$i$-incidental $N$-naturalness

TL;DR

The paper extends the $N$-naturalness framework by introducing an $i$-incidental realization in which our Standard Model resides in a heavier SM-like sector ($i_{ m SM}>0$). It shows that resonant reheaton–Higgs mixing generically selects the sector to be preferentially reheated, yielding a viable cosmology with a light SM-like sector as the SM and subdominant additional sectors. The authors analyze cosmological constraints from $\Delta N_{ m eff}$ and heavy relic overclosure, finding substantial allowed parameter space for $i_{ m SM}$ up to a few thousand, and predict a rich stochastic gravitational wave background from many QCD-like phase transitions in the light exotic sectors. They also discuss collider prospects for the reheaton, concluding that direct detection is difficult due to the required tiny mixing, but highlight potential indirect signatures and UV completions that could alter this outlook. Overall, the work identifies a robust, testable cosmological and gravitational-wave-rich pathway to address the hierarchy problem via resonant reheating across a multi-sector landscape.

Abstract

$N$-naturalness is a novel solution to the electroweak hierarchy problem which posits $N$ copies of the Standard Model with varying Higgs mass-squared parameters. Reheating proceeds through a "reheaton" particle that deposits most of its energy density into the Standard Model and small but potentially measurable fractions into the other copies. Typically the sector with the lightest negative Higgs mass-squared is identified as the Standard Model. We demonstrate that $N$-naturalness admits a broader class of realizations in which the Standard Model is identified with a heavier sector, rather than being restricted to the lightest. This is made possible by resonant mixing between the reheaton and the Higgs, which generically causes one sector to be preferentially reheated and to acquire the largest share of the energy density, singling it out as the Standard Model. We demonstrate that this scenario is consistent with current cosmological bounds on new relativistic degrees of freedom and overclosure constraints from heavy stable relics, while future cosmic microwave background and high redshift surveys will probe significant portions of the remaining parameter space. Furthermore, we highlight the possibility of a novel stochastic gravitational wave spectrum from the many cosmological first order QCD phase transitions occurring across the other sectors.

Figures (5)

• Figure 1: Parameter space for the original $N$-naturalness model ($i_{\rm SM} = 0)$. Colored regions denote the sectors that acquire the largest share of the energy density after reheating, which are labeled by sector index. For low reheaton masses, the SM $(i = 0)$ dominates for much of the parameter space (yellow). The lightest exotic sector $(i = -1)$ dominates at large values of $r$ and low $m_\phi$ where on-shell two-body reheaton decays to pairs of Higgs doublets are allowed (gray). As the reheaton mass increases, it will resonantly mix with the Higgs boson of one of heavier SM-like sectors ($i > 0)$, efficiently populating that sector (red bands).
• Figure 2: Parameter space for $i$-incidental $N$-naturalness for $i_{\rm SM} = 10$ (left panel) and $i_{\rm SM} = 100$ (right panel). As in Fig. \ref{['fig:iSM-0']}, colored regions denote the sectors that acquire the largest share of the energy density after reheating, which are labeled by sector index. The reheaton predominantly populates our SM due to resonant mixing with the SM Higgs when $m_\phi \approx 125$ GeV (yellow). As the reheaton mass is varied away from the SM Higgs mass, it undergoes strong resonant mixing with the Higgs boson of another SM-like sector $i$ (red bands), thus dominantly populating that sector. Only for relatively light reheatons can it dominantly populate the lightest SM-like sector ($i = 0$) or the lightest exotic sector ($i = -1$) (gray).
• Figure 3: Constraints on $N$-naturalness from $\Delta N_{\rm eff}^{\rm CMB}$ measurements for $i_{\rm SM} = 5$ (top left), $10$ (top right), $100$ (bottom). In each plot we display the current ${\rm Planck+Lensing+BAO}$ bound, $\Delta N_{\rm eff}^{\rm CMB} < 0.3$Planck:2015fie (allowed region lies between the solid blue lines) and the projected sensitiviy from future CMB and large redshift surveys $\Delta N_{\rm eff}^{\rm CMB} < 0.02$Sailer:2021yzmMacInnis:2023vifCMB-S4:2016ple (dashed blue lines). As in Figs. \ref{['fig:iSM-0']},\ref{['fig:iSM-10-100']}, colored shaded regions denote the sectors that acquire the largest share of the energy density after reheating, which are labeled by sector index.
• Figure 4: Fraction of the SM band that satisfies the bound $\Delta N_{\rm eff}^{\rm CMB}<0.3$ from ${\rm Planck+Lensing+BAO}$ (blue dots) and the sensitivity of future surveys $\Delta N_{\rm eff}^{\rm CMB}<0.02$ (red dots) Also shown is the approximate semi-analytic estimate of the band valid for large $i_{\rm SM}$, Eq. (\ref{['eq:fraction']}) (corresponding solid blue and red curves).
• Figure 5: GW spectra from $N$-naturalness for $i_{\rm SM} = 50$. The figures display the spectra from the 25 lightest exotic sectors, with colors ranging from red to orange to indicate increasingly heavy sectors. In each figure, the blue curve denotes the total observed GW spectrum, Eq. (\ref{['eq:GW-total']}). Top left: GW spectra in the minimal $N$-naturalness model for the runaway scenario, Eq. \ref{['eq:runaway']}. Top right: Same as top left, but for the non-runaway scenario, Eq. \ref{['eq:nonrunaway']}. Bottom left: GW spectra in a scenario with non-universal $SU(3)_c$ gauge couplings in each sector for the runaway benchmark, Eq. \ref{['eq:runaway']}. Bottom right: Same as bottom left, but for the non-runaway scenario. For the non-universal scenarios, random values of $\delta_i = \delta \alpha_{s_i}/\alpha_s \in \{-0.2,0.2\}$ are chosen, leading to correspondingly different sector confinement scales and peak frequencies according to Eqs. (\ref{['eq:modified_lambdaQCDi']}) and (\ref{['eq:f-peak-0']}), respectively. Also shown are sensitivities from various current and future GW observatories; see the main text for further details. The Higgs-reheaton mass splitting (\ref{['eq:h-phi-splitting']}) is fixed to be $\Delta m_{h\phi} = 90$ MeV ($110$ MeV) for the runaway (non-runaway) scenario. Each of these benchmarks predict $\Delta N^{\rm CMB}_{\rm eff} \approx 0.3$.