Standard Model tests with smeared experiment and theory

Abstract

For Standard Model processes in which on-shell intermediate hadronic states contribute - including inclusive semileptonic decays and long-distance effects in rare exclusive decays such as $D\to π\ell\ell$ and $B\to K^{(\ast)}\ell\ell$ - spectral-reconstruction techniques provide a promising route to model-independent lattice QCD predictions for use in phenomenological predictions. The central ingredient is the computation of the energy-smeared spectral density. Following the continuum and infinite-volume limits, the physical amplitude is recovered as the limit of vanishing smearing width. However, achieving sufficiently small smearing for a controlled extrapolation remains a significant challenge for current lattice simulations. In this paper, we therefore propose Standard Model tests, in which both experimental results and theory predictions are smeared with finite width, similar to what has previously been done in the literature for experimental and lattice $R$-ratio data in the context of the muon $(g-2)_μ$. As concrete examples, we discuss the cases of inclusive meson decay and long-distance contributions to rare semileptonic meson decay.

Figures (2)

• Figure 1: Plots of the kernel $\omega^n \theta(\omega_{\rm max}-\omega)K_\epsilon(\omega-q_0)$ for $n=0,1,2$ (from left to right) with $\epsilon=100$ MeV and $\epsilon=300$ MeV with truncation 20 and 40 (see box in plot), for $q_0=0.5,1.0,1.5,2.0$ GeV and edge of the phase space at $\omega_{\rm max}=2$ GeV (vertical line). The dashed curve shows the kernel itself, while the solid lines show the approximation in terms of shifted Chebyshev polynomials that map $[\omega_{\rm min},\infty]\to [-1,1]$.
• Figure 2: Plots of the $\phi=0$ (top) and $\phi=\pi$ (bottom) part of the kernel as defined in Eq. (\ref{['eq:excluive kernel']}) for $\epsilon=0.3{\rm GeV}^2$. The plots on the left (right) show the kernel for the time (a spatial) component of the projector $\bar{t}_\mu$. The dashed lines show the kernel and the solid lines are Chebyshev approximations (mapping energies $[\sqrt{s_{\rm th}},\infty]\to [-1,1]$) of order $N=40$. We show the cases $s=0.5,1.0,1.5,2.0,2.5,3.0\,{\rm GeV}^2$. Vertical lines indicate (from left to right) $s_{\rm th}=0.25{\rm GeV}^2$, $M_\rho^2$, $M_\omega^2$, $M_\phi^2$ and $q^2_{\rm max}=(M_D-M_\pi)^2$. Note that $s_{\rm th}$ can be chosen freely between 0 and the value corresponding to the energy of the lightest on-shell intermediate state.