Running Inflation in the Standard Model

TL;DR

The paper tests whether the Standard Model Higgs field, non-minimally coupled to gravity, can drive inflation when quantum corrections are included via renormalization group methods. They derive the RG-improved effective action, showing that the running of the effective Planck mass and SM couplings affects inflationary observables, notably producing a Higgs-mass–dependent spectral index n_s. Their results yield n_s around 0.968 for typical parameters, with a sharp increase toward ~0.98 as the Higgs mass approaches vacuum instability, and predict small tensor-to-scalar ratio r ~ 0.003. The work connects high-energy SM parameters to Planck-scale cosmology, offering testable predictions for PLANCK and LHC and discussing EFT validity and possible generalizations.

Abstract

An interacting scalar field with largish coupling to curvature can support a distinctive inflationary universe scenario. Previously this has been discussed for the Standard Model Higgs field, treated classically or in a leading log approximation. Here we investigate the quantum theory using renormalization group methods. In this model the running of both the effective Planck mass and the couplings is important. The cosmological predictions are consistent with existing WMAP5 data, with 0.967 < n_s < 0.98 (for N_e = 60) and negligible gravity waves. We find a relationship between the spectral index and the Higgs mass that is sharply varying for m_h ~ 120-135 GeV (depending on the top mass); in the future, that relationship could be tested against data from PLANCK and LHC. We also comment briefly on how similar dynamics might arise in more general settings, and discuss our assumptions from the effective field theory point of view.

Figures (5)

• Figure 1: The spectral index $n_s$ as a function of the Higgs mass $m_h$ for a range of light Higgs masses. The 3 curves correspond to 3 different values of the top mass: $m_t=169$ GeV (red curve), $m_t=171$ GeV (blue curve), and $m_t=173$ GeV (orange curve). The solid curves are for $\alpha_s(m_Z)=0.1176$, while for $m_t=171$ GeV (blue curve) we have also indicated the 2-sigma spread in $\alpha_s(m_Z)=0.1176\pm0.0020$, where the dotted (dot-dashed) curve corresponds to smaller (larger) $\alpha_s$. The horizontal dashed green curve, with $n_s\simeq 0.968$, is the classical result. The yellow rectangle indicates the expected accuracy of PLANCK in measuring $n_s$ ($\Delta n_s\approx 0.004$) and the LHC in measuring $m_h$ ($\Delta m_h\approx 0.2$ GeV). In this plot we have set $N_e=60$.
• Figure 2: The potential in the Einstein frame $V_E$, normalized to a reference value $V_0\equiv \lambda m_{\hbox{\tiny Pl}}^4/4\xi^2$, as a function of the Higgs field $\psi=\sqrt{\xi}\,\phi/m_{\hbox{\tiny Pl}}$. The dashed green curve is the classical case (independent of Higgs mass), the solid blue (red) curve is the quantum case with Higgs mass $m_h=126.5$ GeV ($m_h=128$ GeV). We have set $m_t=171$ GeV and $\alpha_s(m_Z)=0.1176$ for this plot. The inset focusses on the slow-roll inflationary regime.
• Figure 3: Some representative Feynman diagrams. Top row: renormalization of the conformal coupling $\xi$ with Higgs in loop (a), and renormalization of top quark's Yukawa coupling with gauge boson (b) and Higgs (c) across vertex. Bottom row: renormalization of quartic coupling $\lambda$ with Higgs (d), top quark (e), and gauge boson (f) in loop.
• Figure 4: This plot summarizes some of the most important effects of the renormalization group flow. The red curve shows the running of the quartic coupling $\lambda(t)/\lambda(0)$ for a light Higgs $m_h=126.5$ GeV. The dotted purple curve is the top running $y_t(t)/y_t(0)$ and the dot-dashed cyan curve is the commutator function $s(t)$, with $\xi= 2.3\times 10^3$ and $\mu=m_t$. The right-hand region is the slow-roll inflationary regime; here $\lambda$ rises (and so $n_s$ does too), as highlighted by the inset.
• Figure 5: The running of the spectral index $\alpha\,(\times 10^4)$ (left panel) and the tensor to scalar ratio $r\,(\times 10^3)$ (right panel) as a function of the Higgs mass $m_h$. The 3 solid curves correspond to 3 different values of the top mass: $m_t=169$ GeV (red curve), $m_t=171$ GeV (blue curve), and $m_t=173$ GeV (orange curve). The horizontal dashed green curve, with $\alpha\simeq -5.2\times 10^{-4}$ and $r\simeq 3.0\times 10^{-3}$, is the classical result. We have set $\alpha_s=0.1176$ and $N_e=60$ in this plot.