Unsolved problems in qcda lattice approach

Zusammenfassung

Quantum Chromo-Dynamics ($QCD$) is a challenging theory: the strong $CP$ problem or its behaviour for very high densities are some of the yet-unsolved problems that worry the $QCD$ practitioners all along the world. Even some of its fundamental properties, like the confinement of quarks, remain unproved\footnote{At this moment, only numerical 'proof' of the confinement exists. In fact, the confinement of quarks is one of the \emph{millenium problems}, awarded with a substantial prize by the \emph{Clay Mathematics Institute}.}. One of the points that makes the analysis of $QCD$ so intrincate is its non-perturbative behaviour at low energies. Contrary to what happens in Quantum Electro-Dynamics ($QED$), whose coupling constant $\alpha_{QED}$ is small, $QCD$ is only accessible to perturbative techniques in the high energy region, where its coupling constant $\alpha_{QCD}$ becomes small, due to the property of asymptotic freedom. For low energies, $\alpha_{QCD}$ takes very high values, and the perturbative expansion does not converge any more: we need an infinite number of Feynmann diagrams to compute any observable. Moreover, perturbation theory is unable to give account for all the topological properties of $QCD$. Since the topology plays a fundamental role in the theory, the availability of a non-perturbative tool to study $QCD$ is very desirable. The most prominent tool to investigate non-perturbative theories is the lattice. Logically, in this work, which is about $QCD$, the lattice is the underlying basis for every development. Lattice $QCD$ and numerical simulations usually come together: in most cases, lattice theories are very complex to study analitically, but surprisingly easy to put in a computer. Taking into account the increasingly high computer power available nowadays, lattice $QCD$ simulations are becoming a powerful and accurate tool to measure the relevant quantities of the theory. Unfortunately, the lattice is far from perfect, and there are still obstacles (like the sign problem) which prevent some simulations to be performed. The title of this thesis summarizes the aim seeked along these four years of research: to tackle some of the yet-unsolved problems of $QCD$, to explore or create techniques to study the nature and the origin of the problem, and in some isolated cases, to find the desired solution. As the topics developed in this work have been the responsible of the headaches and frustrations of many scientist during the last decades, only in very few cases a reasonable solution to the problem exposed is found. In any case, the effort put into these pages constitute valuable information to those who decide to embark themselves in such a similar task. The work exposed here can be divided in two parts. The first one (comprising chapters one to five) is related to the chiral anomaly, the strong $CP$ problem, and the discrete symmetries of $QCD$, mainly Parity. All these topics are introduced very briefly in chapter one, to make the manuscript as self contained as possible. The second chapter is devoted to the key tool used to study all the aforementioned phenomena: the Probability Distribution Function ($p.d.f.$) formalism, a well-known technique used to study the spontaneous symmetry breaking (SSB) phenomena in statistical mechanics, brought to quantum field theories. Of particular interest is the extension of the domain of application of this technique to fermionic bilinears, well explained at the end of the second chapter. This chapter thus establish the theoretical framework required to understand the analyses performed in the following pages. The third chapter is the first one containing original contributions. It begins with an introduction to the Aoki phase, explaining its origin and properties from several points of view. The Aoki phase is a consequence of the explicit chiral symmetry breaking of Wilson fermions. In this phase, Parity and Flavour symmetries are spontaneously broken, and even if the Aoki phase is not physical, its existence represents an obstacle to the proof of Parity and Flavour conservation in $QCD$. The new research performed here is the application of the $p.d.f.$ to the study the breaking of Parity and Flavour. Surprisingly we obtain unusual predictions: new undocumented phases appear, that seem to contradict the standard picture of the Aoki phase; more exactly, the $\chi PT$ and the $p.d.f.$ predictions clash and become difficult to reconcile. Since $\chi$PT is one of the pillars of the lattice $QCD$ studies, it is quite important to find if one of the approaches is giving wrong results. The resolution to the contradiction is seeked --without much success- in a dynamical simulation of the Aoki phase without a twisted mass term, an outstanding deed never done before, due to the small eigenvalues of the Dirac operator that cause the Aoki phase. The chapter four is, in some way, a continuation of chapter three. It tries to answer a question that arises after a careful examination of the Aoki phase: are Parity and Flavour spontaneously broken in $QCD$? The old Vafa and Witten theorems on Parity and Flavour conservation are reviewed and critiziced, and a new and original proof for massive quarks, within the frame of the $p.d.f.$, is developed. This proof makes use of the Ginsparg-Wilson regularisation, which lacks the problems of the Wilson fermions, despite of being capable of reproducing the anomaly on the lattice without doublers: its nice chiral properties depletes the spectrum of small eigenvalues for massive fermions and forbids the appearance of an Aoki-like Parity or Flavour breaking phase. The fifth chapter drifts from the trend established in chapters three and four, in order to deal with the simulations of physical systems with a $\theta$ term. Although the main goal behind this chapter is to study $QCD$ with a $\theta$ term, it is devoted entirely to the antiferromagnetic Ising model within an imaginary magnetic field, which is used as a testbed for our simulation algorithms. The first half of the chapter introduces the Ising model and the techniques used to avoid the sign problem inherent to the imaginary field. The second part, which contains the new contributions to the scientific world, applies the aforementioned algorithms to the two- and three-dimensional cases with success. The complete phase diagram of the model is sketched with the help of a mean-field calculation and, despite the simplicity of the model, its $\theta$ dependence proved to be more complicated than what one expects for $QCD$, featuring a phase transition for $\theta<\pi$ for some values of the coupling. The chapter ends with the commitment to apply these methods to $QCD$, in order to find out what would be the behaviour of nature if $\theta$ departed from zero, and to try to find out what it behind the strong $CP$ problem. The possibility to overcome the sign problem in the numerical simulations of $QCD$ with a $\theta$ term inspired us to deal with another long-standing problem of $QCD$ suffering from the same handicap: finite density $QCD$. This difficulties take us to the second part of this work, which deals with the polimerization of fermionic actions. The polimerization as a technique was born around thirty years ago, with the purpose of performing simulations of fermions on the lattice. Nowadays, fermion simulations are carried out by means of the effective action, obtained after the integration in the full action of the Grassman variables\footnote{Direct simulation of the Grassman fields in the computer is not feasible at this moment.}. This integration yields the fermionic determinant, characterized by its high computational cost, for it usually reaches large dimensions\footnote{The most impresive numerical simulations involving dynamical fermions invert a $\sim10^8\times\sim10^8$ fermionic matrix.}. The polymeric alternative transforms the quite expensive computationally determinant of the effective theory into a simple statistical mechanics system of monomers, dimers and baryonic loops, easy to simulate. Although this formulation naively displays a severe sign problem, it can be solved in some cases by clustering configurations, being the most notorious one the $MDP$ model of Karsch and Mütter for $QCD$ in the strong coupling limit. That is why the sixth chapter of this work is completely devoted to the $MDP$ model of Karsch and Mütter. When this model appeared, it was claimed that it allowed the performance of numerical simulations of finite density $QCD$ in the strong coupling limit, for the sign problem within this model was mild in the worst case. Some doubts were raised when the analysis of Karsch and Mütter was carefully revised and extended, and in analyzing these objections we found a serious problem with the original simulation algorithm, which was not ergodic for small masses. In particular, the chiral limit $m\rightarrow0$ was not simulable at all. Our contribution to the field is the improvement of the simulation algorithm to enhance its ergodicity, matching the good properties of the new so-called worm algoriths, to whom it is compared. The results are disappointing: the sign problem is recovered once the ergodicity is restored, so it was the lack of ergodicity the responsible of the reduction of the sign problem's severity. This result opposses that of Philippe de Forcrand in his last papers, were he claims to solve the sign problem in the same model by using a worm algorithm. Theoretical arguments explaining why the sign problem for high values of the chemical potential can not be solved within the $p.d.f.$ framework are then given. On the other hand, the $MDP$ model should suffer from a severe sign problem even at zero chemical potential, but from the experience of Karsch and Mütter, it seems that a clever clustering might be able to remove the sign problem completely, or at least to reduce its severity. This fact takes us to the seventh chapter, which successfully tries to apply the polimerization technique to a pure, abelian Yang-Mills action\footnote{Although we regarded at first this application of the polymerization as an original contribution never done before, we discovered afterwards that we were wrong. The polymerization for abelian gauge theories was proposed some years ago, but it was never simulated in a computer.}, in an attempt to take the $MDP$ model beyond the strong coupling limit. The polymeric abelian gauge model is simulated and it is found to perform at least as well as the standard heat-bath procedure, besting it in some areas. A reduction of the critical slowing down for some systems featuring a second order phase transition is also observed. Nevertheless, the complete system featuring fermions plus Yang-Mills fields has a severe sign problem, even at zero chemical potential, and can not be simulated in the computer. In the last chapter, which can be viewed as an extension of chapter seven, the polymerization procedure is generalized and applied to other systems, mixing failure with success, being the most important of the latter the Ising model under an imaginary field $h=i\pi/2$ for any value of the dimension, which takes us back to the fifth chapter and the theories with a $\theta$ term. Hence, the underlying topic of these four years of research has been the sign problem inherent to complex actions, in particular finite density $QCD$ and $QCD$ with a $\theta$ term, and any deviation from this point (mainly chapters two, three, four, seven and eight) is a side-work which ended up giving interesting results. These are long-standing problems to which hundreds of scientist have devoted half their lives without too much success. My collaborators and me are no exception, and although some significant advances were made within these four years, we are still very far from finding a solution to these problems.