  By considering an onsite attraction between a spin-$\uparrow$ and a
spin-$\downarrow$ fermion in a multiband tight-binding lattice, here we study
the two-body spectrum, and derive an exact relation between the inverse of the
effective-mass tensor of the lowest bound states and the quantum-metric tensor
of the underlying Bloch states. In addition to the intraband (or the so-called
conventional) contribution that depends only on the single-particle spectrum
and the interband (or the so-called geometric) contribution that is controlled
by the quantum metric, our generalized relation has an additional interband
contribution that depends on the so-called band-resolved quantum metric. All of
our analytical expressions are applicable to those multiband lattices that
simultaneously exhibit time-reversal symmetry and fulfill the condition on
spatially-uniform pairing. As a nontrivial illustration we analyze the two-body
problem in a Kagome lattice with nearest-neighbor hoppings, and show that the
exact relation provides a perfect benchmark.
