  We investigate local quantum field theories for one-dimensional (1D) Bose and
Fermi gases with contact interactions, which are closely connected with each
other by Girardeau's Bose-Fermi mapping. While the Lagrangian for bosons
includes only a two-body interaction, a marginally relevant three-body
interaction term is found to be necessary for fermions. Because of this
three-body coupling, the three-body contact characterizing a local triad
correlation appears in the energy relation for fermions, which is one of the
sum rules for a momentum distribution. In addition, we apply in both systems
the operator product expansion to derive large-energy and momentum asymptotics
of a dynamic structure factor and a single-particle spectral density. These
behaviors are universal in the sense that they hold for any 1D scattering
length at any temperature. The asymptotics for the Tonks-Girardeau gas, which
is a Bose gas with a hardcore repulsion, as well as the Bose-Fermi
correspondence in the presence of three-body attractions are also discussed.
