Note that this sum runs over all poles and zeros in the variable $s$ in the amplitude. Such poles can even arise from long-range force carriers in the EFT: for instance, massless exchanges in the $s$ channel, which generate $-1/s$, or massless exchanges in the $t$ channel at fixed $u$, which generate $1/(s+u)$.

See Ref. [7] for an analogous construction of an amplitudelike object from the logarithmic derivative of the Landau-Riemann $\mathrm{\Xi }$ function, encoding the locations of the zeta function’s zeros.

Cauchy’s argument principle states that $$\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f^{\prime }(z)}{f(z)}dz=d_{\text{zeros}}-d_{\text{poles}},$$where $f(z)$ is meromorphic and $\gamma$ is a closed curve encircling $d_{\text{zeros}}$ zeros and $d_{\text{poles}}$ poles of $f(z)$.

A higher-degree zero can be accommodated by allowing $\rho$ to run over repeated elements or, equivalently, by rescaling the corresponding term in Eq. (10) by the degree of degeneracy. The same applies for poles, though higher-degree poles are forbidden by locality.

For $k=0$ and $-1$, this sum must, of course, be handled with care.

In the case of higher-degree zeros, we keep the ${\lambda }_n$ distinct but absorb the degree of degeneracy into the magnitude of ${\sigma }_n$. Meanwhile, if a pole and a zero happen to coincide, then their contributions cancel in $c_k$, the associated ${\sigma }_n$ effectively vanishes, and the corresponding ${\lambda }_n$ can be excluded from the sum. In such cases, the Vandermonde matrix $[\mathit{\lambda }{]}_{\mathrm{in}}={\lambda }_n^i$ remains left-invertible.

A similar mathematical construction was used in Ref. [3] to extract the spectrum, albeit in the context of scattering in the forward limit. In the present Letter, we directly extract both the poles and zeros of the amplitude at general kinematics.