Search for a scalar partner of the $X(3872)$ via $\psi (3770)$ decays into $\gamma \eta {\eta }^{\mathrm{\prime }}$ and $\gamma {\pi }^{+}{\pi }^{-}J\mathrm{/}\psi$

Using a data sample corresponding to an integrated luminosity of 2.93 fb${}^{-1}$ collected at a center-of-mass energy of 3.773~GeV with the BESIII detector at the BEPCII collider, we search for a scalar partner of the $X(3872)$, denoted as $X(3700)$, via $\psi (3770)\to \gamma \eta {\eta }^{\mathrm{\prime }}$ and $\gamma {\pi }^{+}{\pi }^{-}J\mathrm{/}\psi$ processes. No significant signals are observed and the upper limits of the product branching fractions $\mathcal{B}(\psi (3770)\to \gamma X(3700))\cdot \mathcal{B}(X(3700)\to \eta {\eta }^{\mathrm{\prime }})$ and $\mathcal{B}(\psi (3770)\to \gamma X(3700))\cdot \mathcal{B}(X(3700)\to {\pi }^{+}{\pi }^{-}J\mathrm{/}\psi )$ are determined at the 90% confidence level, for the narrow $X(3700)$ with a mass ranging from 3710 to 3740 MeV/$c^2$, which are from 0.8 to 1.8 $(\times {10}^{-5})$ and 0.9 to 3.4 $(\times {10}^{-5})$, respectively.