Star spectroscopy in the constant B-field background

In this paper we calculate the spectrum of Neumann matrix with zero modes in the presence of the constant B-field in Witten's cubic string field theory. We find both the continuous spectrum inside [-1/3, 0) and the constraint on the existence of the discrete spectrum. For generic θ, -1/3 is not in the discrete spectrum but in the continuous spectrum. For each eigenvalue in the continuous spectrum there are four twist-definite degenerate eigenvectors except for -1/3 at which the degeneracy is two. However, for each twist-definite eigenvector the twist parity is opposite among the two spacetime components. Based upon the result at -1/3 we prove that the ratio of brane tension to be one as expected. Furthermore, we discuss the factorization of star algebra in the presence of B-field under the zero-slope limit and comment on the implications of our results to the recent proposed map of Witten's star to Moyal's star.