  Photometric data can be analyzed using the three-dimensional correlation
function $\xi_{\rm p}$ to extract cosmological information via e.g.,
measurement of the Baryon Acoustic Oscillations (BAO). Previous studies modeled
$\xi_{\rm p} $ assuming a Gaussian photo-$z$ approximation. In this work we
improve the modeling by incorporating realistic photo-$z$ distribution. We show
that the position of the BAO scale in $\xi_{\rm p}$ is determined by the
photo-$z$ distribution and the Jacobian of the transformation. The latter
diverges at the transverse scale of the separation $s_\perp $, and it explains
why $\xi_{\rm p } $ traces the underlying correlation function at $s_\perp $,
rather than $s$, when the photo-$z$ uncertainty $ \sigma_z / (1+ z) \gtrsim
0.02$. We also obtain the Gaussian covariance for $\xi_{\rm p}$. Due to
photo-$z$ mixing, the covariance of $\xi_{\mathrm{p}}$ shows strong
off-diagonal elements. The high correlation of the data causes some issues to
the data fitting. Nonetheless, we find that either it can be solved by
suppressing the largest eigenvalues of the covariance or it is not directly
related to the BAO. We test our BAO fitting pipeline using a set of mock
catalogs. The data set is dedicated for Dark Energy Survey Year 3 (DES Y3) BAO
analyses and includes realistic photo-$z$ distributions. The theory template is
in good agreement with mock measurement. Based on the DES Y3 mocks, $\xi_{\rm
p}$ statistic is forecast to constrain the BAO shift parameter $\alpha$ to be
$1.001 \pm 0.023$, which is well consistent with the corresponding constraint
derived from the angular correlation function measurements. Thus $\xi_{\rm p}$
offers a competitive alternative for the photometric data analyses.
