  We present a new on-shell method for the matching of ultraviolet models
featuring massive states onto their massless effective field theory. We employ
a dispersion relation in the space of complex momentum dilations to capture, in
a single variable, the relevant analytic structure of scattering amplitudes at
any multiplicity. Multi-variate complex analysis and crossing considerations
are therefore avoided. Remarkably, no knowledge about the infrared effective
field theory is required in dimensional regularisation. All matching
information is extracted from the residues and discontinuities of the
ultraviolet scattering amplitudes, which unitarity expresses in terms of
lower-point and lower-loop results, respectively. This decomposition into
simpler building blocks could deliver new insights in the structure of the
effective field theories obtained from classes of ultraviolet scenarios and
facilitate computations at higher loop orders.
