Descent relations and the oscillator level-truncation method

We reexamine the oscillator level-truncation method in the bosonic string field theory by calculating the descent relation $\langle V_3\mid V_1\rangle=\widetilde{Z}_3\langle V_2|$. We use the fermionic vertices in the standard oscillator basis for the ghost sector. We propose two new calculation schemes: in the first, we assume that the insertion satisfies the overlap equation for the vertices; in the second, we use direct calculations. In both schemes, we obtain the correct structures of the exponent and preexponent of the vertex $\langle V_2|$, but we find different normalization factors $\widetilde Z_3$ .