The world is not enough (neither the Standard Model)

.... according to our standard way to look at it

Leaving for a while my tentative defence and illustration of spectral noncommutative geometry I elect to focus today on a specific Standard Model extension called SMASH by its authors that I regard as worthy of report for their discussing with some detail a quite minimal inflation scenario relying on a complex singlet scalar with a ultra-high scale vacuum expectation value involved both in the  seesaw mechanism and a Peccei-Quinn symmetry breaking:

... new physics beyond the Standard Model (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) [1]. 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 [2]. The SM also lacks enough CP violation to explain why the Universe contains a larger fraction of baryonic matter than of antimatter. 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 θ-parameter of quantum chromodynamics (QCD) is constrained from measurements of the neutron electric dipole moment 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 10¹¹ GeV. The SM extension we consider consists just of the KSVZ axion model [3, 4] and three right-handed (RH) heavy SM-singlet neutrinos [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 [46]. 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 [61].]. This extra matter content was recently proposed in [6], where it was emphasized 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 stabilize the effective potential of the SM at high energies thanks to a threshold mechanism [7, 8]. This extension also leads to successful primordial inflation by using the modulus of the KSVZ SM singlet scalar field [9]. 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...

THE SMASH MODEL

We extend the SM with a new complex singlet scalar field σ and two Weyl fermions Q and Q in the 3 and 3 representations of SU(3)c and with charges −1/3 and 1/3 under U(1)Y . With these charges, Q and Q can decay into SM quarks, which ensures that they will not become too abundant in the early Universe. We also add three RH fermions N_i. The model is endowed with a new Peccei-Quinn (PQ) global U(1) symmetry, which also plays the role of lepton number. The charges under this symmetry are: q(1/2), u(−1/2), d(−1/2), L(1/2), N(−1/2), E(−1/2), Q(−1/2) Q(−1/2), σ(1); and the rest of the SM fields (e.g. the Higgs) are uncharged. The new Yukawa couplings are: L ⊃ −[F_ijL_iHN_j + 1/2 Y_ijσN_iN_j + yQσQ + y_QdiσQd_i + h.c.]. The two first terms realise the seesaw mechanism once σ acquires a vacuum expectation value (VEV) <σ> = v_σ/√2, giving a neutrino mass matrix of the form m_v = −FY ^-1F^t v²/(√2 v_σ), with v = 246 GeV. The strong CP problem is solved as in the standard KSVZ scenario, with the role of the axion decay constant, f_A, played by v_σ = f_A. Due to non-perturbative QCD effects, the angular part of σ = (ρ + v_σ) exp(iA/f_A)/√2, the axion field A, gains a potential with an absolute minimum at A = 0. At energies above the QCD scale, the axion-gluon coupling is L ⊃ −(α_S/8π)(A/f_A)GG, solving the strong CP problem when <A> relaxes to zero. The latest lattice computation of the axion mass gives m_A=(57.2 ± 0.7)(10¹¹GeV/f_A) µeV [23].

INFLATION

The scalar sector of the model has the potential 

V(H,σ) = λ_H ( H^†H − v²/2)² + λ_σ (|σ|² − v_σ² /2)² + 2λ_Hσ (H^†H − v²/2)(|σ|² − v_σ² /2). (1) 

In the unitary gauge, there are two scalar fields that could drive inflation: h, the neutral component of the Higgs doublet H^t=(0, h)/√2, and the modulus of the new singlet, ρ = √2|σ|.

... inflation in SMASH is mostly driven by ρ, with a non-minimal coupling 2×10^-3≲ξ_σ≲1. The upper bound on ξ_σ ensures that the scale of perturbative unitarity breaking is at M_P (provided that also ξ_H ≲1), whereas the lower bound on ξ_σ corresponds to a tensor-to-scalar ratio r ≲ 0.07 (as constrained by the Planck satellite and the BICEP/Keck array [1, 29]). Neglecting ξ_H, predictive slow-roll inflation in SMASH in the Einstein frame can be described by a single canonically normalized field χ with potential  

V(χ) = (λ/4)ρ(χ)⁴ (1 + ξ_σ ρ(χ)²/M_P²)^-2 , (2)  

where λ can be either λ_σ or  λ_σ =λ_σ −λ_Hσ ²/λ_H, with the second case being possible only if λ_Hσ < 0, corresponding to an inflationary valley in a mixed direction in the plane (ρ, h). The field χ is the solution of Ω²dχ/dρ≃ (bΩ² + 6ξ_σ² ρ²/M²_P)^1/2, where Ω≃ 1 + ξ_σ ρ²/M_P² is the Weyl transformation into the Einstein frame and b=1 for λ=λ_σ  or b=1+|λ_Hσ  /λ_H|∼ 1 for λ=λ_σ. The value of b determines the angle in field space described by the inflationary trajectory: h²/ρ²≃ b −1. The predictions in the case λ = λ_σ (or b → 1) for r vs the scalar spectral index n_s are shown in FIG. 1 for various values of ξ_σ. The running of n_s is in the ballpark of 10^-4–10^-3, which may be probed e.g. by future observations of the 21 cm emission line of Hydrogen [30]. 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 r and n_s, fully consistent (and predictive) inflation in SMASH occurs if 10^-13≲ λ≲10^-9.

FIG. 1. The tensor-to-scalar ratio, r, v_s the scalar spectral index, n_s, at k=0.002Mpc^-1 for the SMASH inflationary potential (2), assuming λ_Hσ≪ λ_H. The color coded contours represent current observational constraints at 68% and 95% CL from [1]. The threading of thin continuous lines indicates the number e-folds N from the time the scale k = 0.002 Mpc^-1 exits the horizon to the end of inflation. Lines of constant ξσ are shown dotted. The thick black line takes into account the fact that after inflation the Universe enters a radiation era. The line identified as “quartic inflation” shows the prediction of N for a purely quartic monomial potential (ξ_σ → 0), which is ruled out by the data.

STABILITY

For the measured central values of the Higgs and top quark masses, the Higgs quartic coupling of the SM becomes negative at h = Λ_I ∼ 10¹¹ GeV [31]. If no new physics changes this behaviour, Higgs inflation is not viable, since it requires a positive potential at Planckian field values...

Remarkably, the Higgs portal term ∝ λ_Hσ  in (1) allows absolute stability (even when the corresponding low-energy SM potential would be negative if extrapolated to large h) via the threshold-stabilisation mechanism of [7, 8, 22]. In SMASH, instabilities could also originate in the ρ direction due to quantum corrections from the RH neutrinos and KSVZ fermions. For λ_Hσ  > 0, absolute stability requires  

 λ_H, λ_σ   > 0, for h < √2 Λ_h and λ_H, λ_σ  > 0, for h > √2 Λ_h , (3) 

where we define Λ_h² = λ_Hσ  v_σ² /λ_H, λ_H = λ_H − λ_Hσ ²/λ_σ  and λ_σ  = λ_σ  −λ_Hσ ²/λ_H . Instead, for λ_Hσ < 0, the stability condition is λ_H, λ_σ > 0, for all h [33]. 

An analysis based on two-loop renormalization group (RG) equations for the SMASH couplings and one-loop matching with the SM [22] shows that stability can be achieved for δ ≡ λ_Hσ ²/λ_σ  between 10^-3 and 10^-1, depending on m_t ... The Yukawas must satisfy the bound 6y⁴ + ∑Y_ii⁴ ≲ 16π² 2λ_σ / log(30M_P/√2 λ_σv_σ). It will prove convenient to define SMASH benchmark units:  

λ₁₀= λ_σ  /10^-10 ; δ₃ = δ /0.03 ; v₁₁= v_σ /10¹¹ GeV. (4)...

DARK MATTER

For λ_Hσ>0, the PQ symmetry is restored nonthermally after inflation and then spontaneously broken again before reheating. On the other hand, for λ_Hσ<0 and efficient reheating, the restoration and breaking are thermal. In the phase transition, which happens at a critical temperature T_c ≳ λ_σ ^1/4 v_σ  , a network of cosmic strings is formed. Its evolution leads to a low-momentum population of axions that together with those arising from the realignment mechanism [43–45] constitute the dark matter in SMASH. Requiring that all the DM is made of axions restricts the symmetry breaking scale to the range 3×10¹⁰GeV≲v_σ≲1.2×10¹¹GeV, which translates into the mass window 50 µeV≲m_A≲200 µeV, (6) updating the results of [46] with the latest axion mass data [23]. The main uncertainty now arises from the string contribution [46, 47], which is expected to be diminished in the near future [48, 49]. Importantly, the SMASH axion mass window (6) will be probed in the upcoming decade by axion dark matter direct detection experiments such as CULTASK [50], MADMAX [51], and ORPHEUS [52], see also [23, 53] and FIG. 3 for our estimates of their future sensitivity...

FIG. 3. SMASH predictions for the axion-photon coupling (thick solid horizontal line) with current bounds on axion DM (ADMX,BRF) and prospects for next generation axion dark matter experiments, such as ADMX2(3) [54], CULTASK [50], MADMAX [51], ORPHEUS [52], X3 [55], and the helioscope IAXO [56].

BARYOGENESIS

The origin of the baryon asymmetry of the Universe is explained in SMASH from thermal leptogenesis [57]. This requires massive RH neutrinos acquiring equilibrium abundances and then decaying when production rates become Boltzmann suppressed. If λ_Hσ <0, then T_reheating > T_c for stable models in the DM window (5). The RH neutrinos become massive after the PQ phase transition, and those with masses M_i < Tc retain an equilibrium abundance. The stability bound on the Yukawas Yii enforces Tc > M₁, 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 M₁ ≳ 5×10⁸ GeV [58, 59]. However, for v_σ as in (5), this is just borderline compatible with stability. Nevertheless, leptogenesis can occur with a mild resonant enhancement [60] for a less hierarchical RH neutrino spectrum, which relaxes the stability bound and ensures that all the RH neutrinos remain in equilibrium after the phase transitions.