Why we use a mean-field theory in Bose-Eisntein condensate? Is there another formulation that does not use a mean-field theory?
Mean-field theory is a good ansatz for a number of simple cases. Neglecting all interactions and assuming full "connectedness" of all particles, the Hartree-Fock separation into a product state and assuming a pseudo-potential should be exact. When it is not exact, the ansatz leads to a non-linear Schrödinger equation where the higher powers are perturbative corrections, i.e. a Gross-Pitaevskii equation. There are other ways to do perturbation theory of course.
There are many other ways to treat a condensate wavefunction. These methods can be better because they take condesate geometry, size and time evolution into account as well as interactions within the condensate and with non-condensed atoms. These methods either rely on some other perturbation theory or stochastic methods, to my knowledge. This paper gives a brief introduction: https://arxiv.org/pdf/0706.3541.pdf