Unifying inflation with the axion, dark matter, baryogenesis and the seesaw mechanism

# Unifying inflation with the axion, dark matter, baryogenesis and the seesaw mechanism

## Abstract

A minimal extension of the Standard Model (SM) with a single new mass scale and providing a complete and consistent picture of particle physics and cosmology up to the Planck scale is presented. We add to the SM three right-handed SM-singlet neutrinos, a new vector-like color triplet fermion and a complex SM singlet scalar that stabilises the Higgs potential and whose vacuum expectation value at  GeV breaks lepton number and a Peccei-Quinn symmetry simultaneously. Primordial inflation is produced by a combination of (non-minimally coupled to the scalar curvature) and the SM Higgs. Baryogenesis proceeds via thermal leptogenesis. At low energies, the model reduces to the SM, augmented by seesaw-generated neutrino masses, plus the axion, which solves the strong CP problem and accounts for the dark matter in the Universe. The model predicts a minimum value of the tensor-to-scalar ratio , running of the scalar spectral index , the axion mass and cosmic axion background radiation corresponding to an increase of the effective number of relativistic neutrinos of . It can be probed decisively by the next generation of cosmic microwave background and axion dark matter experiments.

DESY 16-049, IPPP/16/25

## I Introduction

The Standard Model of particle physics (SM) describes with exquisite precision the interactions of all known elementary particles. In spite of intensive searches, no significant deviation from the SM has been detected in collider or other particle physics experiments Patrignani (2016). However, several long-standing problems indicate that new physics beyond the SM is needed to achieve a complete description of Nature. First of all, there is overwhelming evidence, ranging from the cosmic microwave background (CMB) to the shapes of the rotation curves of spiral galaxies, that nearly 26% of the Universe is made of yet unidentified dark matter (DM) Ade et al. (2015). Moreover, the SM cannot generate the primordial inflation needed to solve the horizon and flatness problems of the Universe, as well as to explain the statistically isotropic, Gaussian and nearly scale invariant fluctuations of the CMB 1. The SM also lacks enough CP violation to explain why the Universe contains a larger fraction of baryonic matter than of anti-matter. Aside from these three problems at the interface between particle physics and cosmology, the SM suffers from a variety of intrinsic naturalness issues. In particular, the neutrino masses are disparagingly smaller than any physics scale in the SM and, similarly, the strong CP problem states that the -parameter of quantum chromodynamics (QCD) is constrained from measurements of the neutron electric dipole moment Pendlebury et al. (2015); Schmidt-Wellenburg (2016) to lie below an unexpectedly small value: .

In this Letter we show that these problems may be intertwined in a remarkably simple way, with a solution pointing to a unique new physics scale around GeV. The SM extension we consider consists just of a KSVZ-like axion model Kim (1979); Shifman et al. (1980) and three right-handed (RH) heavy SM-singlet neutrinos 2. This extra matter content was recently proposed in Dias et al. (2014), where it was emphasised that in addition to solving the strong CP problem, providing a good dark matter candidate (the axion), explaining the origin of the small SM neutrino masses (through an induced seesaw mechanism) and the baryon asymmetry of the Universe (via thermal leptogenesis), it could also stabilise the effective potential of the SM at high energies thanks to a threshold mechanism Lebedev (2012); Elias-Miro et al. (2012). This extension also leads to successful primordial inflation by using the modulus of the KSVZ SM singlet scalar field Fairbairn et al. (2015). Adding a cosmological constant to account for the present acceleration of the Universe, this Standard Model Axion Seesaw Higgs portal inflation (SMASH) model offers a self-contained description of particle physics from the electroweak scale to the Planck scale and of cosmology from inflation until today. Although some parts of our SMASH model have been considered separately Langacker et al. (1986); Shin (1987); Shaposhnikov and Tkachev (2006); Bezrukov and Shaposhnikov (2008); Lerner and McDonald (2009); Lebedev and Lee (2011); Fairbairn et al. (2015); Boucenna et al. (2014); Bertolini et al. (2015); Salvio (2015); Kahlhoefer and McDonald (2015); Clarke and Volkas (2016); Ahn and Chun (2016), a model incorporating all of them simultaneously had not been proposed until now. Remarkably, SMASH can accommodate the constraints from cosmological observations and Higgs stability, successfully reheat the Universe, provide the correct dark matter abundance and explain the origin of the baryon asymmetry. In this Letter, we present the most important aspects and predictions of SMASH. Further details are given in Ballesteros et al. (2016).

## Ii The SMASH model

We extend the SM with a new complex singlet scalar field and a Dirac fermion , which can be split in two Weyl fermions and in the and representations of with charges and under . This ensures that can coannihilate and decay into SM quarks, thereby evading possible overabundance problems Nardi and Roulet (1990); Berezhiani et al. (1992). We also add three RH fermions . The model is endowed with a new Peccei-Quinn (PQ) global symmetry Peccei and Quinn (1977), which also plays the role of lepton number in our case. Using left-handed Weyl spinors, we denote by , and the SM quark doublet and the conjugates of the right-handed quarks of each generation ; and by and the corresponding lepton doublet and the conjugate of the right-handed lepton. Denoting the Higgs by , the charges under the PQ symmetry are: , , , , , , , , , . The most general Yukawa couplings involving the new fields are: , where is the two-component antisymmetric symbol. The Yukawa couplings and realise the seesaw mechanism once acquires a vacuum expectation value (VEV) , giving a neutrino mass matrix of the form , with GeV. The strong CP problem is solved as in the standard KSVZ scenario, with the role of the axion decay constant, , played by . Due to non-perturbative QCD effects, the angular part of , the axion field Weinberg (1978); Wilczek (1978), gains a potential with an absolute minimum at . At energies above the QCD scale, the axion-gluon coupling is , solving the strong CP problem when relaxes to zero3. The latest lattice computation of the axion mass gives  Borsanyi et al. (2016).

## Iii Inflation

Given the symmetries of SMASH, the most general renormalisable tree-level potential is

 V(H,σ) =λH(H†H−v22)2+λσ(|σ|2−v2σ2)2 Missing or unrecognized delimiter for \left (1)

In the unitary gauge, there are two scalar fields that could drive inflation: , the neutral component of the Higgs doublet , and the modulus of the new singlet, . In the context of the SM, it was proposed in Bezrukov and Shaposhnikov (2008) that could be the inflaton if it is non-minimally coupled to the scalar curvature through a term Salopek et al. (1989), with . Such a large value of is required by the constraint to fit the amplitude of primordial fluctuations and it implies that perturbative unitarity breaks down at the scale Burgess et al. (2009); Barbon and Espinosa (2009), where is the reduced Planck mass. This raises a serious difficulty for Higgs inflation, which requires Planckian values of and an energy density of order . Since new physics is expected at or below to restore unitarity, the predictivity of Higgs inflation is lost, because the effect of this new physics on inflation is undetermined. This issue affects some completions of the SM such as the MSM Asaka et al. (2005); Asaka and Shaposhnikov (2005) and the model proposed in Salvio (2015). Instead, inflation in SMASH is mostly driven by , with a non-minimal coupling , where ensures that the scale of perturbative unitarity breaking is at (provided that also ). Neglecting 4, predictive slow-roll inflation in SMASH can happen along two directions in field space: the -direction for and the line for . We call them hidden scalar inflation (HSI) and Higgs-hidden scalar inflation (HHSI), respectively. In both cases, inflation can be described in the Einstein frame by a single canonically normalised field with potential

 ~V(χ)=λ4ρ(χ)4(1+ξσρ(χ)2M2P)−2, (2)

where stands for in HSI and for in HHSI. The field is the solution of , being the Weyl transformation into the Einstein frame; and (for HSI) or (for HHSI). The small value of required for stability (see below) typically means that in HHSI, which makes impossible distinguishing in practice between HSI and HHSI from the inflationary potential. However, even a small Higgs component in the inflaton is relevant for reheating, as we will later discuss. The predictions of the potential (2) in the case (or in HHSI) for the tensor-to-scalar ratio vs the scalar spectral index are shown in FIG. 1 for various values of .

In SMASH, the equation of state (EOS) of the Universe after inflation is (like radiation) uninterruptedly until the standard epoch of matter-radiation equality is reached; see the reheating section below. This allows to compute the number of e-folds of inflation, , for any comoving scale, , matching precisely the predictions for the inflationary spectrum with the observations of the CMB Liddle and Leach (2003). This determines the thick line of FIG. 1 as the SMASH prediction for and at the fiducial scale Mpc, which we use through the Letter for all the primordial inflationary parameters. The prediction spans , depending on , and its width ( e-fold) quantifies the small uncertainty on the transient regime from the end of inflation to radiation domination.

Note that the the condition corresponds to , which is within the planned sensitivities of PIXIE Kogut et al. (2011), LiteBird Matsumura et al. (2013), CMB-S4 Abazajian et al. (2016) and COrE+ (which will measure with an error of ). The joint constraints of the Planck satellite and the BICEP/Keck array Ade et al. (2015, 2016b) give at 95% CL, corresponding in SMASH to . Taking into account the former constraints, the spectral index at lies in the interval , and its running lies in the range , which may be probed e.g. by future observations of the 21 cm emission line of Hydrogen Mao et al. (2008). Since inflation is effectively single-field slow-roll, non-Gaussian features are suppressed by Acquaviva et al. (2003); Maldacena (2003). These values of the primordial parameters are perfectly compatible with the latest CMB data, and the amount of inflation that is produced solves the horizon and flatness problems. Given the current bounds on and , and the fact that fitting the amplitude of primordial scalar fluctuations requires , fully consistent (and predictive) inflation in SMASH occurs if .

## Iv Stability

For the measured central values of the Higgs and top quark masses Patrignani (2016), the Higgs quartic coupling of the SM becomes negative at GeV 5. If no new physics changes this behaviour, Higgs inflation is not viable, since it requires a positive potential at Planckian field values. Moreover, this instability is a problem even if another field drives inflation. This is because scalars that are light compared to the Hubble scale, , acquire fluctuations of order . These can make the Higgs field move into the instability region of the potential, which would contradict the present electroweak vacuum 6. Remarkably, the Higgs portal term in (III) allows stability of the SMASH potential via the threshold-stabilisation mechanism of Lebedev (2012); Elias-Miro et al. (2012), which relies on a nontrivial matching with the SM potential at low energies. The matched Higgs quartic in the SM is , where the threshold correction is . Even if the running of in the SM makes it negative, the actual Higgs quartic coupling in the UV theory, , can remain positive provided that is large enough. A more detailed analysis 7 shows that, for , absolute stability requires Ballesteros et al. (2016)

 {~λH,~λσ>0,forh<√2ΛhλH,λσ>0,forh>√2Λh, (3)

where and all the couplings run with the beta functions of SMASH, not the SM. The scale arises as the divide between large and small field values of , for which cannot be neglected and the quadratic interactions are relevant, as can be seen from (III). Instead, for , the stability condition is just , for all . The Higgs direction is the one most prone to be destabilised (from top loops) and the potential must remain positive beyond the values needed for inflation. A one-loop analysis shows that a value of above (depending on the top mass, see FIG. 2) ensures stability up to for a Higgs mass of  GeV. Finally, in SMASH, instabilities could also originate in the direction of due to quantum corrections from and . Stability in this direction, requires 8.

## V Reheating

SMASH provides a complete model of cosmology for which the evolution after inflation can be calculated. The PQ symmetry is spontaneously broken during inflation by the large evolving value of . Slow-roll inflation ends at , where the effect of is negligible. Since , the inflaton starts to undergo Hubble-damped oscillations in a quartic potential.

The first oscillations of the inflaton constitute a phase of so-called preheating Kofman et al. (1997), during which fluctuations of in the direction orthogonal to the inflaton increase exponentially. The post-inflationary background can be understood as a homogeneous condensate of particles with energy given by the oscillation frequency , where is the scale factor of the Universe and denotes cosmic time 9. In SMASH, is the weakest coupling and thus SM particles coupled to the inflaton have effective masses , which are much larger than except when . Higgs particles and electroweak bosons could in principle be produced by parametric resonance Greene et al. (1997) at these crossings but they either have large self-interactions or decay very efficiently into SM fermions. In contrast, the effective mass of excitations is , which allows them to grow by parametric resonance. The growth of fluctuations of a complex inflaton field in a quartic potential was studied analytically in Greene et al. (1997) and numerically in Tkachev et al. (1998). Our own numerical simulations Ballesteros et al. (2016) corroborate their results. After the first oscillations after inflation, the fluctuations of become as large as the inflaton amplitude , so the PQ symmetry is non-thermally restored. Only if were larger than would the field get trapped around its minimum before the non-thermal restoration can occur. However, such high values of are ruled out by CMB axion isocurvature constraints Fairbairn et al. (2015) 10.

Aside from these common features, reheating progresses differently for HSI and HHSI. The reason is that the small Higgs component of the inflaton in HHSI (which is lacking in HSI) accelerates in that case the production of SM particles. We will now discuss the two cases separately.

Reheating for HSI (): During preheating, Higgs bosons are non-resonantly produced during inflaton crossings because of the large value of the Higgs self-coupling Anisimov et al. (2009), as well as the fast decay of Higgses into tops and gauge bosons. When the PQ symmetry is non-thermally restored, the induced Higgs mass stabilises around a large value , thus blocking Higgs production. Efficient reheating has to wait until the spontaneously symmetry breaking (SSB) of the PQ symmetry, i.e. when becomes . We have simulated numerically the phase transition, finding that the energy initially stored in fluctuations becomes equipartitioned into axions and particles. The latter can soon decay into Higgses and reheat the SM sector. The corresponding reheating temperature is , where we introduce SMASH benchmark values:  GeV), , 11. The accompanying axions are relativistic and remain decoupled from such a low temperature SM thermal bath Graf and Steffen (2011). They contribute to the late Universe expansion rate as extra (relativistic) neutrino species. We estimate above the SM value Mangano et al. (2002). Current CMB and baryon acoustic oscillation data give at 68% CL Ade et al. (2015), disfavouring HSI.

Reheating for HHSI (): As in HSI, the direct production of Higgs excitations stops when the PQ symmetry is non-thermally restored. However, the Higgs component of the inflaton continues to oscillate around so that and gauge bosons can still be produced during crossings. The fast decay of into light fermions when moves away from zero prevents their exponential accumulation but makes the comoving energy in light fermions increase. When light particles thermalise, a population of bosons is created by the thermal bath during crossings (when their mass is below the temperature) and decays when their mass grows with . This mechanism enhances the drain of energy from the inflaton to the SM bath. Using Boltzmann equations with thermal and non-thermal sources, and accounting for the energy loss of the background fields, we have calculated numerically the reheating temperature, finding GeV) for the values of and satisfying the requirements for inflation and stability.

The critical temperature for the PQ phase transition is Ballesteros et al. (2016). For SMASH benchmark values , and requiring the previous stability bound on the Yukawa couplings of the new fermions, . Therefore, the PQ symmetry, which had been non-thermally restored by preheating, is also restored thermally at the end of reheating. A few Hubble times after, the temperature drops below and the PQ symmetry becomes spontaneously broken, this time for good. We thus predict a thermal abundance of axions, which decouple at where Masso et al. (2002); Graf and Steffen (2011); Salvio et al. (2014). Considering relativistic degrees of freedom at axion decoupling we get , which is much smaller than in HSI and in good agreement with current data. This small value of could be probed with future CMB polarisation experiments Abazajian et al. (2015); Errard et al. (2016). As discussed in Baumann et al. (2016), a non-detection of new thermal relics with future CMB probes reaching will imply that if such relics exist they were never in thermal equilibrium with the SM.

Finally, we remark that the EOS of the Universe is both in the period of inflaton oscillations in a quartic potential Shtanov et al. (1995) and the non-thermally PQ restored phase because the evolution is conformal in a quartic potential. This is so both for HHSI and HSI. However, in HSI, there is a small period of matter domination before the particles decay to reheat the SM, whose effects on are within the uncertainties.

## Vi Dark matter

At the spontaneous breaking of the PQ symmetry, a network of cosmic strings is formed both in HHSI and HSI. In the first case, this happens by the standard Kibble mechanism in thermal equilibrium Kibble (1980) and in the second, non-thermally Tkachev et al. (1998). The evolution of the network leads to a population of low-momentum axions that together with those arising from the realignment mechanism Preskill et al. (1983); Abbott and Sikivie (1983); Dine and Fischler (1983) constitute the dark matter in SMASH. Requiring that all the DM is made of axions demands

 3×1010GeV≲vσ≲1.2×1011GeV, (4)

which translates into the mass window

 50μeV≲mA≲200μeV, (5)

where we have updated the results of Kawasaki et al. (2015) with the latest axion mass data Borsanyi et al. (2016). The main uncertainty arises from the string contribution Kawasaki et al. (2015); Fleury and Moore (2016a), which we estimate as 3-4 times larger than the misalignment one; the uncertainty is expected to be diminished in the near future Moore (2016); Fleury and Moore (2016b). The SMASH axion mass window (5) will be probed in the upcoming decade by direct detection experiments such as MADMAX Redondo (2016); Caldwell et al. (2016) and ORPHEUS Rybka et al. (2015). A sizeable part of the DM in this scenario may be in the form of axion miniclusters Hogan and Rees (1988), which offer interesting astrophysical signatures Kolb and Tkachev (1996); Tinyakov et al. (2016).

## Vii Baryogenesis

The origin of the baryon asymmetry of the Universe is explained in SMASH from thermal leptogenesis Fukugita and Yanagida (1986). This requires the massive RH neutrinos, , acquiring equilibrium abundances and then decaying when their production rates become Boltzmann suppressed. As we have seen, in HHSI, for stable models in the DM window (5). The RH neutrinos become massive after the PQ SSB, and those with masses retain an equilibrium abundance. The stability bound on the Yukawa couplings enforces , so that at least the lightest RH neutrino stays in equilibrium. Moreover, the annihilations of the RH neutrinos tend to be suppressed with respect to their decays. This allows for vanilla leptogenesis from the decays of a single RH neutrino, which demands GeV Davidson and Ibarra (2002); Buchmuller et al. (2002). However, for as in (4), this is just borderline compatible with stability. Nevertheless, leptogenesis can occur with a mild resonant enhancement Pilaftsis and Underwood (2004) for a less hierarchical RH neutrino spectrum, which relaxes the stability bound and ensures that all the RH neutrinos remain in equilibrium after the PQ SSB.

## Viii Future perspectives

SMASH provides very clear predictions, which will be tested by the next generation of CMB, large scale structure and axion DM experiments. The model predicts a correlation between , and a small negative value of , as well as tiny non-Gaussianities. It also implies the existence of a cosmic background of relativistic axions which may be detected with future CMB polarisation experiments. In SMASH, the totality of the DM in the Universe is made of cold axions with mass in the range (5), which will be explored in the next decade. If all these features are met simultaneously, it will be a very compelling hint in favor of SMASH. If only one is not, the model will be ruled out. We recall that the cosmological predictions of SMASH are reliable; as opposed to those of incomplete models such as Higgs inflation, which suffers from an early breaking of perturbative unitarity.

SMASH provides an explanation for five of the most pressing problems in particle physics and cosmology: inflation, DM, baryogenesis, the strong CP problem and the smallness of neutrino masses; some of which are naturalness issues. However, the model does not solve the hierarchy problem nor the cosmological constant problem. It would be interesting to explore if e.g. some relaxation mechanism along the lines of Abbott (1985); Graham et al. (2015); Alberte et al. (2016); Arvanitaki et al. (2016) could be embedded in SMASH to solve also these problems while maintaining its minimality.

Acknowledgments. We thank F. Bezrukov, A. G. Dias, J. R. Espinosa, D. Figueroa, F. Finelli, J. Garcia-Bellido, J. Jaeckel, F. Kahlhöfer, B. Kniehl, J. Lesgourgues, K. Saikawa, M. Shaposhnikov, B. Shuve, S. Sibiryakov and A. Westphal for discussions. The work of G.B. is funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement number 656794 and was partially supported by the German Science Foundation (DFG) within the Collaborative Research Center SFB 676 âParticles, Strings and the Early Universe.â G.B. thanks the DESY Theory Group and the CERN Theory Department for hospitality. J.R. is supported by the Ramon y Cajal Fellowship 2012-10597 and FPA2015- 65745-P (MINECO/FEDER). G.B. and C.T. thank the Mainz Institute for Theoretical Physics for hosting them during a workshop. C.T. thanks MIAPP for hospitality while attending a programme.

### Footnotes

1. If the Higgs field has a large coupling to the curvature , inflation might be obtained Bezrukov and Shaposhnikov (2008). However, as we will argue, this requires physics beyond the SM.
2. One may also chose alternatively the DFSZ axion model. The inflationary predictions in this model stay the same, but the window in the axion mass will move to larger values Kawasaki et al. (2015). Importantly, in this case the PQ symmetry is required to be an accidental rather than an exact symmetry in order to avoid the overclosure of the Universe due to domain walls ?
3. Since the quarks have hypercharge and the PQ charge assignments are different than in the standard KSVZ scenario, the axion has a non-standard coupling to the photon, as well as a coupling to neutrinos Ballesteros et al. (2016)
4. Taking into account radiative corrections to and one can check that the window ensures that can be neglected with respect to
5. is very sensitive to small variations of the top mass, to the extent that the potential may be completely stable for sufficiently low (but still allowed) values.
6. The fluctuations can be suppressed if , which induces a large effective mass for the Higgs during inflation. In that case it would still be necessary to check that the classical trajectory of the fields does not fall in the negative region of the potential
7. At large field values, stability demands positivity of quartic couplings, but at intermediate field values, where negative quadratic interactions are important, one has to ensure positivity of the potential along the potential energy valleys
8. In this expression and in FIG. 2, we demand stability up to a large RG scale , which is sufficiently higher than during inflation..
9. We use natural units for which .
10. Reference Fairbairn et al. (2015) does not mention parametric resonance and non-thermal restoration of the PQ symmetry, because it is model dependent. The authors of Fairbairn et al. (2015) showed that a large inflaton VEV suppresses the isocurvature constraints. We have redone the analysis finding stronger bounds Ballesteros et al. (2016). Probably, ref. Fairbairn et al. (2015) used the Jordan frame VEV instead of the effective axion decay constant in the Einstein frame, . In any case, is safely excluded.
11. We will see that in this scenario axion DM requires GeV.

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