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 $\alpha_\mathrm{s} = \mathrm{d}n_{\mathrm{s}} /
\mathrm{d}\log k$ and the running of the running $\beta_{\mathrm{s}} =
\mathrm{d}\alpha_{\mathrm{s}} / \mathrm{d}\log k$ of the spectral index
$n_{\mathrm{s}}$ of primordial scalar fluctuations. We find
$\alpha_\mathrm{s}=0.011\pm0.010$ and $\beta_\mathrm{s}=0.027\pm0.013$ at
$68\%\,\mathrm{CL}$, 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 $143\,\mathrm{GHz}$, $A^{PS}_{143}$, which shifts by
half sigma when the running of the running is considered. We further study the
cosmological implications of such preference for
$\alpha_\mathrm{s},\beta_\mathrm{s}\sim0.01$ by including in the analysis the
lensing amplitude $A_L$, the curvature parameter $\Omega_k$, 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 $\Omega_k = 0$ 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
$\beta_\mathrm{s}$.
