The story of the anomalous magnetic moment of the muon, $a_{\mu}$, has been a long and absorbing read, with several unexpected plot twists. The theme: is new physics visible in $a_{\mu}$ or not? is clearly an important one. As we head towards what may be the final chapters, the pace of the narrative has quickened and lattice QCD is providing a critical new perspective on some of the key characters. I will review the story so far, focussing on recent episodes but (spoiler alert) depending on your point of view, it may not be heading to a happy ending.

In this paper, we investigate a digitised SU$(2)$ lattice gauge theory in the Hamiltonian formalism. We use partitionings to digitise the gauge degrees of freedom and show how to define a penalty term based on finite element methods to project onto physical states of the system. Moreover, we show for a single plaquette system that in this framework the limit $g\to0$ can be approached at constant cost.

Momentum conservation in the nucleon is examined in terms of continuous flow of the momentum density current (or in short, momentum flow), which receives contributions from both kinetic motion and interacting forces involving quarks and gluons. While quarks conduct momentum flow through their kinetic motion and the gluon scalar (anomaly) contributes via pure interactions, the gluon stress tensor has both effects. The quarks momentum flow encodes the information of the force density on them, and the momentum conservation allows to trace its origin to the gluon tensor and anomaly ("negative pressure"). From state-of-the-art lattice calculations and experimental fits on the form factors of the QCD energy-momentum tensor, we exhibit pictures of the momentum flow and forces on the quarks in the nucleon. In particular, the anomaly contributes a critical attractive force with a strength similar to that of a QCD confinement potential.

Quantum simulation of lattice gauge theories is an important avenue to gain insights into both particle physics phenomena and constrained quantum many-body dynamics. There is a growing interest in probing analogs of high energy collision phenomena in lattice gauge theories that can be implemented on current quantum simulators. Motivated by this, we characterize the confined mesons that originate from a local high energy excitation in a particle-conserving 1D $Z_2$ lattice gauge theory. We focus on a simple, experimentally accessible setting that does not require preparation of colliding wavepackets and isolates the effects of gauge field confinement strength and initial state energy on the nature of propagating excitations. We find that the dynamics is characterized by the propagation of a superposition of differently sized mesons. The linear confinement leads to meson size oscillations in time. The average meson size and oscillation frequency are controlled by the strength of the gauge field confinement. At a constant confinement field, the average meson length is controlled by the initial excitation's energy. Higher energies produce longer mesons and their effective mass depends strongly on their size: longer mesons propagate more slowly out of the central excitation. Mesons of different sizes get spatially filtered with time due to different speeds. We show that this phenomenology is a consequence of linear confinement and remains valid in both the strong and weak confinement limit. We present simple explanations of these phenomena supported by exact numerics.