TABLE IV: Summary of the systematic uncertainties on the branching fractions (%). 1 The actual value of the statistical uncertainty of B ( τ − → π − π $^{+}$π − ν τ ) is 0 . 003 × 10 $^{$_{−}$2}$. 22 Using the results of the branching fractions and the correlation matrix, the ratios of the branching fractions involving kaons to the branching fraction of τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ decay are The covariance matrices for the branching fraction measurements are given in Table V in terms of the correlation coefficients ρ $^{BF}$( i, j ) = cov( i, j ) / √ cov( i, i ) · cov( j, j ), where i and j correspond to one of the four three-prong decay modes. There is a strong correlation between τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ and τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ decays, which mostly comes from the effect of kaon identification. B ( τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ ) = (8 . 42 ± 0 . 00(stat . ) +0 . 26 − 0 . $_{25}$(sys . )) × 10 − 2 , 1 B ( τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ ) = (3 . 30 ± 0 . 01(stat . ) +0 . 16 − 0 . $_{17}$(sys . )) × 10 − 3 , B ( τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ ) = (1 . 55 ± 0 . 01(stat . ) +0 . 06 − 0 . $_{05}$(sys . )) × 10 − 3 , and B ( τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ ) = (3 . 29 ± 0 . 17(stat . ) +0 . 19 − 0 . $_{20}$(sys . )) × 10 − 5 . τ → τ → τ → τ → Source πππν Kππν KKπν KKKν Tracking efficiency +2 . 2/ − 2 . 0 +2 . 1/ − 2 . 0 +2 . 1/ − 2 . 0 +2 . 1/ − 1 . 9 Particle ID ± 1 . 9 +4 . 0/ − 4 . 1 ± 2 . 3 +5 . 4/ − 5 . 6 Mass spectrum ± 0 . 1 ± 0 . 1 ± 0 . 1 +0 . 5/ − 0 . 8 Trigger efficiency ± 0 . 5 ± 0 . 5 ± 0 . 6 ± 0 . 6 γ veto ± 0 . 9 ± 1 . 5 ± 1 . 7 ± 0 . 5 K 0 S veto ± 0 . 2 ± 0 . 2 ± 0 . 1 ± 0 . 3 Background estimation ± 0 . 3 ± 1 . 3 ± 0 . 3 ± 0 . 4 Leptonic branching fraction ± 0 . 2 ± 0 . 2 ± 0 . 2 ± 0 . 2 Total +3 . 1/ − 3 . 0 ± 5 . 0 ± 3 . 6 +5 . 9/ − 6 . 0