Momentum Distribution and Contact Parameters of a mass-imbalanced three-body system across the Efimov-Unatomic transition
We investigate the single-particle momentum distribution and contact parameters of mass-imbalanced three-body systems at the critical dimension Dc, where the transition between discrete and continuous scale invariance takes place as the spatial dimension is tuned between three and two dimensions. We show that the asymptotic momentum distribution at Dc is governed by a distinct logarithmic scaling structure, which differs fundamentally from both the log-periodic behavior of Efimov states and the power-law scaling of the unatomic regime. This structure requires the introduction of an additional three-body contact parameter associated with a quadratic logarithmic contribution, leading to a finite and well-defined description of the momentum tail at the transition. This additional three-body parameter depends sensitively on the mass imbalance, changing sign across different mass configurations and vanishing for identical particles. As a consequence, the three-body contribution to the momentum distribution can be suppressed at a characteristic momentum scale, leaving the asymptotic tail entirely determined by the two-body contact. We further analyze the narrow intermediate region connecting the Efimov and unatomic regimes, here identified as an intermediate scaling regime, whose extent and properties are strongly controlled by the mass ratio. These results establish the critical dimension as a regime with emergent scaling properties and provide experimentally accessible signatures for probing the transition between discrete and continuous scale invariance in few-body quantum systems.