Running the running

We use the recent observations of Cosmic Microwave Background temperature and polarization anisotropies provided by the Planck satellite experiment to place constraints on the running $\alpha_$\mathrm{s}$ = $\mathrm{d}$n_{$\mathrm{s}$} / $\mathrm{d}$\log k$andtherunningoftherunning$\beta_{$\mathrm{s}$} = $\mathrm{d}$\alpha_{$\mathrm{s}$} / $\mathrm{d}$\log k$ofthespectralindex$n_{$\mathrm{s}$}$ofprimordialscalarfluctuations\mathrm{.}Wefind$\alpha_$\mathrm{s}$=0.011\pm0.010$and$\beta_$\mathrm{s}$=0.027\pm0.013$at$68%,$\mathrm{C}\mathrm{L}$$, suggesting the presence of a running of the running at the level of two standard deviations. We find no significant correlation between ${\beta }_{\mathrm{s}}$ and foregrounds parameters, with the exception of the point sources amplitude at $143\mathrm{G}\mathrm{H}\mathrm{z}$, $A_{143}^{PS}$, which shifts by half sigma when the running of the running is considered. We further study the cosmological implications of such preference for ${\alpha }_{\mathrm{s}},{\beta }_{\mathrm{s}}\sim 0.01$ by including in the analysis the lensing amplitude $A_L$, the curvature parameter ${\mathrm{\Omega }}_k$, and the sum of neutrino masses $\sum m_{\nu }$. We find that when the running of the running is considered, Planck data are more compatible with the standard expectations of $A_L=1$ and ${\mathrm{\Omega }}_k=0$ but still hint at possible deviations. The indication for ${\beta }_{\mathrm{s}}>0$ survives at two standard deviations when external datasets such as BAO and CFHTLenS are included in the analysis, and persists at $\sim 1.7$ standard deviations when CMB lensing is considered. We discuss the possibility of constraining ${\beta }_{\mathrm{s}}$ with current and future measurements of CMB spectral distortions, showing that an experiment like PIXIE could provide strong constraints on ${\alpha }_{\mathrm{s}}$ and ${\beta }_{\mathrm{s}}$.