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 Particle Physics 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 naturalness, 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 fermion 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 baryons 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 Standard Model (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 boson 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 fermions 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 baryon 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 axion 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 þÁÁþ : ð8Þ generation of both the observed dark matter abundance as 3j 3 3N well as the baryon asymmetry.