Beyond thresholds: reconstructing UV physics from IR expansions

Abstract

We show that ultraviolet information can be extracted from low-energy expansion coefficients, assuming analyticity and the absence of massless singularities. By reorganizing the low-energy expansion through an inverse Laplace transform and a controlled coarse-graining procedure, we make ultraviolet behavior accessible beyond the cutoff of the effective field theory. In particular, we determine the sign of the beta function and the associated dynamical scale directly from the low-energy expansion of a physical observable below the mass thresholds in QED and QCD-like theories.

Figures (12)

• Figure 1: Integration contours in the complex $z$-plane.
• Figure 2: The UV behavior of $S$ reconstructed from the IR expansion by fitting $\tilde{S}$ (solid lines) in the QED model. We take $\tau_{\rm max}=10 m_e^2$ and $n_{\rm max}=10$. The colors denote the interpolation orders ($1,3,5,7,9,10$). The exact result is shown as a black dot-dashed line, while the IR expansion is shown as a dotted-dashed line, which overlaps with the red solid curve for interpolation order $10$.
• Figure 3: $d \ln \tilde{S}|_{n_{\rm max}}/d \ln \tau$. See Eq. \ref{['dStl']}. The truncation order of the low-energy expansion is indicated by $n_{\rm max}$. The curves are cut off considering their validity ranges. The $\overline{\rm MS}$-scheme result for $k \beta(\alpha)/\alpha = k/\log(\Lambda_{\rm QED}^2/\tau)$ with $k=1$ is also shown as the blue dashed line.
• Figure 4: Extracted $S(Q^2)$ (blue) with an error band, compared with the exact result (dashed) obtained in the UV theory.
• Figure 5: $d \ln \tilde{S}|_{n_{\rm max}}/d \ln \tau-1$. See Eq. \ref{['betafnCPN']}. The truncation order of the low-energy expansion is indicated by $n_{\rm max}$. The curves are cut considering their validity ranges. The $\overline{\rm MS}$ scheme result is also shown by the blue dashed line for $k=1$ and $k=2$.