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We analyze the paradigmatic competition between intra-band and cross-bandCooper-pair formation in two-band superconductors, neglected in most works todate. We derive the phase-sensitive gap equations and describe the crossoverbetween the intraband-dominated and the crossband-dominated regimes, delimitedby a ``gapless'' state. Experimental signatures of crosspairing comprisenotable gap-splitting in the excitation spectrum, non-BCS behavior of gapsversus temperature, as well as changes in the pairing symmetry as a function oftemperature. The consequences of these findings are illustrated on the examplesof MgB$_2$ and Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$.

Cross-band versus intra-band pairing in superconductors: signatures and consequences of the interplay