TABLE II: The number of reconstructed events (second column), the number of true events (third column), the number of background events other than three-prong cross-feeds (fourth column), and the main source of other backgrounds with its fraction of the total (fifth column). where N rec j is the number of events for the j -th reconstructed decay mode and N other j is the number of background events in the j -th mode. Here E ij is the efficiency for detecting the j -th mode as the i -th one. The values of N rec and N other are listed in the second and third columns of Table II, respectively. We have determined the efficiency (migration) matrix iteratively. First, it is determined using the mass spectra as modeled in the current TAUOLA MC program. (We refer to this efficiency matrix as E (1) ij ). As is discussed in the next paragraph, we then unfolded the mass spectra of each mode to obtain the true mass distribution. The resultant unfolded mass spectra are then used to determine the efficiency matrix E (2) ij using a weighting procedure. The matrix elements E (2) ij are given in Table I, where the values given in parentheses are the percentage differences between E (1) ij and E (2) ij ; ( E (2) ij −E (1) ij ) / E (1) ij . Since the efficiency is smooth as a function of hadronic mass, the difference between E (1) ij and E (2) ij is very small ( ∼ 1%) in most cases, except for the off-diagonal entries related to the τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ mode. The source of these large discrepancies is the mass dependence of the fake rates, together with the considerable differences between the unfolded spectrum and the generated spectrum. The unfolding procedure extracts the true mass spectra for all four modes, denoted by the vector x , by solving the matrix equation ˆ A x = b . In this equation, b is the vector of the reconstructed mass spectrum with backgrounds other than the three-prong cross-feed subtracted, and ˆ A is the response matrix. The matrix ˆ A takes into account the cross-feed between different modes caused by particle misidentification, as well as the effects of finite resolution and the limited acceptance of the detector. The response matrix is obtained by merging 16 sub-response matrices as shown in 13 Decay mode N rec N true N other Main component in N other τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ 8 . 86 × 10 6 3 . 52 × 10 7 9 . 35 × 10 5 (10 . 6%) τ − → π $^{−}$π $^{+}$π $^{−}$π $^{0}$ν$_{τ}$ (64.2%) τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ 7 . 94 × 10 5 1 . 38 × 10 6 9 . 60 × 10 4 (12 . 1%) τ − → π $^{−}$π $^{+}$π $^{−}$π $^{0}$ν$_{τ}$ (34.4%) τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ 1 . 08 × 10 5 6 . 47 × 10 5 7 . 16 × 10 3 (6 . 66%) e $^{+}$e − → q ¯ q (30.3%) τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ 3 . 16 × 10 3 1 . 37 × 10 4 1 . 71 × 10 2 (5 . 41%) e $^{+}$e − → q ¯ q (53.0%)