broadening of the linewidth [@Pellegrino2014ObservationEnsemble; @Jenkins2016CollectiveExperiment; @Jenkins2016OpticalGases; @Jennewein2016PropagationAtoms; @Jennewein2018CoherentTheory], including a linear scaling with density [@Jennewein2016CoherentCloud; @Corman2017TransmissionAtoms]. A maximum index of $n\approx 1.26$ has also been observed in experiments involving dense, hot atomic vapours [@Keaveney2012MaximalNanolayer], which has been attributed to atomic collisions. However, while complex collision dynamics necessitate semi-phenomenological models [@Allard1982TheLines], here, our mechanism for saturation is quite fundamental, and occurs even for perfectly identical, stationary atoms.

Renormalization group approach
==============================

Our RG theory is based upon the key intuition gained in the collective scattering of just two atoms, to build up an understanding of the many-atom problem in a hierarchical manner. To be specific, let us consider the problem of two identical atoms, whose distance is much smaller than a wavelength, $\rho_{12} \equiv k_0 |{\mathbf{r}}_1 - {\mathbf{r}}_2|\ll 1$. Applying [Eq. \[eq:CDS\_input-output\_eq\_single\_mode\]]{} and [Eq. \[eq:CDS\_refractive\_index\_def\_from\_t\]]{}, we can calculate the imaginary part of the “index" of the two-atom system, as illustrated in [Fig. \[fig:RG\_plots\]]{}-a. One can see that the characteristic two-atom spectrum (blue line) is not twice the response of a single, isolated atom (green dashed curve), but instead consists of two, well-separated peaks with different linewidths and shifted resonances.

To understand this behavior, we consider the normal modes of the 