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In this talk, I will present how scattering amplitudes involving background dilaton and graviton fields can serve as powerful probes of trace anomalies along four-dimensional renormalization group (RG) flows. By coupling these fields to quantum field theory degrees of freedom while preserving the Weyl anomaly structure, I will show that dilaton-dilaton and graviton-dilaton scattering amplitudes directly encode the changes in trace anomaly coefficients between ultraviolet (UV) and infrared (IR) conformal field theories (CFTs). This framework will be tested through explicit perturbative examples, demonstrating its robustness across controlled settings. I will then describe how these ideas can be extended non-perturbatively using the S-matrix bootstrap, which leverages analyticity, unitarity, crossing symmetry, and Regge behavior to place universal lower bounds on the UV a-anomaly coefficient for gapped RG flows arising from relevant deformations of UV CFTs. This study provides a concrete step toward addressing a fundamental question in quantum field theory: for a given massive QFT, what is the space of possible UV CFTs that can flow to it?
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