Unconventional probes into the QCD phase diagram using Rational approximations and Lefschetz thimbles

Abstract

In this thesis, we present two methods of probing quantum chromodynamics (QCD) at finite densities on a lattice. This theory studies the dynamics of the interactions between quarks and gluons. Being a strongly interacting theory, it is hard to study it using perturbative approaches. We need non-perturbative methods to study this theory. Lattice QCD, a numerical approach in which space-time is discretized and quarks and gluons are put on four dimensional space-time lattices is one such non-perturbative method which has been very successful in studying QCD at zero chemical potential. However, at finite densities we encounter the numerical sign problem, which hinders progress in simulating QCD via lattice methods. One of the main reasons we want to study QCD at finite densities is to understand its phase diagram in the temperature - chemical potential plane. Currently, most of the phase diagram is a conjecture, although some regions of the phase space are well studied both by theoretical methods and heavy-ion collision experiments. In the early stages of the Universe, the temperatures were so high that quarks and gluons existed in a de-confined phase called the quark gluon plasma (QGP). At some point in time when the temperatures dropped below a certain value (transition temperature), quarks and gluons combined to form hadrons like protons and neutrons. A particularly active field of research, currently, is the search for the transition from the de-confined to confined phases at finite densities. Through lattice methods, it has been successfully shown that at zero chemical potentials, this transition is an analytic crossover. It has also been seen that up to small chemical potentials, it continues to remain a crossover. However, at larger chemical potentials, this crossover line is expected to terminate at a critical end point. This search still remains an open problem to this day and forms a very active field of research. The goal of this thesis was to make progress towards the understanding of the QCD phase diagram at finite densities by using methods that minimise/evade the numerical sign problem to facilitate this goal. To this end we have developed a rational approximation method to study some thermodynamic variables associated with QCD simulated at imaginary chemical potentials, in the hope of finding its singularities in the complex chemical potential plane. Apart from this, some progress in the direction of studying Lefschetz thimbles within the scope of single thimble simulations and regularising non-abelian gauge theories on thimbles have also been made.

We begin by a general introduction to the importance of making progress in the QCD phase diagram in Chapter 1. This thesis is then divided into two parts. The first part describes an unconventional method of re-summation of the Taylor series expansions of thermodynamic variables simulated at zero and purely imaginary values of chemical potential to look for non-analyticities (singularities) of the partition function in the complex chemical potential plane. The scaling of these complex singularities can give very important hints about the phase diagram. In Chapter 2 of this thesis we describe in detail the method of re-summation used by us. This is followed by Chapter 3 where we give a general overview of phase transitions in QCD. Chapter 4 contains an introduction to the complex singularities mentioned above, called Lee Yang edge singularities, with a section containing a case study on the 2D Ising model performed by us recently to demonstrate the validity of our analysis. In Chapter 5 we discuss the most significant work of this thesis, i.e., studying the phase diagram of 2+1 flavour QCD at imaginary values of chemical potential using the rational-function re-summation techniques to find the Lee-Yang edge singularities. We then study the scaling of these singularities in the vicinity of the Roberge-Weiss transition and show the consistency of our results. We also briefly mention another singularity that we found in the vicinity of a chiral transition. \\ We then move on to the second part of the thesis that is more directly focused on the numerical sign problem and the current status of its possible solution using Lefschetz thimbles. In Chapter 6 we describe the numerical sign problem and introduce the Lefschetz thimble approach and a method to do single thimble simulations. In Chapter 7 we formulate a method of applying Lefschetz thimbles to non-abelian gauge theories which is currently an unsolved problem. We further discuss the problem of having extra zero modes when regularising gauge theories and how they can be removed by using different boundary conditions. We end the discussion on non-abelian gauge theories by introducing some ideas as to why studying the topological charge using Lefschetz thimble regularisation can be a good idea. We finally conclude and provide a summary of our results and perspective on future directions in Chapter 8.