This work develops a framework to apply normalizing-flow transformations of field configurations for all-orders Quantum Electrodynamics (QED) corrections in lattice field theory. This opens a new possibility to determine all-order corrections without the need for additional Monte Carlo sampling, generally bypassing the complexity in Wick-contraction diagrams needed at fixed order. The new method is applied to lattice scalar QED in two, three, and four spacetime dimensions, using both analytical and machine-learned flows, with both approaches yielding estimates with significantly reduced variance with respect to standard methods. It is further shown that flows can be trained using small lattice geometries and subsequently evaluated on much larger lattice geometries while maintaining good efficiency. A generalization to theories with fermions is envisaged, suggesting a path to applications in challenging field theories including lattice Quantum Chromodynamics.

We present the first complete lattice QCD calculation of the four structure-dependent form factors governing the rare charged kaon decay $K^- \to \ell^- \bar{\nu}_\ell \ell'^+ \ell'^-$, with fully controlled statistical and systematic uncertainties. Our calculation is based on gauge ensembles generated by the Extended Twisted Mass Collaboration (ETMC) with $N_f = 2+1+1$ flavors of Wilson-clover twisted-mass fermions. Simulations are performed directly at the physical values of the light and strange quark masses, and include an estimate of the quark-disconnected contributions in which the virtual photon couples to sea quarks. All four form factors are determined across the kinematical region probed by experiments. The Spectral Function Reconstruction (SFR) method of Ref. [1] is employed to overcome the analytic continuation problem for dilepton invariant masses above the two-pion threshold. Finite-volume effects are investigated using ensembles with spatial extents $L\simeq [3.8,7.6]~\mathrm{fm}$, while the continuum limit is obtained from three lattice spacings in the range $a\in[0.057, 0.08]~\mathrm{fm}$. Our results for the form factors enable the evaluation of decay rates and differential observables for all four channels, $K^- \to e^- \bar{\nu}_e e^+ e^-$, $K^- \to e^- \bar{\nu}_e \mu^+ \mu^-$, $K^- \to \mu^- \bar{\nu}_\mu e^+ e^-$, and $K^- \to \mu^- \bar{\nu}_\mu \mu^+ \mu^-$, thereby providing first-principles Standard Model predictions against which existing and upcoming measurements can be directly compared. A detailed phenomenological analysis of the decay rates and associated observables is presented in a companion paper [2].

Weak decays of charged kaons with an additional lepton-antilepton pair, $K^- \to \ell^- \bar{\nu}_\ell \ell'^{+} \ell'^{-}$ ($K_{\ell2\ell'}$), are suppressed at order $O(G_{F}^{2}\alpha_{\rm em}^{2})$ in the Standard Model (SM) and provide sensitive probes of its flavour structure, as well as independent determinations of the Cabibbo angle $|V_{us}|$. In this Letter we present the SM predictions for all four channels with $\ell,\ell' =e,\mu$, based on the first complete lattice QCD calculation of the structure-dependent form factors reported in a companion paper [1]. Using the PDG value [2] $|V_{us}|^{\rm PDG}=0.22431(85)$, we obtain branching fractions with controlled uncertainties and precisions ranging from $2\%$ to $7\%$, depending on the channel. For the three modes with published measurements, our results agree with experiment. For the $K_{\mu2\mu}$ mode, for which no published experimental result is available, we compare our prediction with the preliminary NA62 result, finding agreement at the $1.4\sigma$ level. Conversely, the measured decay rates can be used together with our results to extract $|V_{us}|$ from these modes. A weighted average over the two most precise channels, $K_{\mu2e}$ and $K_{\mu2\mu}$, yields $|V_{us}|=0.2283(42)$, corresponding to a $1.8\%$ determination. These results pave the way for using $K_{\ell2\ell'}$ decays as precision probes of the SM.

A novel theoretical framework, the inverse problem approach, is proposed to calculate non-perturbative quantities in quantum chromodynamics (QCD). Based on the dispersion relation of quantum field theory, this approach determines unknown low-energy non-perturbative quantities from known high-energy perturbative inputs via solving an inverse problem. The resulting inverse problem is rigorously proven to be ill-posed, with the solutions being unique but unstable. To address this instability, the well-established Tikhonov regularization is employed, yielding stable approximate solutions that converge to the true values as input errors vanish. The key features of this approach are illustrated through three toy models, demonstrating that solution precision can be systematically improved through reduced input errors and optimized regularization strategies.