The Rosetta Stone Project. II. The correlation between star formation efficiency and L/M indicator for the evolutionary stages of star-forming clumps in post-processed radiative magnetohydrodynamics simulations

Context. The evolution of massive star-forming clumps that are progenitors of high-mass young stellar objects are often classified based on a variety of observational indicators ranging from near-infrared to radio wavelengths. Among them, the ratio of the bolometric luminosity to the mass of their envelope, $L/M$, has been observationally diagnosed as a good indicator for the evolutionary classification of parsec-scale star-forming clumps in the Galaxy. Aims. We developed the Rosetta Stone project$\unicode{x2013}$an end-to-end framework designed to enable an accurate comparison between simulations and observations for investigating the formation and evolution of massive clumps. In this study, we calibrate the $L/M$ indicator in relation to the star formation efficiency (SFE) and the clump age, as derived from our suite of simulations. Methods. We performed multi-wavelength radiative transfer post-processing of radiative magnetohydrodynamics (RMHD) simulations of the collapse of star-forming clumps fragmenting into protostars. We generated synthetic observations to obtain far-infrared emission from $70$ to $500\,\mu$m, as was done in the Hi-GAL survey, and at $24\,\mu$m in the MIPSGAL survey, which were then used to build the spectral energy distributions (SEDs) and estimate the $L/M$ parameter. An additional $1.3\,$mm wavelength in ALMA Band 6 was also produced for the comparison with observational data. We applied observational techniques$\unicode{x2013}$commonly employed by observers$\unicode{x2013}$to the synthetic data in order to derive the corresponding physical parameters. Results. We find a correlation between $L/M$ and the SFE, with a power-law form $L/M\propto {\rm SFE}^{1.20^{+0.02}_{-0.02}}$. This correlation is independent of the mass of the clumps and the choice of initial conditions of the simulations in which they formed. (Abridged)