PHYSICAL REVIEW D 100, 055003 (2019)

All-in-one relaxion: A unified solution to five particle-physics puzzles

R. S. Gupta, J. Y. Reiness, and M. Spannowsky Institute for Phenomenology, Durham University, South Road, Durham DH1 3LE, United Kingdom

(Received 24 May 2019; published 5 September 2019)

We present a unified relaxion solution to the five major outstanding issues in particle physics: Higgs mass , dark matter, matter-antimatter asymmetry, neutrino masses and the strong CP problem. The only additional field content in our construction with respect to standard relaxion models is an up-type vectorlike pair and three right-handed neutrinos charged under the relaxion shift symmetry. The observed dark matter abundance is generated automatically by oscillations of the relaxion field that begin once it is misaligned from its original stopping point after reheating. The matter-antimatter asymmetry arises from spontaneous baryogenesis induced by the CPT violation due to the rolling of the relaxion after μ μ reheating. The CPT violation is communicated to the and leptons via an operator, ∂μϕJ , where J consists of right-handed neutrino currents arising naturally from a simple neutrino mass model. Finally, the strong CP problem is solved via the Nelson-Barr mechanism, i.e., by imposing CP as a symmetry of the Lagrangian that is broken only spontaneously by the relaxion. The CP breaking is such that although an Oð1Þ strong Cabibbo-Kobayashi-Maskawa (CKM) phase is generated, the induced strong CP phase is much smaller, i.e., within experimental bounds.

DOI: 10.1103/PhysRevD.100.055003

I. INTRODUCTION We show in this paper that the relaxion construction has Recently, particle physics research has been driven to a many interesting built-in features that can provide solutions large extent by the expectation of physics beyond the to multiple other BSM puzzles in a way that is completely (BSM) at the TeV scale. While there are different from the other examples referred to above. These many theoretical and observational reasons to extend the features are: spontaneous CPT violation during its rolling, Standard Model (SM)—such as Higgs mass naturalness, spontaneous CP violation when it stops and oscillations dark matter, matter-antimatter asymmetry, neutrino masses about its stopping point after reheating. The spontaneous and the strong CP problem—only the first of these issues CPT violation leads to spontaneous baryogenesis during necessarily requires TeV-scale new physics. In fact, if the rolling of the relaxion after reheating [8]; the sponta- Higgs mass naturalness is ignored and new physics scales neous CP violation leads to a Nelson-Barr solution [9,10] far beyond the TeV scale are allowed, the other issues can of the strong CP problem [11,12]; and the relaxion be solved by very minimal extensions of the SM [1–6]. oscillations generate the observed dark matter abundance It is arguably far more challenging to find an explanation [13]. The spontaneous baryogenesis mechanism requires (apart from tuning or anthropics) for a light Higgs mass that baryons and/or leptons are charged under the relaxion with a high new physics scale. While conventional wisdom shift symmetry. In this work the relaxion shift symmetry is says this is impossible, the recently proposed cosmological identified with a Froggatt-Nielsen symmetry [14], under relaxation (or relaxion) models [7] aim to find just such an which three new right-handed (RH) neutrino states (but no explanation. In these models the rolling of the so-called SM states) are charged. This satisfies the requirement of relaxion field during inflation leads to a scanning of the spontaneous baryogenesis while also giving an explanation Higgs mass squared from positive to negative values. Once for the smallness of neutrino masses. the Higgs mass squared becomes negative it triggers a Thus, we achieve a unified solution to five BSM puzzles, backreaction potential that stops the scanning soon after, at namely the lightness of the Higgs in the absence of a value much smaller than the new physics scale. TeV scale new physics, dark matter, matter-antimatter asymmetry, neutrino masses and the strong CP problem.

Published by the American Physical Society under the terms of II. REVIEW AND BASIC SETUP the Creative Commons Attribution 4.0 International license. In relaxion models, the Higgs mass squared parameter is Further distribution of this work must maintain attribution to 2 the author(s) and the published article’s title, journal citation, promoted to a dynamical quantity μ ðϕÞ, which varies due and DOI. Funded by SCOAP3. to its couplings to the relaxion field, ϕ,

2470-0010=2019=100(5)=055003(7) 055003-1 Published by the American Physical Society GUPTA, REINESS, and SPANNOWSKY PHYS. REV. D 100, 055003 (2019)

ϕ matter requires no additional ingredient. This is due to the V ¼ μ2ðϕÞH†H þ λ ðH†HÞ2 − r2 M4 cos ; ð1Þ roll H roll F fact that during the second phase of rolling, the relaxion gets misaligned from its original stopping point by an with, angle [13], ϕ 2 2 2 2 Δϕ 1 mϕ ϕ μ ðϕÞ¼κM − M cos : ð2Þ Δθ ¼ ≃ 0 ð Þ F 20 ð Þ tan : 6 f H Tc f

Here, H is the SM Higgs doublet, λH is its quartic coupling, M is the UV cutoff of the Higgs effective theory and κ ≲ 1 As shown in Ref. [13], this sets off relaxion oscillations that [15]. The rolling starts from a relaxion field value, can give rise to the observed dark matter relic abundance, ϕ ϕ ¼ −j −1κj μ2 0 < c Fcos , such that > . After crossing 4 3 2 Λ 100 GeV the point ϕ ¼ ϕ , μ becomes negative, prompting electro- Ω 2 ≃ 3Δθ2 d ð Þ c h 1 : 7 weak symmetry breaking. This in turn activates the back- GeV Tosc reaction potential, which induces periodic “wiggles” on top of the linear envelope, Note that the correct relic density can always be reproduced ϕ0 by choosing an appropriate value of tan f . While there is ϕ some room for this in the relaxion mechanism, as the V ¼ Λ4 cos : ð3Þ br c f relaxion is spread across multiple vacua at the end of its k rolling, the probability distribution of the relaxion field ϕ 4 n 4−n Oð1Þ 0 Here, Λc ¼ m vðϕÞ , is an increasing function of the peaks for values of tan f [20]. Thus the extent to Higgs vacuum expectation value (VEV). These wiggles ϕ0 which tan f deviates from unity can be interpreted as a cause the relaxion field to come to a halt soon after, measure of the tuning required to get the correct relic generating a large hierarchy between the Higgs VEV and abundance. the cutoff M. As discussed in [7], the cutoff, M, cannot be It was shown in [8] that with just one additional raised to an arbitrarily high value because of cosmological ingredient, this second phase of rolling can also give requirements, spontaneous relaxion baryogenesis (SRB). One requires that some with B þ L charge are charged under M 1=2 Λ4 1=6 M ≲ P c : ð4Þ the relaxion shift symmetry. This leads to the presence of μ μ rroll fk the operator, ∂μϕJ =f, where J contains the B þ L current. This operator generates a chemical potential for The relaxion mechanism must be complemented by a new B þ L violation once the second phase of relaxion rolling mechanism at the scale M (for e.g., supersymmetry [16] or results in a CPT-breaking expectation value for ∂μϕ.A Higgs compositeness [17]) to solve the full hierarchy asymmetry is consequentially generated via problem up to the Planck scale. (B þ L)-violating sphaleron transitions. Let us now discuss what happens after inflation. For the As shown later, generation of the observed baryon backreaction sector we adopt the non-QCD model of [7], asymmetry requires a hierarchy f ≪ f . This and the fact where ϕ is the of a new strong sector. If the reheating k that the relaxion, in any case, requires a large hierarchy temperature is greater than the critical temperature of the between f and its field excursion during rolling, f ≪ F, chiral phase transition of the new sector, i.e., if k k pffiffiffiffiffiffi are problematic as explained in [21]. The solution to ∼ 4π 0 Tr >Tc fπ , the wiggles disappear and the relaxion generating the latter hierarchy is embedding the relaxion 0 “ ” starts rolling again. Here fπ is the pion decay constant of construction in a so-called clockwork model [22–24]; this the new sector. When the universe cools below Tc again, can easily be extended to also generate the former hier- the backreaction potential reappears and the rolling even- archy, giving f ≪ fk ≪ F. In clockwork models there is a tually stops provided, mϕ ≲ 5HðT Þ. This condition is c system of interacting complex scalars, Φi, all of which get a f π obtained by demanding that the relaxion does not have hΦ i¼pffiffi i i=f VEV such that i 2 e . There is an approximate enough kinetic energy to overshoot the barriers once the ð1Þ backreaction potential reappears [13,18,19]. If satisfied, the Abelian symmetry, U i, at each site which is sponta- relaxion enters a slow-roll-like regime with, neously broken to give rise to a corresponding pseudo- Goldstone mode πi. Explicit breaking effects give the 0 angular fields, π , a mass matrix such that the lightest V ðϕÞ¼5Hϕ_ : ð5Þ i state is a massless (Goldstone) mode given by, It is this second phase of rolling that can lead to a X π j π1 πN ϕ ∝ ¼ π0 þ þþ : ð8Þ generation of both the observed dark matter abundance as 3j 3 3N well as the baryon asymmetry. The explanation for dark j

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O (i) Instead of introducing the operator, SB, by hand, we propose a simple neutrino mass model at site 0, which generates this operator with a current con- taining only three new right-handed (RH) neutrino fields. The operator arises because the RH fields are charged under the relaxion shift symmetry, which in turn is identified with a Froggatt-Nielsen symmetry. This also explains the observed smallness of neu- trino masses. Only the SM-singlet RH neutrinos are charged under the relaxion shift symmetry, which is thus automatically anomaly-free with respect to both QCD and electromagnetism. (ii) We show that the rolling potential can be generated by the addition of a single up-type vector-like pair at FIG. 1. Schematic representation of our set-up with the vertical site N. This modification can also solve the strong line representing the clockwork system. The three right-handed CP problem via the Nelson-Barr mechanism [9,10]. neutrinos at the 0th site, ni, couple to the SM in the usual way, In Nelson-Barr models, CP is a good symmetry in generating neutrino masses; they also provide the current in the the UV and is spontaneously broken at an inter- all-important operator for spontaneous baryogenesis, mediate scale to generate an Oð1Þ Cabibbo-Kobaya- O ¼ ∂ ϕ μ SB μ J . At the kth site we introduce a new strong sector shi-Maskawa (CKM) phase but a much smaller which couples to SM via its fermionic matter content, strong CP phase (within allowed constraints). We ð c cÞ L; L ;N;N . This sector generates the backreaction wiggles borrow the Nelson-Barr relaxion sector from [11], and relaxion oscillations inside these wiggles generate the where the relaxion phase upon stopping results in an observed dark matter abundance. Finally, at site N there is a Oð1Þ Nelson-Barr sector that radiatively generates the rolling potential CKM phase. while also providing a solution to the strong CP problem. This Also, spontaneous baryogenesis is an attractive potential sector couples to the SM up sector via a new vector-like feature from the point of view of the Nelson-Barr relaxion pair ðψ; ψ cÞ. model, as it does not require explicit CP violation.

III. NEUTRINO MASSES AND SPONTANEOUS −k Given that the mixing angle hπkjϕi ∼ 3 , any Lagrangian BARYOGENESIS term where the angular field πk couples with a decay A. Getting the operator OSB constant f translates to an interaction of ϕ with an exponentially enhanced effective decay constant, 3kf,in At site 0 we introduce a sector that simultaneously the mass basis. Coming back to our setup, the hierarchy generates small neutrino masses and an operator suitable ≪ ≪ O for spontaneous baryogenesis. We introduce three RH f fk F can be obtained by having the operator, SB, at the 0-th site, the backreaction sector at the intermediate neutrinos, ni, that are charged under the 0th site Abelian kth site and the rolling potential at the Nth site, such that symmetry, Uð1Þ0. We fix the charge of Φ0 to be −1 under k N this symmetry and take all SM fields to be neutral. The fk ¼ 3 f and F ¼ 3 f. This is schematically shown in Fig. 1. The rolling and backreaction potentials eventually Lagrangian for the couplings of these RH neutrinos is lift the flat direction in Eq. (8), giving a mass to the given by, relaxion. q q þq Φ0 nj † Φ0 ni nj The SRB setup reviewed above is incomplete in two L ⊃ yij l Hn þ Mˆ ijn n ; ð9Þ O FN n Λ i j Λ n i j respects: the operator, SB, and the rolling potential are FN FN nonrenormalizable and introduced in a somewhat ad hoc O † way. While SB arises naturally if baryons and/or lepton where li are right handed spinors denoting the SM lepton are charged under the Abelian symmetry of which the doublets, qnj are the Abelian charges for the sterile relaxion is a Goldstone boson, charging the SM fermions neutrinos and Mij is the Majorana mass matrix. Given O n seems to have no purpose other than generating SB. that we will eventually use the Nelson-Barr solution to the Furthermore, the charge assignments have to be carefully strong CP problem, we impose CP as an exact symmetry of chosen such that they are anomalyfree with respect to QCD the Lagrangian so that all couplings above are real. (to avoid generating a strong CP phase) and preferably also f iπ0=f Substituting, hΦ0i¼pffiffi e , we obtain exponentially- with respect to electromagnetism (to avoid the generation 2 of a ϕγγ coupling that rules out most of the parameter space suppressed effective Yukawa couplings and Majorana ij ij q ij ij q þq ¼ ðϵ Þ nj ¼ ˆ ðϵ Þ ni nj in this set-up [8]). Here we complete the SRB set-up as masses, Yn ypnffiffiffi FN and Mn Mn FN , follows: where ϵFN ¼ f= 2ΛFN < 1.

055003-3 GUPTA, REINESS, and SPANNOWSKY PHYS. REV. D 100, 055003 (2019) iπ0=f 2 2 3 −1 Factors of e , that appear upon substitution of Φ0 in ϕ_ 2π 15 ϕ_ η ≡ nB ¼ T gT ¼ gSB ð Þ Eq. (9), can be rotated away by the field redefinition gSB × 2 : 14 s f 6 45 4π g fT −iπ0qn =f nj → nje j , which, through the redefinition of the kinetic terms for the RH neutrinos, yields the desired The equilibrium distribution changes after electroweak O operator, SB, symmetry breaking, when there is no longer a need to _ conserve T3. This gives μQ ¼ −ð4=11ÞμB−L ¼ −Qnϕ=12f q † q † ni ð∂ π Þ σ¯ μ → ni ð∂ ϕÞ σ¯ μ ð Þ g ¼ 3Q =4 μ 0 ni ni μ ni ni; 10 and, once again, SB n . However, species such as f f the RH fermions, which are coupled very weakly to the thermal plasma, would not be able to reequilibrate on the where we ignore an Oð1Þ factor corresponding to hπ0jϕi. timescale of the electroweak phase transition. The precise value of g is thus hard to compute without considering O SB B. Getting baryon asymmetry from SB the full dynamics of the process and may be different from O The presence of SB can lead to spontaneous baryo- the value obtained above. Given that these subtleties only Oð1Þ genesis, an idea developed in [25,26]. The essential feature lead to an ambiguity in gSB, for definiteness we stick of this mechanism is the presence of a rolling field that to the value derived in Eq. (13). η ¼ ¼ 130 breaks CPT, a role played here by the relaxion [8]. The value of is frozen at T Tsph GeV, the O During the second phase of the rolling, the operator SB temperature at which the sphaleron processes decouple causes equal and opposite shifts in the energies of particles [27]. Demanding the observed baryon asymmetry, −11 versus antiparticles, implying, η0 ¼ 8.7 × 10 , we obtain,

rffiffiffi 3 2 μ ¼ −μ¯ ¼ ϕ_ þð − Þμ þ μ þ μ 2π3 = 3 η −2 i i qi =f Bi Li B−L Qi Q T3i T3 ; f 2 g T 0 mϕ k ¼ sph ∼ 109 ; ð Þ 5 9 2 −5 15 f gSBmϕMPl 10 eV where μi (μ¯ i) is the chemical potential for (anti-) particles of ð1Þ the ith specie; qi is its charges under U 0 (which is 0 _ 4 using V ¼ 5Hϕ ∼ Λc=fk (see Sec. II). It is crucial that the nonzero only for the RH neutrinos); Qi is the electromag- relaxion keeps rolling with a nonzero ϕ_ when the value of η netic charge; T3i is the charge corresponding to the ð2Þ is frozen at T ¼ T . To ensure this, we require that the diagonal generator of SU L; and the chemical potentials sph μ critical temperature for the phase transition of the strong Q;T3;B−L have been introduced to enforce conservation of Q; T3, and B − L. In the presence of (B þ L)-violating sector, Tc, is lower than Tsph. sphaleron processes, we find, We have assumed so far that the RH neutrinos are relativistic and in equilibrium at the temperatures relevant ni − n¯ i ¼ fðT;μÞ − fðT;μ¯Þ to the calculation. The first condition requires that the ≲ ij ∼ T2 T2 seesaw scale, Mn Tsph, where Mn Mn. The second ¼ μ μ ð Þ n gi i 6 ;gi i 3 ; 11 requirement implies that the interaction rate of i with SM Γð Þ ð Þ Γð Þ ∼ 2 2 particles satisfies, n >H Tsph . Taking, n g YNT ð μÞ ij for fermions and respectively, where f T; is the [28], where g is the weak coupling and Yn ∼ Yn and Fermi-Dirac (Bose-Einstein) distribution for fermions Γ ð Þ ≳ 10−8 requiring >H Tsph , we get Yn . (bosons). We have taken μ ≪ T and the factor gi denotes the number of degrees of freedom for each species. μ C. Neutrino masses The quantities T3;Q;B−L can be obtained by imposing ¼ ¼ ¼ 0 The Lagrangian in Eq. (9) generates masses for the SM nT3 nQ nB−L . For temperatures above the critical 2 2 neutrinos mν ∼ Ynv =Mn ≲ 0.1 eV. Given that spontane- temperature for the electroweak phase transition, we obtain ≲ ous baryogenesis demands Mn Tsph, we require small the following chemical potentials, −6 effective Yukawas Yn ≲ 10 , which can be naturally 3 ϕ_ 3 ϕ_ 33 ϕ_ obtained via the Froggatt-Nielsen mechanism, as explained μ ¼ − Qn μ ¼ Qn μ ¼ Qn ð Þ Q T3 B ; 12 above. 14 f 14 f L 112 f The constraints derived in the previous subsections imply that our model requires a finite range for both the taking all the q ¼ Qn. We subsequently obtain a baryon ni effective Yukawa coupling and effective Majorana mass number density, 10−8 ≲ ≲ 10−6 30 ≲ ≲ scale: Yn , MeV Mn Tsph. Note that ϕ_ T2 sterile neutrinos with masses below 500 MeVare in tension n ¼ −n ¼ g ; ð13Þ B L SB f 6 with big-bang nucleosynthesis (BBN) [29]. Masses around a few GeV may be within reach of future experiments such ¼ 3 4 ϵ ¼ 0 1 where gSB Qn= and finally for its ratio with the as SHiP [29].For FN . the above range of the Yukawa entropy density, couplings can be obtained for 6 ≤ Qn ≤ 8, where we have

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¼ 2 taken all qni Qn. Given that all the couplings in Eq. (9) turn is a direct consequence of the Z symmetry. This θ¯ are real, a concrete prediction of our model is a CP-even ensures that at tree level there is no contribution to QCD neutrino mass matrix, which is still comfortably allowed by from the phase, θN,asArgðdetðMuÞÞ ¼ 0, where we use the μ μ2 þ ≫ 2 neutrino experiments [30]. fact that ψ is real. On the other hand, for BiBi v , we can integrate out the vectorlike pair to give an effective IV. NELSON-BARR SECTOR AND THE ROLLING 3 × 3 mass squared matrix of the SM sector with POTENTIAL an Oð1Þ phase. This gets translated into an Oð1Þ phase in † the CKM matrix V ¼ Vu V (see [11]). Radiative A. Generating the rolling potential CKM L dL effects can spoil the solution to the strong CP problem As shown in [11], the rolling potential in Eq. (1), (2) can be −2 unless yψ ≲ 10 [11]. generated by a minimal modification of the SM up sector, Before going to the next section we would like to namely the addition of a vectorlike pair, ðψ; ψ cÞ, where ψ has comment that there appears to be no obvious difficulty the same quantum numbers as an up-type singlet. in extending our model along the lines of [12] to also Furthermore, as shown in [31], the same modification can address the SM flavor puzzle for the charged leptons and also give a Nelson-Barr solution [9,10] to the strong CP . At the cost of complicating our model, this can be problem, provided we impose an additional Z2 symmetry. achieved by identifying one of the intermediate sites of the The Lagrangian terms for the relevant interactions are, clockwork chain with the flavon for the charged fermions L ¼ u ˜ c þ i ψΦ c þ ˜i ψΦ c and the Abelian symmetry at this site with a Froggatt- NB YijQHu yψ Nui yψ Nui Nielsen flavor symmetry. In order not to generate a θ¯ , þ μ ψψc þ ð Þ QCD ψ H:c: 16 the charge assignment of the SM fermions must be c anomalyfree with respect to QCD as emphasised in [12] The ψ, ψ ,andΦN are odd under the Z2 symmetry, which forbids the term QHψ c. Recall that an exact CP symmetry where an example charge assignment was also presented. ð1Þ We do not explore this direction further and stick to our has been imposed and thus all couplings are real. The U N i i more minimal setup here. symmetry is collectively broken by yψ and y˜ψ , leading to breaking of the relaxion shift symmetry and radiative gen- eration of the rolling potential in Eq. (1), (2) with, V. PARAMETER SPACE qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi First, let us consider the constraints that arise on the SRB i ˜j ð u† uÞ yψ yψ Y Y ij scenario for the right-handed neutrino current introduced in M ∼ f; ð17Þ 4π Sec. III. We fix the fk=f ratio according to Eq. (15) such qffiffiffiffiffiffiffiffiffiffi that each point in the plot gives the correct baryon k k 4π yψ y˜ψ asymmetry. The vertical green band shows the region that r ∼ : ð18Þ is ruled out by requiring, mϕ < 5HðTcÞ, the condition in roll i ˜j ð u† uÞ yψ yψ Y Y ij Eq. (5) that the relaxion does not overshoot the barriers of the backreaction potential once they reappear after reheat- Φ → Φ The 1-loop N N diagram gives the first term ing. Here we have taken the maximal value T ¼ T (see Φ2 † c sph whereas the NH H box diagram gives the second term. Sec. III). The blue shaded region corresponds to the region For the loop diagram generating the first term, we have Λ2 > 16π2v2 ruled out by the requirement (derived in [7]) ψ c c taken the cutoff for the u loop to be the mass of the that the Higgs-dependent parts of the backreaction potential ∼ clockwork radial modes, mρ f. dominate over any Higgs-independent contribution. The dashed lines show the required value of tanðϕ0=fkÞ to B. Nelson-Barr solution to the strong CP problem reproduce the correct dark matter density in Eq. (7).As ϕ0 The Lagrangian in Eq. (16) also provides a solution to explained below Eq. (7), the extent to which tan f deviates the strong CP problem. Once the relaxion stops, the phase, from unity can be interpreted as a measure of the required θ ¼hπ i ∼ ϕ 4 4 N N =f =F, enters the × matrix for the up tuning. The orange region shows fifth force constraints that sector, arise due to the fact that the relaxion mixes with the Higgs Λ4 θ ∼ c ðμψ Þ1 1 ðBÞ1 3 boson with a mixing angle (see [32]), sin f vm2 . M ¼ × × ; ð19Þ k h u ð0Þ ð uÞ The rest of the constraints arise from the implementation 3×1 vY 3×3 pffiffiffi of the Nelson-Barr mechanism in Sec. IV. First of all, from i iθN i −iθN where Bi ¼ fðyψ e þ y˜ψ e Þ= 2. The phase, θN,is Eq. (18) we see that for a given value of f and yψ one can nothing but the phase of the cosine of the rolling potential at fix the value of the Higgs mass cutoff, M, giving us the the relaxion stopping point. Note that the bottom-left scale on the right-hand side of the frame. The red band at element in the above mass matrix is zero due to the the top shows the region where the value of the cutoff, M, absence of the QHψ c term in the Lagrangian, which in exceeds the upper bound imposed in Eq. (4). The grey band

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VI. DISCUSSION AND CONCLUSION We have presented a simultaneous solution to five BSM puzzles: namely the lightness of the in the absence of TeV scale new physics, dark matter, matter- antimatter asymmetry, neutrino masses and the strong CP problem. While our construction is admittedly more involved than some other attempts to solve BSM puzzles in a unified way [1,2,5,6], this is because we also use cosmological relaxation to achieve the challenging task of obtaining a light Higgs boson without adding any TeV scale states that cutoff the Higgs mass divergence. Our construction has all the ingredients of a standard relaxion model—such as a chain of clockwork scalars and a TeV- scale strong sector—but beyond this we only make minimal modifications by adding three RH neutrinos and FIG. 2. The parameter space for an all-in-one relaxion in the an up-type SUð2Þ singlet vectorlike quark pair. ð Þ L mϕ;f plane. For each value of f, taking the maximal value Our all-in-one relaxion setup gives a diverse set of ¼ 10−2 yψ , the value of the Higgs mass cut-off, M, is fixed as observational predictions. Our construction predicts the shown on the right-hand side of the frame [see Eq. (17)]. The red absence of a CP-violating phase in the neutrino mass band shows the region violating the bounds imposed by cosmo- matrix and GeV-scale sterile neutrinos potentially close to logical requirements in [7]; blue denotes the region where the the reach of future experiments such as SHiP [29]. The Higgs-independent contributions to the backreaction are no longer subdominant [7]; the orange region shows the fifth-force strong CP phase in our model is nonzero and may be exclusions due to mixing of ϕ with the Higgs [32]; green denotes detectable in future experiments. The finite allowed param- the region in which the relaxion overshoots the backreaction eter space in Fig. 2 can also be probed by future atomic barriers after reheating (see Sec. II) for the maximal value Tc ¼ physics experiments [33,34] or future improvements in fifth Tsph and the purple dashed lines show the tuning needed to force experiments. Finally, we would like to point out the reproduce the correct relic density [8]. The grey band at the interesting trade-off that exists in Fig. 2 between the Higgs bottom shows the region where the relaxion mechanism is unable mass cutoff scale and the tuning required to reproduce the to raise the Higgs cut-off beyond 2 TeV. Finally, the region to the correct abundance of dark matter. The least tuned regions left of the dashed line may be probed in future atomic physics correspond to cutoff values smaller than 100 TeV, a scale experiments if there is an overdensity of relaxion dark matter where top partners in a full solution to the hierarchy around the Earth [33,34]. problem can be seen in future high energy colliders. We conclude by mentioning an interesting future direc- at the bottom shows the region where the relaxion mecha- tion. Our construction involves the standard relaxion nism is unable to raise the Higgs cutoff beyond 2 TeV. mechanism, which utilises Hubble friction to stop the Finally, the region to the left of the dashed line may be relaxion. It will be interesting to see if some of our ideas probed in future atomic physics experiments if there is an can be implemented in alternative models involving particle overdensity of relaxion dark matter around the earth production [35–38], which are attractive because they [33,34]. We see from Fig. 2 that after all the constraints decouple the relaxion mechanism from inflation. are imposed, a finite allowed region remains that remark- ably contains the region in which tuning to obtain the ACKNOWLEDGMENTS correct relic density for dark matter is minimal. We will comment in the next section on how this allowed We would like to thank S. Abel, J. Scholtz and A. Titov region can be probed further by future experiments. for very useful discussions.

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