Diffuse $\gamma$-ray emission from BL Lac blazars

13 pages, 6 figures. Version 2: some rewording in the introduction, results and conclusions unchangedInternational audienceBlazars represent the most abundant class of high-energy extragalactic $\gamma$-ray sources. The subset of blazars known as BL Lac objects is on average closer to Earth and characterized by harder spectra at high energy than the whole sample. The fraction of BL Lacs which results too dim to be detected and resolved by current $\gamma$-ray telescopes is therefore expected to contribute to the high energy isotropic diffuse $\gamma$-ray background (IGRB). The IGRB has been recently measured over a wide energy range by the Large Area Telescope (LAT) on-board the Gamma-ray Space Telescope ({\it Fermi}). We present a new prediction of the diffuse $\gamma$-ray flux due to the unresolved BL Lac blazar population. The model is built upon the spectral energy distribution and the luminosity function derived from the fraction of BL Lacs detected (and spectrally characterized) in the $\gamma$-ray energy range. We focus the attention on the ${\cal O}(100)$ GeV energy range, predicting the emission up to the TeV scale and taking into account the absorption on the extragalactic background light. In order to better shape the BL Lac SED, we combine the {\it Fermi}-LAT data with Imaging Atmospheric Cherenkov Telescopes measurements of the most energetic sources. Our analysis is carried on separately for low and intermediate synchrotron peaked BL Lacs on one hand, and highly synchrotron peaked BL Lacs on the other one: we find in fact statistically different features for the two. The diffuse emission from the sum of both BL Lac classes grows from about $10\%$ of the measured IGRB at 100 MeV to $\sim$100$\%$ of the data level at 100 GeV. At energies greater than 100 GeV, our predictions naturally explain the IGRB data, accommodating their softening with increasing energy. Uncertainties are estimated to be within of a factor of two of the best fit flux up to 500 GeV