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Using a realistic multi-band model for two holes doped into a CuO$_2$ layer,we devise a method to turn off the magnon-mediated interaction between theholes. This allows us to verify that this interaction is attractive, andtherefore could indeed be (part of) the superconducting glue. We derive itsanalytical expression and show that it consists of a novel kind ofpair-hopping+spin-exchange terms. Its coupling constant is fitted from theground state energy obtained with variational exact diagonalization, and itfaithfully reproduces the effect of the magnon-mediated attraction in theentire Brillouin zone. For realistic parameter values, this effectiveinteraction is borderline strong enough to bind the holes into preformed pairs.

The magnon-mediated attraction between two holes doped in a CuO$_2$ layer