The production of the doubly charmed baryon in deeply inelastic $ep$ scattering at the Large Hadron Electron Collider

In this paper, we carry out a detailed study on the production of the doubly charmed baryon in deeply inelastic $ep$ scattering (DIS) for $Q^2\in[2, 100]~\textrm{GeV}^2$, at the Large Hadron Electron Collider (LHeC) with $E_e=60(140)$ GeV and $E_p=7000$ GeV. To exclude the contributions from the diffractive productions and the $b$ hadron decays, we impose the kinematic cuts $0.3<z<0.9$, and $p_{t,\textrm{baryon}}^{\star2}>1~\textrm{GeV}^2$ in the center-of-mass (CM) frame of $\gamma^{*}p$. Based on the designed LHeC luminosity, by detecting the decay channel $\Xi^{+}_{cc}(\Xi^{++}_{cc}) \to \Lambda_c^{+}$ with the subsequent decay $\Lambda_c^{+} \to pK^{-}\pi^{+}$, we predict that about 1880 (2700) $\Xi^{+}_{cc}$ events and 3750 (5400) $\Xi^{++}_{cc}$ events can be accumulated per year, which signifies the prospect of observing them via the DIS at the forthcoming LHeC. In addition, we also predict the distributions of a rich variety of physical observables in the laboratory frame and the $\gamma^{*}p$ CM frame, including $Q^2$, $p_t^{2}$, $Y$(rapidity), $p_t^{\star2}$, $Y^{\star}$, $W$, and $z$ distributions, respectively, which can provide helpful references for studying the doubly charmed baryon. In conclusion, we think that in addition to the LHC, the LHeC is also a helpful platform for studying the properties of the doubly charmed baryon.