Equation and solution for the 4-point correlation function of galaxies in the 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 wave number kJ, like the equations of the 2PCF and 3PCF. We obtain the analytical solution of 4PCF that consists of four portions, η=ηodd⁰+ηₑᵥₑₙ⁰+ηFP+ηI, and has a very rich structure. ηodd⁰ and ηₑᵥₑₙ⁰ form the homogeneous solution and depend on boundary conditions. The parity-odd ηodd⁰ is more interesting and qualitatively explains the observed parity-odd data of BOSS CMASS, the parity-even ηₑᵥₑₙ⁰ contains the disconnected 4PCF ηᵈⁱˢᶜ (arising from a Gaussian random process), and both ηodd⁰ and ηₑᵥₑₙ⁰ are prominent at large scales r10 Mpc, and exhibit radial oscillations determined by the Jeans wave number. ηFP and ηI are parity even, and form the inhomogeneous solution. ηFP is the same as the Fry-Peebles ansatz for 4PCF, and dominates at small scales r10 Mpc. ηI is an integration of the inhomogeneous term, subdominant. We also compare the parity-even 4PCF with the observation data.