Erice: INTERNATIONAL SCHOOL OF NUCLEAR PHYSICS, 46th COURSE

I will discuss the present state with regards to the search for a QCD critical point, especially in view of the new high statistics data from the RHICH beam energy scan II. I will also discuss possible next steps needed to either confirm or rule out the existence of a CP in the region accessible by experiment.

I will discuss the motivation for investigating the QCD phase diagram in the domain of large baryochemical potential and how we phenomenologically access this region, focusing on the statistical hadronization model. I will review briefly the existing measurements and will describe the CBM experiment in construction at FAIR, introducing as well its expected physics performance.

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I will review recent progress in understanding cold and dense QCD matter, with a particular focus on the equation of state and its applications to neutron stars, including contributions from my own work.

Anisotropic phases are hypothesised to play an important role in the QCD phase diagram for small temperatures and large densities. I will present a study of such a phase, the chiral density wave (CDW), within a nucleon-meson model, taking also into account the nucleonic vacuum fluctuations. In particular, the main goal is exploring the possible existence of the CDW in the interior of neutron stars. With that in mind, imposing beta equilibrium and electric charge neutrality, I will discuss whether, and for what parameter choices, this phase is energetically preferred. I will then demonstrate the construction of compact stars with an anisotropic interior and, comparing with observations, I will show that realistic stars do not have an anisotropic (CDW) core.

Dense QCD matter can feature a moat regime, where the static energy of mesons is minimal at nonzero momentum. We elucidate various features of this regime in this work. To capture the main effects, we use a two-flavor quark-meson model and put forward an efficient renormalization scheme to account for the nontrivial momentum dependence of meson self-energies in the moat regime.
We show that the extent of the moat regime critically depends on the interaction of quarks and mesons, and analyze quark and meson correlation functions in this regime. Since it arises from particle-hole fluctuations of quarks in a dense medium, the resulting spatial modulations manifest in various different meson correlations, including scalar, pseudoscalar, vector and axialvector mesons. We clarify the nature of these modulations based on the analytic structure of meson self-energies, demonstrating in particular their distinction from Friedel oscillations which also occur in the presence of a Fermi surface. In contrast, while quark correlations are enhanced in the moat regime, they do not show oscillatory behavior.

The Parity Doublet Model is a effective theory describing nucleon and negative parity resonances associated with each other in the parity doubling framework.
We present a renormalization group invariant formulation of the model, describing baryons and their negative-parity chiral partners in dense hadronic matter. By constructing the mesonic potential in terms of fermion masses and absorbing all divergences into a single running coupling, we define a fully multiplicatively renormalizable mean-field framework.
We investigate the thermodynamics of symmetric and asymmetric nuclear matter, including the impact of fermionic vacuum fluctuations, the role of chiral invarian mass of fermions and chiral symmetry breaking. Physical parameters are fitted to hadron vacuum properties and empirical nuclear matter observables at saturation. We show that vacuum fluctuations delay and soften the chiral transition, turning it into a smooth crossover in β-equilibrated matter at densities above 7n₀. Consequences for the QCD phase diagram and neutron star cores are discussed.

In this talk I will show how the functional renormalization group (FRG) can be used to constrain low energy model for quark-matter, with a focus on color superconductivity. Working in vacuum, I will explain how the FRG can be used describe both the non-pertubative behavior of the glue sector, and how low-energy degrees of freedom (in particular the sigma, pion and scalar diquark) can be included. I will explain why this approach is valuable at finite temperature and densities, and give outlooks regarding applications at asymptotic chemical potentials.

The equation of state (EoS) of neutron star matter encodes the relationship between pressure and density at supranuclear densities, fundamentally governing the star’s structure and observable macroscopic properties, such as mass, radius, and tidal deformability. In this work, we apply Neural Posterior Estimation (NPE) with conditional normalising flows to infer the EoS from synthetic observational data. We consider a model-agnostic EoS family and train our models on mock mass-radius and mass-radius–tidal deformability datasets with varying noise levels. We evaluate reconstruction performance in terms of pressure and squared speed of sound across baryonic densities, and quantify the impact of including tidal deformability information. Our results demonstrate that tidal measurements significantly reduce inference uncertainty, particularly for pressure, and confirm that NPE-based models can accurately capture physical constraints. The framework also generalises well to previously unseen EoS parametrisations, highlighting the robustness of the approach for future multimessenger astrophysical analyses.

While symmetric nuclear matter has been studied in laboratories, neutron star matter is characterized by high asymmetry. Therefore, by examining the strongly interacting matter properties in a wide range of densities and isospin asymmetry we confront two regimes to understand how the enforced electric neutrality and beta equilibrium alter the onset density of quark matter. Particularly, we demonstrate the dependence of the onset density of deconfined quarks in the electrically neutral beta-equilibrated matter on the onset density for symmetric matter. This allows us to map the phase diagram of cold strongly interacting matter in the plane of baryon density vs isospin asymmetry which is important in modeling hybrid stars based on the nuclear and low-energy heavy-ion collision experiments.

Strangeness production is a key signature of the formation of a hot and dense medium in heavy-ion collisions. Understanding the enhancement of strange particles in such environments remains a central challenge. Hybrid approaches—combining transport theory and hydrodynamics within the core–corona framework—have been successful in describing this behavior.

At the same time, signs of collective behavior and enhanced strangeness production have also been observed in small systems such as proton–proton collisions. This raises the intriguing possibility that the phenomena observed in heavy-ion collisions could emerge from the coherent superposition of multiple nucleon–nucleon interactions.

One such approach is provided by the Pythia/Angantyr model, where overlapping color flux tubes (strings) lead to an increased color field strength (string tension), supported by lattice QCD calculations on Casimir scaling. This, in turn, enhances the production of strange quarks via the Schwinger mechanism. The resulting hadronization process is referred to as rope hadronization.

Earlier transport models, such as RQMD, have explored the use of color ropes to explain strangeness enhancement. In this work, we revisit the concept using the modern SMASH transport framework, implementing a dynamic string tension mechanism that responds to local string overlap, motivated by the Pythia/Angantyr approach.

We compare the rope-based mechanism to a hydrodynamical evolution using SMASH+vHLLE within the same transport framework, assessing whether microscopic string interactions alone can account for the strangeness production observed by NA49.

I will discuss recent progress in the study of the properties of heavy flavor particles in lattices QCD, including the in-medium masses and width of quarkonia, open charmed hadrons and heavy quark diffusion constant.

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The HEFTY collaboration has been created to address, in a comprehensive way, the production, transport and hadronization of heavy-flavor particles in heavy-ion collisions, by bringing together state-of-the-art expertise on the various components needed to achieve that. In this talk we focus on recent determinations of nonperturbative transport coefficients in the strongly coupled QGP, for both open and hidden HF particles, by utilizing quantum many-body theory constrained by lattice QCD. We also discuss their implementation into transport models for heavy-ion collisions and with first comparisons to experimental data from the LHC.

Understanding the space–time evolution of strongly interacting matter created in heavy-ion collisions requires precise probes of its collective expansion. While anisotropic flow has long been established as a key signature of the quark–gluon plasma (QGP), the isotropic component, radial flow, has until now been inferred only indirectly through spectral slopes and blast-wave fits.

A recently introduced observable, v0(pT), provides the first direct measurement of radial-flow fluctuations and thus opens a new way to study the isotropic expansion of the system. Results from ALICE and ATLAS demonstrate that v0(pT) exhibits all hallmarks of collectivity—long-range correlations, factorization, and near-universal scaling with centrality. Moreover, identified-particle measurements reveal mass ordering at low pT and baryon–meson separation at intermediate pT, linking radial flow to hadronization mechanisms.

Crucially, v0(pT) is sensitive to the bulk viscosity and the equation of state of QCD matter, making it a powerful new probe of transport properties. This talk will review the conceptual development of the observable, present the first experimental results, and discuss future opportunities at RHIC, the LHC, and next-generation facilities.

The High-Acceptance-Di-Electron-Spectrometer (HADES) at GSI, Darmstadt, measures heavy-ion and elementary collisions at beam energies of a few GeV, thereby accessing the QCD phase diagram at high densities, around 2-3 times the saturation density, and moderate temperatures in the order of tens of MeV.

Such conditions are similar to those expected in neutron star mergers, which provides a meaningful opportunity to study the nuclear Equation-Of-State in the laboratory, not only fundamentally, but also specifically to deepen our understanding of such astrophysical phenomena.

One of the essential pillars of the HADES program is the reconstruction of rare electromagnetic probes. Due to their penetrating nature, they encode unique and direct information from all stages, including the hottest and densest, and can probe the evolution of the colliding system.

In this contribution, we present measurements of dielectrons reconstructed from Ag+Ag and Au+Au collisions at $\sqrt{s_{NN}}=2.55$ GeV and $\sqrt{s_{NN}}=2.42$ GeV. For one, we focus on the isolation of the thermal excess which entails information on the lifetime and temperature of the fireball. For another, we extend the analysis with an investigation of collective phenomena, in particular the measurement of the azimuthal anisotropy. This gives new experimental hints about the origin and development of the collective flow over time at beam energies of a few GeV.

We explore some regularization prescriptions and how they impact the effective potential and the gap equation. We also explore the role of the integration measure/geometry in momentum space in symmetry breaking. We conclude that every regularization prescription and geometry greatly affect the properties of symmetry breaking/restoration. We also propose a modified representation of proper time, that does not interchange divergences in the effective potential. Finally, with this representation we propose a way to include thermal effects that can be extended to other agents like magnetic field and finite volume.

Summary: The Nambu-Jona-Lasinio model phase diagram for a 2 + 1 flavor quark system will be described using the temperature and chemical potential as thermodynamic properties. Various susceptibilities will be used in order to obtain several qualitatively different phase diagrams, and each will be analyzed in both the light and strange quark sectors. A crossover region will be defined for the light quarks and another for the heavier strange quark. We observe that, because of the 't Hooft flavor mixing interaction term, the strange quark transition is heavily influenced by the chiral restoration of the light sector and, as a consequence, two different crossover regions can be observed when constructing the phase diagram for the strange quark.

We investigate the spectrum and wave functions of heavy mesons in terms
of the number of light flavours and mass, upon introducing a heavy quark
in the perturbative regime of the QCD conformal window. We then compute
the conformal Isgur-Wise function which is a central quantity of the
dynamics of heavy mesons. We further determine the impact of the
residual low energy confining dynamics. The impact of radiative
corrections and first-order mass corrections is discussed.
As a working framework, we adapt the heavy quark effective theory to the
perturbative conformal window dynamics. The investigation can be easily
extended to other gauge representations of the colour group and
multi-quark bound states.
Our work lays the foundations to systematically go beyond the infinite
mass defect approximation in conformal field theories. Furthermore, our
results can be used when constructing bright or dark extension of the
Standard Model, for example dark matter models featuring dark mesons.

Understanding the phase structure of Quantum Chromodynamics (QCD) remains the main goal of both experimental and theoretical studies of strongly interacting matter. Lattice QCD calculations at zero $\mu_B$ predict a smooth crossover between the hadronic phase and the quark-gluon plasma (QGP). One of the key experimental signatures of the formation of the QGP is the enhancement of strange particle production. In particular, the non-monotonic behaviour- commonly referred to as the “horn”- observed in the K+/π+ ratio as a function of collision energy is interpreted as a possible signal of the onset of deconfinement and chiral symmetry restoration.
In this work, we investigate the appearance and evolution of the horn structure in the context of strangeness enhancement in Au+Au collisions within the energy range of 3.5–14.5 GeV. The analysis is performed using the Parton-Hadron Quantum Molecular Dynamics (PHQMD) transport model, which combines the Parton-Hadron-String Dynamics (PHSD) for modeling partonic degrees of freedom with the Quantum Molecular Dynamics (QMD) approach for baryonic interactions. The model offers a dynamical treatment of QGP formation in regions with local energy densities exceeding 0.5 GeV/fm³.
We compare results obtained with the QGP phase turned ON and OFF in PHQMD to explore the role of deconfinement dynamics in shaping the observed strange hadron yields. The dependence of the horn feature on both collision energy and centrality is studied. In addition to the strangeness-related observables, we also analyze overall particle production and particle ratios, providing a comprehensive view of the system's evolution across different energy regimes. Our results offer new insights into the interplay between chiral symmetry restoration, QGP formation, and strangeness enhancement, contributing to a deeper understanding of the QCD phase diagram at intermediate baryon densities.

An understanding of the properties of the quark-gluon plasma (QGP) is important for the interpretation of experimental data on the bulk observables, as well as on jet and heavy-quark attenuation in heavy-ion collisions. However, gaining this knowledge is a challenging task, since it pertains to the non-perturbative regime of QCD, for which only limited information from lattice QCD is currently available.

To overcome these difficulties, we employ the dynamical quasi-particle model (DQPM), which describes the non-perturbative nature of the strongly-interacting QGP at finite temperature and baryon chemical potential based on a propagator representation of massive off-shell quarks and gluons, the properties of which are adjusted to reproduce the lattice QCD equation of state for the QGP in thermodynamic equilibrium. From the DQPM propagators, we can explicitly derive the scattering amplitudes for both elastic ($2\to 2$) and inelastic ($2\to 3$) interactions (without resorting to additional approximations), allowing us to investigate not only the properties of the thermalized medium itself, but also jet and heavy-quark energy loss.

In this work, we present several key findings. First, we show how total elastic and radiative cross sections vary with energy and temperature, highlighting the dominance of elastic scattering at low energies and high temperatures, while radiative processes become increasingly relevant at high energies and low temperatures. Second, we obtain the interaction rate and relaxation time in the QGP and show that their values are primarily governed by elastic scatterings. Third, we evaluate the jet transport coefficient $\hat{q}$ and reveal its strong dependencies on the medium temperature, jet momentum and the choice of the strong coupling in thermal, jet-parton, and radiative vertices. We also examine the ratio of $\hat{q}$ to shear viscosity $\eta$, identifying regimes in which the commonly used scaling $\eta/s \approx 1.25 T^3/\hat{q}$​ either holds or is violated. Finally, we obtain the heavy-quark diffusion coefficient and explore its temperature and mass dependencies.

References:
1. I. Grishmanovskii et al, Phys.Rev.C 106, 014903
2. I. Grishmanovskii et al, Phys.Rev.C 109, 024911
3. I. Grishmanovskii et al, Phys.Rev.C 110, 014908
4. I. Grishmanovskii et al, arXiv:2503.22311

... for your entertainment.

Research on hypernuclei plays an essential role in answering how the hierarchy of nuclei is constructed from quarks. We are going to review the recent achievements in hypernuclear programs in J-PARC. One of the recent achievements is the realization of an accurate hyperon-nucleon scattering experiment. The differential cross sections of the Σ+p, Σ−p elastic scatterings and Σ−p → Λn inelastic scattering have been measured with drastically improved accuracy. These new data will become essential inputs to improve the theories of the two-body baryon-baryon interaction. Now, we have launched a new Λp scattering experiment at SPring-8. I'll also mention the current status. Another achievement is the big progress of research on the double hypernuclei. A lot of information on Λ hypernuclei and Ξ hypernuclei has been accumulated through the experiments at J-PARC, such as gamma-ray spectroscopy and emulsion experiments. In this talk, the progress of the hypernuclear program in J-PARC is presented with a focus on these experimental results. Future prospects are also discussed briefly.

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We present a novel relativistic density functional approach for QCD matter, motivated by a nonlocal medium-screened confining interaction among quarks. The approach suggests a phenomenological confining mechanism equivalent to suppressing excitations of quark quasiparticles by their large self-energies already at the mean-field level. Chirally symmetric form of the functional provides spontaneous breaking and dynamical restoration of chiral symmetry of QCD and allows representing the approach as a chiral quark model with self-consistently derived medium-dependent couplings. Hadrons are systematically introduced to the approach within a generalized Beth-Uhlenbeck framework as color-singlet (anti)quark correlations. The approach explains why the abundances of hadrons produced in ultrarelativistic heavy-ion collisions (HIC) are well described by the hadron resonance gas model with a sudden chemical freeze-out at a well-defined hadronization temperature, defined by the chiral symmetry restoration driven Mott dissociation of hadrons, despite the fact that state of the art results of lattice QCD indicate a smooth chiral crossover. At high baryon densities, when repulsion and pairing among quarks play a significant role, the density functional is applied for modeling neutron stars (NS) and constructing equation of state for supernova explosions and mergers of NS. It is shown that color superconductivity drives trajectories of evolution of the QCD matter in these dynamical processes toward the high temperatures typical for HIC.

We review the role of primordial black holes (PBHs) for illuminating the dark ages of the cosmological evolution and as dark matter candidates. We elucidate the role of phase transitions for primordial black hole formation in the early Universe and focus our attention to the cosmological QCD phase transition within a recent microscopical model.We explore the impact of physics beyond the Standard Model on the cosmic equation of state and the probability distribution for the formation of primordial black holes which serve as dark matter (DM) candidates. We argue that besides primordial black holes also droplet-like quark-gluon plasma inhomogeneities may become gravitationally stabilized for a sufficiently long epoch to distill baryon number and form nuclear matter droplets which upon their evaporation may enrich the cosmos locally with heavy r-process elements already in the early Universe.

The density functional renormalization group (density-fRG) is proposed to investigate the density fluctuations within the functional renormalization group approach, which allows us to quantify the medium effect and study physics of high densities. The density-fRG is applied to the nucleon-meson effective field theory, also known as the Walecka model, to study the properties of nuclear matter at high baryon densities. It is found that both the attractive and repulsive nucleon meson interactions are screened by the high density medium, which results in a stiffer equation of state (EoS) of nuclear matter in the regime of $\rho_0 \lesssim \rho \lesssim 2.5 \rho_0$, then a softer EoS when $\rho \gtrsim 2.5 \rho_0$. Furthermore, a new phenomenon called the locking of Fermi surface is found. In the locking of Fermi surface the effective energy of quasi-nucleon is always close to the Fermi surface, which are both running with the renormalization group scale.

The observables computed in simulations of heavy ion collisions are known to be highly sensitive to the initial state of the evolution. Despite advances in modeling, fully understanding this initial state remains a challenge. In this work, we propose a method to relate final observables to the initial state, described as the sum of an average state and a random linear combination of independent fluctuation modes. We illustrate the approach on Pb-Pb events in fixed centrality classes, and quantify how the modes contribute to final-state observables. We show the importance of both linear and quadratic responses in this evolution, and we highlight how statistical noise can affect simulations including hadronic interactions. Our results provide deeper insight into the impact of fluctuations for heavy ion collisions.

Spin hydrodynamics is one of the leading techniques employed to understand spin polarization in heavy ion collisions. Given the topic is still in its infancy, there is a lot of debate about its most basic aspects like the form of the equilibrium distribution function. Current prevalent forms suffer from significant drawbacks, including issues with the normalization of mean spin polarization. To this end, we propose a new form of the equilibrium distribution function using the general form of a $2\times 2$ matrix and demonstrate how it overcomes the aforementioned issues of normalization of the mean spin polarization that plagued the previous forms, furthermore, this form shows an exact agreement with thermodynamical relations for perfect spin hydrodynamics which were found before using the classical concept of spin. Beyond this, we also explore the range of applicability of this form of the distribution function and show that it works well with the parameters used in the heavy-ion simulations.

One of the challenging tasks for the heavy-ion community is the construction of spin hydrodynamics. As various formal issues require clarification, we investigate the application range of perfect spin hydrodynamics with different treatments of spin [1]. By considering a quantum spin density matrix (Wigner function) and a classical spin description, we derive fine constraints that connect the components of the spin polarization tensor, particle mass, temperature, and hydrodynamic flow. Along with the arbitrary reference frame, we consider the rest frame of the fluid and provide mutual relations between the conditions obtained in these cases. The outcome of our study will be advantageous for hydrodynamic modeling of heavy-ion collisions.

[1] Z. Drogosz, W. Florkowski and V.M., arXiv:2506.01537 [hep-ph].

The study of relativistic heavy-ion collisions serves as a valuable tool for investigating the structure of nuclear matter and its behaviour under extreme conditions. Over the past two decades, this domain of nuclear physics has led to several significant discoveries, including the quark-gluon plasma at RHIC, in 2005, the Higgs boson at CERN, in 2012, and exotic hadrons by the LHCb collaboration. In such collisions, a multitude of different types of particles are produced. By analysing their properties, such as energy and momentum, we can gain insight into the evolution of the system formed during the collision. An important phenomenon observed at kinetic freeze-out is the development of a transverse collective flow of matter. The characteristics of the flow are highlighted through the study of the average transverse momentum $⟨p_T⟩$ obtained from the transverse momentum spectra of the particles. This work presents an analysis of $⟨p_T⟩$ of identified strange hadrons ($K_S^0$, $\Lambda$, $\bar{\Lambda}$, $\Xi^-$, $\bar{\Xi}^+$, $\phi$, $\Omega^-$, $\bar{\Omega}^+$) and bulk hadrons obtained in Au-Au collisions at RHIC-BES energies ($\sqrt{s_{NN}} = 7.7$ GeV, 11.5 GeV, 19.6 GeV, 27 GeV and 39 GeV). Special emphasis is placed on the dependence of mean transverse momentum on particle species and event centrality at various incident energies. In order to enhance our interpretation of the experimental results, comparisons were made with model calculations using AMPT (A Multi-Phase Transport) simulations, taking into consideration two distinct scenarios: the string melting version and the default version, which does not include the quark-gluon plasma phase. These combined approaches will be presented and discussed in order to enhance our understanding of the properties of the system produced in relativistic heavy-ion collisions.

Two particle correlations have shown the presence of long-range rapidity correlations in small collision systems. Several other measurements provided insight into the unexpected collective behaviour similar to the one exhibited in heavy-ion collisions. These properties can be explained by several models, which consider a microscopic description like PYTHIA 8 and a macroscopic treatment as EPOS4. Balance functions have been regarded in the past as a method of investigation the late-stage hadronization found in the presence of a strongly-coupled medium. We present balance functions confronting EPOS 4 and PYTHIA 8 in pp collisions at $\sqrt{s} = 13.6$ TeV to distinguish between these models.

Hybrid models are used to describe the evolution of the quark-gluon plasma produced in ultrarelativistic heavy-ion collisions. These models combine viscous relativistic hydrodynamics with pre-equilibrium dynamics and hadronic cascade models. The connection from the hydrodynamics stage to the particlization stage is made through a particlization model, based on the Cooper-Frye formalism. This stage is responsible for translating the hydrodynamics continuous degrees of freedom to the discrete degrees of freedom of a hadron resonance gas. In this project, we analyse the numerical implementation of a new relaxation time approximation (RTA) based viscous correction to the Cooper-Frye formalism using the iSS particlization model, focusing on its impact on final observables of heavy-ion collisions. An important parameter $\gamma$ appears in the new RTA; it is a new free parameter that can be used to better describe these final observables. We investigate how varying values of $\gamma$ affect particle spectra and yields. The analysis was performed for PbPb collisions using parameters from a Bayesian study from the JETSCAPE collaboration and for pPb collisions with parameters based on results by the Duke group. We saw that while the multiplicity of charged particles and the average transverse momentum of the total charged particles remain largely unaffected, a significant dependence on $\gamma$ is observed when analyzing the identified particle yields. These results highlight the importance of a detailed treatment of viscous corrections in improving the precision of hybrid model predictions.

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Tetraquark systems are composed of two quarks and two antiquarks which could be understood
as diquark-antidiquark bound states or meson-meson molecules. In the latter case, the
antisymmetrisation of the molecular wave function against quark exchange becomes necessary.
We discuss the resulting Pauli-blocking effect for the all-charm tetraquark Tcc(6900) as a J/ψ-J/ψ
molecule [1] and the light tetraquark a0(500) as a two-pion molecule of the ππ interaction [2,3].
In the case of the heavy-light tetraquark χc1(3872) (formerly X(3872)), we demonstrate that
double quark exchange in a J/ψ-ρ or J/ψ-ω molecule leads to an energy-dependent kernel of the
Bethe-Salpeter equation for the T-matrix of the two-meson molecule which has a pole at the D-
D threshold that leads to a bound state with the mass of X(3872) [4]. In this way we can explain
by a dynamical quark exchange kernel that the X(3872) is a well-defined long-lived resonance
close to the D-D threshold, because it is a quasi-unitary bound state with a very small binding
energy. It follows also in accordance with observations that it dominantly decays into a J/ψ and
two or three pions under strong CP violation. We speculate about modifications of the X(3872)
when it is created in the hot and dense medium of a heavy-ion collision.

The finite-size effects on the phase diagram, fluctuations, and thermodynamics are investigated using the explicitly volume-dependent partition function. We reproduce the expected finite-size scaling behavior while discussing different physical quantities in a finite volume.

I will present the calculation of the propagator of a charged vector boson in the presence of an external magnetic field using the Ritus Eigenfunction method. Although this is not a novel result, the methodology used throughout the calculation is relevant in the sense that the calculation is done with a diagonalized equation of motion. This advantage allows a more direct interpretation of the energy and Landau levels present in the methodology and their physical scenarios. Moreover, I present the path from the propagator in the Ritus Eigenfunction method representation to the propagator in the Schwinger proper time parameter representation.

In most fields of science, visualizations help to understand concepts and to identify potential issues. For these reasons, animations of relativistic heavy-ion collisions provide interesting visual insights into the simulation software used as well as the time evolution of the collisions. In our work, we produce animations for collisions at several beam energies below 200 GeV for different impact parameters and collision systems. The underlying software for our animations are the relativistic transport approach SMASH (Simulating Many Accelerated Strongly-interacting Hadrons) and the SMASH-vHLLE-Hybrid approach. SMASH constitutes an effective solution of the relativistic Boltzmann equation and is suitable for the dynamical description of collisions at low beam energies. The SMASH-vHLLE-Hybrid approach mainly combines SMASH, which is used to describe the non-equilibrium phases of a collision, with vHLLE, a software for the simulation of relativistic hydrodynamics. The latter is employed to describe the fluid phase of the hot and dense fireball. This approach is suitable for collisions at higher beam energies. Each visualization is based on the output of a single event from SMASH or the SMASH-vHLLE-Hybrid and is animated using the open-source software ParaView.

The violation of the conformal limit for the speed of sound, $c_s^2=1/3$, has emerged as a critical feature of dense strongly interacting matter. Astrophysical observations—including gravitational wave data from LIGO/Virgo and precise neutron star radius measurements from NICER—suggest that the equation of state must exhibit a rapid stiffening at intermediate densities. This behavior is commonly associated with a peak in the speed of sound, potentially signaling a phase transition in the dense QCD regime.

Direct lattice QCD simulations at finite baryon density are hindered by the sign problem. However, effective QCD models, as well as lattice-accessible theories such as two-color QCD and QCD at finite isospin chemical potential, provide valuable insights into the nonperturbative dynamics of dense quark matter. These approaches support the existence of a peak in $c_s^2$ offering qualitative agreement with astrophysical expectations.

We present and analyze recent numerical and analytical results obtained from effective models of QCD, emphasizing the behavior of the equation of state and its stiffening at intermediate densities. These developments provide an essential bridge between effective theoretical and phenomenological descriptions of QCD and observable signatures in astrophysics.

We study one-flavor $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ lattice QCD in $1+1$ dimensions at zero temperature and finite density using matrix product states and the density matrix renormalization group. We compute physical observables such as the equation of state, chiral condensate, and quark distribution function as functions of the baryon number density. As a physical implication, we discuss the inhomogeneous phase at nonzero baryon density, where the chiral condensate is inhomogeneous, and baryons form a crystal. We also discuss how the dynamical degrees of freedom change from hadrons to quarks through the formation of quark Fermi seas.

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Strong magnetic fields can be found in all systems where we believe the quark-gluon plasma is formed, such as the early universe, compact stars, and heavy-ion collisions. The most intense magnetic field estimated in the lab is in non-central heavy-ion collisions, which can reach up to 10^{19} G. The scale of the strong interaction is much larger than the electromagnetic interaction, and the latter is usually suppressed. However, in cases where the intensity of the fields is very high, electromagnetic effects may influence strongly interacting matter. The goal of the project is to perform a renormalization group approach to estimate the QCD beta function in an electromagnetic background at one loop.
In this talk, I will focus on how a background magnetic field modifies the dispersion of fermions, leading to the quantization of energy levels known as Landau levels. I will show how these modifications affect the fermionic propagator, self-energy, and vertex function in quantum electrodynamics.