Phase Correlations in Non-Gaussian Fields

We present the general relationship between phase correlations and the hierarchy of polyspectra in the Fourier space, and the new theoretical understanding of the phase information is provided. Phase correlations are related to the polyspectra only through the non-uniform distributions of the phase sum $ heta_{sbfm{k}_1} + ... + heta_{sbfm{k}_N}$ with closed wave vectors, $fm{k}_1 + ... + fm{k}_N = 0$. The exact relationship is given by the infinite series, which one can truncate in a consistent manner. The method to calculate the series to arbitrary order is explained, and the explicit expression of the first-order approximation is given. A numerical demonstration proves that the distribution of the phase sum is a robust estimator and provides an alternative statistic to search for the non-Gaussianity.