Removability criterion for radiative corrections to the Starobinsky attractor in weakly nonlocal gravity

Abstract

Radiative corrections at the percent level can shift the predictions of $\alpha$-attractor and Starobinsky-like models into the region of the $(n_s,r)$ plane preferred by ACT DR6. We show that the decisive property is not the size of a correction but whether it is removable. In the slow-roll $N$-formalism the observables are functionals of $\eps(N)$, and equality of $\eps(N)$ on an interval reconstructs the potential up to an amplitude rescaling and a shift of the canonical field. A larger class, selected by $(V/V^{\prime }){\Delta }^{\prime }=\mathrm{const}$, preserves the leading predictions and provides a criterion for any proposed deformation. In weakly nonlocal completions of $R+R^2$ the improvement scale $\mu =H(\varphi )$ makes the renormalization-group time one half of the logarithm of the tree potential, so the local one-loop flow passes the test and evolves the model onto the E-model family $V\propto(1-e^{-\sqrt{2/3}\,ϕ/\mpl})^{p}$ with $p=2+K$. Since the divergent local beta functions of these super-renormalizable completions are one-loop exact, no higher-loop local running modifies this picture. The induced shifts are $\ord(\ln N/N^{2})$ in $n_s$, obtained in closed form for the removable part and numerically for the rest. The number of $e$-folds, which the criterion does not protect, contributes at the \red{$22\%$} level with an opposite sign. Matter-induced corrections fail the same test at leading order, which is why they shift the predictions. Among the deformations proposed since ACT DR6, an $R^n$ term in $f(R)$ gives $(V/V')Δ'=\ord(N^{n-1})$, so only $n=2$ is removable, and no deformation that increases $n_s$ satisfies the criterion. For Standard Model matter, the removable channel is fixed by the Higgs nonminimal coupling alone and is $\ord(10^{-11})$, because the measured amplitude sets the coefficient of $R^2$ to $5.1\times 10^8$.

20 pages, 5 figures, 5 tables