  We apply the renormalisation group (RG) to analyse scattering by short-range
forces in systems with coupled channels. For two S-wave channels, we find three
fixed points, corresponding to systems with zero, one or two bound or virtual
states at threshold. We use the RG to determine the power countings for the
resulting effective field theories. In the case of a single low-energy state,
the resulting theory takes the form of an effective-range expansion in the
strongly interacting channel. We also extend the analysis to include the
effects of the Coulomb interaction between charged particles. The approach is
then applied to the coupled $p+{^7}$Li and $n+{^7}$Be channels which couple to
a $J^P=2^-$ state of $^8$Be very close to the $n+{^7}$Be threshold. At
next-to-leading order, we are able to get a good description of the $p+{^7}$Li
phase shift and the ${^7}$Be(n,p)${^7}$Li cross section using four parameters.
Fits at one order higher are similarly good but the available data are not
sufficient to determine all five parameters uniquely.
