Theoretical particle physicists continue to push the envelope in both high performance computing and in managing and analyzing large data sets. For example, the goals of sub-percent accuracy in predictions of quantum chromodynamics (QCD) using large scale simulations of lattice QCD and in finding signals of rare events and new physics in exabytes of data produced by experiments at the high luminosity large hadron collider (LHC) require new tools beyond just developments in hardware. Machine learning and artificial intelligence offer the promise of dramatically reducing the computational cost and time. This chapter reviews selected areas where AI/ML tools could have a major impact, provides an overview of the challenges, and discusses how new ideas such as normalizing flows can speed up the generation of gauge configurations needed in lattice QCD calculations; the growth of ML in surrogate models and pattern matching to reduce the cost of event generators and in the analysis of experimental data; and in the search for viable vacua in the landscape of string theories. While such approaches transform aspects of particle theory into computational problems, and thus black boxes, we argue that physics-aware development of these tools combined with algorithms that ensure that the results are bias free will continue to require a deep understanding of the physics. We see this broader transformation as akin to formulating and extracting observables from simulations of lattice QCD, a numerical integration of the path integral formulation of QCD that nevertheless requires a deep understanding of the underlying quantum field theory, the standard model of particle physics and effective field theory methods.

In commonly used Monte Carlo algorithms for lattice gauge theories the integrated autocorrelation time of the topological charge is known to be exponentially-growing as the continuum limit is approached. This $\mathit{topological}\,\,\textit{freezing}$, whose severity increases with the size of the gauge group, can result in potentially large systematics. To provide a direct quantification of the latter, we focus on $\mathrm{SU}(6)$ Yang--Mills theory at a lattice spacing for which conventional methods associated to the decorrelation of the topological charge have an unbearable computational cost. We adopt the recently proposed $\mathit{parallel}\,\,\mathit{tempering}\,\,\mathit{on}\,\,\mathit{boundary}\,\,\mathit{conditions}$ algorithm, which has been shown to remove systematic effects related to topological freezing, and compute glueball masses with a typical accuracy of $2-5\%$. We observe no sizeable systematic effect in the mass of the first lowest-lying glueball states, with respect to calculations performed at nearly-frozen topological sector.

We consider Yang-Mills theory with a compact gauge group $G$ on Minkowski space ${\mathbb R}^{3,1}$ and compare the introduction of masses of gauge bosons using the Stueckelberg and Higgs mechanisms. The Stueckelberg field $\phi$ is identified with a $G$-frame on the gauge vector bundle $E$ and the kinetic term for $\phi$ leads to the mass of the gauge bosons. The Stueckelberg mechanism is extended to the Higgs mechanism by adding to the game a scalar field describing rescaling of metric on fibres of $E$. Thus, we associate Higgs fields as well as running coupling parameters with conformal geometry on fibres of gauge bundles. In particular, a running coupling tending to zero or to infinity is equivalent to an unbounded expansion of $G$-fibres or its contraction to a point. We also discuss scale connection, space-time dependent Higgs vacua and compactly supported gauge and quark fields as an attribute of confinement.

Some sigma models which admit a theta angle are integrable at both $\vartheta=0$ and $\vartheta=\pi$. This includes the well-known $O(3)$ sigma model and two families of coset sigma models studied by Fendley. We consider the ground state energy of these models in the presence of a magnetic field, which can be computed with the Bethe ansatz. We obtain explicit results for its non-perturbative corrections and we study the effect of the theta angle on them. We show that imaginary, exponentially small corrections due to renormalons remain unchanged, while instanton corrections change sign, as expected. We find in addition corrections due to renormalons which also change sign as we turn on the theta angle. Based on these results we present an explicit non-perturbative formula for the topological susceptibility of the $O(3)$ sigma model in the presence of a magnetic field, in the weak coupling limit.

In this paper, we propose a general implementation of the Virasoro generators and Kac-Moody currents in generic tensor network representations of 2-dimensional critical lattice models. Our proposal works even when a quantum Hamiltonian of the lattice model is not available, which is the case in many numerical computations involving numerical blockings. We tested our proposal on the 2d Ising model, and also the dimer model, which works to high accuracy even with a fairly small system size. Our method makes use of eigenstates of a small cylinder to generate descendant states in a larger cylinder, suggesting some intricate algebraic relations between lattice of different sizes.

We focus on studying, numerically, the scalar curvature tensor in a two-dimensional discrete space. The continuous metric of a two-sphere is transformed into that of a lattice using two possible slicings. In the first, we use two integers, while in the second we consider the case where one of the coordinates is ignorable. The numerical results of both cases are then compared with the expected values in the continuous limit as the number of cells of the lattice becomes very large.

The IKKT matrix model (or the type IIB matrix model) is known as a promising candidate for a nonperturbative formulation of superstring theory in ten dimensions. As a most attractive feature, the model admits the emergence of (3+1)-dimensional space-time associated with the spontaneous breaking of the (9+1)-dimensional Lorentz symmetry. Numerical confirmation of such a phenomenon has been attempted for more than two decades. Recently it has been found that the sign problem, the main obstacle in simulating this model, can be overcome by the complex Langevin method. It has been shown that the Lorenzian version of the model is smoothly connected with the Euclidean version, in which the SO(10) symmetry is found to be spontaneously broken to SO(3). Here we propose to add a Lorentz invariant "mass" term to the original model and discuss a scenario that (3+1)-dimensional expanding space-time with Lorentzian signature appears at late times. Some numerical results supporting this scenario are presented.

We present $I=1/2$ $D^\ast\pi$ scattering amplitudes from lattice QCD and determine two low-lying $J^P=1^+$ axial-vector $D_1$ states and a $J^P=2^+$ tensor $D_2^\ast$. Computing finite-volume spectra at a light-quark mass corresponding to $m_\pi=391$ MeV, for the first time, we are able to constrain coupled $J^P=1^+$ $D^\ast\pi$ amplitudes with $^{2S+1}\ell_J\,=\,^3S_1$ and $^3\!D_1$ as well as coupled $J^P=2^+$ $D\pi\{^1\!D_2\}$ and $D^\ast\pi \{^3\!D_2\}$ amplitudes via L\"uscher's quantization condition. Analyzing the scattering amplitudes for poles we find a near-threshold bound state, producing a broad feature in $D^\ast\pi\{^3\!S_1\}$. A narrow bump occurs in $D^\ast\pi\{^3\!D_1\}$ due to a $D_1$ resonance. A single resonance is found in $J^P=2^+$ coupled to $D\pi$ and $D^\ast\pi$. A relatively low mass and large coupling is found for the lightest $D_1$, suggestive of a state that will evolve into a broad resonance as the light quark mass is reduced. An earlier calculation of the scalar $D_0^\ast$ using the same light-quark mass enables comparisons to the heavy-quark limit.

We establish how the Breitenlohner-Freedman (BF) bound is realized on tilings of two-dimensional Euclidean Anti-de Sitter space. For the continuum case and for scalar modes, the BF bound states that on Anti-de Sitter spaces, fluctuation modes remain stable for small negative mass squared. We solve the Klein-Gordon equation both analytically and numerically for finite cutoff. We then numerically calculate the BF bound for both cases. The results agree and are independent of the specific tiling. We also propose a model for a hyperbolic electric circuit and find again numerical agreement with the modified BF bound. This circuit is readily accessible in the laboratory, allowing for the experimental verification of our results.

Recently, based on a novel analysis of the Planck satellite data, a hint of a uniform rotation of the polarization of cosmic microwave background photons, called isotropic cosmic birefringence, has been reported. The suggested rotation angle of polarization of about $0.2-0.4$ degrees strongly suggests that it is determined by the fine structure constant, which can be naturally explained over a very wide parameter range by the domain walls of axion-like particles. Interestingly, the axion-like particle domain walls predict not only isotropic cosmic birefringence but also anisotropic one that reflects the spatial distribution of the axion-like particle field on the last scattering surface. In this Letter, we perform lattice simulations of the formation and evolution of domain walls in the expanding universe and obtain for the first time the two-point correlation function and power spectrum of the scalar field that constitutes the domain walls. We find that while the power spectrum is generally consistent with analytical predictions based on random wall distributions, there is a predominant excess on the scale corresponding to the Hubble radius. Applying our results to the anisotropic cosmic birefringence, we predict the power spectrum of the rotation angles induced by the axion-like particle domain walls and show that it is within the reach of future observations of the cosmic microwave background.

We generalize the idea of the quantized Hall current to count gapless edge states in topological materials in any number of dimensions. This construction applies equally well to theories without any continuous symmetries in the bulk or chiral anomalies on the boundaries. The generalized Hall current is related to the phase space index of the Euclidean fermion operator and can be calculated via one-loop Feynman diagrams. The relevant momentum-space topology is shown to be governed in each case by the homotopy group $\pi_n(S^n) = {\mathbb Z}$, where in $n$ is the number of spacetime dimensions. This holds even for topological matter which is characterized by a ${\mathbb Z}_2$ invariant, since the generalized Hall current is governed by topology in phase space, not just momentum space. We elucidate these ideas by explicit examples in free relativistic field theories in various spacetime dimensions with various symmetries, and argue that this approach may work in interacting theories as well, including the interesting cases where the interactions gap the edge states.

The properties of the $B_c$-meson family ($c\bar b$) are still not well determined experimentally because the specific mechanisms of formation and decay remain poorly understood. Unlike heavy quarkonia, i.e. the hidden heavy quark-antiquark sectors of charmonium ($c\bar c$) and bottomonium ($b\bar b$), the $B_c$-mesons cannot annihilate into gluons and they are, consequently, more stable. The excited $B_c$ states, lying below the lowest strong-decay $BD$-threshold, can only undergo through radiative decays and hadronic transitions to the $B_c$ ground state, which then decays weakly. As a result of this, a rich spectrum of narrow excited states below the $BD$-threshold appear, whose total widths are two orders of magnitude smaller than those of the excited levels of charmonium and bottomonium. In a different article, we determined bottom-charmed meson masses using a non-relativistic constituent quark model which has been applied to a wide range of hadron physical observables, and thus the model parameters are completely constrained. Herein, continuing to our study of the $B_c$ sector, we calculate the relevant radiative decay widths and hadronic transition rates between $c\bar b$ states which are below $BD$-threshold. This shall provide the most promising signals for discovering excited $B_c$ states that are below the lowest strong-decay $BD$-threshold. Finally, our results are compared with other models to measure the reliability of the predictions and point out differences.

In these lecture notes the basics of QED corrections to hadronic decays are reviewed with special emphasis on conceptual (e.g. counting and tracking of infrared sensitive logs) rather than numerical aspects. General matters are illustrated for the cases of increased complexity and decreased inclusiveness: $e^+ e^- \to hadrons$, the leptonic decay $\pi^+ \to \ell^+ \bar \nu$ and the semileptonic decay $B \to \pi \ell^+ \bar \nu$. The non-trivial and ongoing efforts of including structure dependence are very briefly outlined.