# Inflection-point inflation in hyper-charge oriented U(1)$_X$ model

## TL;DR

This paper explores inflection-point inflation within a U(1)$_X$ extended Standard Model, linking inflationary dynamics to collider phenomenology and predicting a specific running spectral index that can be tested soon.

## Contribution

It demonstrates how inflection-point inflation can be realized in a minimal U(1)$_X$ extension of the Standard Model, connecting inflationary parameters with collider constraints.

## Key findings

- Successful inflation requires specific mass ratios among new particles.
- The model predicts a running spectral index of approximately -2.7×10^{-3}.
- Compatibility with LHC Run-2 di-lepton resonance searches is analyzed.

## Abstract

Inflection-point inflation is an interesting possibility to realize a successful slow-roll inflation when inflation is driven by a single scalar field with its value during inflation below the Planck mass ($\phi_I \lesssim M_{Pl}$). In order for a renormalization group (RG) improved effective $\lambda \phi^4$ potential to develop an inflection-point, the running quartic coupling $\lambda(\phi)$ must exhibit a minimum with an almost vanishing value in its RG evolution, namely $\lambda(\phi_I) \simeq 0$ and $\beta_{\lambda}(\phi_I) \simeq 0$, where $\beta_{\lambda}$ is the beta-function of the quartic coupling. In this paper, we consider the inflection-point inflation in the context of the minimal gauged U(1)$_X$ extended Standard Model (SM), which is a generalization of the minimal U(1)$_{B-L}$ model, and is constructed as a linear combination of the SM U(1)$_Y$ and U(1)$_{B-L}$ gauge symmetries. We identify the U(1)$_X$ Higgs field with the inflaton field. For a successful inflection-point inflation to be consistent with the current cosmological observations, the mass ratios among the U(1)$_X$ gauge boson, the right-handed neutrinos and the U(1)$_X$ Higgs boson are fixed. Focusing on the case that the extra U(1)$_X$ gauge symmetry is mostly aligned along the SM U(1)$_Y$ direction, we investigate a consistency between the inflationary predictions and the latest LHC Run-2 results on the search for a narrow resonance with the di-lepton final state. %In addition, the inflection-point inflation provides a unique prediction for the running of the spectral index $\alpha \simeq - 2.7 \times %10^{-3}\left(\frac{60}{N}\right)^2$ ($N$ is the e-folding number), which can be tested in the near future.