A lattice QCD determination of the $\Lambda_c \to N$ vector, axial vector,
and tensor form factors is reported. The calculation was performed with $2+1$
flavors of domain wall fermions at lattice spacings of $a\approx 0.11\:{\rm
fm},\:0.085\:{\rm fm}$ and pion masses in the range $230\:{\rm MeV} \lesssim
m_\pi \lesssim 350$ MeV. The form factors are extrapolated to the continuum
limit and the physical pion mass using modified $z$ expansions. The rates of
the charged-current decays $\Lambda_c \to n\, e^+ \nu_e$ and $\Lambda_c \to n\,
\mu^+ \nu_\mu$ are predicted to be $\left( 0.405 \pm 0.016_{\,\rm stat} \pm
0.020_{\,\rm syst} \right)|V_{cd}|^2 \:{\rm ps}^{-1}$ and $\left( 0.396 \pm
0.016_{\,\rm stat} \pm 0.020_{\,\rm syst} \right)|V_{cd}|^2 \:{\rm ps}^{-1}$,
respectively. The phenomenology of the rare charm decay $\Lambda_c \to p\,
\mu^+ \mu^-$ is also studied. The differential branching fraction, the fraction
of longitudinally polarized dimuons, and the forward-backward asymmetry are
calculated in the Standard Model and in an illustrative new-physics scenario.Comment: 21 pages, 7 figures, form factor parameters included as ancillary
file