  Starting with the density field equation of a self-gravity fluid in a static
Universe, using the Schwinger functional differentiation technique, we derive
the field equation of the 4-point correlation function (4PCF) of galaxies in
the Gaussian approximation, which contains hierarchically 2PCF and 3PCF. By use
of the known solutions of 2PCF and 3PCF, the equation of 4PCF becomes an
inhomogeneous, Helmholtz equation, and contains only two physical parameters:
the mass $m$ of galaxy and the Jeans wavenumber $k_J$, like the equations of
the 2PCF and 3PCF. We obtain the analytical solution of 4PCF that consists of
four portions, $\eta= \eta^0_{odd} + \eta^0_{even} +\eta^{FP} +\eta^I$, and has
a very rich structure. $\eta^0_{odd}$ and $\eta^0_{even}$ form the homogeneous
solution and depend on boundary conditions. The parity-odd $\eta^0_{odd}$ is
more interesting and qualitatively explains the observed parity-odd data of
BOSS CMASS, the parity-even $\eta^0_{even}$ contains the disconnected 4PCF
$\eta^{disc}$ (arising from a Gaussian random process), and both $\eta^0_{odd}$
and $\eta^0_{even}$ are prominent at large scales $r\gtrsim 10$Mpc, and exhibit
radial oscillations determined by the Jeans wavenumber. $\eta^{FP}$ and $
\eta^I$ are parity-even, and form the inhomogeneous solution. $\eta^{FP}$ is
the same as the Fry-Peebles ansatz for 4PCF, and dominates at small scales $r
\lesssim 10$Mpc. $\eta^I$ is an integration of the inhomogeneous term,
subdominant. We also compare the parity-even 4PCF with the observation data.
