary to have an e cient annihilation channel. For instance, as will be discussed below, if mS1 < mN the dominant annihilation is DM DM ! S1S1, which proceeds even if the quartic couplings vanish. In the opposite situation the quartic couplings are necessary in order to induce Si-h and/or Ci-h mixing, leading to annihilation into a pair of h or into h+Si or h+Ci. Similarly, such quartic couplings are interesting because they lead to possibilities of direct detection. Thus, important ingredients along this scenario 1 are the e ective Yukawa couplings of N with the Higgs boson h and the clockwork gears. In the 4-component notation for the spinors they are given by where Thus h 2 X UjLiyUiR 1; 0 = N0 X UjLiyUiR0 = N0 N i=1 j 2 r 2 N + 1 N X sin i=1 r r 2 2 sin N + 1 r 2 N + 1 N + 1 q 1 sin j N + 1 N X sin i=1 j N + 1 : h 2 ij N + 1 1 qi 1 ij N + 1 1 qi r 2 N + 1 sin j N + 1 whereas the UiR0 overlap between NR and the Ri eld is approximately as UiR0 in (2.9). This leads to the couplings and, in particular, to the e ective coupling to the Higgs boson, which is relevant for direct detection: L L mmqN2 N cN ySS1 1 q yC C1 + h.c. mmqN2 N cN p h 2 p mN 2mq2 N cN h + h.c. For the same reason, the clockwork gears are short lived and do not constitute a thermal relic, or spoil Big-Bang Nucleosynthesis predictions. Although these formulas are obtained from the diagonalization of the clockwork mass Lagrangian in the limit mN ! 0, numerically we nd that they are rather accurate as long as mN . 0:7{0:8 m . An exact diagonalization gives that the Dirac spinors i split into pseudo-Dirac Majorana spinors, and the clockwork mechanism is active as long as mN < (q 1) m m . The exact diagonalization also gives that the light Majorana state N Rc acquires an overlap UiL0 with the LH eld Li L Ui0 mN m qi+1 ; Note that, whereas the exact amount of clockwork suppression and the precise masses of the states in the band depend on the details of the couplings along the whole clockwork chain (which, as said above, we assumed to be universal for simplicity), for q 1 the couplings relevant for DM freeze-out and direct detection essentially depend only on the coupling of N with the rst clockwork chain state, L1, therefore being quite insensitive to the exact pro le of the clockwork. Scenario 2. scalar eld The second possibility for annihilations that we will consider is from a single N to couple directly with R0 (with all Si and Ci being non-dynamical) L0 = yN 2 N R0cR0 + h.c. N yN N HyH ; (2.28) whose vev induces both the R0 soft mass of eq. (2.4), mN = yN h N i, and mixing of N with the Higgs boson (through the N interaction). For what concerns the DM annihilation, this scenario is similar to a standard Majorana DM Higgs-portal scenario. The annihilation may proceed through N N ! N N or into SM elds through h- N mixing. 3 Dark matter phenomenology In this section, we discuss the basic phenomenology of the above DM model, in particular its abundance set by thermal freeze-out and the constraints from direct detection through the Higgs coupling. Some comments on collider limits are discussed at the end of this section. { 6 { (2.25) and the scalars h and Si (we assume in the sequel that mh < mSi ). Here and in what follows, with a slight abuse of notation, Si denote the (canonically normalized) real part of the respective complex elds. Scenario 1A: mS1 < mN . In this regime, the dominant channel that drives the relic abundance is annihilation into the scalar S1 with t- and u-channel exchange of the i, the same ones involved in the DM decay. Even if kinematically allowed, all other channels (into other S or C states) are suppressed by HJEP07(21)4 extra powers of 1=q. The relevant coupling of S1 is L p S 2 j S1 j PR N + h.c. ; is found to be (using the Higgs-nucleon coupling of [9]) with as given in eq. (2.24), fN = 0:30, mnuc = 0:946 GeV and the direct-detection nucleon cross-section is GSI = p 2mN 2mq2 v m2 h fN mnuc ; SI = 1 m2N m2nuc (mN + mnuc) 2 G2SI : with and the cross-section is In gure 1 we plot the coupling yS (solid black lines) required for the relic density, for mS = 150 GeV and q 1 (so that one is in the regime of validity of (3.4) and (3.6)). In the region mN & 0:8 m , as discussed above, the relations given in the previous section become less accurate, whereas the clockwork mechanism stops working altogether for mN m (hence the green/shaded region is excluded). Moreover, the plot should be considered valid as long as the phase-space is su ciently open so that processes other than N N ! S1S1 may be neglected for determining the relic abundance of DM. Direct detection is excluding { 7 { (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) 3000 0 .0 0 S θ /q= 8 1. We also show direct-detection limits (blue, continuous) and future prospects (red, dashed), from LUX 2016 and XENON1T, respectively. For a xed value of S=q, the excluded (probed) region is below the blue (red) lines. regions below the blue lines, from LUX 2016 [10]. Future prospects for XENON1T [11] scenario is compatible with the perturbativity bound yS < p 4 are also given, this probing the regions below the red dashed lines. One observes that this 3:5 for mN ; m < 2 TeV. For the benchmark model parameters of gure 1, XENON1T will test completely the allowed region, for S=q > 0:0085. Note also that in the presence of quartic couplings, the S and C states decay into SM elds via mixing with h and thus do not provide any relic on top of the DM. Scenario 1B: mN < mSi and 2mN < mSi + mh. When mN > mh, we are far from the Higgs resonance and the relic density is mainly determined by the process N N ! hh ; with t- and u-channel exchange of i. The relevant couplings are given in (2.18) and the cross-section can be obtained from (3.4) with the replacements mS1 ! mh ; yS ! = yS S : This is the main di erence with respect to scenario 1A: the e ective coupling is now suppressed by mixing with the SM Higgs. The direct-detection cross section is through the Higgs boson, as in scenario 1A, and so is as in eq. (3.6). In gure 2 we show in the mN m plane the e ective coupling (black solid lines) required to have the observed relic density, as well as the direct-detection exclusion limit from LUX 2016. Again, in the green region mN & 0:8 m our approximate expressions become less accurate, while the clockwork mechanism is active up to mN m . Directdetection exclusion limits require a large value of q. As gure 2 reveals, for q & 50 or so the parameter space above the line mN 0:8 m is a priori acceptable. Now, relic { 8 { (3.7) (3.8) 1.8 1.7 1B. Direct-detection limits from LUX 2016 are shown in blue. abundance requires that the e ective coupling is & 0:8. For 130 GeV < mS < 300 GeV the phenomenological bound on the mixing of a single scalar singlet with the Higgs boson is [12{14] . 0:3{0:4. By assuming universal mixing of all the Si with h, this implies S . p 0:4 N ; p 4 with N & 16 to account for the lifetime of the DM particle, eq. (2.13). Thus > 0:8 corresponds to non-perturbative values of yS. Therefore, for mN > mh, scenario 1B does not work, unless one gives up the assumption of universality and the mixing is bigger than the average of S;i. For instance, by taking S;1 = 0:3 (0:4), perturbativity requires . 0:3 (0:4) ' 1:1(1:4), which is compatible with all the constraints, for mN . 225 (350) GeV, m . 275 (425) GeV. We see that this scenario has a smaller parameter space available, as compared to scenario 1A, due to the extra Higgs boson mixing suppression, which in scenario 1A only a ects direct detection. Alternatively, this scenario works near the Higgs resonance mN mh=2, also for universal mixing. To see this, notice that in this regime the relevant coupling for both the relic density and direct detection is (2.27), i.e. essentially the one of a Higgs-portal Majorana DM. By adapting the results of [15], by means of the replacement the results of [15] give 55 GeV < mN < 62:5 GeV and h ! q 4mN p2vqm ; where cres varies between 1 (for mN ' 55 GeV) and 0.2 (for mN ' 62 GeV). Finally, this scenario also works near the S1 resonances, i.e. for 2mN in this case the phenomenology depends strongly on the unknown values of mSi , this option mSi but, since will not be discussed further here. { 9 { (3.9) S;1 (3.10) (3.11) 5.5 5 this scenario interpolates between the previous two, away from the resonances. For universal mixing of all the Si with h, we nd that only mN < 240 GeV (with mS1 < 300 GeV) is compatible with perturbativity, although in a narrow region of the parameter space. Relaxing the universality assumption, the parameter space opens up. For instance, in gure 3 we show the results for S = 0:1, mS1 = 200 GeV. We see that in this case direct-detection constraints require q ' 13, which is a signi cantly milder constraint, with respect to Scenario 1B. Scenario 2: non-dynamical spurions. If the spurions m and qm in (2.6) do not have dynamical origin (or the elds Si and Ci are decoupled), one can still have a successful channel for the annihilation of the DM if, for instance, mN has a dynamical origin, being generated by the vev of a scalar N , as given in eq. (2.28). For heavy N , this scenario e ectively reduces to a Higgs-portal model, which works only near the Higgs resonance [13]. For light N , this scenario can work, for instance, when the annihilation into two N is kinematically open. However, we will not discuss this scenario in detail, since the relevant phenomenology is decoupled from the clockwork mechanism. In particular, in scenario 2 the clockwork chains do not need to be at reach of current or near-future experiments, even if this is a perfectly viable possibility. Other existing limits and future prospects. In the Scenario 1 discussed above, the clockwork gears i need to be at TeV or lighter. In particular, in Scenario 1B and 1C, the combination of perturbativity considerations and the existing limits on Higgs-singlet mixing requires m to be at most at a few hundreds GeV, whereas in Scenario 1A the clockwork gears can be as heavy as TeV, if one allows large values of yS at the boundary of the perturbativity region. On the other hand, in the philosophy of the clockwork mechanism the coupling to the SM doublet y should not be too small either. Therefore, a natural consequence of the framework discussed in this note is the presence of observable e ects of the states i . The coupling of the mass eigenstates i to LSM proceeds via their (large) overlap with the clockwork state RN . Thus, as long as q 1, we may reason in terms of a single state with mass m and Yukawa coupling y to the SM. If the relevant experiments y=p are instead able to resolve the di erent eigenstates i, one should think in terms of a collection of N states with couplings N . The intermediate regime, in which the e ective description of the band of gears in terms of a single state is not valid, but at the same time the gears are not well separated, is much more involved and a detailed analysis goes beyond the purposes of this work. Phenomenologically, the states i are essentially a collection of pseudo-Dirac heavy neutrinos. Their collider signatures are studied in detail [16]. The most stringent existing constraint come from electroweak precision tests (EWPT) and gives jBl j2 v2 jylj2 avour, and slightly weaker for the electron avour. The most sensitive channel for their direct production at LHC is the trilepton plus missing energy [16]. In s = 14 TeV data with 300 fb 1 integrated luminosity will probe couplings allowed by EWPT up to m 200 GeV, which is the region required in Scenarios 1B A stronger indirect bound is provided by LFV in processes like conversion in nuclei. The nal MEG limit [17] on the branching ratio for ! e and ! e ! e is given by BR( ! e ) 8 10 4 jBe j2 jB j2 < 4:2 Although this limit is stronger than the one from EWPT for jBe j ' j B j, it crucially depends on both the muon and electron couplings being sizeable. This result will be improved in the near future by 4 orders of magnitudes by the ! e conversion experiments Mu2e [18] and COMET [19]. Thus, under the reasonable assumption of not having a large hierarchy between the muon and electron couplings, LFV provides the best prospects to probe the framework in the near future. If mN is not much smaller than m , the pseudo-Dirac states split into Majorana pairs with di erent masses and couplings to the SM leptons. In this case, the most sensitive additional channel is provided by same-sign dilepton plus jets with no missing energy. The 300 fb 1 dataset will allow to probe couplings allowed by EWPT up to m 300 GeV [20]. Finally, future electroweak precision tests at lepton colliders would provide a strong indirect probe of the mechanism. As for the scalar sector, Scenario 1B and 1C require a large mixing between the scalar S1 and the Higgs boson, whereas in Scenario 1A (2) the mixing of h with S1 ( N ) can be either large or small. A detailed analysis of the relevant constraints is given in [12{ 14]. For mS1 < 500 GeV the most stringent constraint comes from the direct searches of Clockwork WIMP from deconstruction We now want to investigate whether the scenario we advocate can be related to the deconstruction of an extra dimension [4] in a simple way. In this section we show how this can be achieved in a at spacetime for a fermionic clockwork chain, di erently from the construction in [4], which makes use of a curved metric. We start from the Lagrangian of a free massive fermion in 5D ! M i M ; = (L R)T and we get where Z is the coordinate in the extra dimension, compacti ed with length R, Z L5 L5 1 2 i M LR + h.c. : Higgs-like resonances, giving S < 0:3{0:4. For higher masses the most stringent limit is again given by EWPT: S / 0:3. Therefore, the scalar sector of our framework can give observable signatures at Higgs searches, for mS1 of few hundreds GeV, or at future electroweak precision measurements, for larger mS1 . We conclude this section with a brief comment regarding indirect detection. In the scenarios discussed here, the annihilation of DM is p-wave suppressed, and thus it is relevant only at the time of thermal freeze-out. However, through the clockwork gears, the DM particle can decay on a time scale that may be relevant for indirect searches, As the decay is through the coupling with the SM Higgs and leptons, the most notable potential spectral feature is the presence of mono-energetic neutrinos from the DM decay into h + , a signature that has been studied in the literature (see [21] and references therein). a 2 N 1 X 1 i=1 a (4.1) i 5, (4.2) (4.3) (4.4) In a theory with fermions, it is well known that a naive discretization gives rise to the famous doubling problem on the lattice (see e.g. [22]), which may be removed using a Wilson term (see the discussion in [4]): L a 2 where a = R=N ! 0 is the lattice spacing in the extra dimension. This term, which vanishes in the continuum limit, guarantees that the discrete theory tends to the continuum dimension and rede ning conveniently the elds1 we nd one (4.1) in the appropriate limit N ! 1. By discretizing (4.2) and (4.3) along the extra L N X i i i=1 N X i=1 i + LiRi+1 + h.c. + M LiRi + h.c. : 1To obtain a form that matches exactly (2.6), we have added boundary counterterms, following [4]. However, this is not necessary for the clockwork mechanism to work, but only for convenience of the presentation. The last ingredient to obtain the clockwork chain as in (2.6) is the addition of a chiral eld R0 localized at Z = 0, with a Majorana mass mN , interacting with the bulk fermion , whereas the SM is localized at Z = R. Notice that the precise value of the coupling of the interaction term LR0 (Z) does not a ect signi cantly the clockwork mechanism since, although the overall clockwork suppression is proportional to this coupling, the exponential part of the suppression does not depend on it. Equivalently, one can instead impose the Dirichlet boundary condition L(0) = 0 which e ectively removes L0 and leaves one unpaired right-chiral mode R0 in the discretized theory. We thus end up with the clockwork Lagrangian in (2.6) with the identi cations HJEP07(21)4 qm + y h i = m + y h i : In the N ! 1 we obtain a nite clockwork suppression: m 1 a ; 1 a qm + M : q N = ! e RM : with : L5 y ; The spectrum of the clockwork states in the continuum limit consists of the DM, with a mass mN and a tower of Kaluza-Klein modes (the clockwork gears) starting at the mass M . Since, for the e ectiveness of the clockwork, M 1=R, the Kaluza-Klein modes will have mass di erences M 1=R M , i.e. they will not be largely separated one from each other. The clockwork setup that we envision super cially resembles that of domain wall fermions (see e.g. [23]), with a chiral mode localized on one wall, and the (chiral) SM degrees of freedom localized on another wall. The correspondance is only approximate because we assume only one chiral mode, strictly located at the Z = 0 wall, which furthermore must have a Majorana mass. As for realizing the dynamical scalar elds needed for DM annihilation, we distinguish between the two scenarios discussed in the previous sections. Scenario 1. One possibility is to add a scalar in the bulk, with a Yukawa interaction which can also generate the Dirac bulk mass M , if it acquires a vev. After discretization, this will give rise, e ectively, to the interaction with the elds Ci discussed in section 2. Thus the Ci are dynamical elds, unlike the Si which all come from the discretization of the fth dimension, Si = 1=a. This framework predicts the universality of the clockwork chain. In this case obviously the annihilation cannot proceed into Si states but it can into Ci states, just in the same way as considered in sections 2 and 3, replacing S1 by C1 and yS by yC =q in eqs. (3.2){(3.4). The scenario is perfectly viable provided that q is not too large. Note that in the limit N ! 1, i.e. q ' 1, these Ci interactions are not suppressed with respect to the ones of Si in eq. (3.2) and in this case all the DM DM ! CiCi processes are relevant. If the bulk mass term M is generated by the vev of , eq. (4.5) gives (4.5) (4.6) (4.7) (4.8) con ned to this brane.2 rise to couplings of the form As for the quartic couplings of the Ci and the Higgs boson in (2.17), these can be sizeable if the eld has a sizeable overlap with the brane at Z = R, or if the Higgs eld is not Another possibility is to gauge the fermionic eld with an abelian group, which gives LiUiRi+1 with Ui = exp(iAi) ; (4.9) where Ai is the component of the gauge eld (technically, averaged along the link between sites i and i + 1 times the lattice spacing a). Under a gauge transformation, i ! ei i i, i). This is of course well known. The relevant point is that here the Ci are spurions generated by the M mass term and the link variables Ui play the role of the elds Si in section 2. The relevant Ui degrees of freedom are nevertheless not real elds, like instead the Si considered in eqs. (3.2){(3.4), but phases, which from the 4D perspective are pseudo-scalar elds. In this case one expects that their mixing with the Higgs boson is at least suppressed, and therefore only scenario IA would work (in the same way as in section 3). Finally, notice that in the case of a curved spacetime along the extra dimension, as considered in [4], with a metric of the form ds2 = e 43 kjZj(dx2 + dZ2) originating from a model with a dilaton S, a similar scenario could also be realized, provided that the UV completion of the model ful lls certain conditions. For instance, in order to obtain a Yukawa interaction with a bulk scalar not exponentially-suppressed for the DM light mode, one has to choose conveniently the spurion charge under dilatations of the (dimensionful) bulk Yukawa coupling. In particular, one can show that the unique successful choice is to take y ! e y , under the global Weyl transformation [4] gMN ! e 2 gMN , S ! S + 3 . With this choice, the warping coming from the metric and the interaction with the dilaton combine to give an unsuppressed interaction with the DM mode, localized at Z ' R in the construction of [4]. Scenario 2. In this case one only needs a scalar eld localized on the brane Z = 0, with a \Majorana" interaction with R0: L5 N R0cR0 (Z) : (4.10) To generate the quartic scalar coupling N in (2.28), which need to be nonzero (to provide decay channels of N into the SM) but not necessarily sizeable, one may assume that another (free) scalar eld is present in the bulk, with a signi cant overlap with both the branes. In this case the structure consists of two branes where interactions occur, connected by free elds in the bulk. In alternative, one may take the Higgs eld as living in the bulk and no extra bulk scalar. Finally, notice that this scenario 2 would also work in the extra-dimensional construction of [4] because the interaction is essentially 4D, and therefore not exponentially suppressed by the metric, provided that a decay channel of N into the SM is present. 2In the latter case, the dimension-5 scalar doublet leads to N copies of the Higgs doublet at the di erent sites in theory space. At this stage, we may simply assume that the SM Higgs is the lightest mass eigenstate. As the SM left-handed neutrinos have large Yukawa couplings with the TeV-scale clockwork gears i, one could worry that these couplings induce far too large SM neutrino masses. However, this is not the case because, if there were no R0 at the other end of the clockwork chain, the left-handed neutrinos would have no chiral partner to get a mass from. Since the interactions of R0 with the SM fermions have to proceed along the whole clockwork chain, the light-neutrino masses receive a suppression from the clockwork mechanism m m2D ' q2N mN ; (5.1) where mD = yv, and too large neutrino masses are not generated. On the contrary, since the suppression needed for the DM lifetime is much larger than for the neutrino masses (i.e. typically, in the ordinary language of e ective operators, a dimension-6 versus dimension-5 GUT scale suppression) the mass induced in the above framework is far below the masses needed: m 10 40 eV for mN However, it is worth to point out that a fermion clockwork chain can induce a Majorana neutrino mass in agreement with data through eq. (5.1) if this chain involves a smaller q2N factor. For instance for q = 10, m 10 1 eV requires 2N 15 log10(mN =GeV). Alternatively if a R0 has no Majorana mass, a Dirac mass is induced as in [4], which for q = 10 requires N 12. As one could have expected from the start, the Dirac case needs more clockwork suppression than the Majorana case. Experimentally we know that there are at least two nonzero neutrino masses, and to generate at least two of them there are essentially two simple options, as we discuss now. Option 1. If one assumes that only the last site (i.e. RN ) couples to the SM leptons, in both the Majorana and Dirac cases a single clockwork chain is not enough, because it can only induce one neutrino mass, given that the chain couples to a single combination of Le; ; . In order to induce 2 (3) neutrino masses one therefore needs 2 (3) clockwork chains, including a di erent chiral partner for each chain. Notice that the N R0 eigenstates of the two neutrino mass chains could be used to generate the observed baryon asymmetry of the Universe via resonant leptogenesis [24]. If one assumes that the Majorana breaking is universal for two neutrino mass chains, the mass splitting between the two unstable N R0, necessary for resonant leptogenesis, can be generated by the interaction with the clockwork chains, since the exact mass eigenvalues di er slightly from mN and are function of the q and N of both chains, which are not necessarily the same. By literally repeating the standard argument for leptogenesis, one can gure out that the washout induced by the Yukawa couplings with the SM neutrinos is not large, and therefore leptogenesis is in principle possible. From the continuum 5D point of view, in the at spacetime construction given in the previous section, the clockwork suppression is essentially given by the mass M . By assuming 3 chains as above, i.e. 3 fermions in the bulk and 3 chiral partners (i.e. 3 neutrino chiral partners R0 on the other end of the chain with respect to the SM elds), one could induce 2 non-negligible neutrino masses and account for DM. One of the R0 is essentially the DM particle. The 2 others are necessary to induce 2 non-negligible neutrino masses. Such a 2-brane setup requires 2 of the bulk fermions to have similar masses (involved in the neutrino mass generation), whereas the third one (related to DM) with a larger mass. Instead, in the curved spacetime construction of [4], since the clockwork suppression is universally determined by the metric, such a setup appears not to be feasible (unless one introduces an extra mass term for the bulk eld as above). If one allows the chiral partners to be not all located on the rst brane, but placed on di erent sites (for instance one on the rst site for DM and 2 much closer to the other end of the chain for neutrino masses) more possibilities show up. In this case, in both the at and curved cases, 2 bulk fermions coupling to di erent lepton avour combinations might be enough to generate 2 neutrino masses and account for DM, provided that there are still 3 chiral partners. A more detailed discussion of this may be given elsewhere. Option 2. If one assumes that the SM elds interact in di erent avour combinations with several sites along the chain, a single bulk fermion can be enough to generate two or three nonzero neutrino masses and account for DM, provided that again there are 3 chiral partners (with the one responsible for DM placed far from the 2 others as well as from the SM elds, so that the clockwork suppression for the neutrino masses gets e ectively reduced compared to the DM one). This holds for the at case considered above, as well as for the curved case considered in [4]. 6 Conclusions In this work we have proposed an implementation of a WIMP dark matter candidate within the clockwork framework. The candidate is a Majorana singlet particle that is stable over cosmological time because its e ective Yukawa couplings to the SM Higgs and leptons are suppressed through a clockwork mechanism. For appropriate choices of the parameters, it can be made very long lived, e.g. & 1026 s. The very same particles that suppress this coupling (the clockwork gears) have unsuppressed couplings to the DM candidate, so that its relic abundance may be set by the standard freeze-out mechanism. Concretely, as proof of existence, we have considered a simple clockwork chain of fermions, with a single chiral mode at one end of the chain, and the SM leptons at the other end. In section 3 we have studied the basic DM phenomenology (i.e. relic abundance, direct detection constraints, collider and low-energy limits and prospects) for various scenarios. These constraints, in particular the relic density one, require the clockwork gears to lie at most at the TeV scale, i.e. accessible at colliders. Future LFV experiments will also be able to probe e ciently their e ects. Depending on the assumption one makes about the nature and relative mass hierarchy of the scalar-sector, many viable candidates may be found. Interestingly, a similar (albeit parametrically distinct) clockwork construction may be relevant for SM neutrino phenomenology and in particular it can explain the smallness of the SM neutrinos, see section 5. We pointed out that, in order to have several nonvanishing neutrino masses, one needs several clockwork chains, if only the last site of the chains couples to the SM elds. Intuitively, the clockwork mechanism imposes that the degrees of freedom at the two edges of the chain have little overlap, a picture that is supported by an embedding of the clockwork chain in a 5D spacetime. Interestingly, our implementation departs from the 5D continuum construction of [4], where the bulk eld is massless and the mass gap between the massless mode and the clockwork gears is generated by a curved metric. In the construction of section 4, instead, the metric is at and the bulk eld is massive. The gap between the massless mode, introduced on one of the edges (\branes") and the clockwork gears is, in our case, generated by the mass term of the bulk eld. Although we have explicitly shown how the clockwork chain can originate from a at spacetime setup in the fermionic case, a very similar analysis can be carried out for a scalar clockwork: again, one needs a massive bulk scalar and extra brane terms. 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Thomas Hambye, Daniele Teresi, Michel H.G. Tytgat. A clockwork WIMP, Journal of High Energy Physics, 2017, 1-19, DOI: 10.1007/JHEP07(2017)047