BRANCHING FRACTION AND MASS SPECTRUM CALCULATION The numbers of candidate events in the τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ , τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ , τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ , and τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ modes, after applying all selection criteria, are sum- marized in Table II. Possible backgrounds in these signal candidate samples include (a) cross-feed from the signal modes and (b) other processes such as τ − → π $^{−}$π $^{+}$π $^{−}$π $^{0}$ν$_{τ}$ , τ − → K 0 $_{S}$π $^{−}$ν$_{τ}$ , as well as continuum e $^{+}$e − → q ¯ q . The fractions of backgrounds coming from other processes are 5% to 12%, as summarized in Table II (fourth column) for each signal mode. The fifth column shows the fraction of the main background component as determined from MC. The mode τ − → π $^{−}$π $^{+}$π $^{−}$π $^{0}$ν$_{τ}$ dominates for the τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ and τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ modes, while the background from continuum e $^{+}$e − → q ¯ q is dominant for the τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ and τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ modes. To take into account the cross-feed between the decay channels, the true yield N true i ( i is the mode number and the values from 1 to 4 correspond to τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ , τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ , τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ , and τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ ), is obtained using the following equation: TABLE I: Summary of the efficiencies and the fractions of cross-feed backgrounds. The values in parentheses are the differences of the efficiencies or fake rates from those evaluated directly from the MC with the TAUOLA decay model, ( E (2) ij − E (1) ij ) / E (1) ij in percent (see text). N true i = ∑ j ( E − $^{1}$)$_{ij}$ ( N rec j − N other j ) , (2) Generated decay mode Reconstructed τ − → τ − → τ − → τ − → decay mode π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ 0 . 22 ( − 0 . 1%) 0 . 079 ( − 0 . 1%) 0 . 022 ( − 1 . 1%) 6 . 4 × 10 − 3 (+2 . 6%) τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ 0 . 012 ( − 0 . 2%) 0 . 18 ( − 0 . 5%) 0 . 047 (+1 . 4%) 0 . 019 ( − 3 . 1%) τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ 3 . 9 × 10 − 4 ( − 0 . 3%) 4 . 7 × 10 − 3 ( − 4 . 4%) 0 . 12 ( − 1 . 0%) 0 . 051 ( − 0 . 4%) τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ 5 . 0 × 10 − 6 (+4 . 5%) 1 . 3 × 10 − 4 ( − 9 . 1%) 2 . 3 × 10 − 3 ( − 14 . 0%) 0 . 081 (+1 . 2%) 12