§ CONCLUSIONS

In this work we have extended previous global fits of two-site Minimal 4D Composite Higgs Models by including composite partners for the third-generation leptons in various representations of $SO(5)$, and imposing relevant constraints on the added lepton parameters. We considered all quark partners as embedded in the $\mathbf{5}$ representation due to its favourable phenomenology, and chose to embed the respective partners of the left-handed lepton doublet and right-handed tau lepton into the $\mathbf{5}-\mathbf{5}$ and $\mathbf{14}-\mathbf{10}$ representations (giving the so-called LM4DCHM$^{5-5-5}_{5-5}$ and LM4DCHM$^{5-5-5}_{14-10}$ models) to study how these might affect the fine-tuning and viability of the theory.

Our fits revealed that the LM4DCHM$^{5-5-5}_{14-10}$ is decisively preferred from a Bayesian standpoint over the LM4DCHM$^{5-5-5}_{5-5}$, having the greater Bayesian evidence by several orders of magnitude. This conclusion is robust against both the choice of prior distribution (uniform or logarithmic) and the prior bounds imposed on the parameter spaces. For all choices of prior, the LM4DCHM$^{5-5-5}_{14-10}$ had the favourable posterior-averaged log-likelihood, owing primarily to the SM mass and oblique constraints which it can more easily satisfy. However, the main factor favouring the LM4DCHM$^{5-5-5}_{14-10}$ is its lower fine-tuning as measured by the Kullback-Leibler divergence between the prior and posterior distributions. The exact fine-tuning of the models is strongly prior dependent, although always in favour of the LM4DCHM$^{5-5-5}_{14-10}$. This is to be expected from theoretical considerations, since the LM4DCHM$^{5-5-5}_{5-5}$ suffers from a double-tuning necessary to achieve a viable Higgs potential. Hence, the inclusion of lepton partners is seen to have a significant effect on the fine-tuning of the model, despite the relatively low compositeness of the $\tau$ compared to third generation quarks.

We also analysed the phenomenology of the two models in their viable regions of parameter space. The phenomenology of the composite gauge bosons and quark partners remains unchanged from what was found in the quark-only M4DCHM$^{5-5-5}$ in Ref. <cit.>, but now there are also heavy lepton partners. These lepton partners are generically predicted to lie well above $1$ TeV, with the most probable decay channels being $N_4 \rightarrow W \tau$, $L_4 \rightarrow H \tau$, $L_4 \rightarrow Z \tau$ and $E2 \rightarrow W \tau$. In both models, the lightest composite lepton tended to be $L_4$. In the LM4DCHM$^{5-5-5}_{14-10}$, this predominantly decays through both the $H \tau$ and $Z \tau$ channels, whereas in the LM4DCHM$^{5-5-5}_{5-5}$, it decays almost always into a $Z \tau$ pair. However, conservative estimates of the production cross sections of the heavy leptons revealed that only in a small fraction of the parameter space could any heavy lepton resonance of the LM4DCHM$^{5-5-5}_{14-10}$ possibly be discovered at the HL-LHC, while those of the LM4DCHM$^{5-5-5}_{5-5}$ all lie far beyond the reach of the HL-LHC.

Predictions for the Higgs signal strengths for both models generally align well with experimental bounds. However, more natural values of the NGB decay constant $f \lesssim 1.25$ TeV are currently slightly disfavoured by current experimental measurements compared to higher values. When compared to the Higgs phenomenology of the quark-only M4DCHM$^{5-5-5}$, values $2.5$ TeV $\lesssim f \lesssim 3.5$ TeV are disfavoured as a result of including the third-generation leptons.

While the results presented in this paper pose challenges for the LM4DCHM$^{5-5-5}_{5-5}$, it is important to note that these findings do not definitively exclude the model. Naturalness plays a central role in CHM searches, but it is only by experiment that models can be ruled out. Current prospects for the precision of Higgs signal strength measurements at the HL-LHC <cit.> promise to provide effective tests of this model.

It would be interesting to consider possible modifications to the framework considered here, such as alternative quark embeddings in LM4DCHMs, as both Ref. <cit.> and this work have shown promising results for 14 representation leptons. Furthermore, there may also be room for including right-handed neutrinos, which are well-motivated due to their ability to provide a potential dark matter candidate and to explain the observed baryonic asymmetry <cit.>. The complexity of the resulting parameter spaces, however, may make it difficult to obtain consistent convergent results.

§ ACKNOWLEDGEMENTS

We thank Peter Stangl for allowing us to use  for this work. MJW, AGW and KG are funded by the ARC Centre of Excellence for Dark Matter Particle Physics CE200100008 and are further supported by the Centre for the Subatomic Structure of Matter (CSSM). ENC is supported by the Clarendon Fund Scholarship in partnership with the Oxford-Berman Graduate Scholarship.