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Diphoton excess at 750 GeV and LHC constraints in models with vector-like particles

Recently, the ATLAS and CMS collaborations report excesses around $750$ GeV in the diphoton channels. This might be the evidence which reveals new physics beyond the Standard Model. In this paper, we consider models with a $750$ GeV scalar and vector-like particles, which couple each other through Yukawa couplings. The diphoton decay of the scalar is enhanced by the loop diagrams involving the extra colored particles. We investigate not only the setup required by the excesses, but also the LHC constraints, especially concerned with the vector-like particles. We consider the scenario that the extra colored particles decay to quarks and a dark matter (DM) via Yukawa couplings. Then, the signals from the vector-like particles are dijet, $b\overline{b}$ and/or $t\overline{t}$ with large missing energy. We discuss two possibilities for the setups: One is a model with vector-like fermions and a scalar DM, and the other is a model with vector-like scalars and a fermionic DM. We suggest the parameter region favored by the excess in the each case, and study the constraints based on the latest LHC results at $\sqrt{s}=8$ TeV and $13$ TeV. We conclude that the favored region is almost excluded by the LHC bounds, especially when the $750$ GeV scalar dominantly decays to DMs. The mass differences between the vector-like particles and the DM should be less than ${\cal O}(100)$ GeV (${\cal O}(10)$ GeV) to realize the large diphoton signal and the large decay width, if the extra colored particle only decays to a top (bottom) quark and a DM. Otherwise, these scenarios are already excluded by the latest LHC results.