Note on the Intermediate Field Representation of φ 2k Theory in Zero Dimension

This note is a sequel to Rivasseau and Wang (J. Math. Phys. 51 , 092304, 2010). We correct the intermediate field representation for the stable φ 2 k field theory in zero dimension introduced there and extend it to the case of complex conjugate fields. For k = 3 in the complex case we also provide an improved representation which relies on ordinary convergent Gaussian integrals rather than oscillatory integrals.