Smeared spectral functions from lattice QCD: opportunities and challenges

Abstract

A broad class of physical observables can be expressed in terms of smeared spectral functions. Important examples include the hadronic vacuum polarization contribution to the muon $g-2$, hadronic $τ$ decays, inclusive semileptonic $B$ and $D$ decays, and amplitudes for rare processes such as $B\to K^{(*)}\ell^+\ell^-$. In recent years, a variety of methods have been proposed to reconstruct or constrain such spectral functions from Euclidean correlators and related inputs. However, the inverse problem is intrinsically ill-posed, making it difficult to obtain fully reliable, model-independent results with controlled uncertainties. In this contribution, I summarize recent developments, discuss the opportunities offered by smeared spectral observables, and highlight the main challenges that remain.