  Background: The halo effective field theory (Halo-EFT) provides a very
efficient description of loosely-bound nuclei in models of reaction. It offers
a very systematical ranking of the significance of nuclear-structure
observables in reaction calculations. This greatly helps to infer reliable
structure information from reaction cross sections. However, for a meaningful
analysis, the Halo-EFT scheme needs to have converged. Purpose: In a previous
study [P. Capel, D. R. Phillips, and H.-W. Hammer, Phys. Rev. C 98, 034610
(2018)], NLO descriptions of 11Be have been developed and lead to excellent
agreement with existing breakup data. However, the convergence of the scheme at
NLO was not fully demonstrated. Moreover, a significant dependence on the
regulator of the effective 10Be-n interaction has been observed. Method: We
develop Halo-EFT descriptions of 11Be at N2LO and use them in an accurate
breakup-reaction code. We compare our theoretical cross sections with
experiment on Pb and C targets at about 70 MeV/nucleon. Results: On Pb, the
N2LO descriptions of 11Be lead to little change to the NLO results of the
previous study, confirming the convergence of that scheme. On C, the reaction
is significantly affected by the presence of d resonances in the low-energy
spectrum of 11Be. In the Halo-EFT power counting these resonances appear only
at N2LO; our new descriptions include them naturally. Going to N2LO removes
also the cutoff dependence observed in the previous study. Conclusions: We
demonstrate the convergence of the Halo-EFT description of 11Be at NLO for
Coulomb breakup and at N2LO for nuclear-dominated dissociation. The reliability
of the nuclear-structure information inferred in the previous study is thus
confirmed.
