## Research topics for Master and Ph.D. theses

## Quantum Chromodynamics (QCD) is the microscopic theory of strong interactions, describing the dynamics of quarks and gluons. While confined within hadrons at zero and low temperatures, at high temperatures quarks and gluons are liberated in the so-called quark-gluon plasma. The nature of the transition from the hadronic to the plasma phase, and the properties of the quark-gluon plasma, are relevant to a variety of problems, ranging from the physics of the early Universe to that of heavy-ion collisions. QCD belongs to the class of Quantum Field Theories known as gauge theories, characterised by their symmetry under transformations acting locally in spacetime, which requires the appearance of so-called gauge fields (called gluons in QCD). The study of more general gauge theories can help in shedding light on the properties of real-world QCD. The interaction of quarks and gluons is encoded in the spectrum and eigenvectors of the Dirac operator. These provide microscopic information on the system, from which one can in principle reconstruct most of the observable features of hadrons and of their interactions.

## 1. Localisation of Dirac eigenmodes in gauge theories

## The finite-temperature transition from the confined, hadronic phase to the deconfined, plasma phase is accompanied by a change in the localisation properties of the low-lying Dirac modes, in all cases where this issue has been investigated. In practice, low modes turn from being extended over the whole system to being localised in finite spatial regions. This is analogous to what happens to the wave functions of an electron in a crystal at the centre of the energy band, if the amount of impurities in the crystal becomes too large ("Anderson localisation"). Exploiting this analogy to understand the close connection between Dirac-mode localisation and deconfinement, could lead one to unveil the microscopic mechanisms behind confinement and the deconfinement transition, and to understand the relation between deconfinement and restoration of chiral symmetry (see below).

## 2. QCD at finite temperature in the chiral limit and the Dirac spectrum

## If the masses of the up and down quarks are set zero, QCD acquires an exact SU(2)xSU(2) chiral symmetry. At zero temperature, this symmetry is spontaneously broken to its vector part, leading to the appearance of three massless bosons (corresponding to the three light pions in real-world QCD). At higher temperatures this symmetry gets restored through a phase transition. The nature of this transition is, however, still not fully understood. Useful insight on this question can be obtained by studying the effects of chiral symmetry restoration on the Dirac spectrum and eigenvectors. The spontaneous breaking of chiral symmetry is in fact characterised by the accumulation of Dirac modes near zero. On the other hand, characterising the restoration of chiral in terms of the Dirac spectrum turns out to be more nuanced. As the chiral phase transition is a nonperturbative phenomenon, its study requires nonperturbative techniques. The method of choice is the lattice formulation of QCD, which allows one to investigate the relation between chiral symmetry restoration and the Dirac spectrum numerically, and to some extent also analytically.

## Requirements

## Both lines of research require previous knowledge of the basics of Quantum Field Theory. Previous knowledge of Lattice Field Theory is useful, but it can also be acquired along the way. Work on these topics requires learning the basics of the theory of localisation and of random matrix theory. Numerical works requires at least basic coding skills in FORTRAN or C, knowledge of the basics of data analysis, and saint-like patience in debugging code. Analytical work requires the usual linear algebra and calculus, plus some group theory, a steady hand, and total disregard for one's mental well-being.

Home## Matteo Giordano -- giordano at bodri dot elte dot hu Elméleti fizikai tanszék -- room 6.54B