The two-dimensional metallic quantum critical problem is a long-standing puzzle that is widely believed to hold the key to resolving ubiquitous non-Fermi-liquid behavior in strongly correlated electronic systems. In this study, we present a nonperturbative renormalization group analysis of the metallic Ising-nematic quantum critical point in two dimensions, formulated directly around an intrinsically nonlocal infrared boson propagator. Rather than treating the anomalous dynamical critical exponent $a$ as a fixed phenomenological parameter, we regard it as an intrinsic component of the fixed-point data to be determined from the internal consistency of the low-energy patch field theory under highly anisotropic scaling dimensions ($\left[k_0\right]=a+1$, $\left[k_x\right]=2$, $\left[k_y\right]=1$). While the leading two-loop diagrammatics vanish identically due to kinematic pole configurations, our three-loop evaluation reveals a profound structural asymmetry between the sectors: The fermion self-energy and Yukawa vertex receive nonvanishing logarithmic corrections, whereas the corresponding bosonic counterterm remains strictly zero. Consequently, we find that no self-consistent, intersecting fixed-point solution for the exponent $a$ exists within the three-loop truncation, failing to reproduce the physical value of $a\approx 1.85$ observed in quantum Monte Carlo simulations. We conjecture that the cross-linked topology of the four-loop boson self-energy diagrams is exactly marginal and yields the minimal, mandatory bosonic counterterm required to restore multisector self-consistency. Our framework establishes a rigid multiloop matching scheme necessary to uniquely pin down the critical exponent and uncovers a stable phase space for field anomalous dimensions.

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• Critical phenomena
• Quantum master equation
• Quantum phase transitions
• Renormalization group
• Resummation methods
• Many-body techniques