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Using a sample of $1.31\times10^{9} ~J/\psi$ events collected with the BESIIIdetector, we perform a study of $J/\psi\to\gamma K\bar{K}\eta'$. The $X(2370)$is observed in the $K\bar{K}\eta'$ invariant-mass distribution with astatistical significance of 8.3$\sigma$. Its resonance parameters are measuredto be $M = 2341.6 \pm 6.5\text{(stat.)} \pm 5.7\text{(syst.)}$~MeV/$c^{2}$ and$\Gamma = 117 \pm 10 \text{(stat.)} \pm 8 \text{(syst.)}$~MeV. The productbranching fractions for $J/\psi\to \gamma X(2370), X(2370) \to K^{+}K^{-}\eta'$ and $J/\psi\to \gamma X(2370), X(2370) \to K_{S}^{0}K_{S}^{0}\eta'$ are determined to be $(1.79 \pm 0.23 \text{(stat.)} \pm 0.65\text{(syst.)}) \times 10^{-5}$ and $(1.18 \pm 0.32\ text{(stat.)} \pm 0.39\text{(syst.)}) \times 10^{-5}$, respectively. No evident signal for the$X(2120)$ is observed in the $K\bar{K}\eta'$ invariant-mass distribution. Theupper limits for the product branching fractions of $\mathcal{B}(J/\psi \to\gamma X(2120)\to\gamma K^{+} K^{-} \eta')$ and $\mathcal{B}(J/\psi\to\gammaX(2120)\to\gamma K_{S}^{0} K_{S}^{0} \eta')$ are determined to be$1.49\times10^{-5}$ and $6.38\times10^{-6}$ at the 90% confidence level,respectively.

Observation of the $X(2370)$ and search for the $X(2120)$ in $J/\psi\to\gamma K\bar{K} \eta'$