  We study the $^{11}\mathrm{Li}$ and $^{22}\mathrm{C}$ nuclei at leading order
(LO) in halo effective field theory (Halo EFT). Using the value of the
$^{22}\mathrm{C}$ rms matter radius deduced in Ref. [1] as an input in a LO
calculation, we simultaneously constrained the values of the two-neutron (2$n$)
separation energy of $^{22}\mathrm{C}$ and the virtual-state energy of the
$^{20}\mathrm{C}-$neutron system (hereafter denoted $^{21}$C). The 1$-\sigma$
uncertainty of the input rms matter radius datum, along with the theory error
estimated from the anticipated size of the higher-order terms in the Halo EFT
expansion, gave an upper bound of about 100 keV for the 2$n$ separation energy.
We also study the electric dipole excitation of 2$n$ halo nuclei to a continuum
state of two neutrons and the core at LO in Halo EFT. We first compare our
results with the $^{11}\mathrm{Li}$ data from a Coulomb dissociation experiment
and obtain good agreement within the theoretical uncertainty of a LO
calculation. We then obtain the low-energy spectrum of $B(E1)$ of this
transition at several different values of the 2$n$ separation energy of
$^{22}\mathrm{C}$ and the virtual-state energy of $^{21}\mathrm{C}$. Our
predictions can be compared to the outcome of an ongoing experiment on the
Coulomb dissociation of $^{22}\mathrm{C}$ to obtain tighter constraints on the
two- and three-body energies in the $^{22}\mathrm{C}$ system.
