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Following arXiv: 1506.00612, an effective field theory approach has been introduced to understand the distortion of the Higgs $p_T$ and other excesses observed in Run I LHC data by considering two hypothetical particles $H$ and $\chi$, with the masses $2 m_h < m_H < 2 m_t$ and $m_\chi < m_h/2$ where $m_h$ is mass of the SM Higgs, $h$ and $\chi$ is considered as a dark matter candidate. A fit with the observed $p_T$ spectrum of the Higgs boson at the LHC and a statistical combination of the different relevant processes results $m_H = 272^{+12}_{9}$~GeV with $m_\chi \approx 60$~GeV. In this study we introduce a real scalar $S$ with mass $m_h \lesssim m_S \lesssim m_H - m_h$ in a effective theory to explain large branching ratios of $H \to h\chi\chi$. By introducing an intermediate $S$ further simplifies the coupling structure with comparatively less branching fraction.
Further we introduce a two Higgs doublet model (THDM) where we assume the particle spectrum of the THDM, $h$ as the SM Higgs, $H$ as heavy scalar as in the effective theory, $A$ as a $CP$-odd scalar with $m_A > 2 m_t$ and charged Higgs $H^\pm$ with $m_H^\pm < m_A$. A proper theory with THDM in addition with $S$ and $\chi$ as a real scalar assuming $\chi$ as a dark matter candidate is formulated to describe associated phenomenology with these particles. An explanation in multi-lepton final states with same-sign leptons expected to be observed in different processes $p p > H > h S$, $h, S \to W^+ W^-$; $p p > H^- t + h.c$ with possible decay of $H^\pm$ and $t$-quark in leptonic final states. A full analyses associated with these scalars for few benchmark scenarios have been presented in this work.
