These lecture notes are based on a course given by Mark Hindmarsh at the 24th Saalburg Summer School 2018 and written up by Marvin Lüben, Johannes Lumma and Martin Pauly. The aim is to provide the necessary basics to understand first-order phase transitions in the early universe, to outline how they leave imprints in gravitational waves, and advertise how those gravitational waves could be detected in the future. A first-order phase transition at the electroweak scale is a prediction of many theories beyond the Standard Model, and is also motivated as an ingredient of some theories attempting to provide an explanation for the matter-antimatter asymmetry in our Universe. Starting from bosonic and fermionic statistics, we derive Boltzmann's equation and generalise to a fluid of particles with field dependent mass. We introduce the thermal effective potential for the field in its lowest order approximation, discuss the transition to the Higgs phase in the Standard Model and beyond, and compute the probability for the field to cross a potential barrier. After these preliminaries, we provide a hydrodynamical description of first-order phase transitions as it is appropriate for describing the early Universe. We thereby discuss the key quantities characterising a phase transition, and how they are imprinted in the gravitational wave power spectrum that might be detectable by the space-based gravitational wave detector LISA in the 2030s.

The KOTO II experiment is proposed to measure the branching ratio of the decay $K_L\to\pi^0\nu\bar{\nu}$ at J-PARC. With a beamline to extract long-lived neutral kaons at 5 degrees from a production target, the single event sensitivity of the decay is $8.5\times 10^{-13}$, which is much smaller than the Standard Model prediction $3\times 10^{-11}$. This allows searches for new physics beyond the Standard Model and the first discovery of the decay with a significance exceeding $5\sigma$. As the only experiment proposed in the world dedicated to rare kaon decays, KOTO II will be indispensable in the quest for a complete understanding of flavor dynamics in the quark sector. Moreover, by combining efforts from the kaon community worldwide, we plan to develop the KOTO II detector further and expand the physics reach of the experiment to include measurements of the branching ratio of the $K_L\to\pi^0\ell^+\ell^-$ decays, studies of other $K_L$ decays, and searches for dark photons, axions, and axion-like particles. KOTO II will therefore obtain a comprehensive understanding of $K_L$ decays, providing further constraints on new physics scenarios with existing $K^+$ results.

We argue that the finiteness of quantum gravity amplitudes in fully compactified theories (at least in supersymmetric cases) leads to a bottom-up prediction for the existence of non-trivial dualities. In particular, finiteness requires the moduli space of massless fields to be compactifiable, meaning that its volume must be finite or at least grow no faster than that of Euclidean space. Moreover, we relate the compactifiability of moduli spaces to the condition that the lattice of charged objects transform in a semisimple representation under the action of the duality group. These ideas are supported by a wide variety of string theory examples.

We study the moduli space of 4d N=1 supersymmetric QCD in the Veneziano limit using Hilbert series. In this limit, the numbers of colours and flavours are taken to be large with their ratio fixed. It is shown that the Hilbert series, which is a partition function of an ensemble of gauge invariant quantities parametrising the moduli space, can also be realised as a partition function of a system of interacting Coulomb gas in two dimensions. In the electrostatic equilibrium, exact and asymptotic analyses reveal that such a system exhibits two possible phases. Physical quantities, such as charge densities, free energies, and Hilbert series, associated with each phase, are computed explicitly and discussed in detail. We then demonstrate the existence of the third order phase transition in this system.

Species thermodynamics has been proposed in analogy to black hole thermodynamics. The entropy scales like an area and is given by the mere counting of the number of the species. In this work, we $\textit{derive}$ the constitutive relations of species thermodynamics and explain how those $\textit{originate}$ from standard thermodynamics. We consider configurations of species in thermal equilibrium inside a box of size $L$, and show that the temperature $T$ of the system, which plays a crucial role, is always upper bounded above by the species scale $\Lambda_{\rm sp}$. We highlight three relevant regimes: (i) when L^{-1}< T<\Lambda_{\rm sp}, and gravitational collapse is avoided, the system exhibits standard thermodynamics features, for example, with the entropy scaling like the volume of the box; (ii) in the limit $L^{-1}\simeq T\rightarrow \Lambda_{\rm sp}$ we recover the rules of species thermodynamics with the entropy scaling like the area of the box; (iii) an intermediate regime with L^{-1}\simeq T< \Lambda_{\rm sp} that avoids gravitational collapse and fulfills the Covariant Entropy Bound; this interpolates between the previous two regimes and its entropy is given simply in terms of the counting of the species contributing to the thermodynamic ensemble. This study also allows us to find a novel and independent bottom-up rationale for the Emergent String Conjecture. Finally, we present the $\textit{Black Hole - Tower Correspondence}$ as a generalization of the celebrated Black Hole - String Correspondence. This provides us with a robust framework to interpret the results of our thermodynamic investigation. Moreover, it allows us to qualitatively account for the entropy of black holes in terms of the degrees of freedom of the weakly coupled species in the tower.

A Palatini-type action for Einstein and Gauss-Bonnet gravity with non-trivial torsion is proposed. Three-form flux is incorporated via a deformation of the Riemann tensor, and consistency of the Palatini variational principle requires the flux to be covariantly constant and to satisfy a Jacobi identity. Studying gravity actions of third order in the curvature leads to a conjecture about general Palatini-Lovelock-Cartan gravity. We point out potential relations to string-theoretic Bianchi identities and, using the Schouten-Nijenhuis bracket, derive a set of Bianchi identities for the non-geometric Q- and R-fluxes which include derivative and curvature terms. Finally, the problem of relating torsional gravity to higher-order corrections of the bosonic string-effective action is revisited.

The recent recomputation of the neutrino fluxes from nuclear reactors relaxes the tension between the LSND and MiniBooNE anomalies and disappearance data when interpreted in terms of sterile neutrino oscillations. The simplest extension of the Standard Model with such fermion singlets is the addition of right-handed sterile neutrinos with small Majorana masses. Even when introducing three right-handed neutrinos, this scenario has less free parameters than the 3+2 scenarios studied in the literature. This begs the question whether the best fit regions obtained can be reproduced by this simplest extension of the Standard Model. In order to address this question, we devise an exact parametrization of Standard Model extensions with right-handed neutrinos. Apart from the usual 3x3 neutrino mixing matrix and the 3 masses of the lightest neutrinos, the extra degrees of freedom are encoded in another 3x3 unitary matrix and 3 additional mixing angles. The parametrization includes all the correlations among masses and mixings and is valid beyond the usual seesaw approximation. Through this parametrization we find that the best fit regions for the LSND and MiniBooNE anomalies in a 3+2 scenario can indeed be reproduced despite the smaller number of degrees of freedom.

The NA62 experiment at the CERN SPS reports the first detection of a tagged neutrino candidate based on the data collected in 2022. The candidate consists of a $K^+ \rightarrow \mu^+ \nu_\mu$ decay where the charged particles are reconstructed and the neutrino is detected through a charged-current interaction in a liquid krypton calorimeter.

Extending the Standard Model (SM) with sterile ('right-handed') neutrinos is one of the best motivated ways to account for the observed neutrino masses. We discuss the expected sensitivity of future lepton collider experiments for probing such extensions. An interesting testable scenario is given by 'symmetry protected seesaw models', which theoretically allow for sterile neutrino masses around the electroweak scale with up to order one mixings with the light (SM) neutrinos. In addition to indirect tests, e.g. via electroweak precision observables, sterile neutrinos with masses around the electroweak scale can also be probed by direct searches, e.g. via sterile neutrino decays at the Z pole, deviations from the SM cross section for four lepton final states at and beyond the WW threshold and via Higgs boson decays. We study the present bounds on sterile neutrino properties from LEP and LHC as well as the expected sensitivities of possible future lepton colliders such as ILC, CEPC and FCC-ee (TLEP).

If quantum chromodynamics (QCD) is renormalized by minimal subtraction (MS), at higher orders, the strong coupling constant alpha_s and the quark masses m_q exhibit discontinuities at the flavour thresholds, which are controlled by so-called decoupling constants, zeta_g and zeta_m, respectively. Adopting the modified MS (MS-bar) scheme, we derive simple formulae which reduce the calculation of zeta_g and zeta_m to the solution of vacuum integrals. This allows us to evaluate zeta_g and zeta_m through three loops. We also establish low-energy theorems, valid to all orders, which relate the effective couplings of the Higgs boson to gluons and light quarks, due to the virtual presence of a heavy quark h, to the logarithmic derivatives w.r.t. m_h of zeta_g and zeta_m, respectively. Fully exploiting present knowledge of the anomalous dimensions of alpha_s and m_q, we thus calculate these effective couplings through four loops. Finally, we perform a similar analysis for the coupling of the Higgs boson to photons.

The sign problem of QCD prevents standard lattice simulations to determine the phase diagram of strong interactions with a finite chemical potential directly. Complex Langevin simulations provide an alternative method to sample path integrals with complex weights. We report on our ongoing project to determine the phase diagram of QCD in the limit of heavy quarks (HDQCD) using Complex Langevin simulations.

We investigate how non-standard neutrino interactions (NSIs) with matter can be generated by new physics beyond the Standard Model (SM) and analyse the constraints on the NSIs in these SM extensions. We focus on tree-level realisations of lepton number conserving dimension 6 and 8 operators which do not induce new interactions of four charged fermions (since these are already quite constrained) and discard the possibility of cancellations between diagrams with different messenger particles to circumvent experimental constraints. The cases studied include classes of dimension 8 operators which are often referred to as examples for ways to generate large NSIs with matter. We find that, in the considered scenarios, the NSIs with matter are considerably more constrained than often assumed in phenomenological studies, at least ${\cal O}(10^{-2})$. The constraints on the flavour-conserving NSIs turn out to be even stronger than the ones for operators which also produce interactions of four charged fermions at the same level. Furthermore, we find that in all studied cases the generation of NSIs with matter also gives rise to NSIs at the source and/or detector of a possible future Neutrino Factory.

Monte Carlo methods cannot probe far into the QCD phase diagram with a real chemical potential, due to the famous sign problem. Complex Langevin simulations, using adaptive step-size scaling and gauge cooling, are suited for sampling path integrals with complex weights. We report here on tests of the deconfinement transition in pure Yang-Mills SU(3) simulations and present an update on the QCD phase diagram in the limit of heavy and dense quarks.

Accretion-induced collapse (AIC) or merger-induced collapse (MIC) of white dwarfs (WDs) in binary systems is an interesting path to neutron star (NS) and magnetar formation, alternative to stellar core collapse and NS mergers. Such events could add a population of compact remnants in globular clusters, they are expected to produce yet unidentified electromagnetic transients including gamma-ray and radio bursts, and to act as sources of trans-iron elements, neutrinos, and gravitational waves. Here we present the first long-term (>5s post bounce) hydrodynamical simulations in axi-symmetry (2D), using energy- and velocity-dependent three-flavor neutrino transport based on a two-moment scheme. Our set of six models includes initial WD configurations for different masses, central densities, rotation rates, and angular momentum profiles. Our simulations demonstrate that rotation plays a crucial role for the proto-neutron star (PNS) evolution and ejecta properties. We find early neutron-rich ejecta and an increasingly proton-rich neutrino-driven wind at later times in a non-rotating model, in agreement with electron-capture supernova models. In contrast to that and different from previous results, our rotating models eject proton-rich material initially and increasingly more neutron-rich matter as time advances, because an extended accretion torus forms around the PNS and feeds neutrino-driven bipolar outflows for many seconds. AIC and MIC events are thus potential sites of r-process element production, which may imply constraints on their occurrence rates. Finally, our simulations neglect the effects of triaxial deformation and magnetic fields, yet they provide valuable reference cases for comparison with future long-term magneto-hydrodynamic and three-dimensional AIC studies.

We present the first simulations of core-collapse supernovae in axial symmetry with feedback from fast neutrino flavor conversion (FFC). Our schematic treatment of FFCs assumes instantaneous flavor equilibration under the constraint of lepton-number conservation individually for each flavor. Systematically varying the spatial domain where FFCs are assumed to occur, we find that they facilitate SN explosions in low-mass (9-12 solar masses) progenitors that otherwise explode with longer time delays, whereas FFCs weaken the tendency to explode of higher-mass (around 20 solar masses) progenitors.

We study, from the perspective of supersymmetry and space-time Killing spinors, the local brane densities involved in 1/4-BPS intersecting brane systems. In particular, we classify the possible local brane structures that have maximal (16) supersymmetries in 1/4-BPS intersecting brane backgrounds. Applied to BPS black holes, this classification reveals the allowed local microstructure for pure microstates. We further use these structures with local 16 supersymmetries as building blocks to generalise to 1/8-BPS systems. Moreover, we give examples of 1/8-BPS black holes for which the local supersymmetries are compatible with the combination of different entropy-generating effects from brane interaction. Finally, applying our classification to BPS domain walls, we illustrate how our formalism may possibly describe the local picture of the Hanany-Witten effect.

The species scale provides a lower bound on the shortest possible length that can be probed in gravitational effective theories. It may be defined by the size of the minimal black hole in the theory and, as such, it has recently been given an interpretation along the lines of the celebrated black hole thermodynamics. In this work, we extend this interpretation to the case of charged species. We provide working definitions of minimal black holes for the case of uncharged and charged species constituents. Then, examining the modifications in the thermodynamic properties of near-extremal charged species compared to the uncharged case, we uncover interesting implications for the cosmology of an expanding universe, particularly within the context of the Dark Dimensions Scenario. Finally, we explore possible microscopic constructions in non-supersymmetric string theories in which towers of charged near-extremal species may arise.

It has been previously advocated that the presence of the daily and annual modulations of the axion flux on the Earth's surface may dramatically change the strategy of the axion searches. The arguments were based on the so-called Axion Quark Nugget (AQN) dark matter model which was originally put forward to explain the similarity of the dark and visible cosmological matter densities $\Omega_{\rm dark}\sim \Omega_{\rm visible}$. In this framework, the population of galactic axions with mass $ 10^{-6} {\rm eV}\lesssim m_a\lesssim 10^{-3}{\rm eV}$and velocity$\langle v_a\rangle\sim 10^{-3} c$ will be accompanied by axions with typical velocities $\langle v_a\rangle\sim 0.6 c$ emitted by AQNs. Furthermore, in this framework, it has also been argued that the AQN-induced axion daily modulation (in contrast with the conventional WIMP paradigm) could be as large as $(10-20)\%$, which represents the main motivation for the present investigation. We argue that the daily modulations along with the broadband detection strategy can be very useful tools for the discovery of such relativistic axions. The data from the CAST-CAPP detector have been used following such arguments. Unfortunately, due to the dependence of the amplifier chain on temperature-dependent gain drifts and other factors, we could not conclusively show the presence or absence of a dark sector-originated daily modulation. However, this proof of principle analysis procedure can serve as a reference for future studies.

In this brief note we consider the interaction between high spin excitations in string theory along the Regge trajectory and the Higuchi bound in de Sitter space. There is always a point along the Regge trajectory where the Higuchi bound is violated. However, this point precisely coincides with a string whose length is of order the de Sitter Hubble scale. String theory therefore manifests no immediate inconsistency as long as the string scale $M_s$ is above the Hubble scale $H$. However, an implication is that the Regge trajectory must be significantly modified at some ultraviolet scale. Insisting that this modification should occur no earlier than the Planck scale would lead to a bound on the string scale of $M_s > \sqrt{H M_p}$.