content_image/363302.jpg
<img src='content_image/363302.jpg'>

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 . 

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. 

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 

