  A renormalization method that introduces an auxiliary field to represent a
three-body bound state is studied in a three-boson system with a triboson
field. A cutoff dependence in the three-boson system emerges as a limit cycle,
and the cyclic singularity is renormalized by employing two methods: a standard
method and the auxiliary field method. For each method, different sets of
diagrams are involved for renormalization, and thus we numerically study three
quantities: counter term for renormalization, scattering length of $s$-wave
boson-diboson scattering, and normalized wavefunction. We confirm that the two
methods would lead to the same result.
