Naturalness, the Hyperbolic Branch and Prospects for the Observation of Charged Higgs at High Luminosity LHC and 27 TeV LHC

One of the early criterion proposed for naturalness was a relatively small Higgs mixing parameter $\mu$ with $\mu/M_Z$ order few. A relatively small $\mu$ may lead to heavier Higgs masses ($H^0, A, H^{\pm}$ in MSSM) which are significantly lighter than other scalars such as squarks. Such a situation is realized on the hyperbolic branch of radiative breaking of the electroweak symmetry. In this analysis we construct supergravity unified models with relatively small $\mu$ in the sense described above and discuss the search for the charged Higgs boson $H^{\pm}$ at HL-LHC and HE-LHC where we also carry out a relative comparison of the discovery potential of the two using the decay channel $H^{\pm} \to \tau \nu$. It is shown that an analysis based on the traditional linear cuts on signals and backgrounds is not very successful in extracting the signal while, in contrast, machine learning techniques such as boosted decision trees prove to be far more effective. Thus it is shown that models not discoverable with the conventional cut analyses become discoverable with machine learning techniques. Using boosted decision trees we consider several benchmarks and analyze the potential for their $5\sigma$ discovery at the 14 TeV HL-LHC and at 27 TeV HE-LHC. It is shown that while the ten benchmarks considered with the charged Higgs boson mass in the range 373 GeV- 812 GeV are all discoverable at HE-LHC, only four of the ten with Higgs boson masses in the range 373 GeV-470 GeV are discoverable at HL-LHC. Further, while the model points discoverable at both HE-LHC and HL-LHC would require up to 7 years of running time at HL-LHC, they could all be discovered in a period of few months at HE-LHC.