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This chapter reviews the solution to the U(1)A-problem, from a quantitative perspective. We discuss the ‘t Hooft large-Nc limit of QCD, which removes the axial anomaly, and its reappearance under 1/ Nc corrections. This provides mass to the η’ meson, which is related to the topological susceptibility by the Witten–Veneziano formula. This susceptibility is now well-defined on the lattice, based on the index theorem. It has a controlled continuum limit, which substantiates the Witten–Veneziano formula.
The longitudinal fields of a tightly focused Laguerre–Gaussian (LG) laser can be used to accelerate electron pulse trains when it is reflected from a solid plasma. However, the normal transverse mode of laser beams in high-power laser systems is approximately Gaussian. A routine and reliable way to obtain high-intensity LG lasers in experiments remains a major challenge. One approach involves utilizing a solid plasma with a ‘light fan’ structure to reflect the Gaussian laser and obtain a relativistic intense LG laser. In this work, we propose a way to combine the mode transformation of a relativistic laser and the process of electron injection and acceleration. It demonstrates that by integrating a nanowire structure at the center of the ‘light fan’, electrons can be efficiently injected and accelerated during the twisted laser generation process. Using three-dimensional particle-in-cell simulations, it is shown that a circularly polarized Gaussian beam with ${a}_0=20$ can efficiently inject electrons into the laser beam in interaction with the solid plasma. The electrons injected close to the laser axis are driven by a longitudinal electric field to gain longitudinal momentum, forming bunches with a low energy spread and a small divergence angle. The most energetic bunch exhibits an energy of 310 MeV, with a spread of 6%. The bunch charge is 57 pC, the duration is 400 as and the divergence angle is less than 50 mrad. By employing Gaussian beams, our proposed approach has the potential to reduce experimental complexity in the demonstrations of twisted laser-driven electron acceleration.
Scalar quantum electrodynamics is constructed by promoting a global U(1) symmetry to a local one. We address electrically charged infraparticles, and the corresponding superselection sectors, in infinite volume and in finite volume with two kinds of boundary Conditions.
By means of an ion crystal model, we illustrate the concepts of a particle in the sense of quantum mechanics and of quantum field theory. The latter describes reality in particle physics, but in order to avoid confusion, we temporarily denote it as a “wavicle”.
Chiral symmetry of free fermions is studied in the continuum and on the lattice. In the latter case, we review the fermion doubling problem and the Nielsen–Ninomiya theorem, then we construct Wilson fermions and finally several types of Ginsparg–Wilson fermions, which are endowed with an exact, lattice modified chiral symmetry.
This chapter deals with the renormalization group in Wilson’s spirit. General concepts, like fixed points, are illustrated with examples, such as block-variable transformations, perfect lattice actions, the Wilson–Fisher fixed points, the Callan–Symanzik equation, and various scenarios for running couplings.
The free electromagnetic field is quantized canonically and with the functional integral. We emphasize the roles of the Gauss law, helicity, and gauge fixing in the continuum. We also derive Planck’s formula for black-body radiation and apply it to the cosmic microwave background.
In high-energy scattering processes, hadrons can be described as a set of partons. This picture is compatible with QCD, where the partons are identified as quarks, anti-quarks, and gluons. In this picture, we consider electron–positron annihilation, which can lead to hadrons or a muon–anti-muon pair. The R-ratio of the cross sections for these scenarios allows us to identify the number of colors, Nc = 3, experimentally. Next we discuss deep inelastic electron–nucleon scattering, which leads to the concepts of the Bjorken variable, structure functions, the parton distribution function, Bjorken scaling, the Callan–Gross relation, and the DGLAP evolution equation. The hadronic tensor takes us to the scaling functions, where high-energy neutrino–nucleon scattering provides further insight, in particular a set of constraints which are expressed as sum rules.