These lectures aim to provide a basic introduction to dispersive methods and their modern applications to the phenomenology of the Standard Model at low energy. This approach exploits analyticity properties of Green functions and scattering amplitude, often combined with unitarity constraints. To find a logically coherent set of topics in this vast subject, I start with the two-point Green's function, show that this needs the three-point function as input which in turn needs the four-point function. The sequence stops here, just like these lectures, because the four-point function is related only to itself (if one ignores inelastic effects), I will discuss these dispersion relations both in the case of toy models, simple scalar theories, as well as in the phenomenologically relevant case of QCD.

Making use of a dimensionally-reduced effective theory at high temperature, we perform a nonperturbative study of the electroweak phase transition in the Two Higgs Doublet model. We focus on two phenomenologically allowed points in the parameter space, carrying out dynamical lattice simulations to determine the equilibrium properties of the transition. We discuss the shortcomings of conventional perturbative approaches based on the resummed effective potential - regarding the insufficient handling of infrared resummation but also the need to account for corrections beyond 1-loop order in the presence of large scalar couplings - and demonstrate that greater accuracy can be achieved with perturbative methods within the effective theory. We find that in the presence of very large scalar couplings, strong phase transitions cannot be reliably studied with any of the methods.

We present the first end-to-end nonperturbative analysis of the gravitational wave power spectrum from a thermal first-order electroweak phase transition (EWPT), using the framework of dimensionally reduced effective field theory and pre-existing nonperturbative simulation results. We are able to show that a first-order EWPT in any beyond the Standard Model (BSM) scenario that can be described by a Standard Model-like effective theory at long distances will produce gravitational wave signatures too weak to be observed at existing and planned detectors. This implies that colliders are likely to provide the best chance of exploring the phase structure of such theories, while transitions strong enough to be detected at gravitational wave experiments require either previously neglected higher-dimension operators or light BSM fields to be included in the dimensionally reduced effective theory and therefore necessitate dedicated nonperturbative studies. As a concrete application, we analyze the real singlet-extended Standard Model and identify regions of parameter space with single-step first-order transitions, comparing our findings to those obtained using a fully perturbative method. We discuss the prospects for exploring the electroweak phase diagram in this model at collider and gravitational wave experiments in light of our nonperturbative results.

Phase transitions in a non-perturbative regime can be studied by ab initio Lattice Field Theory methods. The status and future research directions for LFT investigations of Quantum Chromo-Dynamics under extreme conditions are reviewed, including properties of hadrons and of the hypothesized QCD axion as inferred from QCD topology in different phases. We discuss phase transitions in strong interactions in an extended parameter space, and the possibility of model building for Dark Matter and Electro-Weak Symmetry Breaking. Methodological challenges are addressed as well, including new developments in Artificial Intelligence geared towards the identification of different phases and transitions.

New field content beyond that of the Standard Model of particle physics can alter the thermal history of electroweak symmetry breaking in the early universe. In particular, the symmetry breaking may have occurred through a sequence of successive phase transitions. We study the thermodynamics of such scenario in a real triplet extension of the Standard Model, using nonperturbative lattice simulations. Two-step electroweak phase transition is found to occur in a narrow region of allowed parameter space with the second transition always being first order. The first transition into the phase of non-vanishing triplet vacuum expectation value is first order in a non-negligible portion of the two-step parameter space. A comparison with 2-loop perturbative calculation is provided and significant discrepancies with the nonperturbative results are identified.

We present the third-order QCD prediction for the production of top-anti-top quark pairs in electron-positron collisions close to the threshold in the dominant S-wave state. We observe a significant reduction of the theoretical uncertainty and discuss the sensitivity to the top quark mass and width.

Laboratory experiments searching for galactic dark matter particles scattering off nuclei have so far not been able to establish a discovery. We use data from the XENON100 experiment to search for dark matter interacting with electrons. With no evidence for a signal above the low background of our experiment, we exclude a variety of representative dark matter models that would induce electronic recoils. For axial-vector couplings to electrons, we exclude cross-sections above 6x10^(-35) cm^2 for particle masses of m_chi = 2 GeV/c^2. Independent of the dark matter halo, we exclude leptophilic models as explanation for the long-standing DAMA/LIBRA signal, such as couplings to electrons through axial-vector interactions at a 4.4 sigma confidence level, mirror dark matter at 3.6 sigma, and luminous dark matter at 4.6 sigma.

A novel light detection scheme has been tested for use in medium-pressure gas TPCs, in view of rare events searches in low energy particle physics. It has the advantage of minimal interference with the ionization collection system, used for track imaging. It provides an absolute time reference, which allows an absolute determination of the Z coordinate of events, along the direction of the drift field. This makes possible a fiducial cut along the Z-axis, allowing to reduce the background from the ends of the drift volume.

We show that a holographic description of four-dimensional asymptotically locally flat spacetimes is reached smoothly from the zero-cosmological-constant limit of anti-de Sitter holography. To this end, we use the derivative expansion of fluid/gravity correspondence. From the boundary perspective, the vanishing of the bulk cosmological constant appears as the zero velocity of light limit. This sets how Carrollian geometry emerges in flat holography. The new boundary data are a two-dimensional spatial surface, identified with the null infinity of the bulk Ricci-flat spacetime, accompanied with a Carrollian time and equipped with a Carrollian structure, plus the dynamical observables of a conformal Carrollian fluid. These are the energy, the viscous stress tensors and the heat currents, whereas the Carrollian geometry is gathered by a two-dimensional spatial metric, a frame connection and a scale factor. The reconstruction of Ricci-flat spacetimes from Carrollian boundary data is conducted with a flat derivative expansion, resummed in a closed form in Eddington-Finkelstein gauge under further integrability conditions inherited from the ancestor anti-de Sitter set-up. These conditions are hinged on a duality relationship among fluid friction tensors and Cotton-like geometric data. We illustrate these results in the case of conformal Carrollian perfect fluids and Robinson-Trautman viscous hydrodynamics. The former are dual to the asymptotically flat Kerr-Taub-NUT family, while the latter leads to the homonymous class of algebraically special Ricci-flat spacetimes.

We present the first results of searches for axions and axion-like-particles with the XENON100 experiment. The axion-electron coupling constant, $g_{Ae}$, has been probed by exploiting the axio-electric effect in liquid xenon. A profile likelihood analysis of 224.6 live days $\times$ 34 kg exposure has shown no evidence for a signal. By rejecting $g_{Ae}$, larger than $7.7 \times 10^{-12}$ (90\% CL) in the solar axion search, we set the best limit to date on this coupling. In the frame of the DFSZ and KSVZ models, we exclude QCD axions heavier than 0.3 eV/c$^2$ and 80 eV/c$^2$, respectively. For axion-like-particles, under the assumption that they constitute the whole abundance of dark matter in our galaxy, we constrain $g_{Ae}$, to be lower than $1 \times 10^{-12}$ (90\% CL) for mass range from 1 to 40 keV/c$^2$, and set the best limit to date as well.

A dispersive analysis of $\eta\to 3\pi$ decays has been performed in the past by many authors. The numerical analysis of the pertinent integral equations is hampered by two technical difficulties: i) The angular averages of the amplitudes need to be performed along a complicated path in the complex plane. ii) The averaged amplitudes develop singularities along the path of integration in the dispersive representation of the full amplitudes. It is a delicate affair to handle these singularities properly, and independent checks of the obtained solutions are demanding and time consuming. In the present article, we propose a solution method that avoids these difficulties. It is based on a simple deformation of the path of integration in the dispersive representation (not in the angular average). Numerical solutions are then obtained rather straightforwardly. We expect that the method also works for $\omega\to 3\pi$.

We present a lattice QCD calculation of the up, down, strange and charm quark masses performed using the gauge configurations produced by the European Twisted Mass Collaboration with Nf = 2 + 1 + 1 dynamical quarks, which include in the sea, besides two light mass degenerate quarks, also the strange and charm quarks with masses close to their physical values. The simulations are based on a unitary setup for the two light quarks and on a mixed action approach for the strange and charm quarks. The analysis uses data at three values of the lattice spacing and pion masses in the range 210 - 450 MeV, allowing for accurate continuum limit and controlled chiral extrapolation. The quark mass renormalization is carried out non-perturbatively using the RI-MOM method. The results for the quark masses converted to the bar{MS} scheme are: mud(2 GeV) = 3.70(17) MeV, ms(2 GeV) = 99.6(4.3) MeV and mc(mc) = 1.348(46) GeV. We obtain also the quark mass ratios ms/mud = 26.66(32) and mc/ms = 11.62(16). By studying the mass splitting between the neutral and charged kaons and using available lattice results for the electromagnetic contributions, we evaluate mu/md = 0.470(56), leading to mu = 2.36(24) MeV and md = 5.03(26) MeV.

We study $SU(2)$ Chern-Simons theories at level $k$ coupled to a scalar on $T^2\times \mathbb{R}$ at large baryon number. We find a homogeneous but anisotropic ground state configuration for any values of $k$ on the IR fixed-point of those models. This classical analysis is valid as long as we take the baryon number large. As a corollary, by comparing the symmetry breaking pattern at large chemical potential, we find that the theory does not reduce to the singlet sector of the $O(4)$ Wilson-Fisher fixed-point at large-$k$, as expected from general grounds. This paper will be one primitive step towards quantitative analysis of Chern-Simons-matter dualities using the large charge expansion.

We apply the large-charge limit to the first known example of a four-dimensional gauge-Yukawa theory featuring an ultraviolet interacting fixed point in all couplings. We determine the energy of the ground state in presence of large fixed global charges and deduce the global symmetry breaking pattern. We show that the fermions decouple at low energy leaving behind a confining Yang-Mills theory and a characteristic spectrum of type I (relativistic) and type II (non-relativistic) Goldstone bosons. Armed with the knowledge acquired above we finally arrive at establishing the conformal dimensions of the theory as a triple expansion in the large-charge, the number of flavors and the controllably small inverse gauge coupling constant at the UV fixed point. Our results unveil a number of noteworthy properties of the low-energy spectrum, vacuum energy and conformal properties of the theory. They also allow us to derive a new consistency condition for the relative sizes of the couplings at the fixed point.

In this thesis, I present dispersive treatments of two hadronic processes: the semileptonic kaon decay $K_{\ell4}$ and hadronic light-by-light scattering. The $K_{\ell4}$ decay is one of the best sources of information on some of the parameters of chiral perturbation theory. The dispersion relation for $K_{\ell4}$ provides a resummation of $\pi\pi$- and $K\pi$-rescattering effects. In contrast to a pure chiral treatment, it reproduces the observed curvature of one of the form factors. The matching of the dispersion relation to the chiral representation of the form factors allows the extraction of the values of three low-energy constants. Hadronic light-by-light scattering appears as a virtual process in the calculation of the anomalous magnetic moment of the muon $(g-2)_\mu$. For more than a decade, a discrepancy of about $3\sigma$ has persisted between the experimental determination and the standard-model prediction of the $(g-2)_\mu$. It is expected that within a few years hadronic light-by-light scattering will dominate the uncertainty of the theory prediction of the $(g-2)_\mu$. So far, only model calculations of the hadronic light-by-light contribution are available. However, in view of forthcoming $(g-2)_\mu$ experiments at Fermilab and J-PARC it is crucial that the hadronic light-by-light calculation can be improved systematically. The dispersive description presented here provides a formalism for a data-driven determination of hadronic light-by-light scattering and hence opens up an avenue towards a model-independent evaluation of the $(g-2)_\mu$.

Many low-threshold experiments observe sharply rising event rates of yet unknown origins below a few hundred eV, and larger than expected from known backgrounds. Due to the significant impact of this excess on the dark matter or neutrino sensitivity of these experiments, a collective effort has been started to share the knowledge about the individual observations. For this, the EXCESS Workshop was initiated. In its first iteration in June 2021, ten rare event search collaborations contributed to this initiative via talks and discussions. The contributing collaborations were CONNIE, CRESST, DAMIC, EDELWEISS, MINER, NEWS-G, NUCLEUS, RICOCHET, SENSEI and SuperCDMS. They presented data about their observed energy spectra and known backgrounds together with details about the respective measurements. In this paper, we summarize the presented information and give a comprehensive overview of the similarities and differences between the distinct measurements. The provided data is furthermore publicly available on the workshop's data repository together with a plotting tool for visualization.

In lattice QCD and other field theories with a mass gap, the field variables in distant regions of a physically large lattice are only weakly correlated. Accurate stochastic estimates of the expectation values of local observables may therefore be obtained from a single representative field. Such master-field simulations potentially allow very large lattices to be simulated, but require various conceptual and technical issues to be addressed. In this talk, an introduction to the subject is provided and some encouraging results of master-field simulations of the SU(3) gauge theory are reported.

In lattice QCD and other field theories with a mass gap, the field variables in distant regions of a physically large lattice are only weakly correlated. Accurate stochastic estimates of the expectation values of local observables may therefore be obtained from a single representative field. Such master-field simulations potentially allow very large lattices to be simulated, but require various conceptual and technical issues to be addressed. In this talk, an introduction to the subject is provided and some encouraging results of master-field simulations of the SU(3) gauge theory are reported.

In this review we study quantum field theories and conformal field theories with global symmetries in the limit of large charge for some of the generators of the symmetry group. At low energy the sectors of the theory with large charge are described by a hybrid form of Goldstone's theorem, involving its relativistic and non-relativistic forms. The associated effective field theory in the infrared allows the computation of anomalous dimensions, and operator product expansion coefficients in a well defined expansion in inverse powers of the global charge. This applies even when the initial theory does not have a reliable semiclassical approximation. The large quantum number expansion complements, and may provide an alternative approach to the bootstrap and numerical treatments. We will present some general features of the symmetry breaking patterns and the low-energy effective actions, and a fairly large number of examples exhibiting the salient features of this method.

We determine the scaling properties of the Yang-Lee edge singularity as described by a one-component scalar field theory with imaginary cubic coupling, using the nonperturbative functional renormalization group in $3 \leq d\leq 6$ Euclidean dimensions. We find very good agreement with high-temperature series data in $d = 3$ dimensions and compare our results to recent estimates of critical exponents obtained with the four-loop $\epsilon = 6-d$ expansion and the conformal bootstrap. The relevance of operator insertions at the corresponding fixed point of the RG $\beta$ functions is discussed and we estimate the error associated with $\mathcal{O}(\partial^4)$ truncations of the scale-dependent effective action.