Higgs inflation and $R^2$ inflation (Starobinsky model) are two limits of the same quantum model, hereafter called Starobinsky-Higgs. We analyze the two-loop action of the Higgs-like scalar $\varphi$ in the presence of (1) nonminimal coupling ($\xi$) and (2) quadratic curvature terms. The latter are generated at the quantum level with $\varphi$-dependent couplings ($\tilde{\alpha }$) even if their tree-level couplings ($\alpha$) are tuned to zero. Therefore, the potential always depends on both Higgs field $\varphi$ and scalaron $\rho$; hence, multifield inflation is a quantum consequence. The effects of the quantum (one- and two-loop) corrections on the potential $\widehat{W}(\varphi ,\rho )$ and on the spectral index are discussed, showing that the Starobinsky-Higgs model is, in general, stable in their presence. Two special cases are also considered: first, for a large $\xi$ in the quantum action, one can integrate $\varphi$ and generate a “refined” Starobinsky model which contains additional terms ${\xi }^2{R}^2{\ln }^p(\xi |R|/{\mu }^2)$, $p=1$, 2 ($\mu$ is the subtraction scale). These generate corrections linear in the scalaron to the “usual” Starobinsky potential and a “running” scalaron mass. Second, for a small fixed Higgs field ${\varphi }^2\ll M_p^2/\xi$ and a vanishing classical coefficient of the $R^2$ term, we show that “usual” Starobinsky inflation is generated by quantum corrections alone, for a suitable nonminimal coupling ($\xi$).

• Received 4 August 2018

DOI:https://doi.org/10.1103/PhysRevD.98.103524

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Published by the American Physical Society