Analog $B\left(M1\right)$ strengths in the $T_z=\pm \frac{3}{2}$ mirror nuclei ${}_{47}\mathrm{Mn}$ and ${}_{47}\mathrm{Ti}$

The lifetimes of the first excited ${\frac{7}{2}}^{-}$ states in the $T_z=\pm \frac{3}{2}$ mirror nuclei ${}_{47}\mathrm{Mn}$ and ${}_{47}\mathrm{Ti}$ have been extracted utilizing the $\gamma$-ray line shape method, giving $\tau =687\left(36\right)$ ps and $\tau =331\left(15\right)$ ps respectively. Since these transitions are essentially pure $M1$ transitions, these results allow for a high-precision comparison of analog $M1$ strengths in mirror nuclei. The two analog $B\left(M1\right)s$ are observed to be identical to a precision of about $10\%$. The expected dependence of the transition matrix element with $T_z$ has been used to extract the separate isoscalar and isovector components of the transition strength, and the results are discussed in the context of predictions, based on the isospin formalism, regarding analog $B\left(M1\right)$ strengths.