Effective proton-neutron interaction near the drip line from unbound states in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi mathvariant="normal">F</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>25</mml:mn><mml:mo>,</mml:mo><mml:mn>26</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>

Background: Odd-odd nuclei, around doubly closed shells, have been extensively used to study proton-neutron interactions. However, the evolution of these interactions as a function of the binding energy, ultimately when nuclei become unbound, is poorly known. The $^{26}\mathrm{F}$ nucleus, composed of a deeply bound $\ensuremath{\pi}0{d}_{5/2}$ proton and an unbound $\ensuremath{\nu}0{d}_{3/2}$ neutron on top of an $^{24}\mathrm{O}$ core, is particularly adapted for this purpose. The coupling of this proton and neutron results in a ${J}^{\ensuremath{\pi}}={1}_{1}{}^{+}\ensuremath{-}{4}_{1}{}^{+}$ multiplet, whose energies must be determined to study the influence of the proximity of the continuum on the corresponding proton-neutron interaction. The ${J}^{\ensuremath{\pi}}={1}_{1}{}^{+},\phantom{\rule{0.16em}{0ex}}{2}_{1}{}^{+},\phantom{\rule{0.16em}{0ex}}{4}_{1}{}^{+}$ bound states have been determined, and only a clear identification of the ${J}^{\ensuremath{\pi}}={3}_{1}{}^{+}$ is missing.Purpose: We wish to complete the study of the ${J}^{\ensuremath{\pi}}={1}_{1}{}^{+}\ensuremath{-}{4}_{1}{}^{+}$ multiplet in $^{26}\mathrm{F}$, by studying the energy and width of the ${J}^{\ensuremath{\pi}}={3}_{1}{}^{+}$ unbound state. The method was first validated by the study of unbound states in $^{25}\mathrm{F}$, for which resonances were already observed in a previous experiment.Method: Radioactive beams of $^{26}\mathrm{Ne}$ and $^{27}\mathrm{Ne}$, produced at about $440A\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ by the fragment separator at the GSI facility were used to populate unbound states in $^{25}\mathrm{F}$ and $^{26}\mathrm{F}$ via one-proton knockout reactions on a ${\mathrm{CH}}_{2}$ target, located at the object focal point of the ${\mathrm{R}}^{3}\mathrm{B}$/LAND setup. The detection of emitted $\ensuremath{\gamma}$ rays and neutrons, added to the reconstruction of the momentum vector of the $A\ensuremath{-}1$ nuclei, allowed the determination of the energy of three unbound states in $^{25}\mathrm{F}$ and two in $^{26}\mathrm{F}$.Results: Based on its width and decay properties, the first unbound state in $^{25}\mathrm{F}$, at the relative energy of 49(9) keV, is proposed to be a ${J}^{\ensuremath{\pi}}=1/{2}^{\ensuremath{-}}$ arising from a ${p}_{1/2}$ proton-hole state. In $^{26}\mathrm{F}$, the first resonance at 323(33) keV is proposed to be the ${J}^{\ensuremath{\pi}}={3}_{1}{}^{+}$ member of the ${J}^{\ensuremath{\pi}}={1}_{1}{}^{+}\ensuremath{-}{4}_{1}{}^{+}$ multiplet. Energies of observed states in $^{25,26}\mathrm{F}$ have been compared to calculations using the independent-particle shell model, a phenomenological shell model, and the ab initio valence-space in-medium similarity renormalization group method.Conclusions: The deduced effective proton-neutron interaction is weakened by about 30--40% in comparison to the models, pointing to the need for implementing the role of the continuum in theoretical descriptions or to a wrong determination of the atomic mass of $^{26}\mathrm{F}$.

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