Gravity Induced Chiral Condensate Formation and the Cosmological Constant

It is well known that the covariant coupling of fermionic matter to gravity induces a four-fermion interaction. The presence of this term in a homogenous and isotropic space-time results in a BCS-like Hamiltonian and the formation of a chiral condensate with a mass gap. We calculate the gap ($\Delta$) via a mean-field approximation for minimally coupled fermionic fields in a FRW background and find that it depends on the scale factor. The calculation also yields a correction to the bare cosmological constant ($\Lambda_0$), and a non-zero vev for $<\psi^\dag\psi>$ which then behaves as a scalar field. Hence we conjecture that the presence of fermionic matter in gravity provides a natural mechanism for relaxation of the $\Lambda_0$ and explains the existence of a scalar field from (almost) first principles.