Particle decays of levels in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi mathvariant="normal">N</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>11</mml:mn><mml:mo>,</mml:mo><mml:mn>12</mml:mn></mml:mrow></mml:mmultiscripts></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi mathvariant="normal">O</mml:mi><mml:mprescripts/><mml:none/><mml:mn>12</mml:mn></mml:mmultiscripts></mml:math> investigated with the invariant-mass method

Particle-decaying states of the light nuclei $^{11,12}\mathrm{N}$ and $^{12}\mathrm{O}$ were studied using the invariant-mass method. The decay energies and intrinsic widths of a number of states were measured, and the momentum correlations of three-body decaying states were considered. A second $2p$-decaying ${2}^{+}$ state of $^{12}\mathrm{O}$ was observed for the first time, and a higher-energy $^{12}\mathrm{O}$ state was observed in the $4p+2\ensuremath{\alpha}$ decay channel. This $4p+2\ensuremath{\alpha}$ channel also contains contributions from fissionlike decay paths, including $^{6}\mathrm{Be}_{\mathrm{g}.\mathrm{s}.}+^{6}\mathrm{Be}_{\mathrm{g}.\mathrm{s}.}$. Analogs to these states in $^{12}\mathrm{O}$ were found in $^{12}\mathrm{N}$ in the $2p+^{10}\mathrm{B}$ and $2p+\ensuremath{\alpha}+^{6}\mathrm{Li}$ channels. The momentum correlations for the prompt $2p$ decay of $^{12}\mathrm{O}_{\mathrm{g}.\mathrm{s}.}$ were found to be nearly identical to those of $^{16}\mathrm{Ne}_{\mathrm{g}.\mathrm{s}.}$, and the correlations for the new ${2}^{+}$ state were found to be consistent with sequential decay through excited states in $^{11}\mathrm{N}$. The momentum correlations for the ${2}_{1}^{+}$ state in $^{12}\mathrm{O}$ provide a new value for the $^{11}\mathrm{N}$ ground-state energy. The states in $^{12}\mathrm{N}/^{12}\mathrm{O}$ that belong to the $A=12$ isobaric sextet do not deviate from the quadratic isobaric multiplet mass equation form.