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The first amplitude analysis of the $B^\pm \to \pi^\pm K^+ K^-$ decay isreported based on a data sample corresponding to an integrated luminosity of3.0 fb$^{-1}$ of $pp$ collisions recorded in 2011 and 2012 with the LHCbdetector. The data is found to be best described by a coherent sum of fiveresonant structures plus a nonresonant component and a contribution from$\pi\pi \leftrightarrow KK$ $S$-wave rescattering. The dominant contributionsin the $\pi^\pm K^\mp$ and $K^{+}K^{-}$ systems are the nonresonant and the$B^\pm \to \rho(1450)^{0}\pi^{\pm}$ amplitudes, respectively, with fitfractions around $30\%$. For the rescattering contribution, a sizeable fitfraction is observed. This component has the largest $CP$ asymmetry reported todate for a single amplitude of $(-66 \pm 4 \pm 2)\%$, where the firstuncertainty is statistical and the second systematic. No significant $CP$violation is observed in the other contributions.

Amplitude analysis of $B^\pm \to \pi^\pm K^+ K^-$ decays