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It is a matter of current debate whether the gate-tunable superconductivityin twisted bilayer graphene is phonon-mediated or arises from electron-electroninteractions. The recent observation of the strong coupling of electrons toso-called $K$-phonon modes in angle-resolved photoemission spectroscopyexperiments has resuscitated early proposals that $K$-phonons drivesuperconductivity. We show that the bandwidth-enhancing effect of interactionsdrastically weakens both the intrinsic susceptibility towards pairing as wellas the screening of Coulomb repulsion that is essential for the phononattraction to dominate at low temperature. This rules out purely$K$-phonon-mediated superconductivity with the observed transition temperatureof $\sim 1$ K. We conclude that the unflattening of bands by Coulombinteractions challenges any purely phonon-driven pairing mechanism, and must beaddressed by a successful theory of superconductivity in moir\'e graphene.

Coulomb-driven band unflattening suppresses $K$-phonon pairing in moir\'e graphene