We study the prospects of observing the non-resonant di-Higgs pair production in the Standard Model (SM) at the high luminosity run of the 14 TeV LHC (HL-LHC), upon combining multiple final states chosen on the basis of their yield and cleanliness. In particular, we consider the $bbar{b}gamma gamma, bbar{b} tau^+ tau^-, bbar{b} WW^*, WW^*gamma gamma$ and $4W$ channels mostly focusing on final states with photons and/or leptons and study 11 final states. We employ multivariate analyses to optimise the discrimination between signal and backgrounds and find it performing better than simple cut-based analyses. The various differential distributions for the Higgs pair production have non-trivial dependencies on the Higgs self-coupling ($lambda_{hhh}$). We thus explore the implications of varying $lambda_{hhh}$ for the most sensitive search channel for the double Higgs production, textit{viz.}, $bbar{b}gammagamma$. The number of signal events originating from SM di-Higgs production in each final state is small and for this reason measurement of differential distributions may not be possible. Furthermore, we consider various physics beyond the standard model scenarios to quantify the effects of contamination while trying to measure the SM di-Higgs signals in detail. In particular, we study generic resonant heavy Higgs decays to a pair of SM-like Higgs bosons or to a pair of top quarks, heavy pseudoscalar decaying to an SM-like Higgs and a $Z$-boson, charged Higgs production in association with a top and a bottom quark and also various well-motivated supersymmetric channels. We set limits on the cross-sections for the aforementioned new physics scenarios, above which these can be seen as excesses over the SM background and affect the measurement of Higgs quartic coupling. We also discuss the correlations among various channels which can be useful to identify the new physics model.
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