momentum scale. The systematic uncertainty varies as a function of the mass of the hadronic system in each mode. The typical uncertainties averaged over all mass bins are evaluated by summing the systematic uncertainties of each bin and ignoring the correlations between bins, and the results are 0.7%, 2.2%, 2.2%, and 9.5%, for 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$_{τ}$ decay modes, respectively. In practice, we have obtained the covariance matrix for each source of the systematic uncertainties and added them to obtain the total uncertainty. The gray bands in Fig. 7 show the square roots of the diagonal components of the covariance matrix. The off-diagonal components of the covariance matrix are shown in Fig. 8 in term of the correlation coefficients ρ $^{Spec}$( i, j ) defined by ρ $^{Spec}$( i, j ) = cov( i, j ) / √ cov( i, i ) · cov( j, j ), where i and j are bin numbers. Figure 7 shows rather large discrepancies between the mass spectra of the genera- tor and our unfolded spectra except for τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ decay. For example, for the τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ mass spectrum (Fig. 7b), it is evident that there are contributions from two resonant peaks, similar to the MC expectations. However, the faster rise of the real data as compared with the MC in the 1 . 0 − 1 . 1 GeV mass region and the fall-off at high masses in the τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ mass spectrum may also be suggestive of a substantial non-resonant component that is not included in the current MC model. The use of the unfolded invariant mass distributions in estimating the efficiencies allows one to compensate for model deficiencies when evaluating the branching fractions. In order to determine the branching fraction, we normalize to the number of pure leptonic decays, where one τ decays to τ → eνν and the other decays to τ → µνν [23] (hereafter such events are referred to as { e, µ } events). The branching fraction for the i -th decay mode can then be written as: where N$_{sig}$$_{,}$$_{e}$$_{µ}$ = 8 . 1 × 10 6 and ε$_{eµ}$ = 0 . 221 ± 0 . 003 are the number of { e, µ } events and the corresponding detection efficiency, respectively. The uncertainty in ε$_{eµ}$ is determined from the lepton identification uncertainties. Here B τ → eνν = 0 . 178 and B τ → µνν = 0 . 174 are the branching fractions of τ → eνν and τ → µνν decays, respectively, and B τ → lνν = B i = N true i · ε$_{eµ}$ N$_{sig}$$_{,}$$_{e}$$_{µ}$ · B τ → eνν · B τ → µνν B τ → lνν , (3) 16