  We consider an alternative to conventional three-point statistics such as the
bispectrum, which is purely based on the Fourier phases of the density field:
the line correlation function. This statistic directly probes the non-linear
clustering regime and contains information highly complementary to that
contained in the power spectrum. In this work, we determine, for the first
time, its potential to constrain cosmological parameters and detect baryon
acoustic oscillations (hereafter BAOs). We show how to compute the line
correlation function for a discrete sampled set of tracers that follow a local
Lagrangian biasing scheme and demonstrate how it breaks the degeneracy between
the amplitude of density fluctuations and the bias parameters of the model. We
then derive analytic expressions for its covariance and show that it can be
written as a sum of a Gaussian piece plus non-Gaussian corrections. We compare
our predictions with a large ensemble of $N$-body simulations and confirm that
BAOs do indeed modulate the signal of the line correlation function for scales
$50$-$100\,h^{-1}\,\mathrm{Mpc}$, and that the characteristic S-shape feature
would be detectable in upcoming Stage IV surveys at the level of $\sim4\sigma$.
We then focus on the cosmological information content and compute Fisher
forecasts for an idealized Stage III galaxy redshift survey of volume $V\sim
10\,h^{-3}\,\mathrm{Gpc}^3$ and out to $z=1$. We show that, combining the line
correlation function with the galaxy power spectrum and a Planck-like microwave
background survey, yields improvements up to a factor of two for parameters
such as $\sigma_8$, $b_1$ and $b_2$, compared to using only the two-point
information alone.
