  We examine the extent to which the properties of three-nucleon bound states
are well-reproduced in the limit that nuclear forces satisfy Wigner's SU(4)
(spin-isospin) symmetry. To do this we compute the charge radii up to
next-to-leading order (NLO) in an effective field theory (EFT) that is an
expansion in powers of $R/a$, with $R$ the range of the nuclear force and $a$
the nucleon-nucleon ($N\!N$) scattering lengths. In the Wigner-SU(4) limit, the
triton and Helium-3 point charge radii are equal. At NLO in the range expansion
both are $1.66$ fm. Adding the first-order corrections due to the breaking of
Wigner symmetry in the $N\!N$ scattering lengths gives a ${}^3\mathrm{H}$ point
charge radius of $1.58$ fm, which is remarkably close to the experimental
number, $1.5978\pm0.040$ fm (Angeli and Marinova in At Data Nucl Data Tables
99:69-95, 2013). For the ${}^3\mathrm{He}$ point charge radius we find $1.70$
fm, about 4% away from the experimental value of $1.77527\pm0.0054$ fm (Angeli
and Marinova 2013). We also examine the Faddeev components that enter the
tri-nucleon wave function and find that an expansion of them in powers of the
symmetry-breaking parameter converges rapidly. Wigner's SU(4) symmetry is thus
a useful starting point for understanding tri-nucleon bound-state properties.
