Acoustic analogy of quantum Baldin sum rule for optimal causal scattering

The mass law is a cornerstone in predicting sound transmission loss, yet it neglects the constraints of causal dispersion. Current causality-based theories, such as the Rozanov limit, are applicable only to one-port reflective absorbers. Here, we derive a universal sum rule governing causal scattering in acoustic systems, establishing a rigorous analogy to the Baldin sum rule in quantum field theory. This relation reveals that the integral of the extinction cross-section is fundamentally locked by the scatterer's static effective mass and stiffness, which is validated numerically using seminal examples of underwater metamaterials. Furthermore, the proposed sum rule predicts an optimal condition for an anomalously broadened transmission loss bandwidth, as experimentally observed through the spectral shaping effect of an acoustic Fano resonator. Our findings open up an unexplored avenue for enhancing the scattering bandwidth of passive metamaterials.