  We present the first five-body calculations of $s$-wave $n$-$^4$He scattering
within leading order and next-to-leading order (NLO) pionless effective field
theory. Using an harmonic oscillator trap technique and pionless effective
field theory fitted to just six well-established experimental parameters, we
predict the $s$-wave $n$-$^4$He phase shifts, scattering length $a^{1/2}_{n
^4\text{He}}(\text{NLO})=2.47(4\ \text{num.})~(17\ \text{theor.})~{\rm fm}$,
and effective range $r^{1/2}_{n ^4\text{He}}(\text{NLO})=1.384(3\
\text{num.})~(211\ \text{theor.})~{\rm fm}$ in agreement with experiment. The
apparent cutoff independence of our results is used to estimate the theoretical
errors coming as an integral part of our final results.
