Pair Kondo effect: A mechanism for time-reversal symmetry breaking superconductivity in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>UTe</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:math>

An important open puzzle in the superconductivity of UTe$_2$ is the emergence of time-reversal broken superconductivity from a non-magnetic normal state. Breaking time-reversal symmetry in a single second-order superconducting transition requires the existence of two degenerate superconducting order parameters, which is not natural for orthorhombic UTe$_2$. Moreover, experiments under pressure (Braithwaite et. al., Comm. Phys. \bf{2}, 147 (2019), arXiv:1909.06074 [cond-mat.str-el]) suggest that superconductivity sets in at a single transition temperature in a finite parameter window, in contrast to the splitting between the symmetry breaking temperatures expected for accidental degenerate orders. Motivated by these observations, we propose a mechanism for the emergence of time-reversal breaking superconductivity without accidental or symmetry-enforced order parameter degeneracies in systems close to a magnetic phase transition. We demonstrate using Landau theory that a cubic coupling between incipient magnetic order and magnetic moments of Cooper pairs (pair-Kondo coupling) can drive time-reversal symmetry breaking superconductivity that onsets in a single, weakly first order transition over an extended region of the phase diagram. We discuss the experimental signatures of such transition in thermodynamic and resonant ultrasound measurements. A microscopic origin of pair-Kondo coupling is identified as screening of magnetic moments by chiral Cooper pairs, built out of two non-degenerate order parameters - an extension of Kondo screening to unconventional pairs.