Study of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>K</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math>reaction in the energy range from 2.6 to 8.0 GeV

The ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ cross section and charged-kaon electromagnetic form factor are measured in the ${e}^{+}{e}^{\ensuremath{-}}$ center-of-mass energy range ($E$) from 2.6 to 8.0 GeV using the initial-state radiation technique with an undetected photon. The study is performed using $469\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data collected with the BABAR detector at the PEP-II2 ${e}^{+}{e}^{\ensuremath{-}}$ collider at center-of-mass energies near 10.6 GeV. The form factor is found to decrease with energy faster than $1/{E}^{2}$ and approaches the asymptotic QCD prediction. Production of the ${K}^{+}{K}^{\ensuremath{-}}$ final state through the $J/\ensuremath{\psi}$ and $\ensuremath{\psi}(2S)$ intermediate states is observed. The results for the kaon form factor are used together with data from other experiments to perform a model-independent determination of the relative phases between electromagnetic (single-photon) and strong amplitudes in $J/\ensuremath{\psi}$ and $\ensuremath{\psi}(2S)\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ decays. The values of the branching fractions measured in the reaction ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ are shifted relative to their true values due to interference between resonant and nonresonant amplitudes. The values of these shifts are determined to be about $\ifmmode\pm\else\textpm\fi{}5%$ for the $J/\ensuremath{\psi}$ meson and $\ifmmode\pm\else\textpm\fi{}15%$ for the $\ensuremath{\psi}(2S)$ meson.