Isgur-Wise symmetry in two dimensions

The approach to the heavy-quark Isgur-Wise limit is examined in (1+1)-dimensional QCD where exact calculations are possible. To facilitate the discussion, an explicit scaling equation for the mesonic Bethe-Salpeter amplitude is derived. We find that the leading finite heavy-quark mass corrections to the decay constant and the elastic form factor are comparable in strength. Nonrelativistic estimates of both of the quantities work reasonably well and generalize to 3+1 dimensions. We find that the corrections to the (3+1)-dimensional form factor at large v⋅v’ are comparable to those estimated for the decay constant by lattice gauge theory and QCD sum rules. Furthermore, the structure function ${\mathit{f}}_{\mathrm{-}}$(v⋅v’) is found to be quite large at v⋅v’=1 and we argue that this will remain true in 3+1 dimensions. This implies that one must exercise caution when attempting to extract Cabibbo-Kobayashi-Maskawa matrix elements from semileptonic decays. Finally, it is shown that the Isgur-Wise symmetry never breaks down in two-dimensional QCD for heavy-quark to heavy-quark transitions.