The Breathing Universe: 708Renormalization and Scale Flow of the Vacuum-Tension Field

This work develops the renormalization structure of the vacuum-tension effective field theory within the Breathing Universe framework, in which the physical vacuum is interpreted as a structured dynamical medium capable of storing and redistributing internal tension. The macroscopic state of this medium is described by a scalar order parameter H (x), representing the local imbalance relative to a zero-line equilibrium configuration. While previous studies established the scalar–tensor formulation of the theory and identified higher-derivative operators as the origin of small corrections to gravitational-wave propagation, the associated parameters were treated as scale-independent. In contrast, effective field theory principles require that these couplings depend on the observational scale. In this paper, a renormalization group framework is introduced by defining a scale-dependent effective redistribution field Hₑff (μ) and running couplings α (H₀, μ), β (H₀, μ), and M* (μ). This leads to a scale-dependent gravitational-wave dispersion relation of the form: ω² = c² k² 1 + α (H₀, μ) + β (H₀, μ) (k/M* (μ) ) ⁿ. The analysis demonstrates that different gravitational-wave observational regimes—ranging from cosmological scales and pulsar timing arrays to space-based and ground-based interferometers—probe distinct regions of the renormalization flow of the vacuum-tension field. As a result, multi-band gravitational-wave astronomy provides a natural experimental framework for constraining the scale dependence of the effective theory. It is further shown that the renormalization flow remains stable across the observational domain: the running couplings remain perturbatively small, and higher-derivative corrections are strongly suppressed below the effective cutoff scale. The framework therefore preserves consistency with General Relativity while allowing small, scale-dependent deviations. These results establish the scale-dependence layer of the Breathing Universe effective field theory and provide a unified interpretation of gravitational-wave constraints across frequency bands. Future multi-band observations may offer direct empirical access to the renormalization flow of vacuum-tension dynamics, opening a new observational window on the structure of the physical vacuum.