In superconductivity, a surge of interests in enhancing $T_{\mathrm{c}}$ is ever mounting, where a recent focus is toward multiband superconductivity. In $T_{\mathrm{c}}$ enhancements specific to two-band cases, especially around the Bardeen-Cooper-Schrieffer to Bose-Einstein condensate crossover considered here, we have to be careful about how quantum fluctuations affect the many-body states, i.e., particle-hole fluctuations suppressing the pairing for attractive interactions. Here, we explore how to circumvent the suppression by examining multichannel pairing interactions in two-band systems. With the Gor'kov-Melik-Barkhudarov (GMB) formalism for particle-hole fluctuations in a continuous space, we look into the case of a deep dispersive band accompanied by an incipient heavy-mass (i.e., quasiflat) band. We find that, while the GMB corrections usually suppress $T_{\mathrm{c}}$ significantly, this in fact competes with the enhanced pairing arising from the heavy band, with the trade-off leading to a peaked structure in $T_{\mathrm{c}}$ against the band-mass ratio when the heavy band is incipient. The system then plunges into a strong-coupling regime with the GMB screening vastly suppressed. This occurs prominently when the chemical potential approaches the bound state lurking just below the heavy band, which can be viewed as a Fano-Feshbach resonance, with its width governed by the pair-exchange interaction. The diagrammatic structure comprising particle-particle and particle-hole channels is heavily entangled, so that the emergent Fano-Feshbach resonance dominates all the channels, suggesting a universal feature in multiband superconductivity and superfluidity.

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• BCS-BEC crossover
• Fano resonance
• Multiband superconductivity
• Transition temperature
• Feshbach resonance
• Feynman diagrams