A new resummation scheme in scalar field theories is proposed by combining
parquet resummation techniques and flow equations, which is characterized by a
hierarchy structure of the Bethe--Salpeter (BS) equations. The new resummation
scheme greatly improves on the approximations for the BS kernel. Resummation of
the BS kernel in the t and u channels to infinite order is equivalent to
truncate the effective action to infinite order. Our approximation approaches
ensure that the theory can be renormalized, which is very important for
numerical calculations. Two-point function can also be obtained from the
four-point one through flow evolution equations resulting from the functional
renormalization group. BS equations of different hierarchies and the flow
evolution equation for the propagator constitute a closed self-consistent
system, which can be solved completely.Comment: 12 pages, 4 figures. v2: title changed, one figure added, final
version in Eur. Phys. J.
