On broken zero modes of a string world sheet, and a correlation function of a 1/4 BPS Wilson loop and a 1/2 BPS local operator

We reconsider a gravity dual of a 1/4 BPS Wilson loop. In the case of an expectation value of the Wilson loop, it is known that broken zero modes of a string world sheet in the gravity side play important roles in the limit $\lambda \to \infty$ with keeping the combination $\lambda \cos^2 \theta_0$ finite. Here, $\lambda$ is the 't Hooft coupling constant and $\theta_0$ is a parameter of the Wilson loop. In this paper, we reconsider a gravity dual of a correlation function between the Wilson loop and a 1/2 BPS local operator with R charge $J$. We take account of contributions coming from the same configurations of the above-mentioned broken zero modes. We find an agreement with the gauge theory side in the limit $J \ll \sqrt{\lambda \cos^2 \theta_0}$.