Here we study two important properties of $2+1$-dimensional QCD—confinement and pseudoscalar glueball spectrum—with holographic approach. We consider the low energy decoupled geometry of the isotropic non-supersymmetric D2 brane. We find the corresponding gauge theory is similar to the $2+1$-dimensional Yang-Mills (YM) theory with the running coupling ${\lambda }^2$. At the extremal limit (i.e., Bogomol’nyi-Prasad-Sommerfield limit), this gauge theory reduces to the super-YM theory. From the Nambu-Goto action of a test string, the potential of a $Q\text{−}\overline{Q}$ pair located on the boundary is calculated. At large $Q\text{−}\overline{Q}$ separation, it gives the tension $\sigma$ of the QCD flux tube. The parametric dependence of $\sigma$ is shown pictorially. It is found that $\sigma$ is a monotonically increasing function of the effective coupling ${\lambda }^2$. In comparison, $\sqrt{\sigma }/g_{\mathrm{YM}}^2{N}_c$ is found to match accurately with the previous results. In the next part, we consider fluctuation of the axion field in the aforementioned gravity background. From the linearized field equation of the fluctuation, we calculate the mass spectrum of $0^{-+}$ numerically using the Wentzel-Kramers-Brillouin approximation. The pseudoscalar mass is found to be related to the string tension approximately as $M_{0^{-+}}/\sqrt{\sigma }\approx 3(n+2)$ for first three energy states, $n=0$, 1, 2.

• Received 3 August 2020
• Accepted 21 January 2021

DOI:https://doi.org/10.1103/PhysRevD.103.066001

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Published by the American Physical Society