  Transport coefficients are typically divergent for quantum integrable systems
in one dimension, such as a Bose gas with a two-body contact interaction.
However, when a one-dimensional system is realized by confining bosons into a
tight matter waveguide, an effective three-body interaction inevitably arises
as leading perturbation to break the integrability. This fact motivates us to
study the thermal conductivity of a Bose gas in one dimension with both
two-body and three-body interactions. In particular, we evaluate the Kubo
formula exactly to the lowest order in perturbation by summing up all
contributions that are naively higher orders in perturbation but become
comparable in the zero-frequency limit due to the pinch singularity.
Consequently, a self-consistent equation for a vertex function is derived,
showing that the thermal conductivity in quasi-one-dimension is dominated by
the three-body interaction rather than the two-body interaction. Furthermore,
the resulting thermal conductivity in the weak-coupling limit proves to be
identical to that computed based on the quantum Boltzmann equation and its
temperature dependence is numerically determined.
