Constraining fine tuning in Composite Higgs Models with partially composite leptons

Minimal Composite Higgs Models (MCHM) have long provided a solution to the hierarchy problem of the Standard Model, yet suffer from various sources of fine tuning that are becoming increasingly problematic with the lack of new physics observations at the LHC. We develop a new fine tuning measure that accurately counts each contribution to fine tuning (single, double, triple, etc) that can occur in a theory with $n_p$ parameters, that must reproduce $n_o$ observables. We then use a novel scanning procedure to perform a comprehensive study of three different two-site, 4D, $SO(5)\rightarrow SO(4)$ MCHMs with all third generation fermions included, distinguished by the choice of the lepton embeddings. These are the MCHM$^{\textbf{5-5-5}}_{\textbf{5-5-5}}$, MCHM$^{\textbf{5-5-5}}_{\textbf{14-14-10}}$ and MCHM$^{\textbf{5-5-5}}_{\textbf{14-1-10}}$, where MCHM$^{q-t-b}_{l-\tau-\nu}$ has the lepton doublet partner in representation $l$, tau partner in representation $\tau$, and so on. We find that embedding at least one massive lepton in the symmetric $\textbf{14}$ of $SO(5)$ reduces the tuning for the case of low top partner masses (in line with previous results), but that this is balanced against the increased complexity of the model when one properly accounts for all sources of fine tuning. We study both the current relative fine-tuning of each scenario, and the future prospects. Noting that the different scenarios behave differently with respect to future improvements in collider measurements, we find that the MCHM$^{\textbf{5-5-5}}_{\textbf{14-1-10}}$ enjoys a relatively low increase in fine tuning even for a future lower bound on the top partner masses of 3.4 TeV (or equivalently a maximum Higgs-fermion or Higgs-gluon coupling deviation of 2%).