The equation and solution of 4-point correlation function of galaxies in Gaussian approximation and its parity-odd part

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.