zur Erlangung des akademischen Grades

Gluon and ghost propagator studies in lattice QCD at finite temperature D I S S E R TAT I O N zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Physik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultät I Humboldt-Universität zu Berlin von Herrn Magister Rafik Aouane Präsident der Humboldt-Universität zu Berlin: Prof. Dr. Jan-Hendrik Olbertz Dekan der Mathematisch-Naturwissenschaftlichen Fakultät I: Prof. Stefan Hecht PhD Gutachter: 1. Prof. Dr. Michael Müller-Preußker 2. Prof. Dr. Christian Fischer 3. Dr. Ernst-Michael Ilgenfritz eingereicht am: 19. Dezember 2012 Tag der mündlichen Prüfung: 29. April 2013 Ich widme diese Arbeit meiner Familie und meinen Freunden v Abstract Gluon and ghost propagators in quantum chromodynamics (QCD) computed in the infrared momentum region play an important role to understand quark and gluon confinement. They are the subject of intensive research thanks to non-perturbative methods based on DYSON -S CHWINGER (DS) and functional renormalization group (FRG) equations. Moreover, their temperature behavior might also help to explore the chiral and deconfinement phase transition or crossover within QCD at non-zero temperature. Our prime tool is the lattice discretized QCD (LQCD) providing a unique ab-initio nonperturbative approach to deal with the computation of various observables of the hadronic world. We investigate the temperature dependence of L ANDAU gauge gluon and ghost propagators in pure gluodynamics and in full QCD. The aim is to provide a data set in terms of fitting formulae which can be used as input for DS (or FRG) equations. We concentrate on the momentum range [0.4, 3.0] GeV. The latter covers the region around O(1) GeV which is especially sensitive to the way how to truncate the system of those equations. Regarding the gluon propagator, we compute its longitudinal (DL ) as well its transversal (DT ) components. For pure gluodynamics in a fixed-scale approach we show DL to react stronger than DT , when crossing the first order deconfinement phase transition. At the same time the ghost propagator G looks nearly insensitive to the temperature. Since the longitudinal component turns out to be most sensitive with respect to the critical behavior we propose some combinations of it playing the role of an indicator for the transition. Major attention is paid to the extraction of the continuum limit as well as to systematic effects, as there are the choice of the right P OLYAKOV loop sector, finite size and G RIBOV copy effects. Fortunately, finitesize and G RIBOV copy effects are found to be weak in the momentum range considered and at temperatures close to the deconfinement phase transition. In a second step we deal with full (N f = 2) LQCD with the twisted mass fermion discretization. We employ gauge field configurations provided by the tmfT collaboration for temperatures in the crossover region and for three fixed pion mass values in the range [300, 500] MeV. The gluon and ghost propagators in the momentum interval [0.4, 3.0] GeV show a smooth temperature dependence. We provide fit formulae and extract D−1 L at zero momentum being proportional to the electric screening mass squared. Finally, within SU(3) pure gauge theory (at T = 0) we compute the L ANDAU gauge gluon propagator according to different gauge fixing criteria. Our goal is to understand the influence of gauge copies with minimal (non-trivial) eigenvalues of the FADDEEV-P OPOV operator (FP). Therefore, we compare the gluon propagator according to two different criteria, namely gauge copies with maximal gauge functional values versus those with minimal FP eigenvalues. Such a study should clarify how the G RIBOV copy problem influences the behavior of the gluon and ghost propagators in the infrared limit. By tending to smaller FP eigenvalues the ghost propagator is expected to become more infrared singular. The main aim is then to see whether the gluon propagator becomes infrared suppressed and therefore whether LQCD may describe a larger manifold of the so-called decoupling solutions as well as the scaling solution of DS equations. In an exploratory study we restricted ourselves to small lattice sizes, for which the influence of those copies at smallest accessible momenta turned out to be small. vii Zusammenfassung Die im infraroten Impulsbereich der Quantenchromodynamik (QCD) berechneten Gluonund Ghost-Propagatoren spielen eine große Rolle für das sogenannte Confinement der Quarks und Gluonen. Sie sind Gegenstand intensiver Foschungen dank nicht-perturbativer Methoden basierend auf DYSON -S CHWINGER- (DS) und funktionalen Renormierungsgruppen-Gleichungen (FRG). Darüberhinaus sollte es deren Verhalten bei endlichen Temperaturen erlauben, den chiralen und Deconfinement-Phasenübergang bzw. das Crossover in der QCD besser aufzuklären. Unser Zugang beruht auf der gitter-diskretisierten QCD (LQCD), die es als ab-initioMethode gestattet, verschiedenste störungstheoretisch nicht zugängliche QCD-Observablen der hadronischen Welt zu berechnen. Wir untersuchen das Temperaturverhalten der Gluonund Ghost-Propagatoren in der L ANDAU-Eichung für die reine Gluodynamik und die volle QCD. Ziel ist es, Datensätze in Form von Fit-Formeln zu liefern, welche als Input für die DS- (oder FRG-) Gleichungen verwendet werden können. Wir konzentrieren uns auf den Impulsbereich von [0.4, 3.0] GeV. Dieses Intervall deckt den Bereich um O(1) GeV mit ab, welcher für den auf verschiedene Weise vorzunehmenden Abbruch des Gleichungssystems sensitiv ist. Für den Gluon-Propagator berechnen wir deren longitudinale (DL ) sowie transversale (DT ) Komponenten. Für die reine Gluodynamik bei fixierter kleiner Gitter-Einheit zeigt sich, dass DL im Vergleich zu DT stärker bei Überschreiten des Phasenübergangs erster Ordnung variiert. Andererseits reagiert der Ghost-Propagator nahezu unempfindlich auf die Temperaturänderung. Da sich die longitudinale Komponente als empfindlich gegenüber dem kritischen Verhalten erweist, schlagen wir einige Kombinationen der Komponenten vor, die die Rolle von Indikatoren für den Phasenübergang spielen können. Große Aufmerksamkeit schenken wir der Extraktion des Kontinuumslimes und den systematischen Effekten, wie der Wahl des richtigen P OLYAKOV-Loop-Sektors, dem Einfluss des endlichen Volumens und der G RIBOV-Kopien. Es erweist sich, dass die Effekte endlichen Volumens und von G RI BOV -Kopien relativ schwach in unserem Impulsbereich sowie für Temperaturen in der Nähe des Deconfinement-Phasenübergangs sind. In einem zweiten Abschnitt beschäftigen wir uns mit der vollen (N f = 2) LQCD unter Verwendung der sogenannten twisted mass-Fermiondiskretisierung. Von der tmfT-Kollaboration wurden uns dafür Eichfeldkonfigurationen für Temperaturen im Crossover-Bereich sowie jeweils für drei fixierte Pion-Massenwerte im Intervall [300, 500] MeV bereitgestellt. Die Gluon- und Ghost-Propagatoren zeigen im Intervall [0.4, 3.0] GeV eine vergleichsweise schwache Temperaturabhängigkeit. Für die Impulsabhängigkeit lassen sich in diesem Intervall relativ gute Fits erhalten. D−1 L bei verschwindendem Impuls, das proportional zum Quadrat der elektrischen Abschirmmasse ist, wird als Funktion der Temperatur dargestellt. Schließlich berechnen wir innerhalb der reinen SU(3)-Eichtheorie (bei T = 0) den L AN DAU Gluon-Propagator unter Verwendung verschiedener Eichfixierungskriterien. Unser Ziel ist es, den Einfluss von Eich-Kopien mit minimalen (nicht-trivialen) Eigenwerten des FADDEEV-P OPOV-Operators (FP) zu verstehen. Eine solche Studie soll klären, wie G RI BOV -Kopien das Verhalten der Gluon- und Ghost-Propagatoren im infraroten Bereich prinzipiell beeinflussen. Durch kleinere FP-Eigenwerte wird der Ghost-Propagator singulärer. Das Hauptziel ist es zu sehen, ob der Gluon-Propagator im Infraroten unterdrückt wird, viii und ob somit die LQCD eine größere Mannigfaltigkeit der sogenannten decoupling- und scaling-Lösungen der DS- Gleichungen zu beschreiben gestattet. In einer explorativen Studie beschränken wir uns auf kleine Gittergrößen, für die sich der Einfluss solcher Kopien bei den von uns erreichbaren kleinen Impulsen als noch relativ gering erwies. CONTENTS 1 General introduction 1 2 Introduction to QCD at finite T 9 2.1 2.2 2.3 2.4 2.5 3 Reviewing QCD . . . . . . . . . . . . . . . . . . . . . 2.1.1 Fields, symmetries and classical action . . . . 2.1.2 The quantization path integral formalism . . . 2.1.3 Regularization and renormalization . . . . . . 2.1.4 The functional method approaches to QCD . . QCD at finite T . . . . . . . . . . . . . . . . . . . . . 2.2.1 Path integrals and the M ATSUBARA formalism Order parameters in QCD at finite T . . . . . . . . . . 2.3.1 The P OLYAKOV loop . . . . . . . . . . . . . . 2.3.2 The chiral condensate . . . . . . . . . . . . . Nature of the phase transition in QCD . . . . . . . . . The gluon and ghost propagators at T > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QCD at T > 0 on the lattice 3.1 3.2 3.3 3.4 General introduction . . . . . . . . . . . . . . . . . . . . . 3.1.1 Gauge fields and gauge symmetries . . . . . . . . . A closer look to our lattice actions . . . . . . . . . . . . . . 3.2.1 The gauge W ILSON action . . . . . . . . . . . . . . 3.2.2 The improved S YMANZIK gauge action . . . . . . . 3.2.3 The improved twisted mass action . . . . . . . . . . How to perform the continuum limit? . . . . . . . . . . . . Fixing the L ANDAU gauge . . . . . . . . . . . . . . . . . . 3.4.1 Gauge fixing and gauge functional . . . . . . . . . . 3.4.2 A new proposal to deal with the G RIBOV ambiguity 9 9 12 15 17 27 27 29 29 33 36 38 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 42 45 45 46 47 48 49 49 51 ix x 4 CONTENTS Lattice observables at T > 0 4.1 4.2 4.3 4.4 5 5.2 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specification of our lattice samples . . . . . . . . . . . . . . . . 5.1.1 Localization of our critical βc . . . . . . . . . . . . . . 5.1.2 Selecting the momenta and the M ATSUBARA frequency 5.1.3 Gauge fixing process . . . . . . . . . . . . . . . . . . . 5.1.4 Fixing the scale . . . . . . . . . . . . . . . . . . . . . . The P OLYAKOV loop results . . . . . . . . . . . . . . . . . . . Results on the gluon and ghost propagators at T > 0 . . . . . . . 5.3.1 The T dependence of the gluon and ghost propagators . 5.3.2 Improving the sensitivity around Tc . . . . . . . . . . . 5.3.3 Study of the systematic effects . . . . . . . . . . . . . . 5.3.4 The P OLYAKOV sector effects . . . . . . . . . . . . . . 5.3.5 Finite volume effects . . . . . . . . . . . . . . . . . . . 5.3.6 The G RIBOV ambiguity investigated . . . . . . . . . . . 5.3.7 Scaling effects study and the continuum limit . . . . . . Lattice setting and parameters . . . . . . . . . . . . . . . . . Results on the gluon and ghost propagators . . . . . . . . . . 6.2.1 Fitting the bare gluon and ghost propagators . . . . . . 6.2.2 The T dependence of the gluon and ghost propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1 2 3 61 62 63 64 65 65 67 67 69 73 73 74 75 78 85 . . . . Correlation between gauge functional and λmin . . . . . . . . . . . . . . . . . The gluon propagator and its zero-momentum value D(0) . . . . . . . . . . . . Conclusion 55 56 57 59 61 Alternative study for the L ANDAU gauge fixing 7.1 7.2 8 . . . . Results for full QCD 6.1 6.2 7 55 . . . . Results in the pure gauge sector of QCD 5.1 6 The P OLYAKOV loop on the lattice The lattice gluon propagator . . . The ghost propagator . . . . . . . Renormalizing the propagators . . 85 86 86 92 97 98 98 103 105 A note on the over-relaxation method . . . . . . . . . . . . . . . . . . . . . . 105 The G ELL -M ANN matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 The gamma matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 CHAPTER 1 General introduction I is nowadays commonly believed that quantum chromodynamics (QCD) is the true theory describing the hadronic world. The elementary degrees of freedom of QCD are quarks and gluons. The quarks are fermions carrying a spin of 21 while gluons are the gauge bosons mediating the strong interaction. The physical gauge group considered throughout this thesis is the physical color symmetry group, namely SU(3). This non-abelian gauge group introduces extra features to QCD which do not exist in the abelian U(1) theory as quantum electrodynamics QED. As an example, the gluons are interacting with themselves as well as with quarks. In fact, QCD becomes an asymptotically free theory at large momenta. Such typical aspect of QCD is often named as asymptotic freedom. This refers actually to the property of the quarks and gluons to behave nearly as a free particles system at small distances and/or high energies exchange. Hence, perturbation theory computing physical observables expanded in powers of the small coupling is of avail at this high energy scale. Indeed, perturbative results were confronted to experiments and proven to be valid in deep inelastic scattering [1, 2, 3]. Another essential aspect to be mentioned is the complexity to describe QCD in terms of elementary degrees of freedom, namely quarks and gluons. Consequently, quarks and gluons, and generally colored states, are not observed in nature as asymptotic states. This is described in a dynamical view by asserting that color-charged particles experience a linear potential if pulled apart, e. g. the quark-anti-quark constituents of a meson, such that only color singlets can form asymptotic states. This peculiar phenomenon is generally known as confinement. In addition, it rises specifically in the infrared region of momenta, i. e. at low momenta, where exactly QCD perturbative theory breaks down. Still, strong investigation efforts were dedicated to look for isolated quarks even if the answer remained negative. Hence, such imposing experimental fact motivated and supported the hypothesis of confinement such that only bound states as hadrons and also glueballs might be in principle observed. Besides experiments QCD as a theory exhibits also a rich phase structures due to several symmetry properties. That is, different degree of freedom according to different phases of the theory should exist. For example, in the confining phase, at low temperature and low chemical T 1 2 CHAPTER 1. GENERAL INTRODUCTION FIGURE 1.1: The expected QCD phase diagram in the T − µB plane, where T and µB refer to the temperature and the baryon chemical potential respectively. Different regimes of energies and densities are subjects of different experiments projects covering different phase diagram regions. In our present work we focus exclusively on the case µB = 0 where lattice QCD is highly effective. potential, hadrons are the degrees of freedom of interest. Therein chiral symmetry is spontaneously broken whereas the color center group symmetry is not. This latter group is nothing but Z(3) supposed to be the underlying symmetry for confinement. On the other hand, at energy densities large compared to the natural scale (ΛQCD ∼ 200 MeV/fm3 ) the situation is completely different, and one should expect a strong increase in the degrees of freedom of the theory. This means that gluons and quarks should in principle behave as a (nearly) free particle system. This latter state of matter is well described through the theoretical framework of QCD. Quantitatively speaking, at higher energies regime the average distance r between quarks and gluons −1 becomes r ≪ ΛQCD ∼ 1 fm ∼ size of a hadrons exactly where QCD predicts a weak interaction, a phenomena, as already said, known as asymptotic freedom [4, 5, 6]. In other words, this is a consequence of the decrease of the coupling with the decrease of the distance between quarks and gluons, or equivalently with the increase of the momenta/energy exchange. Such free state of matter might be reached by increasing the temperature and/or densities. However, just before reaching this state matter should undergo a transition between a purely confined system of bound states to a plasma consisting of confined quarks and gluons. This latter state of matter is called in the literature the quark-gluon plasma, abbreviated as (GPA). GPA studies are the concern of extensive experiments using heavy ions collisions at different scales of energies probing matter structures at extreme conditions. First experiments with √ s moderate center mass energy per nucleon A of 2 GeV to 18 GeV were performed at the Alter- 3 nating Gradient Synchrotron (AGS) in Brookhaven and in the Super Proton Synchrotron (SPS) at CERN. More interesting results were also collected thanks to the Relativistic Heavy Ion Col√ s lider (RHIC), with A = 200 GeV. This race for √higher energies goes on with the promising Large Hadron Collider (LHC) at CERN reaching As = 5.5 TeV and with the heavy-ion detector ALICE. These multitude projects with their corresponding different phase diagram regions of interest are presented in Fig. 1.1. As a matter of fact, the main goal of the experimental efforts go into the direction of reaching high energy regimes as well as checking the reliability of many experimental models determining signatures of the QGP. In this thesis, we focus exclusively on the case µB = 0 (zero baryon chemical potential) using lattice QCD (LQCD) as our prime tool. In a nutshell, the most problematic aspect of LQCD with finite chemical potential is the presence of the so-called “sign problem”. This latter traces back to the fact that the fermion determinant becomes complex, and no standard LQCD computation might be possible. One of the most interesting applications of QCD under extreme conditions is the study of the thermodynamics of the universe. It seems that during the evolution of the early universe a quarkhadron transition took place. Therefore a QGP formation seemingly happened shortly (around ∼ 10−5 sec) after the Big Bang. The early universe exposed a very hot state of matter (temperatures up to & 1012 K) and looked quite different from actual observed universe. In fact, it was likely dominated by a total pressure of QCD degrees of freedom for temperatures larger than the transition temperature Tc . Further consequences of this are the actual rate expansion of the universe, and other physical phenomena as gravitational waves and dark matter. Other cases of matter under extreme conditions are the compact stellar objects whose the neutron stars are good examples. Since these stars expose high density regime 1016 − 1017 g/cm3 a production of QGP within these objects is very likely expected. Therefore, a proper understanding of basic thermodynamic quantities as the pressure among others as function of the temperature is essential for a proper understanding of our universe. As said before, highly non trivial phenomena as confinement or chiral symmetry restoration are out of the reach of perturbation theory. As a result, perturbation theory which might be applied only at high temperatures (T ) (or high densities µB ), where essential non-perturbative effects has faded away, is of no avail. Thereby, one needs to consider non-perturbative methods to tackle such phenomena. However, successful predictions of perturbation theory (PT) suggests that the effective coupling must decrease with the increase of T and/or µB as g(T ) ∼ 1 , (11Nc − 2N f ) log(T 2 /Λ2QCD ) where, Nc and N f are the number of colors and flavors respectively and ΛQCD is the QCD scale. In fact, PT is successfully describing the hadronic matter in this regime. Nevertheless, it encounters serious problems at the stage of moderate temperatures around Tc . For example, in the case of zero temperature (T = 0) for a massless renormalizable theory the renormalization scale Λ is the only scale of the theory. Computing the self-energy correction(s) Π(p) to the relevant propagator(s) of such theory -dimensional arguments and L ORENTZ invariance taken into account- 4 CHAPTER 1. GENERAL INTRODUCTION 2 will provide a behavior like Π(p) = g2 p2 f ( Λp 2 ), where f is a dimensionless function. One points out that this correction is small compared to the scale introduced by external momentum p for small g, and no divergence happens when resumming in the propagator. At finite T (T > 0) the situation is fundamentally different. There, the temperature is introduced as a new scale, and it follows effects on the integrals for soft modes (p < T ) which are dominated by momenta k ∼ T . Note that within this scheme the self-corrections reads Π ∼ g2 · T 2 . Indeed, this means that these corrections become as large as the inverse propagator itself for soft modes ∼ gT , and this is in fact this point which makes the PT framework break down [7]. For an overview of such non-perturbative problems we refer to [8] for example. In order to overcome non-perturbative problems many strategies have been developed and proposed on the market. One might first talk briefly about the Hard Thermal Loop (HTL) resummation methods. Those latter provide a way out by improving the infrared behavior of the theory thanks to a consistent resummation of all loops dominated by hard thermal fluctuations [9, 10, 11, 12]. In general, these effects manifest themselves in gauge theories through the appearance of a thermally generated mass. It is worth to note, that a virtue of this scheme HTL is being manifestly gauge invariant and a consistent picture of QCD can be reproduced. One other strategy is the use of the so-called chiral perturbation theory (χPT) [13] which applies at low temperatures and low chemical potential. This method accounts for the smallness of the up and down quarks masses and for the broken chiral symmetry in a systematic way. The drawback, however, is the non predictability as the hadron resonances start to influence the properties of strong interacting matter. In general, such models base their study on phenomenology ingredients similar to that of QCD as there are W ILSON lines or bound states. Apart from the aforementioned methods Lattice QCD (LQCD) provides an ab-initio method to handle physical problems along the whole axis of energies/temperatures. For a comprehensive account of LQCD we refer to excellent standard books [14, 15, 16, 17]. In addition, simulations of LQCD with the help of Monte Carlo (MC) techniques provide a vast amount of data, and brings insights into the structure of the QCD phase diagram. In fact, Fig. 1.1 shows different regions of the phase diagram with different energy regimes. Actually, it is the crossover region close to the temperature axis which is explored by the experiments at RHIC and LHC at CERN. Note also the conjectured existence of a superconducting phase transition, in principle, reached by heavy ion collision depending on low temperature regime. Such superconducting behavior has been studied thanks to models yielding temperatures on the order of 50 MeV. In order to study phase transitions within LQCD one needs to construct order parameters on the lattice in order to detect the passage from the confined to the deconfined regime. To illustrate, in pure gauge theory LQCD, when neglecting the fermion loops, the P OLYAKOV loop plays the role of an order parameter for both SU(2) and SU(3) gauge groups [18, 19]. This is based on the fact that the center group Z(3) (for SU(3)) is spontaneously broken crossing the phase transition temperature T = Tc . Hence, in the broken phase region a reduced number of flips are observed between different sectors of the P OLYAKOV loop, and the transition expected in the pure gauge sector is of first order. In contrast, the situation dealing with full QCD is more 5 involved, and different observables might help. The chiral condensate is a good example of such order parameters whereas the P OLYAKOV loop is not an exact order parameter. This chiral condensate signals the restoration of the chiral symmetry above the phase transition. The observables of interest in this thesis are mainly the gluon and ghost propagators at finite temperature. These two-point G REEN functions represent building blocks of the DYSON S CHWINGER (DS) equations [20, 21, 22, 23] as well as basic components for the functional renormalization group equations (RGE) investigations [24, 25]. Therefore, our goal is to provide data serving as input for these non-perturbative methods. Moreover, the propagator behavior data might also serve to confirm or to reject confinement scenarios as proposed by G RIBOV and Z WANZIGER [26, 27, 28] and K UGO and O JIMA [29, 30]. On the other hand, the zero temperature gluon and ghost propagators were intensively investigated within LQCD mainly for the L ANDAU gauge, see [31, 32, 33, 34, 35, 36, 37, 32] and references therein. However, these propagators are less investigated at finite temperature. In fact, the SU(2) gauge group for the L ANDAU gauge was the scope of a few papers [38, 39, 40, 41, 42, 43, 44, 45]. The SU(3) gauge group is less studied within pure gauge theory, see [46, 47, 48, 43, 49]. Our results are published in [49] aiming to fill this gap providing valuable data for pure gauge theory. Furthermore, the SU(3) gluon and ghost propagators in the presence of dynamical fermions are even less studied [50, 51]. Therefore, we decided to study the fermionic case with the help of the lattice twisted mass discretization, as presented in our paper [52]. Regarding the ghost propagator, most of the papers support a temperature independent behavior for the ghost propagator as in [53, 54, 55, 56] and references therein. Still, we show in [49] that small fluctuation appear in the region of small momenta and at higher temperatures. Our full QCD results at finite temperature for the gauge group SU(3) are in qualitative agreements with [42, 43]. We have also studied the impact of the G RIBOV problem on the (SU(3) at T = 0) gluon propagator using a new criteria to select uniquely the gauge. Namely, we select gauge copies with minimal FADDEEV-P OPOV eigenvalues. These gauge copies aim to make the ghost propagator more singular in comparison to gauge copies with maximum gauge functional. In principle, this comparative study aim to clarify which solution of DS equations might be supported by LQCD when the G RIBOV ambiguity is removed. The structure of this thesis is as follows: Firstly in chapter II, the necessary theoretical background connected to QCD at finite temperature is introduced after a basic review of the framework of QCD. In brief, we introduce the classical formalism of QCD with the corresponding fields and symmetries. Then, we move on to the finite temperature case by virtue of the M ATSUBARA formalism. Furthermore, a discussion of the nature of the QCD phase transition is given thanks to the concept of order parameters. At the end of this chapter the present status of art of the gluon and ghost propagator is presented. In chapter III we present the lattice and mathematical definitions of the gauge fields and the different lattice action discretizations used throughout this work. Moreover, we define the G RIBOV problem and discuss our strategy to deal with it. Lattice definitions of the observables of interest, namely the P OLYAKOV loop, the gluon and ghost propagators are given in chapter IV. Our results are presented in chapters V, VI and VII. Actually, in chapter V we focus on different aspects of the gluon and ghost propagator in 6 CHAPTER 1. GENERAL INTRODUCTION pure gauge QCD at finite temperature. For this we relied on the standard W ILSON pure gauge action. First and foremost we study different lattice artifacts as momenta pre-selection and even the P OLYAKOV loop sector effects. We were also able to locate the critical temperature Tc thanks to the P OLYAKOV loop susceptibility. In fact, the critical (inverse) coupling βc used in this study is suggested by an extrapolation function proposed in [57], and the P OLYAKOV loop study locating Tc is namely a check for the value of βc . These prerequisites prepared the ground to multiple upcoming studies as the finite volume and finite lattice spacing effects. The G RIBOV ambiguity was also a target of investigations in order to understand how our momenta regime is affected by such a problem. After that we extrapolate our data to the continuum limit a = 0. At the end of the day we establish the continuum limit and get the continuum data in hand. We found out that our data reach indeed the continuum limit being at the same time important input data to the DYSON -S CHWINGER and the renormalization group equations. Quite recently authors in [58] took advantage of our data to compute the effective potential of P OLYAKOV loop. Hence, F UKUSHIMA et al. were able to perform a thorough thermodynamic study taking into account our parametrization of the gluon and ghost propagators at finite T . Regarding sensitivity issues around the critical temperature Tc we propose “new order” parameters constructed out the longitudinal component of gluon propagator, namely DL . These new objects turn out to react stronger than DL around Tc , and hence might be of interest for further investigations using different volumes and critical temperatures. Our second concern was to present results in chapter VI for full QCD, i. e. including fermions, for the special case of two number of flavors NF = 2. In order to achieve this investigation we considered configurations provided by the tmfT collaboration, see [52]. These configurations are thermalized thanks to a combination of the S YMANZIK action as an improved pure gauge action and the so-called twisted mass action for the fermion part. One advantage of such twisted mass actions is to provide an automatic O(a) improvement when tuning the hopping parameter κ to its critical value κc . Thanks to these configurations we are able to compute the gluon and ghost propagators as functions of the momentum and temperatures. Moreover, we fit again the gluon propagators data with G RIBOV-S TINGL fitting formula giving good χ 2 . This latter fit allowed us to show the gluon propagators as function of the temperatures for a few interpolated momenta. In fact our temperatures are chosen in a way to cover the crossover region where the expected temperatures where deconfinement and chiral symmetry breaking might happen. On the other hand the ghost propagator does show a weak reaction to temperatures variation as expected from its scalar tensorial structure. Within chapter VII, and as a third subject of investigations, we get a closer look into the G RI BOV problem in SU(3) pure gauge QCD at T = 0. It is already a notorious problem that the multiple DYSON -S CHWINGER solutions for the gluon and ghost propagators generally need to be confirmed thanks to lattice results. In general, these latter lattice results support the so-called decoupling solution [59, 60, 61]. Still, this solution does not satisfy important confinement scenarios as the KOGU -O JIMA scenario. We believe that this situation might be clarified thanks to a careful study of the G RIBOV ambiguity thanks to a new approach. Our goal here is to give some indications that namely standard gauge fixing prescriptions (using e. g. methods as simulated an- 7 nealing) might be confronted to our new criteria giving at the end different results on the lattice. In other words, we compare exclusively the gluon propagator using two different criteria to fix the gauge copies, namely simulated annealing (gauge copies with the highest gauge functional) vs. a new method picking up gauge copies with the smallest FADDEEV-P OPOV (FP) eigenvalue for fixed configuration. This latter choice of the gauge copies using the FP eigenvalues defines the gauge uniquely. Finally, we sum up our results and draw a conclusion. CHAPTER 2 Introduction to QCD at finite T W this introductory chapter we review first basic elements of QCD at finite temperature T . Fields and gauge symmetries are also presented to fix notations and symbols used throughout this thesis. First, we adopt the path integral as the most usual quantization approach of QCD, and provide standard results on how QCD is regularized and renormalized. The second part of this chapter translates QCD concepts to the finite temperature case thanks to the so-called: M ATSUBARA formalism. Finally, we discuss order parameters as there are the P OLYAKOV loop and the chiral condensate, and their relevance within the phase diagram of QCD. To end up we give the status of art of the gluon and ghost propagators at T > 0 and define them mathematically in the continuum space-time. ITHIN 2.1 Reviewing QCD 2.1.1 Fields, symmetries and classical action Strong interactions in nature are described mathematically by the so-called quantum chromodynamics (QCD). Within this theory bound states as hadrons arise as particle excitations of the fundamental constituents, namely: quarks and gluons. In brief, QCD is a quantum field theory (QFT) which accounts for six type of quarks, called quarks flavors: up (u), down (d), strange (s), charm (c), bottom (b) and top (t). To have a basic understanding of their elementary properties please have a look to Table 2.1. Mathematically, QCD is also describing the gluons using eight 4-vector potentials Aaµ , with a = 1, . . . , 8 or in a matrix notation as Aµ = Aaµ λ a , (2.1) where λ a are the 3×3 linearly independent G ELL -M ANN matrices (see Appendix 8), a and µ are the color and L ORENTZ indices respectively. In the fundamental representation of the L IE group 9 10 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T Particle u d s c b t g Mass[MeV/c2 ] 1.5-3.0 3-7 95±25 (1.25±0.09).103 (4.20±0.07).103 (174.2±3.3).103 0 Electric charge 2/3 -1/3 -1/3 2/3 -1/3 2/3 0 Baryon number 1/3 1/3 1/3 1/3 1/3 1/3 0 Spin 1/2 1/2 1/2 1/2 1/2 1/2 1 TABLE 2.1: Some elementary properties of quarks and gluons. The electric charge and the spin are given as multiples of the electron charge |e| and h̄ SU(3) these λ a represent the generators of the group satisfying both relations: Tr(λ a λ b ) = δba /2 and the commutation relation [λ a , λ b ] = i f abc λ c . Here, the quantities f abc are called the constant structure of the gauge group, here SU(3), and δba are the usual K RONECKER symbol. Each quark (fermion) flavor corresponds to three (color) D IRAC 4-spinor fields ψ c with c = 1, 2, 3. The quark fields look like in general as µ,c ψ ≡ ψf . (2.2) Here, f = 1, . . . , 6 (the flavor number) and µ = 1, . . . , 4 (the D IRAC indices). The fundamental principle of QCD is the local gauge invariance. This principle, together with the general requirement of locality, L ORENTZ invariance and renormalizability strictly constraints the form of the Lagrangian LQCD . The simplest form of the Lagrangian in Euclidean four-dimensional space-time reads 1 a aµν F + LQCD = − Fµν ∑ ψ(iDµ γ µ − mψ )ψ, 4 ψ=u,d,s,c,b,t (2.3) where Dµ = ∂µ − ig0 Aµ is the gauge covariant derivative. The sum is defined over all the flavor quarks ψ f and anti-quarks ψ f , and γ µ denotes the D IRAC gamma matrices (see Section 8). The gluon field strength tensor is denoted as Fµν = gi0 [Dµ , Dν ] with g0 is the bare coupling constant and mψ representing the bare mass for each quark flavor. In general, the structure of Fµν looks like a Fµν = ∂µ Aaν − ∂ν Aaµ + g0 f abc Abµ Aνc . (2.4) Here and in the following, the latin indices (a, b, c) represent the color indices taking the values 1, . . . , Nc2 − 1 (adjoint representation), where Nc is in general the number of color (equals to 3 for the SU(3) color gauge group case). On the other hand, the greek indices µ and ν symbolize the usual L ORENTZ indices running from 1 to 4. According to Eq. (2.3) and Eq. (2.4) one may Sec. 2.1. Reviewing QCD 11 observe the gluons self-interaction as well as interactions between quarks and gluons. This selfinteraction of the gluons within Eq. (2.4) is essentially due to the non-abelian structure of SU(3), i. e. the structure constants f abc for the L IE algebra su(3) are not equal to zero. The quark and anti-quark fields representing the matter fields are connected through ψ ≡ ψ † γ0 . (2.5) These latter fields transform as usual under the fundamental representation of the SU(3) color group, i. e. the color indices runs over c = 1, . . . , (Nc = 3) with the γ0 matrix is obviously the D IRAC matrix corresponding to a zero (temporal) L ORENZ index. The classical QCD action may be defined as a four dimensional space-time integral of the Lagrangian density Eq. (2.3) SQCD = Z dt Z d 3 xLQCD . (2.6) Here is (x,t) the space-time point. This action (Eq. (2.6)) is invariant by definition under the following set of SU(3) local gauge transformations of (anti-)quarks and gluon fields † , Aµ −→ Aωµ = gω Aµ gω ψ −→ ψ ω = gω ψ, ω ψ −→ ψ = (2.7) † ψgω . The SU(3) local gauge transformations gω are parametrized by the real functions ω(x) as g(x) = exp(iω a (x)λ a /2), (2.8) where the G ELL -M ANN matrices λ a (see Appendix 8) are acting on the color indices of the (anti)quark field. Due to the previous set of color gauge symmetry the quark-gluon and gluon-gluon interaction strength are determined by the same universal coupling constant g0 . Consequently, this fact constraints the the number of independent Z-factors introduced within the regularization scheme. Regularization and renormalization shall be discussed more in detail in Section 2.1.3. For the sake of completeness, we recall as well the infinitesimal form of the local gauge transformations Eq. (2.7) yielding b δ Aaµ = ∂µa + g0 f abc ω b Acµ = Dab µ ω , δ ψ = −ig0 ω a λ a ψ, a a δ ψ = +ig0 ω λ ψ. (2.9) (2.10) (2.11) 12 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T Here and in general, we define the infinitesimal gauge transformation for the a generic gauge field Ψ → Ψω as follows [62] δ Ψ ≡ ωb ∂ |ω=0 =: ω b δb Ψ(x). ∂ ωb Ψω (2.12) In the next section, we focus first on the quantization of QCD using the path integral approach. Furthermore, we discuss the FADDEEV-P OPOV method in order to introduce the physical fields content for quantizing QCD with a particular interest to the L ANDAU gauge. 2.1.2 The quantization path integral formalism The quantization of QCD using either the path integral or the canonical quantization method is not a trivial task. The complexity comes from the fact that the QCD Lagrangian (Eq. (2.3)) is invariant under local gauge transformations. However, such problems are not present when dealing with gauge invariant observables, and especially on the lattice, where integrals over the compact gauge group (as SU(3)) become automatically finite. In the continuum, one usually generalizes the classical Lagrangian Eq. (2.6) adding extra terms (extra fields). This guarantees that expectation values of gauge invariant observables are independent of the gauge condition. However, along this quantization process, the gauge invariance is lost, and another symmetry takes place, namely, the BRST symmetry. To quantize a classical theory, one needs to choose a quantization procedure suitable to the nature of the physical problem. Different quantizations methods treat the fields and the computation of the n-point functions (the G REEN functions) differently. For example, the canonical quantization method regards the fields as operators, and the G REEN functions are computed as vacuum expectation values. Most interesting for us is the path integral formalism taking the fields as c-numbers, and the G REEN functions defined as functional integrations of products of fields over all of their (weighted) possible functional forms. Within this latter formalism the action remains classical without the introduction of any auxiliary fields. In the following, we concentrate on the path integral formalism. Classical QCD is quantized using the functional integration formalism integrating over the (anti-)quark and the gauge bosons fields. The Grassmannian integral on the (anti)quark fields is Gaussian, and might be performed instantly leaving only integration over gluon fields. Therefore, we concentrate in the following only on integrations over the gauge boson fields A. As well said before, the fields A(ω) and A are related by a gauge transformation Eq. (2.8), and thus they are physically equivalent. We say that the gauge fields are belonging to the same orbit. In fact, this orbit is spanned by all the gauge transformed fields at each space-time point. So, in principle, in order to quantize a gauge theory one performs an integration over gauge transformations belonging to different equivalence classes, i. e. different orbits of the gauge fields. This procedure avoids to take into account the redundancies of the gauge field within the same orbit whose in general present extra difficulties for the quantization. In the literature such quantization method selecting unique representative for each orbit is called the FADDEEV- Sec. 2.1. 13 Reviewing QCD P OPOV (FP) quantization method [63, 64]. In general, one usually constructs the generating functional Z[ j, ω, ω] = Z [DADψDψ] ∆ f [A] δ ( f [A]) e− R 4 d xL QCD + R 4 d x(Aaµ jµa +ωψ+ψω ) , (2.13) where j, ω and ω are the corresponding sources of the gluons, quarks and anti-quarks fields. ∆ f is the FP determinant whose definition is given a bit below in Eq. (2.17). The integration over the representative of each orbit is done using the general relation, i. e. the gauge fixing condition f [A; x] = 0, (2.14) at each space-time point x. In our particular case, we focus on the L ANDAU gauge, i. e. on the gauge condition ∂µ Aµ = 0, as we shall see later on. We assume for the moment that the path integral measure is well defined in Eq. (2.13). In case Eq. (2.14) is satisfied only once for each gauge orbit we call the gauge condition ideal [65]. If this is not the case the gauge condition is called non-ideal1 , and integration over the fields would be ambiguous. Indeed, this problem occurs specially beyond perturbation theory where the coupling constant becomes significant. Different solutions to Eq. (2.14) are called G RIBOV copies, and the space spanned by unique representatives of each orbit is the fundamental modular region (FMR) Λ . Therefore, in principle, an integration over Λ of the type Z Λ [DA] eSQCD [A] (2.15) is well defined. However, an analytical construction of such space is not trivial. On the lattice, for example, such construction is based on the study of maxima (or minima depending on the definition) of the gauge functional in order to get as close as possible to the (unknown) absolute global gauge transformation. The general strategy of the FP method is to start with the introduction of the FADDEEV-P OPOV determinant ∆ f [A] defined with Eq. (2.13) by means of invariant integration Z i  h (2.16) ∆ f [A] Dω(x) ∏ δ f A(ω) (x) = 1, x yielding in the general case ∆−1 f [A] = ∑ i: f [A(ωi ) ]=0 1 One det −1 δ   f A(ωi ) . δω (2.17) needs to note that even popular gauge fixing conditions as the L ANDAU gauge are in fact non-ideal. The success of the L ANDAU gauge in the perturbative regime comes from the fact that the coupling is small yielding small fluctuations around the unit gauge transformation. 14 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T The L ANDAU gauge is a particular case of Eq. (2.14) 2 , namely f [A] : ∂µ Aµ = 0. (2.18) Therefore, in this case, one obtains the following expression for the FP determinant
∆Landau [A] = det ∆ + ig0 ∂µ Aµ = Z [DcDc] e− R 4 4 a ab (x,y)cb (y) d xd yc (x)MFP . (2.19) The anti-commuting fields c and c defined in the adjoint representation of the gauge group are called FADDEEV-P OPOV ghosts, and ab (4) MFP (x, y) = −∂µx Dab (x − y) x,µ [A]δ (2.20) is the so-called Faddeev-Popov matrix. One may generalize the aforementioned generating functional Eq. (2.13) to the general case of covariant gauges 3 as f [A] : ∂ µ Aµ = a(x), (2.21) a(x) ∈ su(NC ). Note, ∆ f is the same as in the L ANDAU gauge case Eq. (2.19). Integrating on a(x) with some Gaussian weight having a dispersion ξ yields for the generating functional Z[ j, ω, ω, σ , σ ] = Z [DADψDψDcDc] e− R 4 d xL e f f [A,ψ,ψ,c,c]+Σ , (∂µ Aµ )2 − ca (x)(δ ab ∆ + ig0 f abc Acµ ∂µ )cb (x) 2ξ Z
Σ = d 4 x Aµ jµ + ωψ + ψω + σ c + cσ . Le f f [A, ψ, ψ, c, c] = LQCD − (2.22) (2.23) (2.24) Putting ξ = 0 corresponds to the L ANDAU gauge. The gauge fixing term in Eq. (2.22) can be expressed as a result of Gaussian integration on an auxiliary field Ba (x). As a result, this bring us to an effective Lagrangian form Le f f ≡ LBRST ξ LBRST = LQCD − (Ba )2 + Ba ∂µ Aaµ + ca (δ ab ∆ − ig0 f abc ∂µ Acµ )cb . 2 (2.25) Hence, we are ending up with an effective Lagrangian invariant under the so-called BRST transformations [66, 67, 68]. The BRST transformations are the remnant of the classical gauge transformations resulting from replacing the gauge parameters by Grassmann variables. These transformations are global ones. The virtue of the BRST transformation is to allow simpler ways of derivation of 2 We suppose for the moment that the L ANDAU gauge is unique. This is nearly the case in perturbation theory since one assumes the coupling to be small in this regime. 3 The L ANDAU gauge is a special case of the family of covariant gauges. Sec. 2.1. 15 Reviewing QCD the S LAVNOV-TAYLOR identities [69, 70] as a direct consequence of the gauge invariance. These identities were the cornerstone of the general proof of the renormalizability of non-abelian gauge theories by ’ T H OOFT and V ELTMAN [6] in 1972. 2.1.3 Regularization and renormalization General approach QCD as defined so far needs special care at the level of perturbative theory (PT). Expanding the G REEN functions of QCD in terms of the coupling within PT brings extra technical difficulties dealing with loop integrals. The involved mathematical expressions actually diverge as the cutoff of the internal momenta is sent to infinity. Hopefully, thanks to the renormalizability of QCD one can absorb all the divergences (at any order) in a suitable redefinition of a finite number of parameters in the Lagrangian, and also into the normalization of the Green functions. Thereby, calculations at any order of PT in QCD would lead to finite results after renormalization. Thus, the renormalized QCD becomes a predictable theory, and results might be confronted to experiments. In order to define completely QCD 4 one needs to compute the whole set of G REEN functions (n-points functions). These functions might be defined through the path integral formalism introduced in Section 2.1.2 as functional derivatives with respect to the general sources Jiai (x) hΦa11 (x1 ) · · · Φann (xn )i = δ n Z[J ] J1a1 (x1 ) · · · Jnan (xn ) J a1 ,...,Jnan =0 1 (2.26) where i counts the number of fields Φ and ai denotes the collection of indices including L OR ENTZ, D IRAC and the flavor indices. Now, using Eq. (2.22), this latter formula may be rewritten in a compact path integral form as hΦa11 (x1 ) · · · Φann (xn )i = 1 Z[0] Z [DΦ] Φa11 (x1 ) · · · Φann (xn ) e−S[Φ] (2.27) where Z[0] stands for the partition function while switching off the sources. In fact, connected n-point functions can be generated from the functional W [J ] = log(Z[J ]) by successive differentiations as in Eq. (2.26). A further step would be to transform W [J ] according to the L EGENDRE transformation yielding the effective action   Z Γ[Φ] := sup −W [J ] + J Φ , (2.28) R where J = J [Φ] is meant to extremize −W [J ] + J Φ with Φ denoting the expectation values hφ i. This last effective action generates the 1PI (one particle irreducible) G REEN functions 4 Not only QCD, but in general any quantum field theory. 16 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T by differentiating with respect to Φ like hΦa11 (x1 ) · · · Φann (xn )i1PI = δ n Γ[Φ] Φa11 (x1 ) · · · Φann (xn ) . (2.29) Φ0 This latter expression needs to be evaluated at vanishing sources at the end, i. e. Φ0i = δW δ Ji . (2.30) J =0 As explained before, these G REEN functions being in general not gauge-invariant pose problem in the perturbative range of QCD where the coupling is supposed to be small. Actually, momenta loops integrations yields infinite quantities, and a renormalization prescription is necessary. Prior to renormalize QCD one needs first the regularize it. Regularizing QCD may be done invoking several regularization schemes, e. g. the PAULI -V ILLAR and the dimensional regularization methods. However, more interesting for us is the lattice regularization of QCD. This method introduces an ultraviolet cutoff Λ = a−1 . This cutoff renders instantly all the momentum loops integrations finite. So, in principle, any computation of the G REEN functions on the lattice should not suffer from such kind of divergences. After regularizing QCD, for example by introducing the lattice cutoff λ , we renormalize our G REEN functions. This is achieved by introducing the so-called Z-factors formally into the bare Lagrangian in Eq. (2.3). Concretely, this amounts to define a renormalized effective Lagrangian Ler f f [71] as Ler f f     1 a 1 2 = Z3 Aµ −∂ δµν − − 1 ∂µ ∂ν Aνa 2 Z3 ξr  
+ Ze3 ca ∂ 2 ca + Ze1 gr f abc ca ∂µ Acµ cb − Z1 gr f abc ∂µ Aaν Abµ Acν
1 + Z4 g2r f abe f cde Aaµ Abν Acµ Aνd + Z2 ψ − γµ ∂µ + Zm mr ψ 4 − Z1F igr ψγµ T a ψ Aaµ (2.31) with the renormalized parameters gr , mr , ξr are connected to their bare values go , mo , ξo with the relations go = Zg gr , (2.32) mo = Zm mr , (2.33) ξo = Z3 ξr . (2.34) Moreover, theses fields appearing in the quantized Lagrangian Eq. (2.25) (in the L ANDAU gauge) Sec. 2.1. Reviewing QCD 17 have also to be rescaled as 1/2 Aaµ → Z3 Aaµ , ψ→ ca → 1/2 Z2 ψ, 1/2 Ze3 ca . (2.35) (2.36) (2.37) We mention also that the renormalized Gr and regularized Greg G REEN functions are related to each other as follows Gr (p1 , . . . , pn ; gr , mr , ξr ) = ZG · Greg (p1 , . . . , pn ; Λ, go , ξo , mo ). (2.38) In this last equation ZG denotes in general some combination of the Z-factors who in general are depending on the cutoff. Still, the renormalized G REEN functions must not depend on the cutoff, but rather on the scale of the theory. This scale might be in principle experimentally determined. In principle, the Z-factors introduced so far are independent from each others a consequence of the S LAVNOV-TAYLOR identities. These latter identities constraint the number of independent Z factors, and is a consequence of the universality of the bare coupling go in QCD. It is worth to note that G REEN functions need as said before to be renormalized, and the Z-factors correspondingly somehow to be computed as well. However, there are different renormalization schemes in order to determine the Z-factors. The difference between these schemes is essentially the way how the divergences are absorbed when rescaling the parameters. We concentrate in this thesis on the so-called MOM scheme 5 . Within the MOM scheme, the Z-factors are determined such that the two and three point function equal their corresponding tree-level expressions at some momentum µ. The momentum point µ is called the renormalization (or sometimes subtraction) point. During this thesis, our results regarding the gluon and ghost propagators are renormalized choosing µ = 5 GeV for the results for pure gauge theory while µ = 2.5 GeV is reserved for our fermionic investigations. For more details on our renormalization procedure we refer to Section 4.4. 2.1.4 The functional method approaches to QCD Beside LQCD, some of the most interesting non-perturbative approaches to study the behavior of the gluon and ghost propagators in QCD (in the continuum) are the so-called functional methods. In particular, the DYSON S CHWINGER equations (DSE) and the functional renormalization group equations (FRGE) are two of such methods. We give hereafter an introduction to the DSE equations. We start with a derivation of these equations, and then we interpret their simple solutions for the case of the gluon and ghost propagators. After that, we give also an overview of the renormalization group techniques emphasizing their role in both infrared and ultraviolet regions of QCD. 5 There are also other renormalization subtraction schemes such as the MS and MS schemes. 18 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T The DYSON -S CHWINGER equations (DSE) approach Basically, the DSE equations correspond to a functional, continuum approach to the quantum theory beyond perturbation theory. These equations are viewed as the equations of motion for exact propagators and vertices. The starting observation to derive the DSE is to make the assumption that the functional integral of a total derivative vanishes 0= Z [dφ ] δ  [−S[φ ]+R J φ ]  e . δφ This latter equation might be rewritten as ! δ S[φ ] Ji − Z[J ] = 0. δ φi φ → δ (2.39) (2.40) (δ J ) Here, the subscript i collects all type of indices as the space time, color and D IRAC degrees of freedom. Moreover, the field φi might correspond to one of the following fields: Aaµ , ca , ca , ψ, ψ. Now, using the following identities Ji = δ Γ[Φ] , δ Φi (2.41) and F[ δ δ ]Z[J ] = F[ ]eW [J ] , δ Ji δ Ji (2.42) with Γ is the effective action defined in Eq. (2.28), and using the derivative with respect to the source terms as   δΦj δ δ δ 2W δ = = , (2.43) δ Ji δ Ji δ Φ j δ Ji δ J j δ Φi one gets the interesting form δ Γ[Φ] δS δ 2W δ ) + Φ j ]. = [( δ Φi δ φi δ J j δ Jk δ Φk (2.44) Furthermore, arbitrary 1PI (one-particle irreducible) correlators correspond to differentiating Eq. (2.44) with respect to the fields, and putting the sources equal to zero at the end. This yields finally a tower of infinite integral equations coupling G REEN functions to each other. In order to be solved one needs to truncate this system of equations at some level. We remark that all the terms of perturbation theory might be totally recovered reiterating these equations indefinitely. The most interesting DSE equations for us are the ones corresponding to the quark, gluon and Sec. 2.1. 19 Reviewing QCD ghost fields. For instance, the quark propagator reads in terms of the effective action Γ as S(x, y) ≡ hT ψ(x)ψ̄(y)iconnected =  δ 2Γ δ Ψ(y)δ Ψ̄(x) Ψ0 −1 (2.45) Moreover, the renormalized S(p, Λ) and unrenormalized S(p, µ) quark propagators are related in momentum space via the Z2 -factor as (2.46) S(p, Λ) = Z2 (µ, Λ) S(p, µ), where λ denotes the cutoff parameter and µ the renormalization point. In order to get the DSE equation for the full renormalized quark propagator we combine Eq. (2.44) (differentiating with respect to Ψ and Ψ) together with Eq. (2.45) and Eq. (2.46). After some algebra6 , the DSE for the quark field is given by S−1 (p, µ) = Z2 (µ, Λ) S0−1 (p, Λ) + Σ(p, µ), (2.47) where the subscript (0) denotes the bare propagator and Σ(p, µ) symbolizes the self energy. In terms of equations the bare quark propagator and its bare mass are S0−1 (p, Λ) = /p + m0 (Λ), (2.48) (2.49) m0 (Λ) = Zm (µ, Λ) mr (µ). Here mr denotes the renormalized quark mass. On the other hand, the self energy Σ(p, µ) describing the quark-gluon interaction is given by 2 Σ(p, µ) = Z1F (µ, Λ) gr (µ) C f Z d4q γµ S(q, µ)Γν (k, l, µ)Dµ,ν (k, µ), (2π)4 2 (2.50) c −1 . This self energy is a composition of the where the C ASIMIR factor is given by C f = N2N c D IRAC matrices γµ (see Appendix 8), the full quark propagator S(q, µ), the 1PI qqg-vertex gs (µ)Γν (k, l, µ) and the full gluon propagator Dµν (k). The gluon propagator momentum is denoted by k = (p − q) and the average momentum by l = (p + q)/2. As usual gr denotes the strong coupling. We observe from Eq. (2.47) the quark DSE are composed of fundamental building blocks as there are the full quark, the gluon propagator and the qqg-vertex. The quark DSE might be viewed diagrammatically in Fig. 2.1. Further functional derivatives of the expression Eq. (2.44) with respect to a suitable number of fields φ , and subsequently setting all sources to zero, lead actually to the DYSON -S CHWINGER equation for any desired full n-point function. Obviously, in order to get the DSE for the gluon and ghost propagators, and also for the ghostgluon vertex function one needs to differentiate with respect to the corresponding set of fields 6 Readers interested in more details are referred to the textbooks [72, 73] or the reviews [74, 71]. Sec. 2.1. Reviewing QCD 23 to clarify this situation. Therefore, it might happen that modifying the gauge fixing method one might end with a solution agreeing with confinement criteria as K UGO -O JIMA’s for example. In this spirit, we have studied the gluon propagator (at T = 0) for moderate small momenta using other gauge fixing criteria. In fact we think that the G RIBOV copies problem mostly present in the infrared might be the source of getting exclusively decoupling solutions on the lattice so far, and at the same time, to be in contradiction with confinement criteria. In fact, the different solutions (deviating around 1 GeV) are results of different boundary conditions at zero momentum, namely J = 0 and J = finite, excluding strongly the G RIBOV effects. As a result, we propose a new procedure to select gauge copies in such a way to avoid the G RIBOV ambiguity. That is why, we take into account gauge copies lying as close as possible to the G RIBOV horizon, i. e. with the lowest FP operator eigenvalues. This process will map uniquely to a unique gauge copy, and therefore, no ambiguity due the fields redundancy need to be taken into account. For a full discussion of this problem we refer to Section 3.4.2. In general, in order to resolve the DSE one needs to make truncations at some level of the theory. That is, in order to manage towers of infinite integral equations one needs to neglect higher order n-point functions. In fact, this truncation is absolutely not trivial as there is no general prescription how to do it. In fact, one needs to proceed taking into account the symmetries of the theory as dictated by WARD -TAKAHASHI and TAYLOR -S LAVNOV identities [69, 70]. So far, we presented the DSE for the case of zero temperature. That is, QCD in vacuum. However, we will see in section Section 2.2 how temperature dependent concepts at finite temperature (T 6= 0) are introduced. The mathematical arena we are basing our finite temperature considerations on the so-called M ATSUBARA formulation of QCD. Within this formalism we show how the partition function is defined, and also how the periodicity of the field influence the energy spectrum. For the sake of completeness and after presenting the formalism, we derive, and comment out the example of the DS equation of the quarks field at T > 0. The functional renormalization group (FRG) methods: The C ALLEN’s equations A physical quantity must be a priori independent of the renormalization point. As said before, the renormalized G REEN functions depend on the renormalization point µ. This subtraction point µ is not unique, and then different renormalized G REEN functions are obtained for different values of µ. This change of the G REEN function comes from the dependency of the the renormalized parameters gr , mr and ξr (via the corresponding Z-factors) on µ. However, two renormalized G REEN functions computed on two different subtraction points µ and µ ′ are related via finite multiplicative factor z

(2.56) Gr pi ; gr (µ ′ ), mr (µ ′ ), ξr (µ ′ ), µ ′ = z(µ ′ , µ) · Gr pi ; gr (µ), mr (µ), ξr (µ), µ , where z is depending on µ and µ ′ . This renormalization factors form an abelian group called the renormalization group (RG). Therefore, within this context, physical observables need to be renormalization group invariants. This is in general not the case for the G REEN functions whose 24 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T depend on the renormalization point. In fact, the key observation to derive the renormalization group equations is to stress that the unrenormalized G REEN function (depending on the bare parameters go , mo and ξo ) does not depend on µ. This last property phrased in equation looks like 0=µ d G(pi ; go , ξo , mo , Λ). dµ (2.57) Now using this last equation together with Gr (p1 , . . . , pn ; gr , mr , ξr ) = ZG · Greg (p1 , . . . , pn ; Λ, go , ξo , mo ), and using the chain differentiation rule7 one obtains   ∂ ∂ ∂ ∂ + βξi Gr = 0. − γ + mr γm µ +β ∂µ ∂ gr ∂ ξr ∂ mr Therefore, one defines the RG functions (see [90]) as   ∂ gr mr β gr , , ξr := µ µ ∂ µ go ,mo ,ξo ,Λ fixed   µ ∂ mr mr γm gr , , ξr := µ mr ∂ µ go ,mo ,ξo ,Λ fixed   ∂ ln ZG mr γ gr , , ξr := µ µ ∂ µ go ,mo ,ξo ,Λ fixed   mr ∂ ξr βξ gr , , ξr := µ µ ∂ µ go ,mo ,ξo ,Λ fixed (2.58) (2.59) (2.60a) (2.60b) (2.60c) (2.60d) The meaning of these equations is obviously to translate how the renormalized G REEN function varies under a change of the renormalization point µ. In particular, one important feature of the L ANDAU gauge is that it is a renormalized group fixed point. It has been already shown that the β -function is gauge independent [91], i. e. β (gr , ξr ) = β (gr ). (2.61) Moreover, one supposes that mµ ≫ mr 8 . The evolution on µ of the RG functions β (gr ), γm (gr ) and γ(gr ) (see Eq. (2.60a), Eq. (2.60b) and Eq. (2.60c)) is determined by the solution of the 7 One must keep in mind that the renormalized G REEN function Gr depends on µ explicitly, but also implicitly because of the dependency of renormalized parameters gr , mr and ξr on µ . 8 As it holds in a mass-independent renormalization scheme for example. Sec. 2.1. 25 Reviewing QCD differential equations (Eq. (2.60a) and Eq. (2.60b)) as  Z g (µ ′ ) r m(µ ′ ) γm (h) dh , = exp m(µ) β (h) gr (µ) Z g (µ ′ )  r µ′ dh = exp . µ gr (µ) β (h) (2.62) (2.63) In general, the G REEN function also transform under a RG transformation with a finite factor [62] ′ z(µ , µ) = exp Z gr (µ ′ ) gr (µ) γ(h) dh β (h)  . (2.64) The RG functions are quite unknown, and approximations to these functions are relying on perturbative expansions in gr . Therefore, these approximations are only valid in the perturbative regime of QCD. To give an example, let us focus on the case of the β -function. As we know, the β -function is depending on gr which is related to the bare coupling go as 3/2 gr = Z3 Z1−1 go . The renormalization constants, Z3 and Z1 , are defined at a subtraction point p = µ in terms of the bare (transverse) gluon propagator and the bare three-point vertex, respectively. Now, relying on the two-loop order expansion of the Z-factors, and plugging it into Eq. (2.60a) for gr (µ) one ends with an expansion for the β -function [92] β [gr (µ)] = −β0 g3r (µ) g5r (µ) β − + O(g7r (µ)), 1 16π 2 128π 4 (2.65) where 2 β0 = 11 − N f , 3 19 β1 = 51 − N f , 3 (2.66a) (2.66b) where N f is the number of quark flavors as usual. Obviously, this expansion is valid only for small gr . Moreover, this procedure to get the form β (gr ) is depending on which scheme the Z-factors are defined in, and also on the definition of running coupling. Still, the first two terms β0 and β1 are shown to be the only renormalization-scheme independent terms. Therefore, we understand from here that most predictions of the RG equations go to the ultraviolet asymptotic momenta regime. As we are interested in predictions related to the gluon an ghost propagators let us focus on the RG equations expectations for these particular G REEN functions. Under the ass- 26 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T umption of large value of the scale λ and setting µ ′ = λ µ one get from Eq. (2.56) and Eq. (2.64) Gr (λ pi , gr , λ µ) = λ D µ D f (pi · p j /µ 2 ) = λ D Gr (pi , gr , µ), where D is the dimension of G and f is dimensionless. Since G is L ORENTZ invariant f must be a function of the scalar product pi · p j . To lowest order in PT the anomalous dimension γ(gr ) is given by the expansion γ(gr ) = c0 g2r + O(rg4r ) [62] where c0 is the zeroth-order coefficient. β0 Using c0 and the corresponding coefficient b0 = 16π 2 from Eq. (2.66b) of the β -function, we obtain from the definitionz(λ ) = z(λ µ, µ) Eq. (2.64) to lowest order in PT [62] z(λ µ, µ) ≃ exp Z ∝ [ln λ ] gr (λ µ) gr (µ) −δ c0 dh b0 h   gr(λ µ) = gr (µ) λ →∞ c0 /b0   1 + O(g2r ) (2.67) (2.68) where δ := c0 /(2b0 ). For the gluon and ghost propagators in the quenched case (N f = 0) these exponents are δD = 13/22 and δG = 9/44, respectively. Therefore, the corresponding dressing functions, Z and J, behave in the far ultraviolet momentum region like  p2 Z(p ) ∼ ln 2 Λ 2 −δD and −δG  p2 . J(p ) ∼ ln 2 Λ 2 Besides the ultraviolet momenta region, FRGE methods are also predicting an infrared behavior for the propagators in the infrared region. Moreover, as seen in Section 2.1.4 DSE also bring useful informations on the propagators in the infrared region. In fact, a very sensible issue is whether the solutions from the DSE and FRGE are compatible in the infrared. We were already observing that solving the DSE imposes somehow to perform truncations. Moreover, it has been shown that the solutions provided by the FRGE and the ones of DSE agree exactly in the infrared as shown in [83, 93] and approximatively at the mid-momentum range [89]. So far, our analysis was always connected to the zero temperature case, and we are not going to describe the FRGE at finite T . Still, we should say that one of the most common use of them (at T 6= 0) is the study of coupling constant as a function of the temperature, see [94, 95, 96, 97, 98, 99] and references therein. In the next section, we move on to the so-called M ATSUBARA formalism of QCD. This is our key tool to analyze temperature effects on the propagators. We will also review briefly how the DSE already developed in the vacuum (T = 0) might be translated into the finite temperature framework. To do this, propagators will be given proper modifications at finite temperature, where the Euclidean space-time symmetry breaks down due to the heat-bath. 27 Sec. 2.2. QCD at finite T 2.2 QCD at finite T 2.2.1 Path integrals and the M ATSUBARA formalism The path integral in the case of (imaginary time) finite temperature is commonly connected to statistical quantum theory. Indeed, for a classical statistical system in a heath bath with a temperature T the partition function looks like Z(T ) = Tr[eĤ/(kB T ) ] = Tr[eβ Ĥ ], (2.69) where Ĥ is the Hamiltonian operator, and β symbolizes the inverse temperature β = 1/(kB T ), with the B OLTZMANN constant kB 9 . We follow the usual notation, and set always in the subsequent developments kB = 1. Therefore, the temperature is given by β = 1/T. (2.70) The trace in Eq. (2.69) restrict the fields to be periodic (bosons) or anti-periodic (fermions) in time. That is, Ψ(~x,t + β ) = ±Ψ(~x,t), (2.71) where the sign is depending whether the field Ψ is a boson (+) or a fermion (-). In addition, in Eq. (2.13) one assumes that the time extent becomes infinite in the partition function10 . This is not anymore the case at finite temperature where one set the time interval to be [0, β ]. Therefore, the Euclidean space is compactified to R3 × [0, β ]. Furthermore, the partition function Eq. (2.69) might be transformed into a path integral over (anti) periodic configurations yielding [100] Z(T ) = Z [DΨ]e−SE [Ψ] , (2.72) where SE [Ψ] = Z β 0 dt Z R3 d 3 xL (Ψ(t,~x), ∂µ Ψ(t,~x)). (2.73) Here, the measure [DΨ] and the action SE [Ψ] are discretized on the lattice. Within this discretization the extent of the time direction is limited up to β . The time extent in units of the lattice spacing is a · Nτ whereas the spatial extent is a · Nσ [100]. Hence, β = a · NT = 9 The 1 . T (2.74) inverse temperature and the inverse gauge coupling both denoted β lead sometimes to a confusion. During this section, and unless stated otherwise, we refer always to β as the inverse temperature 10 This is equivalent to setting T = 0. 28 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T According to this last equation, the case of T −→ 0 corresponds to the limit β −→ ∞. Therefore, the case we were considering in Section 2.1.2 corresponds to zero temperature case where the spatial and time extents are equal, i.e. a symmetric lattice. We note that along our lattice investigations and in order to reduce finite volume effects commonly present on a finite lattice, one takes the aspect ratio Nσ /Nτ to be large. Subsequently, one performs the continuum limit of the theory a −→ 0 at fixed temperature and spatial physical volume, i. e. holding a · Nσ and a · Nτ fixed. Other aspects to mention are the physical implications of the (anti-)periodicity of the fields within the so-called M ATSUBARA formalism defined in Eq. (2.71). In fact, if one defines a F OURIER transformation of a periodic function in imaginary time direction, i. e. satisfying 1 f (τ) = f (τ + ), by T n=+∞ f (τ) = T ∑ e−iωn f (iωn ), (2.75) n=−∞ one finds that only an energy discrete spectrum is allowed with respect to the time direction. That is, only energies values of the form ωn = n · 2π = 2πn · T. β (2.76) The integer multiples ωn are called commonly the M ATSUBARA frequencies. However, there are different M ATSUBARA energy levels for bosons and fermions. For bosons (periodic fields) the M ATSUBARA energies are of the form ωboson = 2πn · T , with an integer n related to the lattice structure −Nτ /2 + 1 ≤ n ≤ Nτ /2. The situation is rather different for fermions ψ (and also ψ) which obey anti-periodicity condition (in time direction) ψ(τ) = −ψ(τ + 1/T ), ψ(τ) = −ψ(τ + 1/T ). (2.77) (2.78) Here the energy levels are of the form ωfermion = (2n + 1) · πT . Therefore, the smallest energy corresponds to πT . As usual, within the M ATSUBARA formalism the fourth component of the Euclidean momenta p4 is identified to multiples of the M ATSUBARA frequencies. In our finite temperature investigations we analyzed exclusively data with zero M ATSUBARA frequency. Hence, only time components momenta with zero values were taken into account. Some conventions in the M ATSUBARA formalism read p4 −→ −ωn , /p = −ωn γ4 + γ · p. (2.79) (2.80) It is also worthwhile to note that within the M ATSUBARA formalism one needs to replace the Sec. 2.3. 29 Order parameters in QCD at finite T integral over the fourth component of the Euclidean four vector with sums over M ATSUBARA (discrete)frequencies as follows Z d4 p f (−ip4 , p) → −T ∑ (2π)4 np Z d3 p f (iωn p , p). (2π)3 (2.81) Based on these rules one can easily convert the zero temperature DSE to the case of finite temperature. In order to illustrate this point let us take the example of the DSE for quark fields 11 in QCD already derived in Eq. (2.47). In fact, at finite temperature the DSE [101] for the quark field looks like S−1 (iωn , p, µ) = Z2 (µ, Λ)S0 (iωn , p, Λ) + Σ(iωn , p, µ), (2.82) with the self-energy defined in Eq. (2.50). To get this formula (valid at T > 0) one apply the following substitution pµ = (p4 , p) = (−ωn , p). Moreover, we have also considered S(Q, µ) = S(iωn , p, µ) and correspondingly Γν (K, L, µ) and Dµν (K, µ). Here, Dµν and Γν are the gluon propagator and the qq̄g-vertex function at finite temperature. Note that we do not present the tensorial structure of the vertex function as being of no interest for us in this thesis, see [101]. However, the gluon propagator is one of the important targets of our study. Therefore, we refer for a proper definition of this quantity (at finite T ) to Section 2.5. In the following section we focus on a study of the order parameters of QCD at finite T . We start first with the P OLYAKOV loop studying its role as an order parameter in pure gauge theory. Next, we move to the chiral limit of QCD, and introduce the chiral condensate as an order parameter for full QCD. A theoretical understanding of these order parameters allows one to have a comprehensive picture of the QCD phase diagram discussed later on. 2.3 Order parameters in QCD at finite T 2.3.1 The P OLYAKOV loop The pure gauge sector of QCD is an interesting area where one supposes the quarks fields to be infinitely heavy. This special situation also called the quenched approximation of QCD is less computing power demanding in order to be investigated within lattice QCD in comparison to the full QCD case. Moreover, confinement might also be present even in this quenched approximation. Therefore, a proper understanding of the deconfinement here should in principle bring up valuable informations and a good start to move to the full case in presence of fermions. Another key property within this context is the existence of a (de)confinement phase transition. This last property is already expected in the continuum, and was the focus of lattice investigations as well. To be able to investigate this transition one needs to rely on order parameters guided by the symmetries of the action. In particular, the ideal order parameter of quenched QCD 11 This particular DSE is also called sometimes the gap equation of QCD in the scientific literature 30 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T is called theP OLYAKOV loop [18]. In order to have an understanding of this quantity let us start with a system without fermions, i. e. pure Yang-Mills theory. First, the canonical Euclidean partition function as seen before reads Z(T ) = Tr[eβ Ĥ ], (2.83) where Ĥ is the Hamiltonian operator, and the trace applied on gauge invariant physical states. At this step it is interesting to observe that the exponential [exp(−β Ĥ)] is similar to the evolution operator [exp(−iĤt)] where β plays the role of the Euclidean time interval it. Now, switching to the path integral formulation one can also write Z= Z [DAµ ] exp(−SEucl [A]), (2.84) with the gauge fields Aµ and the Euclidean action SEucl [A] 1 SEucl [A] = − 2 Z β 0 dτ Z d 3 x Tr(Fµν Fµν ). (2.85) Here and in the following we suppose that measure DAµ to be mathematically defined. Moreover, the fields obey periodic boundary conditions in Euclidean time direction Aµ (~x, τ + β ) = Aµ (~x, τ). (2.86) The action SEucl [A] already defined in Eq. (2.85) is invariant under the transformations Agµ = g(Aµ − ig† ∂µ g)g† , (2.87) where g ∈ SU(Nc ) with Nc is the number of colors. In order to fulfill the boundary conditions in Eq. (2.86) the gauge transformations must be of the form g(~x, τ + β ) = g(~x, τ), (2.88) or more importantly g(~x, τ + β ) = hg(~x, τ), (2.89) with h ∈ SU(Nc ). This last non trivial transformation is called sometimes the twisted gauge transformation and is a direct consequence of the structure of the action. Moreover, if h are global transformations and commuting with gauge fields this would still respect the boundary Sec. 2.3. Order parameters in QCD at finite T 31 conditions as shown Agµ (~x, τ + β ) = g(~x, τ + β )[Aµ (~x, τ + β ) − ig(~x, τ + β )† ∂µ g(~x, τ + β )]g(~x, τ + β )† = hg(~x, τ)[Aµ (~x, τ) − ig(~x, τ)† h† ∂µ (hg(~x, τ))]g(~x, τ)† h† = hAgµ (~x, τ)h† = hh† Agµ (~x, τ) = Agµ (~x, τ). (2.90) There, h is an element of the center group of SU(Nc ), i. e. the set of all elements of SU(Nc ) which commute with each others. Therefore, this is why this symmetry is called sometimes the center group symmetry. Such elements of the center group are proportional to the unit SU(Nc ) matrix I and might be parametrized as h = z · I, (2.91)   2πin with z = exp ∈ ZNc . Therefore, from the mathematically point of view the center of Nc SU(Nc ) is isomorphic to ZNc . The system becomes a bit more involved when treating fermions. Adding dynamical fermions to our previousYang-Mills theory brings extra difficulties to keep the center group as a symmetry of the action. In fact, in order to see that in more details let us consider the quark field 12 and the anti-periodic boundary conditions in Euclidean time direction as ψ(~x, τ + β ) = −ψ(~x, τ), which transform under the gauge transformations g(x) as ψ(x) → ψ(x) → g(x)ψ(x). Therefore, if one wants to keep the gauge transformed quark under Eq. (2.89) always compatible with the anti-periodicity condition, that is ψ(~x, τ + β )g = g(~x, τ + β )ψ(~x, τ + β ) = −zg(~x, τ)ψ(~x, τ) = −zψ(~x, τ)g , (2.92) the only solution would be to take z = I, and thus the center group symmetry is destroyed in the presence of the fermions. In other words, one says that the center symmetry is explicitly broken when including dynamical fermions. Back to YANG -M ILLS theory one defines the P OLYAKOV 12 Here we consider the fermions fields in the fundamental representation of SU(Nc ) 32 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T loop l(~x) through the color operator trace in the fundamental representation given by l(~x) = 1 Trc (L(~x)), Nc (2.93) where L(~x) is the P OLYAKOV loop operator L(~x) = P exp(i Z β 0 dτA4 (x)), (2.94) where P is the path ordering of the exponential, and β stands here for the inverse temperature β = 1/T . An interesting relation to the free energy Fqq might be found computing the thermal expectation value of a product of two P OLYAKOV loop operators, namely hl(~x)l(~y)† iβ = e−β Fqq (~x,~y,T ) . (2.95) We point out that in the limit of large distances between two static color sources q and q, i. e. |x − y| → ∞, the correlation goes to zero. Therefore, at the temperature T one finds that the free energy F(q) of a single static quark F∞ = lim Fqq (r = |~x −~y|, T ) = −T log |hli|2 6= 0, r→∞ (2.96) where the hli = e−β Fq . We interpret physically the equations above as follows: in the confined phase the free energy of a single quark diverges, whereas in the deconfined phase the free energy remains finite. Therefore, the expectation values of the P OLYAKOV loop takes zero values in the confined phase and non-zero otherwise ( 0 ⇒ confinement, (2.97) hli = 6= 0 ⇒ deconfinement. This last property is of not course fulfilled when dealing with fermions having well defined masses. In this case confinement can not be really proven according to this order parameter criteria. This change of situation is mostly due to screening effects tending to produce particleantiparticle quark pairs, and therefore ending with a finite free energy. Thus, the picture of confinement we present here connected to infinitely heavy masses separated static quarks is not valuable when dealing with fermions with finite masses. Still, the P OLYAKOV loop may act as an approximate order parameter in presence of heavy massive quarks. After this brief review of the P OLYAKOV loop one understands that one can probe a system by a test quark with an infinitely heavy static charge. This non trivial behavior of the P OLYAKOV loop is indeed a consequence of their non trivial transformation under the center group Sec. 2.3. Order parameters in QCD at finite T 33 symmetry. That is, Z β 1 dτA4 (x)g )] Trc [P exp(i Nc 0 Z β 1 = dτA4 (~x, τ))g(~x, τ)† ] Trc [g(~x, τ + β )P exp(i Nc 0 Z β 1 dτA4 (~x, τ))g(~x, τ)† ] Trc [zg(~x, τ)P exp(i = Nc 0 = zl(~x). l(~x)g = (2.98) Therefore, the P OLYAKOV loop transforms non trivially under the center group ZNc as soon as the P OLYAKOV loop picks a non-zero expectation values in the deconfined phase. In fact, the deconfined phase correspond to higher temperatures, and therefore the center symmetry is spontaneously broken at this regime. Concretely, what we compute in our lattice investigations are the discretized version of the continuum definition of the P OLYAKOV loop we have see so far, that is L= 1 Nσ3 Nτ −1 ∑ Tr ∏ U4 (~n, n4 ), n (2.99) N4 =0 where Nτ , Nσ are the temporal and spatial lattice extents respectively, and U4 (~n, n4 ) are the link variables at each lattice point (~n, n4 ) pointing in the time direction. Therefore, as L is a complex number we take only its real part to study it as a function of the temperature T . We will introduce in more details these lattice definitions in the next Chap. 3. 2.3.2 The chiral condensate In order to start to discuss the chiral condensate it would be worthwhile to start with a proper understanding of the so-called chiral symmetry. In simple words, this latter symmetry happens in the special case where the quarks masses are supposed equal to zero. This mass limit is called in the literature the chiral limit. What makes this limit physically interesting is the fact that the quarks (u,d and s) masses are small compared to typical hadrons masses. Moreover, the other three quarks, namely the top, charm and bottom quarks are very heavy compared to the first three. Therefore, in the regime of low energies these latter can be considered to be infinitely heavy and their dynamic to be neglected. Hence, the study of the chiral limit corresponding to vanishing quark masses is worth to do, and present a nice first approximation of the real world in this regime. As mentioned before in the chiral limit there is an emerging symmetry of the QCD action called the chiral symmetry. To make things clearer let us start our analysis defining the projection 34 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T operators 1 PR,L = (1 ± γ5 ), 2 (2.100) with γ5 = iγ0 γ1 γ2 γ3 in the M INKOWSKI space, see Appendix 8. Using this last operator one defines left- and right-handed quarks as (2.101) ψR,L = PR,L ψ. The Lagrangian in the case of vanishing quark masses is invariant under the following transformations a ψR,L −→ eiθR,L λ a /2 ψR,L . (2.102) This symmetry is represented by the transformation group SU(NF )R ⊗ SU(NF )L . Moreover, according to N OETHER theorem the corresponding conserved currents look like µ,a JR,L (x) = ψ R,L γ µ λa ψR,L . 2 (2.103) These right and left currents might be combined into a vector and an axial current vectors as follows a a Vµa = JR,µ + JL,µ , (2.104) Aaµ (2.105) = a JR,µ a − JL,µ . The left- and right-handed quarks are mixed into the Lagrangian thanks to the mass term. Therefore, this mass term breaks explicitly the chiral symmetry and the aforementioned currents follow different conservation laws. In fact, the vector current is conserved while the axial one is not. This latter non-conservation property is translated into equations as ∂µ Aµ = iψ{M, λa }γ5 6= 0. 2 (2.106) Furthermore, the chiral symmetry is also spontaneously broken. This is due to the non-invariance of the QCD vacuum under chiral transformations while the Lagrangian remains chiral symmetric. Indeed, if one considers the ground state to be symmetric under the chiral symmetry, this would mean that the vector and axial charges must annihilates the vacuum, i. e. QVa |0i = QAa |0i = 0. (2.107) Therefore, in this case there should be in principle parity partners with equal masses for the vector and axial-vector mesons. However, this is not what happens in nature, and a mass gap for Sec. 2.3. Order parameters in QCD at finite T 35 example between the ρ (mρ = 0.77 GeV) and a1 (ma1 = 1.23 GeV) mesons is well established. These experimental observations justify the fact that the vacuum is annihilated by the vector charge whereas this is not the case for the axial charge, i. e. QVa |0i = 0, QAa |0i (2.108) 6= 0. (2.109) Thereby, the symmetry group of QCD due to the spontaneous breaking of the chiral symmetry boils down as follows SU(N f )L ⊗ SU(N f )R −→ SU(N f )R+L = SU(N f )V . (2.110) The spontaneous breakdown of the chiral symmetry generates according to the G OLDSTONE theorem massless excitations called G OLDSTONE bosons. The quantum states corresponding to these bosons are of the form |φa i = QAa |0i, and since [H, QAa ] = 0 the states |φa i must be energetically degenerate with the vacuum |0i. The number of this states is eight in the case of SU(3) since QAa is an axial charge. Due to the mass term as said before the chiral symmetry is also explicitly broken. Therefore, it results light G OLDSTONE bosons rather than being massless. The order parameter which account for the spontaneous breaking of the chiral symmetry is the so-called the chiral condensate, and is defined as hψψi = h0|ψψ|0i = −i Tr lim SF (x, y), y→x (2.111) where SF (x, y) = −ih0|T ψψ|0i is the F EYNMAN propagator and T is the time ordering operator. After discussing different symmetries within QCD in the previous sections let us now summarize the status of the symmetries present in the real world. Hence, we start with the symmetry group SU(Nc ) ⊗U(N f )V ⊗U(N f )A , (2.112) with SU(Nc ) is the local symmetry gauge group where the two other ones are global symmetries. As already known from group theory U(N f ) might be decomposed into U(N f ) = U(1)⊗SU(N f ) getting then SU(Nc )⊗U(N f )V ⊗U(N f )A (2.113) ⇓ SU(Nc ) ⊗U(1)V ⊗SU(N f )V ⊗U(1)A ⊗ SU(N f )A . Note that U(1)V corresponds to the conservation of the baryon number and is not broken whereas U(1)A is broken and called the U(1)A -anomaly. This particular anomaly is connected to the mass splitting between the η and η ′ . After such analysis, the gauge group shown in Eq. (2.114) boils down to SU(Nc ) ⊗ SU(N f )V where the last part is connected to the chiral symmetry for 36 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T vanishing quark masses. In general, the chiral symmetry is explicitly broken for non-vanishing quark masses, and might be also spontaneously broken. Thereby, taking into account the chiral spontaneous breakdown and the anomaly of U(1)A at the same time one ends with SU(Nc )⊗U(N f )V ⊗U(N f )A (2.114) ⇓ SU(Nc )⊗U(1)V ⊗ SU(N f )V . Moreover, as said before for different non-vanishing quark masses the chiral symmetry is also explicitly broken. This amounts to SU(Nc )⊗U(1)V ⊗ SU(N f )V (2.115) ⇓ SU(Nc ) ⊗U(1)V . Concluding, one understands that at the end there is only the gauge symmetry and the U(1)V which remain. This latter symmetry corresponds according to N OETHER theorem to the conservation of the baryon number. Thus, the baryonic symmetry should be a part of our real world or any realistic QCD model as well as the gauge symmetry SU(Nc ). 2.4 Nature of the phase transition in QCD One important question which comes to one’s mind when dealing with phase transitions of QCD is the nature of these latter. Therefore, it is quite natural to ask which kind of phase transition one expects in QCD. Throughout the present section we review different aspect of the chiral and deconfinement phase transitions. It is quite interesting to note that these latter transitions show dependencies with the quark masses and the flavor numbers. Thus, this leads quite naturally to a rich and a complex picture of the whole image we had so far on chiral symmetry, confinement and their interplay in QCD. Here, we focus on the main transition features summarized in Fig. 2.5. This latter plot (called sometimes in the literature the C OLUMBIA plot) shows the conjectured phase transitions at vanishing potential. Moreover, this plot shows the chiral and deconfinement phase transitions as a function of the flavor and quark masses as discussed in [101]. First, let us concentrate on the chiral symmetry restoration region within the C OLUMBIA plot. Thanks to the universality class properties a first model for chiral symmetry restoration was adressed in [103]. The universality principles guided also to construct models which mimic QCD in many apects. However, such models need to reproduce the restoration of the symmetry SU(NF )R+L −→ SU(NF )R ⊗ SU(NF )L , where the temperature effects are neglected. If one takes into account all effects into considerations, then a first order transition is expected for three degenerate chiral flavors. Moreover, a crossover and a second order transitions are expected for 38 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T even the existence of this point is still unresolved [106]. Arrived at this point, let us discuss the deconfinement according to the Columbia picture. In the quenched approximation of QCD, namely, in the absence of quarks (or in the limit of infinitely heavy static quarks) occurs a first order phase transition in SU(3) gauge theory. This region is located in the upper right part of the Fig. 2.5. This first order transition is well established by lattice gauge theory investigations [107]. Furthermore and always seen from Fig. 2.5, the region of first order deconfinement phase transition ends at a second order deconfinement phase transition line lying in the Z(2) universality class. The corresponding model is the so-called Potts model with a complex scalar field and invariance under 2π/3 rotations in the complex plane. In this thesis, we investigate the case of quenched SU(3) QCD and the phase order transition thanks to the P OLYAKOV loop as an order parameter corresponding to the study of the upper right corner region of the Columbia plot. In a second project, we study also the regime of intermediate quarks masses NF = 2 (quark up and down) using three values of pion masses corresponding to the upper line of the Fig. 2.5. As one might expect from this plot, at moderate quark masses the transitions are crossovers rather than strict phase transitions. Moreover, it is also expected that the physical quark masses lie into the crossover region. Remark, that the chiral condensate and the P OLYAKOV loop in their corresponding limits show typically a rapid change around the critical transition temperatures. Thanks to these order parameters responses it is already stated that the chiral transition might happen at Tχ = 147 − 164 MeV whereas deconfinement occurs at the Tdeconf ∼ 165 MeV , see [108] and references therein. Still, these values and the nature of the phase transition are a hot topic under discussion within the scientific community. 2.5 The gluon and ghost propagators at T > 0 Even if the gluon and ghost propagators are a quite basic quantities in QFT, they remain very useful for plenty of reasons. Apart from that, a QFT is generally soluble only knowing all its G REEN functions. A special case of these latter are the propagators which play also an interesting physical role connected to confinement for example in QCD. Moreover, the gluon and ghost propagators represent the building blocks of the DS equations for example as discussed before. Recently, powerful non-perturbative approach has been developed based on DSE [20, 21, 22, 23] in parallel to the functional RGE investigations [24, 25]. The main focus was first to find a field theoretical, model-independent description of quark and gluon confinement in terms of the infrared behavior of gauge-variant G REEN’s functions, in particular of the L ANDAU or C OULOMB gauge gluon and ghost propagators. The physical prediction of such propagators should confirm or reject confinement scenarios as proposed by G RIBOV and Z WANZIGER [26, 27, 28] and K UGO and O JIMA [29, 30]. The L ANDAU gauge at zero temperature was the focus of the DSE and FRGE (see, e. g. [89] and citations therein) and also the subject of intensive lattice investigations (see [31, 32, 33, 34, 35, 36, 37, 32] and references therein). Sec. 2.5. The gluon and ghost propagators at T > 0 39 At finite temperature the gluon and ghost propagators are less investigated. In fact some papers were devoted the SU(2) pure gauge gluon propagators in L ANDAU gauge (see, e.g. papers [38, 39, 40, 41, 42, 43, 44, 45]). The finite temperature studies started in 1995 with H EL LER and al. [38] who had for example to determine the electric and the magnetic gluon mass at high temperature. There, the phenomenon of the ’dimensional reduction’ appearing at high temperature for the SU(2) gauge group is also investigated. This actually happens at high temperatures reducing a four-dimensional theory to a three-dimensional interacting theory. Latter on, authors in [40] compared the 4D and 3D gluon propagators and found a matching of the two results down to 2 Tc . The infrared limit of the gluon and the ghost propagators show also interesting results thanks to huge volume lattice investigations [45]. There one observes that the longitudinal gluon propagator DL (p) is enhanced, with an apparent plateau value in the infrared, while the transverse propagator DT (p) gets progressively more infrared-suppressed. This result is also supported by our work [49] going up to the physical volume of (2.7 f m)3 for the SU(3) gauge group case. For the case of SU(3) pure gauge theory the gluon propagators are less studied, see [46, 47, 48, 43, 49]. This also applies for the studies of SU(3) gluon propagators in the presence of dynamical fermions [50, 51]. In fact the reference [50] takes into account the number of flavors NF = 2 as we do in our study except that we use a twisted mass fermionic action as our main discretization as described later on. Most of the papers support a temperature independent behavior for the ghost propagator as in [53, 54, 55, 56] and references therein. Still, we were able [49] to observe small fluctuation in the region of small momenta and at higher temperatures. In the case of full QCD at finite temperature for the gauge group SU(3) our results are in qualitative agreements with [42, 43]. After presenting the status of art of the gluon and ghost propagator let us now turn to the theoretical definition of the gluon and ghost propagators in QCD at finite T . The gluon propagator generic form in real space is the time ordered expectation value of two gluon fields hAaµ (x)Aνb (0)i ≡ h0|T (Aaµ (x)Aνb (0))|0i, (2.116) where T denotes the time ordering. The F OURIER transform of this latter gives 2 Dab µν (p ) = −i Z d 4 x · eipx hAaµ (x)Abν (0)i. (2.117) On the other side, the ghost propagator form is hca (x)cb (0)i ≡ h0|T (ca (x)cb (0))|0i, (2.118) and the Fourier transform reads Gab (p2 ) = −i Z d 4 x · eipx hca (x)cb (0)i. (2.119) 40 CHAPTER 2. INTRODUCTION TO QCD AT FINITE T At finite temperature (T 6= 0), the Euclidean symmetry is broken and the gluon propagator splits into two structures [109], namely a transverse DT and a longitudinal DL components to the time direction as follows 2 2 L ab T Dab p ) + Pµν DL (p24 ,~p 2 )), µν (p) = δ (Pµν DT (p4 ,~ (2.120) where the fourth momentum component p4 is the so called M ATSUBARA frequency. The projectors within the Landau gauge are defined as   pµ pν T Pµν = (1 − δµ4 )(1 − δν4 ) δµν − , (2.121) ~p 2   pµ pν L T − Pµν = δµν − 2 . Pµν p In the next chapter, we introduce our lattice framework. We also present our lattice actions used in this work. Next, the continuum theory is commented and how to reach such a limit from our lattice simulations. Last but not least, we discuss the G RIBOV problem and ways to deal with it. CHAPTER 3 QCD at T > 0 on the lattice W introduce basic definitions related to the lattice regularization of QCD. We first introduce gauge fields and gauge transformations on the lattice. Then, we specify our two lattice actions of interest, namely the standard plaquette W ILSON action and also the one used for full QCD consisting of a sum of the S YMANZIK and the twisted mass actions. Next, we discuss the concept of continuum limit and the prescription adopted to reach it on the lattice. At the and, the G RIBOV problem is posed together with the gauge fixing techniques we used. There, we clarify the problem of G RIBOV copies and propose our procedure to deal with it. E 3.1 General introduction Lattice QCD (LQCD) provides the way par excellence to handle problems in the non perturbative regime of QCD. This is basically done by introducing a new scale in the theory, namely a cutoff a−1 , where a is the lattice spacing. Indeed, LQCD is basically a discretization of the Euclidean space-time into a lattice of points separated with distance a. Therefore, this cutoff regularizes QCD which becomes finite. We refer the reader to standard textbooks for more details on the subject [14, 15] and [16, 17]. In fact, perturbation theory (PT) ceases at some regime of energy/temperature to work and purely non-perturbative phenomena as confinement or chiral symmetry restoration are not taken into account. Still, PT was successful to test QCD at high energy regime, i. e. small distances [1, 2, 3], where the asymptotic freedom might happen. Actually, this latter property motivated to consider the quark-gluon plasma as a weakly interacted system [5, 4]. This suggests that the effective QCD potential used in thermodynamical models should be small as a function of high temperature/density. Still, infrared divergences, known as the L INDE problem, might also prohibit the computation of thermodynamic quantities [8]. Therefore, naive perturbation 41 42 CHAPTER 3. QCD AT T > 0 ON THE LATTICE theory might be replaced by improved perturbation techniques, as the dimensional reduction [9, 110, 111] and Hard Thermal Loop (HTL) resummation [9, 10, 11, 12] for example. However, these improved methods seem to work only for relatively higher temperature (& 3Tc ). Therefore, to study the infrared properties of QCD one needs to investigate with the help of LQCD at finite temperature T 1 for example. Moreover, an extra feature of LQCD is the possibility to use Monte-Carlo (MC) simulation techniques to evaluate the observables. These latter are defined with the MC framework in terms of the path integrals are interpreted as sample statistical averages in the equilibrium within LQCD. In the following we introduce the LQCD including basic gauge fields and transformations. Furthermore, we also present the type of lattice actions we used during this present work. This next section is also meant to fix notations and conventions for the upcoming discussions. 3.1.1 Gauge fields and gauge symmetries The general starting point to study QCD at finite temperature is namely to consider the grand canonical partition function for many-particles ensemble at temperature T defined previously as Z = tre−β (H̃−µ Ñ) , (3.1) where, H̃ is H AMILTON operator of the system and β = 1/T 2 . Here, Ñ is the number operator corresponding to some charge as there are the baryon number, the electric charge..., and µ is the chemical potential. Let us assume for this general discussion µ 6= 0 despite our thesis results concern only the zero chemical potential case. the partition function Z Eq. (3.1) might be rewritten in terms of the path integral as Z= Z bc DψDψDAµ exp[− Z β 0 dτ Z d 3 x L ], (3.2) with L is the Lagrangian density and τ is the inverse temperature, i. e. the Euclidean imaginary time. bc denotes the boundary conditions for the fields. That is, periodic boundary conditions for the gauge fields Aµ , while anti-periodic for the quark fields ψ and ψ. In LQCD the Euclidean continuum space-time is discretized to hyper-cubic grid of lattice sites x. The quark fields ψ and ψ in Eq. (3.2) live on the lattice site x ≡ (x0 , x1 , x2 , x3 ). The lattice gauge fields are introduced as links joining two adjacent lattice points x and x+ µ̂, where µ̂ is the unit vector in the xµ direction. We denote these gauge fields as Ux,µ calling them link variables. These fields are introduced in the lattice formulation instead of the gluon field Aµ (x) in order to maintain the explicit gauge invariance [112]. These link variables takes values in a compact L IE group, here SU(3). The link 1 We 2 We recall that we deal in this thesis exclusively with the zero chemical potential case, namely µ = 0. fix the units system to the natural units, i.e. / h = c = kβ = 1 Sec. 3.1. 43 General introduction variables [113] are connected to the continuum gluon field Aµ (x) with 3  Z1  Aµ (x + at µ̂) dt ≃ eiago Aµ (x+µ̂/2) + O(a3 ). Ux,µ ≡ P exp igo 0 (3.3) Here P denotes path ordering of the gluon fields along the integration path, and go is the bare coupling constant as usual. The thermal expectation value of an observable O is defined as hOi = 1 tr[Oe−β (Ĥ−µ N̂) ]. Z (3.4) This latter equations takes the following form in the path integral approach hOi = R 3 Rβ bc DψDψDAµ Oexp[− 0 dτ d xL ] . Rβ R R 3 bc DψDψDAµ exp[− 0 dτ d xL ] R (3.5) LQCD at finite T is a an approach providing essentially prediction for the QGP simulating the partition function. In fact, several observables might be computed within this framework thanks to Eq. (3.5) using Monte-Carlo simulations methods. In the present study we focus first on results using the W ILSON plaquette action described in Section 3.2.1 as the gauge action SG [U] for pure gauge QCD. This amounts to evaluate on the lattice vacuum expectations values [15] of the form hOi = 1 Z Z [DU] e−SG [U] O[U]. (3.6) In fact, Eq. (3.6) provides an evaluation of purely gauge field dependent observables O[U]. However, the situation might be generalized to the case including fermion ψxi and ψxi , where xi are the corresponding lattice sites. Actually, if one considers O = O[ψ, ψ,U] as4 O[ψ, ψ,U] = ψxa11 · · · ψ by11 · · ·Uz1 µ1 · · · . Then, the vacuum expectation values gives 1 hOi = Z Z [DU, Dψ, Dψ] O[ψ, ψ,U] e−SQCD [U,ψ,ψ] . (3.7) In general, the QCD lattice action is a sum of the gauge action part SG [U] and a fermionic part SF [U, ψ, ψ] SQCD = SG [U] + SF [U, ψ, ψ] . 3 This 4 In (3.8) line integral is the lattice version of the parallel transport matrix between adjacent lattice sites, see e.g. [113]. this equation the index a1 · · · b1 denote the color, the spinor and the flavor index. 44 CHAPTER 3. QCD AT T > 0 ON THE LATTICE Moreover, the fermionic part is a bilinear form in the fermion fields ψ and ψ SF [U, ψ, ψ] = ∑ ψ xf Qxy ψyf , (3.9) f ,x,y where, Q is called the fermion matrix acting on fermion fields ψx and ψx . Thereby, the integral in Eq. (3.7) might be performed instantly over the fermionic variables yielding hOi ≡ ψx1 ψ y1 · · · ψxn ψ yn G [U] Z 1 ,...,zn −1 Qz1 ,y1 [U] · · · Q−1 = [DU] e−SG [U] G [U] ∑ εxz11,...,x zn ,yn [U], n Z z1 ,...,zn (3.10) ,...,zn z1 ,...,zn where the antisymmetric tensor is defined as εxz11,...,x n := 1 (εx1 ,...,xn := −1) for even (odd) permutations of the lattice sites x1 , . . . , xn , and zero else. One can re-express the total action called the effective action Se f f in Eq. (3.10) as Se f f [U] = SG [U] − log det Q[U] (3.11) where det Q[U] is called the fermion determinant. Therefore, on remark that the pure gauge case (or the quenched approximation) corresponds actually to a fermion determinant equal to one ending up with SG [U] as the total effective action. What makes things interesting in LQCD is the possibility to apply MC techniques in order to evaluate vacuum expectation values with an effective action Se f f as a statistical averages. The first step would be to generates gauge field configurations U i according to a M ARKOV chain process. These configurations, let us say of number N, are thermalized (brought to equilibrium). That is, these are realizations of the Boltzmann distribution  1 exp −Se f f [U] . Z Actually, we take into account configurations which dominate the exponential according to importance sampling principle. This latter is in general implemented considering mostly configurations minimizing SF , i. e. in the equilibrium. Therefore, one computes in LQCD statistical averages on such sampled configurations as hOiU := 1 N ∑ O[U (i) ]. N i=1 Only in the limit N → ∞ the ensemble estimator hOiU 5 and the expectation value hOi would match. 5 In general, this estimator is afflicted with errors of the form σ / p (N), where σ is the standard deviation. Moreover, additional systematic errors are present due to finite volume and lattice spacing effects. Sec. 3.2. A closer look to our lattice actions 45 Therefore, one concludes that LQCD is performing integrations by taking a limit of sums on discrete points. On the lattice Nσ and Nτ denote the number of points in the spatial and the temporal (or the inverse temperature) directions. The (spatial) volume V and the temperature T are given by V = (Nσ a)3 , (3.12) and T= 1 . Nτ a (3.13) The finite lattice spacing a imposes an ultraviolet cut-off a−1 while the volume V imposes an infrared cut-off at low momenta. Now, let us talk about the gauge transformation on the lattice gx . As seen before, the effective action in Eq. (3.11) is invariant under the following local gauge transformations set gx ∈ SU(3) acting as ψ(x) → ψ g (x) = g(x)ψ(x) g (3.15) gx Ux,µ g†x+µ̂ . (3.16) ψ(x) → ψ (x) = ψ(x)g (x) Ux,µ → gUx,µ = (3.14) † These (arbitrary) gauge transformations need to satisfy the lattice version of the L ANDAU gauge presented later on in order to be fixed. In fact, in our MC process one first generate configurations and their gauge transformations. Secondly, we fix the gauge for example to the L ANDAU gauge. Thirdly and finally, we measure the propagators as statistical averages using the gauge fixed configurations. 3.2 A closer look to our lattice actions 3.2.1 The gauge W ILSON action At this point we start presenting the lattice gauge action we used for our pure gauge theory investigations. This action is called the W ILSON gauge action SG in honor to its discoverer [114]. It reads as a sum over plaquettes x,µν   1 1 − Re Tr x,µν SG [U] := β ∑ ∑ (3.17) Nc x 1≤µ<ν≤4 46 CHAPTER 3. QCD AT T > 0 ON THE LATTICE where the plaquette represent the shortest, non-trivial closed loop6 on the lattice denoted as † † x,µν := Ux,µ Ux+µ̂,ν Ux+ ν̂,µ Ux,ν . (3.18) The trace of this plaquette variable is a gauge-invariant object. This lattice discretization is actually the first formulation of lattice gauge theory [114], and it actually approaches the continuum form, namely the YANG -M ILLS action Eq. (3.20) in the naive limit a → 0. Here and in the following β represents the inverse coupling β := 2Nc g2o (3.19) where Nc = 3 for SU(3) and go is the bare coupling constant. 3.2.2 The improved S YMANZIK gauge action In this section we are going to talk about another type of pure gauge lattice action, namely the S YMANZIK action. Thanks to the tmfT collaboration this kind of action has been used as well as the twisted mass fermionic action to generate configurations for our full QCD investigations. In brief, the S YMANZIK type action belong to a class of actions called improved actions. As noticed before, expanding the gauge W ILSON action Eq. (3.17) in the limit a −→ 0 (small lattice spacings) and using Uµ ∼ e−igAµ (x) the continuum Yang-Mills action is recovered (up to a2 order) SY M = − 1 4 Z β 0 dτ Z µν b Fb + O(a2 ). d 3 xFµν (3.20) As we see from this latter equation the correction to the continuum action is of order O(a2 ). However, a large set of actions might also have the same continuum form, differencing only by irrelevant terms proportional to higher powers of the lattice spacing a. This fact opens the way to define ’improved’ actions, and this is the sense of improvement in this context. The idea behind constructing such actions is to add further counter-terms to SG Eq. (3.17) in order to eliminate order O(a2 ). One might do this operation repeatedly, and therefore eliminate all desired corrections. This brings us to the concept of the so-called perfect actions [115]. Therefore, one understands that the total improved pure gauge action for example is a result of a combination of terms added to the gauge action Eq. (3.17) in order to reduce the discretization effects. Many years ago S YMANZIK in 1985 [116, 117] pointed out that the convergence to the continuum limit might be accelerated thanks to the inclusion of counter-terms of order a (or higher) in the lattice actions and the local operator of interest. For an account of the order a-improvement of lattice QCD we advise the reader to consult [118, 119] and references therein. 6 Into this action the plaquettes are counted with only one spatial orientation. Sec. 3.2. 47 A closer look to our lattice actions The pure gauge S YMANZIK action of interest for us looks like SY M SG = β ∑[c0 x 1 1 1×1 1×2 ) + c1 ∑ (1 − Re Tr Uxµν )], ∑ (1 − 3 Re Tr Uxµν 3 µ<ν (3.21) µ6=ν 1 with c1 = − 12 and the normalization condition c0 = 1 − 8 c1 . Uxµν ∈ SU(3) represent plaquettes 1×1 variables. U represent the usual (1 × 1) W ILSON loop plaquette (denoted x,µν in the previous section) and U 1×2 planar rectangular (1 × 2) W ILSON loops. The second term in this last equation suppresses short-range fluctuations at cut-off scale, and therefore improves the behavior towards the continuum limit. For a comprehensive account on rectangle plaquette actions, and also other generalized improved action we refer to the papers [120, 120, 116, 117]. 3.2.3 The improved twisted mass action We introduce in this section the fermionic part of the improved action used to provide thermalized configuration we relied on to compute the propagators. The action in question is called the improved twisted mass action including dynamical W ILSON twisted fermions. This action represent the fermionic part while the gauge part is given by the S YMANZIK action described in the previous section. More interesting for us, namely the case of two mass-degenerate quarks flavors NF = 2, the twisted mass formulation introduces an isospin structure as a mass term. This twisted mass term provides in fact a useful infrared regulator and can be utilized to obtain O(a) improvement. The twisted mass action for two degenerate fermions consists of the standard W ILSON action augmented by a twisted mass term ψiµ0 γ5 τ 3 ψ. In this context, the hopping parameter κ and the twisted parameter µ0 are connected to the bare quark mass as r 1 1 1 (3.22) ( − )2 + µ02 . m0 = 4 κ κc For a maximal twist we have κ = κc , where κc is the critical hopping parameter. Therefore, the quark mass is defined by µ0 alone. In general, the W ILSON fermions with a twisted mass term evaluated at the maximal twist provide as said before automatic O(a)-improvement, but also the suppression of exceptional configurations. We refer to [121] for a comprehensive report. In general, the twisted mass action looks like
(3.23) SF [U, ψ, ψ] = ∑ χ(x) 1 − κDW [U] + 2iκaµ0 γ5 τ 3 χ(x) . x The W ILSON covariant derivative is given by DW [U]ψ(x) = ∑((r − γµ )Uµ (x)ψ(x + µ̂)(r + γµ )Uµ† (x − µ̂)ψ(x − µ̂)). (3.24) µ Here the twisted mass is expressed into the physical basis {ψ, ψ}. At the maximal twist it is 48 CHAPTER 3. QCD AT T > 0 ON THE LATTICE better to redefine this basis into the so-called twisted basis {χ̄, χ} as 1 ψ = √ (1 + iγ5 τ 3 )χ 2 and 1 ψ = χ √ (1 + iγ5 τ 3 ) . 2 (3.25) In the weak coupling limit, β = 6/g20 → ∞, zero quark mass corresponds to κ = 1/8, setting r = 1. For finite coupling this value of κ gets corrections through mass renormalization. The overall renormalized quark mass M is composed of the twisted and untwisted masses as M 2 = Zm2 (m0 − mcr )2 + Zµ2 µ02 (3.26) where mcr is the critical mass and Zm is the corresponding Z-factor. Our hopping parameters of interest are based on the values of β provided by the European Twisted Mass Collaboration (ETMC) [122]. kc for intermediate lattice spacing a(β ) values were obtained thanks to interpolations as practized in [123, 124]. As usual at finite temperature QCD, the imaginary-time extent corresponds to the inverse temperature T −1 = Nτ a. To be able to set the physical scale for each β we interpolated the data at the β values of 3.90, 4.05, 4.2 provided by the ETMC collaboration [122]. This allowed us to map each β value to a some lattice spacing a in Fermi for example. We consider during the present investigations lattice spacings a < 0.09 f m and three sets of pion masses mπ = 316, 398 and 469 MeV. 3.3 How to perform the continuum limit? In order to extract continuum physics from LQCD computations one needs to extrapolate the outcoming (originally dimensionless) lattice results to the limit of vanishing lattice spacing a −→ 0. This limit makes sense only after fixing the scale of the theory to some physical meaningful quantity as the temperature. That means the lattice spacing needs to be related somehow to the temperature. This latter is expected to remain fixed in the limit of vanishing lattice spacing. One way to achieve this relationship is to consider the two-loop renormalization group equation b1 6b0 − 2b2 β 0 exp(− aΛ ≃ ( ) ), β 12b0 (3.27) 1 2 (11 − N f ) and the relation T = 1/aNτ . Here the two universal coefficients amount to b0 = 16π 2 3 8 1 2 and b1 = ( ) [102 − (10 + )N f ], and Λ is the scale parameter which might be related to 16π 2 3 other regularization schemes. Here, in our thesis we set the scale for pure gauge theory for example using the N ECCO -S OMMER parametrization of ln(a/r0 ) on β [125]. Setting the S OMMER scale to r0 = 0.5 fm allows to determine the physical scale of the lattice spacing a, and employ- Sec. 3.4. Fixing the L ANDAU gauge 49 ing such parametrization one may map β to each a. Regarding our unquenched data, the values of r0 /a were provided to us by the tmfT collaboration. These values result from interpolations of the data provided by the ETMC collaboration [122]. Therefore, after mapping somehow a to β one can in principle perform numerical simulations with β as input, for different Nτ , keeping the aspect ratio (Nσ /Nτ ) and T fixed. That is, we simulate at different lattice spacings keeping T and the (spatial) volume fixed. Hence, once may at the end of day extrapolate results (for example of the propagators) to a −→ 0 and extract the continuum physics. This is the strategy we applied in the study of the continuum limit of quenched QCD as we discuss later. We were able to reach the continuum limit using a lattice size of 483 × 12 and keeping the physical volume fixed above and below the critical temperature Tc . In fact, extrapolations at different a correspond in our case to a quadratic function of a as the W ILSON action has O(a2 ) errors. One other aim was also to extrapolate to the thermodynamic limit V = (aNσ )3 −→ ∞. This is achieved setting the temperature T and Nτ fixed. In fact, such ideal limit is always limited by the finite size of the lattice. We used in this context a maximal lattice size of 643 × 12. This enables us to assess finite size effects which are discussed in the results part of the thesis. To summarize, to extract the continuum physics of an observable O on the lattice one need to perform the following ordered multiple limits hOi = lim lim lim hOiU , Vphys →∞ a→0 N→∞ where N denotes the number of configurations, and Vphys = (Nσ × a)3 is the physical spatial 3d space. 3.4 Fixing the L ANDAU gauge Beside the systematic errors which arise in a typical lattice simulation the G RIBOV ambiguity might be the most problematic aspect one needs to deal with. In this particular section we introduce basic elements on how to gauge fix the fields on the lattice and how the G RIBOV problem comes into play. Two popular gauge-fixing methods are used during our simulations, namely simulated annealing and over-relaxation. Finally, we propose to define the gauge uniquely taking into account the smallest FADDEEV-P OPOV (FP) operator eigenvalues. This way one considers gauge copies as close as possible to the so-called G RIBOV horizon [28]. Results on this new proposed method can be found in Chap. 7. 3.4.1 Gauge fixing and gauge functional In this thesis, we are interested in propagators for the particular case of the L ANDAU gauge fixing condition. In general, during the gauge fixing process on the lattice one needs to search for a gauge transformation g = gx (for a fixed configuration U) which maximizes the gauge 50 CHAPTER 3. QCD AT T > 0 ON THE LATTICE functional FU [g]. This gauge functional might be written for the L ANDAU gauge as FU [g] = 1 4V 4 ∑ ∑ Re Tr gUx,µ (3.28) x µ=1 with V denotes the lattice volume. The continuum form of this lattice form was already presented in Eq. (2.21). Actually, for the trivial case of U = I the gauge functional FU [g] takes the largest value FU [g] = 3 for g = I. Searching a maximum of FU [g] for other U drive all the gUx,µ close to the unity. Therefore, our goal during the gauge-fixing process is to find global maxima of FU [g]. In fact FU [g] has many local maxima whose the number is increasing with the lattice size. The different gauge copies maxima belonging to the same gauge orbit are called G RIBOV copies [26]. These gauge copies are all satisfying the same lattice transversality condition, namely 4 (∇µ Aµ )(x) ≡ (∇ · A)(x) := ∑ µ=1 i h Aµ (x + µ̂/2) − Aµ (x − µ̂/2) = 0 (3.29) Where the lattice gauge potential Aµ (x + µ̂/2) is defined in a mid-point link as Aµ (x + µ̂/2) :=
1 † Ux,µ −Ux,µ |traceless 2i (3.30) Moreover, here and in the following we will assume the notation Ax,µ := Aµ (x + µ̂/2). For the sake of completeness we should also define the gluon fields in the adjoint representation as Aax,µ := Aaµ (x + µ̂/2) = 2 · Im Tr{T aUxµ }. (3.31) This latter equation is needed when computing the gluon propagator. We have used for the implementation of the L ANDAU gauge fixing on the lattice two popular methods, namely simulated annealing (SA) and over-relaxation(OR). In fact we used a combination of both methods (SIM followed by OR) for better efficiency to find gx as close as possible to the global maximum of FU [g] as already practized in [126, 127, 128, 31, 32]. Let us in the next section present such methods in more detail. Simulated annealing and over-relaxation As said in the previous section, gauge fixing on the lattice amounts to find gauge copies being as close as possible to the global maximum of FU [g] [129, 130] as already practized in [33, 35, 34, 36, 31, 32]. This prescription has been shown to provide correct results for L ANDAU gauge photon and fermion propagators within compact U(1) lattice gauge theory Sec. 3.4. 51 Fixing the L ANDAU gauge [131, 132, 133, 134]. Very efficient for this aim is the simulated annealing (SA) algorithm combined with subsequent over-relaxation (OR) iterations [126, 127, 128, 31, 32]. The SA algorithm generates gauge transformations {gx } randomly with a statistical weight ∼ exp(FU [g]/Tsa ). The “temperature” Tsa is a technical parameter which is monotonously lowered in the course of a determined SA simulation sweeps (actually, these are heat-bath updates). Also, for better performance, a few micro-canonical steps are applied after each heat-bath step. Finally, we employ the OR algorithm in order to satisfy the gauge condition Eq. (3.29) with a local accuracy of ε max Re Tr[∇µ Axµ ∇ν A†xν ] < ε. (3.32) x 3.4.2 A new proposal to deal with the G RIBOV ambiguity The ambiguity to find a (absolute) local maxima for the gauge functional corresponds to a problem connected with the nature of L ANDAU gauge itself. Several approaches are developed to study the G RIBOV effects on the propagators. The approach we used to the pure gauge sector of QCD is called first-best copy (fc-bc) strategy. In general, it consists of comparing the gluon propagator (for example) computed on the first (random) gauge copy generated during the MC process with the best copy, i.e. the copy providing the highest maximum of FU [g]. Our alternative to solve the G RIBOV problem is to consider the vicinity of the G RIBOV horizon. That is, to take into account gauge fixed configurations having the minimal FP operator eigenvalues. This latter requirement ensures to get a ghost propagator singular behavior. We will discuss these results in Chap. 7. The FADDEEV P OPOV operator and its eigenvalues Presently we start to introduce basic definitions and concepts we need later on to understand our alternative study of the G RIBOV problem. The master key in our analysis is the FADDEEV P OPOV operator and its eigenvalues on the lattice in the L ANDAU gauge. Let us start first by defining the one-parameter subgroup of local SU(3) gauge group [135] as gω (τ, x) = exp {iτωxc T c } τ, ωxc ∈ R . Here T c are the generators of the SU(3) group. Let us now assume that g is a local maxima of the gauge functional fω (τ) := FU [gω (τ)], then the first derivative with respect to τ should vanish (in fact for any τ) 0= ∂ fω (τ) ∂τ = τ=0   1 ωxc ∑ Acx−µ̂,µ − Acx,µ , ∑ 2 x,c µ therefore one observes from this latter equation that local maxima of the gauge functional automatically satisfy the lattice L ANDAU gauge Eq. (3.29). A second derivation with respect to 52 CHAPTER 3. QCD AT T > 0 ON THE LATTICE τ at τ = 0 defines in the L ANDAU gauge the FADDEEV-P OPOV operator7 as follows ∂2 fω (τ) ∂ τ2 ∑ = τ=0 cd d ωxc Mxy ωy x,y,c,d where   ab ab ab B +C Mxy = Aab δ δ δ − x,y x+ µ̂,y x− µ̂,y ∑ x,µ x,µ x (3.33) µ with h i a b = Re Tr {T , T }(U +U ) Aab ∑ x,µ x−µ̂,µ , µ x h i b a Bab x,µ = 2 · Re Tr T T Ux,µ , h i ab Cx,µ = 2 · Re Tr T a T b Ux−µ̂,µ . (3.34a) (3.34b) (3.34c) M is a symmetric real matrix which satisfy in the L ANDAU gauge M[U] = −∇ · D[U] = −D[U] · ∇ ⇐⇒ ∇·A = 0 where D[U] refers to the covariant derivative [136]. This F-P operator admit eight zero modes. Still, the remaining eigenvalues of the F-P operator are positive. The corresponding gauge field configuration is said to lie within the G RIBOV region. If the lowest non-trivial eigenvalue tends to zero the configuration approaches the so-called G RIBOV horizon. Therefore, the G RIBOV set Ω [28] is defined as a set of local maxima as Ω := {U : U ∈ Γ, M[U] ≥ 0} , where Γ is the set of transverse configurations U, namely satisfying ∇ · A(U) = 0. From this G RIBOV region one might construct the fundamental modular region FMR [136] constituted with all global maxima of FU [g] defined as Λ := {U : FU (I) ≥ FU [g] for all g} . It has been proven on the lattice that such FMR is a G RIBOV-copies free region except on its boundary where G RIBOV copies might be encountered [136]. Still, a prescription to construct such FMR spaces is not a priori a trivial task. The aforementioned F-P eigenvalues have a direct influence on the ghost propagator behavior as this propagator might be given a spectral representation in terms of these eigenvalues. In fact, an enhanced density of eigenvalues near zero causes the ghost propagator to diverge stronger than 1/q2 near zero momentum [28]. Actually, 7 The second derivative is called also the Hessian of FU [g]. Sec. 3.4. Fixing the L ANDAU gauge 53 the ghost propagator might be reconstructed as a spectral representation as n 1~ Φi (k) · ~Φi (−k) i=1 λi Gn (q2 ) = ∑ (3.35) with ~Φi (k) being the Fourier transform of the i-th eigenmode (k denotes the lattice momentum) [137, 138]. Therefore, If all n = 8V eigenvalues and eigenvectors were known, the ghost propagator G(q2 ) would be completely determined. Therefore, considering small FP eigenvalues render the ghost propagator more singular. Therefore, we focus in this investigation only on the gluon propagator at T = 0. An important remark in [137, 138] is that the FP eigenvalues (or the density of lowest eigenvalues) are shifted to zero when moving to higher lattice volumes. This behavior is in agreement with Z WANZIGER paper [28]. We are not going to examine the behavior of the ghost propagator (as a function of the FP eigenvalues) as we already know that it enhances for smaller eigenvalues in the infrared momentum region. How to face the G RIBOV ambiguity? Even if the approach consisting of finding maxima closer to the global one by maximizing the value of FU [g] is quite admitted we believe in fact that the FP eigenvalues might also play a rôle. In fact, we have already seen that these (lower) eigenvalues are dominating the infrared behavior of the ghost propagator in [137, 138], and one observes a singular behavior of the ghost propagator. The conventional method based on maximizing the gauge functional is subject to the G RIBOV ambiguity when different local maxima are taken into account. This situation spoils an unique definition of the gauge, and no criteria is possible to distinguish different G RIBOV copies giving at the end different propagators results (mainly in the infrared). Furthermore, lattice results using the gauge functional approach show a plateau ending with a finite zero momentum gluon propagator even for physical volumes larger than 134 f m4 [61]. On the other hand, solutions of a truncated set of DYSON -S CHWINGER equations (DSE) for gluon and ghost propagators in L ANDAU gauge predict two type of different solutions, namely the scaling and the decoupling solutions [20, 139, 71, 81, 27]. These two types of solutions start to deviate from each other below the region of 1 GeV. In the ultraviolet region of momenta these solution are indistinguishable. These two solutions are the consequence of different boundary conditions on the ghost dressing function at zero momenta [89, 140]. Therefore, fixing these boundary conditions is equivalent to not take the G RIBOV problem seriously. For the moment lattice investigation support coupling solutions contradicting the K UGO -O JIMA confinement criterion [29]. We think that if one is able to define a free G RIBOV problem procedure on the lattice one may observe the ’real’ physical solution of the DSE. In Chap. 7 we propose to concentrate on a new criteria which enable us to choose in an unique way, which gauge transformation needs to be taken into account. We consider in fact only gauge configurations with the lowest positive FP eigenvalues. This set of gauge configuration is actually 54 CHAPTER 3. QCD AT T > 0 ON THE LATTICE a subset of the G RIBOV region, and might intersects with the fundamental modular region. In this respect, we have studied in this respect the correlation between the highest FU [g] and lowest positive FP eigenvalues, and no strict correlation have been observed. Indeed, this is a fortunate situation as this actually means that the ghost propagator might diverge stronger as a result of our new criteria for example. Moreover, this also shows that both criteria (hightest F vs. minimal λ ) might provide independent results. For the gluon propagator it is already known that the influence within the gauge functional approach is weak [137, 138] following the fc-bc method. But still, in this last references no comparison between best copies (in the sense of highest gauge functional] to gauge copies with lowest FP eigenvalues have been done. We shall discuss results of our new approach for the SU(3) gauge group for symmetric lattices (T = 0) in Chap. 7. CHAPTER 4 Lattice observables at T > 0 H we introduce basic definitions of the P OLYAKOV on the lattice. We also define the gluon and ghost propagators at finite temperature. This brings us two define the two independent components of the gluon propagator, namely: the transverse (DT ) and longitudinal (DL ) components on the lattice. We discuss at the end of this chapter renormalization issues for these propagators. ERE 4.1 The P OLYAKOV loop on the lattice As we know confinement is the realization of a global Z(3) center symmetry. When this symmetry does break a phase transition from a confining state to a deconfined plasma occurs. This phenomena was first observed in [141]. The lattice analogous of the P OLYAKOV loop defined previously in the continuum (Eq. (2.93)) looks like L(~n) = Tr Nτ −1 ∏ U4 (~n, n4 ), (4.1) N4 =0 where ~n = (n1 , n2 , n3 , n4 ) is the lattice site and n4 = t. The average of this P OLYAKOV loop is related to the free energy of a static quark Fq as already remarked with Fq hL(~n)i = e− T . (4.2) In fact |hLi| is an order parameter for confinement-deconfinement phase transition since the static q-q̄-potential Fqq̄ (~n) satisfies F~qq (~n) −→ 2Fq , (4.3) 55 56 CHAPTER 4. LATTICE OBSERVABLES AT T > 0 for ~n −→ ∞ in the confining phase. Therefore, |hL(~n)i| satisfies Confinement ⇔ |hL(~n)i| = 0, Deconfinement ⇔ |hL(~n)i| = 6 0. Since the P OLYAKOV line transforms under Z(3) transformation as L(~n) −→ e 2πi 3 j L(~n), j ∈ {0, 2}, (4.4) the free energy diverges as long as the center symmetry is satisfied. Fqq̄ remains finite when the center symmetry is spontaneously broken. We interpret this as a manifestation of deconfinement. 4.2 The lattice gluon propagator On the lattice the gluon propagator (at zero temperature) is defined in momentum space as D E b a e e Dab (q) = (k) A (−k) , (4.5) A µν µ ν eaµ (k) denotes the Fourier transform where h· · · i represents the average over configurations, and A of the gauge-fixed gluon field 1 Aµ (x + µ̂/2) = 1 † ) |traceless (Uxµ −Uxµ 2iag0 (4.6) depending on the integer-valued lattice momentum kµ (µ = 1, . . . , 4), and Uxµ are SU(3) links. The lattice momenta kµ are related to the physical momentum (for the W ILSON plaquette action) as    πkµ 2 qµ (kµ ) = sin , kµ ∈ −Nµ /2, Nµ /2 , (4.7) a Nµ where (Ni , i = 1, 2, 3; N4 ) ≡ (Nσ ; Nτ ) characterizes the lattice size. For non-zero temperature it is convenient to split the propagator into two components, the transverse DT (“chromomagnetic”) (transverse to the heat-bath rest frame) and the longitudinal DL one (“chromoelectric”), respectively, ab T 2 2 L Dab q ) + Pµν DL (q24 ,~q 2 )), µν (q) = δ (Pµν DT (q4 ,~ (4.8) where q4 plays the role of the M ATSUBARA frequency, which will be put to zero later on. For the T,L represent projectors transverse and longitudinal relative Landau gauge, the tensor structures Pµν to the (µ = 4)-direction already defined in Eq. (2.121). For the propagator functions DT,L we 1 In our case satisfying the lattice transversality Landau gauge condition ∇µ Aµ = 0. Sec. 4.3. 57 The ghost propagator find 1 DT (q) = 2Ng * and 1 DL (q) = Ng 3 ∑ i=1 eai (k)A eai (−k) − A q24 ea ea A (k)A4 (−k) ~q 2 4   E q24 D ea ea A4 (k)A4 (−k) , 1+ 2 ~q + (4.9) (4.10) 2 where the number of generators Ng = Ncolor − 1 for Ncolor = 3. The zero-momentum propagator values can be defined as 1 3 D ea ea E ∑ Ai (0)Ai (0) , 3Ng i=1 1 D ea ea E DL (0) = A (0)A4 (0) . Ng 4 (4.11) DT (0) = (4.12) 4.3 The ghost propagator The Landau gauge ghost propagator is given by Gab (q) = a2 ∑he−2πi(k/N)·(x−y) [M −1 ]ab xy i, x,y =δ ab (4.13) G(q), where q 6= 0 and (k/N) · (x − y) ≡ ∑µ kµ (x − y)µ /Nµ . M denotes the lattice FADEEV-P OPOV operator 2 , already defined in Eq. (3.33) and Eq. (3.34). For our simulations and in order to ~ c with coinvert M we use the conjugate gradient (CG) algorithm with plane-wave sources ψ a a ab φ b (y) = lor and position components ψc (x) = δc exp (2π i(k/N) · x) to solve the equations Mxy a ψc (x) [142, 143]. In fact we exploit the formula * +
1 ab . Gab (q2 (k)) = M −1 xy eik·(x−y) V ∑ x,y U Where the ensemble average is translation invariant. Since Gab has the following tensorial struc2 This operator definition is corresponding to the gauge field definition in Eq. (4.6) and the related gauge functional in Eq. (3.28) 58 CHAPTER 4. LATTICE OBSERVABLES AT T > 0 ture Gab (q) = δ ab G(q2 ) [142, 143] we are interested to compute the scalar function G(q2 (k)) = 1 1 ∑ Gaa (q2 (k)) = Nc2 − 1 hTr M −1 (k)iU , Nc2 − 1 a (4.14) where M −1 (k) is the Fourier transform of the inverse FP operator M −1 in momentum space defined as M −1
ab (k) = 1 V ∑ e−ik·x M −1 x,y
ab xy eik·y , (4.15) and 4 k·x ≡ ∑ 2π µ=1 kµ xµ Lµ (4.16) is the product of lattice momentum k and lattice site x. Lµ denotes the lattice extension in direction µ. As we are interested in non-zero momenta we apply the conjugate gradient method to solve the following sparse linear system [Mψb ]cz ≡ ∑ Mcz,ax ψbax = ξbcz (k) (4.17) a,x using a fixed source ξb with 8V complex components ξbcz (k) := δ cb eik·z . Here, c and z label the vector components of ξb , while index b specifies the sources. The inverse FP in Eq. (4.15) might be written as a function of the solutions of the system Eq. (4.17) like M −1
ab (k) = 1 V ∑ e−ik·x · ψbax (k) . (4.18) x with ψbax (k) representing the 8V vector components. Therefore, the problem numerically boils down to solve systems of the form [M cb (k)]cz = δ cb cos(k · z) [M sb (k)]cz = δ cb sin(k · z), (4.19) (4.20) ax where ψbax is decomposed to ψbax = cax b + i sb . Thereby, the inverse FP operator might be rewritten as M −1
ab (k) = 1 V ∑ x,y ax cos(k · x)cax b (k) + sin(k · x)sb (k) ax +i [cos(k · x)sax b (k) − sin(k · x)cb (k)] . (4.21) Sec. 4.4. Renormalizing the propagators 59 Back to the ghost propagator defined in Eq. (4.14) together with Eq. (4.21) and exploiting the property of the symmetry of the FP operator, one gets Tr M −1 (k) = ∑ cos(k · x) · caxa + sin(k · x) · saxa a,x,y ax where cax a and sa are solutions to the two independent linear systems given in Eq. (4.19) and ax (Eq. (4.20)). Therefore, once the values of cax a and sa are found numerically one gets directly the value of ghost scalar function from Eq. (4.14). 4.4 Renormalizing the propagators Throughout this thesis we renormalize our gluon propagator defined as hAaµ (x)Aνb (y)i and the ghost propagator hca (x)c̄b (y)i within the MOM scheme. In fact the MOM scheme defines the Z-factors such that the fundamental two-point and three-point functions equal their corresponding tree-level expressions at some momentum µ, the so-called the renormalization point. The L ANDAU gauge gluon propagator expressed in momentum space looks like   pµ pν Z(p2 , µ 2 ) ab ab µν δ − 2 Dµν (p, µ) = δ (4.22) p p2 with Z is the dressing function of the gluon propagator. This dressing function measure the deviation from the tree-level structure (corresponding to Z ≡ 1). On the other hand, the ghost propagator has the following tensorial structure in Landau gauge Gab (p, µ) = δ ab J(p2 , µ 2 ) . p2 (4.23) Where J is the dressing function of the ghost propagator. As said before, within the MOM scheme the renormalization constants, for instance of the gluon and ghost fields Z3 and Ze3 , are defined by requiring the renormalized expressions to equal their tree-level form at some (large) momentum µ. That is, we renormalized the gluon (DT and DL ) and ghost (G) propagators imposing their dressing functions to be equal to one at the momentum subtraction point p = µ. p2 DT (p)| p=µ = 1, 2 p DL (p)| p=µ = 1, 2 p G(p)| p=µ = 1. (4.24) (4.25) (4.26) 60 CHAPTER 4. LATTICE OBSERVABLES AT T > 0 The Z factors Z3 and Ze3 are defined as Dab µν (p; Λ, go , mo , ξo ) p2 =µ 2 =: Z3 δ ab  δ µν p µ pν − 2 µ  1 , µ2 (4.27) and Gab (p; Λ, go , mo , ξo ) p2 =µ 2 =: Ze3 δ ab 1 , µ2 (4.28) ab where Dab µν (G ) denotes the unrenormalized gluon(ghost) propagator. Actually, it is well expected that at high temperature these two components, DT and DL should behave independently of the temperature. This has a direct consequence on the values of ZT and ZL factors, which should coincide at this high temperature regime. This was observed by us [49] in pure gauge theory confirming a rough matching of the Z values as well as in the full QCD case [52]. We will come back to these points in the results part of this thesis. Hence, the renormalization constants can be determined by calculating the corresponding unrenormalized (regularized) G REEN function. Their values depend on the chosen renormalization point µ and also on the bare parameters of the regularized theory. Therefore, fixing the values of Z3 at some momentum µ thanks to Eq. (4.27) we were able to compute the renormalized gluon propagator (at µ) via multiplicative renormalization according to Eq. (2.38). We took always µ = 5 GeV for our pure gauge investigations based. This choice is done in order to be far enough in the ultraviolet region of momenta (to study the low momenta lattice artifacts conveniently) as well as not being too close to the cutoff. Concerning full QCD we choose the renormalization point to be 2.5 GeV. CHAPTER 5 Results in the pure gauge sector of QCD W this chapter, we report on results in the pure gauge sector of QCD in L ANDAU gauge. In particular, we present gluon and ghost propagators results in the so-called quenched approximation of QCD. Prior to this, we start first by specifying the lattice samples we used during our investigations. Therefore, important different parameters as the critical βc are discussed and identified. Also we comment on how to map to physical units. Lastly, different systematic effects as the momenta preselection, the P OLYAKOV loop sector, finite size and the G RIBOV problem are studied extensively. ITHIN 5.1 Specification of our lattice samples During our lattice investigations of the pure gauge sector of QCD we generated SU(3) gauge configurations according to a Monte-Carlo process. We thermalized the configurations using the standard W ILSON action. In general, each thermalization sweep consist of four micro-canonical over-relaxation steps and one heath-bath. In particular, the system was thermalized with the heat-bath method using the C ABIBBO -M ARINARI trick [144]. Moreover, O(2000) sweeps were discarded between consecutive measurements of observables. In fact, we took this big number of 2000 sweeps in between in order to reduce the auto-correlation between consecutive measurements. Our strategy here in order to vary the temperature is to keep the scale (β ) fixed to the critical value βc = 6.337. This latter value corresponds to a quite big Nτ = 12. Moreover, we choose on purpose a quite high βc to deal with a small lattice spacing, and therefore being close to the continuum limit. Hence, fixing the scale and varying Nτ temperature was able to be changed correspondingly according to Eq. (3.13). In contrast, in order to study finite size effects 61 62 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD T /Tc 0.65 0.74 0.86 0.99 1.20 1.48 1.98 2.97 0.86 0.86 0.86 1.20 1.20 1.20 0.86 0.86 1.20 1.20 Nτ 18 16 14 12 10 8 6 4 8 12 16 6 8 12 14 14 10 10 Nσ 48 48 48 48 48 48 48 48 28 41 55 28 38 58 56 64 56 64 β 6.337 6.337 6.337 6.337 6.337 6.337 6.337 6.337 5.972 6.230 6.440 5.994 6.180 6.490 6.337 6.337 6.337 6.337 a(GeV−1 ) 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.49 0.33 0.24 0.47 0.35 0.23 0.28 0.28 0.28 0.28 a(fm) 0.055 0.055 0.055 0.055 0.055 0.055 0.055 0.055 0.097 0.064 0.048 0.094 0.069 0.045 0.055 0.055 0.055 0.055 ncon f 150 200 200 200 200 200 200 210 200 200 200 200 200 200 200 200 200 200 TABLE 5.1: Temperature values, lattice size parameters, lattice spacing a in units of GeV−1 and fm, and the number ncon f of independent lattice field configurations used throughout this study. we fixed the temperature to two reference values (above and below Tc ), and changed the lattice spacing (β ), and Nσ correspondigly to fix the physical volume according to Eq. (3.12). Our lattice parameters are displayed in Table 6.1. The critical Nτ = 12 provides an interesting range around the critical temperature Tc . The way to localize this critical value is examined in the next section with more detail. 5.1.1 Localization of our critical βc In order to be able to study the phase transition one needs to know the scale where it might happen. We fixed during our first investigations of the gluon and ghost propagator the coupling to the critical value βc = 6.337. This β -value corresponds to a temperature very close to the critical value Tc of the deconfinement phase transition for Nτ = 12 and a linear spatial extent Nσ a(β = 6.337) = 48 a ≃ 2.64 fm. According to Ref. [57] our βc has been fixed using interpolations Sec. 5.1. 63 Specification of our lattice samples 0 f(x) 12/48 βc − βc (Nτ , ∞) -0.001 -0.002 -0.003 -0.004 -0.005 -0.006 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 Nτ /Nσ FIGURE 5.1: The difference βc (Nτ , Nσ ) − βc (Nτ , ∞) as a function of the ratio Nτ /Nσ using Eq. (5.1) presented in [57]. The interpolation is done at Nτ /Nσ = 12/32. (see Fig. 5.1) with the help of the fit formula  Nτ βc (Nτ , Nσ ) = βc (Nτ , ∞) − h Nσ 3 , (5.1) where βc (Nτ , ∞) corresponds to the thermodynamic limit and h denotes a fit parameter (h . 0.1). In fact, our choice Nσ = 48 guarantees a reasonable aspect ratio over the whole temperature range T /Tc ≡ 12/Nτ ∈ [12/18, 12/4] and to reach three-momenta below 1 GeV. 5.1.2 Selecting the momenta and the M ATSUBARA frequency On the lattice, typical lattice artifacts might be reduced by selecting an appropriate set of momenta. For the gluon and ghost propagators the momenta choice is an important issue. We focus first, in order to study systematic hyper-cubic effects, on the influence of the momentum choice on the behavior of the gluon propagator. Thus, both the transverse DT and the longitudinal DL gluon propagators are compared on both on-axis and diagonal momenta. We took as a reference our critical β = 6.337 and found in the lower momentum range only quite small deviations, as observed in Fig. 5.2. In fact, we show the bare transverse and longitudinal gluon propagators as functions of the lattice momenta, where the propagators are computed for the critical inverse coupling βc = 6.337. As the fluctuations between the gluon data for the diagonal and on-axis momenta are small, we conclude that the choice of the momenta pre-selection is irrelevant. Nevertheless to be on the safe side we decided for most of our computations to apply the so-called cylinder cut [113] defined by 1 ∑ k2µ − 4 (∑ kµ )2 ≤ c, µ µ (5.2) 64 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD 1000 1000 On axis momenta Diagonal momenta On axis momenta Diagonal momenta 100 DL .a−2 DT .a−2 100 10 1 10 1 0.1 0.1 0.01 0.1 1 q.a 10 0.1 1 10 q.a FIGURE 5.2: Comparison of the bare transverse (l.h.s.) and the longitudinal (r.h.s.) gluon propagator for β = 6.337, Nσ = 48 on on-axis and diagonal momenta preselections. meaning that we take only into account momenta lying within the cone described with Eq. (5.2). The cut parameter was chosen c = 3. Additionally, we took momenta with k4 = 0. That is, only zero M ATSUBARA frequencies are considered here. This choice of zero M ATSUBARA frequencies is motivated by different reasons. The first reason is to be able to reach low momenta, at least around the physical momenta of 1 GeV . Therefore, concentrating on zero M ATSUBARA frequencies (soft modes) one can access the region of small momenta. The second physical reason to study exclusively the soft modes is that the hard modes (n 6= 0) develop a thermal mass 2π nT , and behave as massive particles, see [145, 54, 109, 9] and references therein. 5.1.3 Gauge fixing process The process we employ here to fix gauge to the L ANDAU gauge was based on two gauge fixing methods, namely: simulated annealing (SA) and over-relaxation(OR). The simulated annealing algorithm generates gauge transformations {gx } randomly with a statistical weight ∼ exp(FU [g]/Tsa ). Tsa is a technical parameter (“temperature”) which is monotonously lowered. In fact, we start with Tsa = 0.45 and decrease this parameter down to Tsa = 0.01 in equal steps applied after each of the 3500 SA simulation sweeps. Fixing the initial and the final SA temperatures to Tsa = 0.45 and Tsa = 0.01 correspondingly rely on our study of the gauge functional as a function of these parameters. We observed actually that starting with 0.45 one might see a transition-like behavior of the gauge functional just above this temperature. The final temperature 0.01 was chosen in order to minimize the OR maximal divergence monotonously with respect to the number of iterations. For better performance a few micro-canonical steps are applied after each SA sweep. On the other hand, over-relaxation (see Appendix 8) is an iterative method transforming each gauge configuration to the L ANDAU gauge looking for a local gauge transformation g ≡ {gx }, which maximizes the gauge functional, and therefore approaching the (unknown) global maximum [146]. The stopping criteria for this iterative gauge fixing process Sec. 5.2. 65 The P OLYAKOV loop results is the violation of the transversality condition. That is, the gauge fixing process is stopped as one satisfies max Re Tr[∇µ Axµ ∇ν A†xν ] < ε , x ε = 10−13 (5.3) Actually, we have always taken a maximum number of OR iterations equal to 80000. Therefore, it has been observed that the OR iterations needed for all temperature cases do not exceed this maximal number. In fact, a combination of SA and OR (simulation annealing finalized with over-relaxation) is proven to be more efficient to bring the gauge fixing process close to the global maximum [126, 127, 128, 31, 32]. 5.1.4 Fixing the scale In order to translate the bare gluon propagator into the physical one needs to fix the scale. To do that, we relied on the SOMMER scale r0 = 0.5 f m [147]. This latter value together with the parametrization of ln(a/r0 ) as a function of β [125] enabled us to physically map all the values of β as displayed on Table 6.1 to the corresponding lattice spacings. In fact, this mapping apply only for the reduced set of beta, namely 5.7 < β < 6.92. Still, this range is enough for us to cover the temperatures range of interest. 5.2 The P OLYAKOV loop results The P OLYAKOV loop is an important order parameter for QCD in the pure gauge sector. In fact, this quantity is an exact order parameter in this sector. That is, it takes different values depending whether the system lies in the confining or the deconfining phase. Ideally, the P OLYAKOV loop takes zero values in the confining phase (below Tc ) and a finite value otherwise. In our lattice study we are interested in the P OLYAKOV loop in order to check whether our localization of βc at 6.337 is exact or not. In Section 5.1.1 such localization has been performed thanks to an interpolating formula proposed in [57]. On the lattice, the P OLYAKOV loop is defined in Eq. (4.1), and due to the translation invariance on the lattice we take the following spatial average into consideration L= 1 Nσ3 ∑ L(~n). (5.4) n In fact, we focus on the real part of the spatial average of the P OLYAKOV in Eq. (5.4). Our results are shown in Fig. 5.3. Here, we show the dependence of the absolute value of the real part as a function of the temperature. In order to cover the region around Tc coveniently, we vary the values of the β around the critical value βc . This allows us to get a good distribution 66 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD 1 3.5 3 0.1 2 χ < |L| > 2.5 0.01 1.5 0.001 1 0.0001 0.5 1e-05 0 1 1.5 2 T /Tc 2.5 3 0.6 0.8 1 1.2 1.4 1.6 T /Tc FIGURE 5.3: The temperature dependence of the absolute value of the P OLYAKOV loop (l.h.s.) and its susceptibility (r.h.s.). The inverse coupling β = 6.337 and the spatial extent nσ = 48. of temperatures around Tc to observe occurately what happens around this latter temperature. In fact, We clearly see a brutal bending of the curve around the critical temperature Tc . Moreover, the (real part of) P OLYAKOV loop values below the inflexion point are nearly close to zero. However, they are not completely equal to zero most probably due to the finite statistics in our analysis. Above Tc the curve take finite values up the maximal temperature investigated around 3 Tc . This P OLYAKOV loop behavior is in agreement with previous studies [141]. Moreover, we are showing in the same previous plot (right side) the susceptibility of the P OLYAKOV loop. This informative quantity is defined as follows χ = Nσ3 (h L2 i − (h L i)2 ). (5.5) This observable is one of the most important tools in statistical physics to study criticality in physical systems. We observe as expected that χ peaks around the critical temperature Tc . This indicates that the localization of our critical βc = 6.337 is satisfying. Still, for higher temperature as Tc we still get error fluctuations due to the statistics used. Actually, we have used the Jackknife method to evaluate the errors. Therefore, from now we will concentrate on the critical beta βc = 6.337 as our reference in the pure gauge sector. One advantage to use such higher critical beta corresponding to a smaller lattice spacing a = 0.055 fm intend to approach the continuum limit of the theory, namely a −→ 0. The continuum limit study will be the focus of the upcoming results. In the next section, we report on the gluon and the ghost propagators as a function of the momentum and temperatures. We concentrate in fact on the components of the gluon, namely: the transverse DT and longitudinal DL propagators. These propagators behave as we will see quite differently showing different responses to the phase transition. Sec. 5.3. Results on the gluon and ghost propagators at T > 0 67 5.3 Results on the gluon and ghost propagators at T > 0 5.3.1 The T dependence of the gluon and ghost propagators We start now discussing our results for the gluon and ghost propagators in the pure gauge sector of QCD. Before starting such analysis we should notice an important fact about the behavior of the gluon propagator for large β . In particular, for such high values of the couplings the L ANDAU gauge gluon propagator is expected to depend on the Z(3)-sectors into which the P OLYAKOV loop spatial averages can fall [148]. This observations goes also in parallel with what we noticed on the influence of the P OLYAKOV sectors on the gluon propagator, and in particular on its longitudinal part. The results of these systematic effects are shown on Fig. 5.9 and discussed there. Therefore, our strategy goes as follows: First, we determine the P OLYAKOV sector to which the produced configuration belongs to, then, if (and only if) the sector measured is not the physical sector 1 one multiply the links Ux,4 in the time direction with a phase rotation exp {±2πi/3} to keep the sector physical. This phase rotation corresponds actually to a Z(3)-flip in time direction before the gauge fixing procedure is started. Such global flips are equivalent to non-periodic gauge transformations and do not change the pure gauge action. As for the observables themselves, namely the gluon and ghost propagators, and as stated before, the gluon propagator might be splittable into two components, namely: the transverse DT (“chromomagnetic”) (transverse to the heat-bath rest frame) and the longitudinal DL one (“chromoelectric”) already defined in Eq. (4.9), Eq. (4.10) on the lattice, respectively, with the zero-momentum propagator values defined in Eq. (4.11). On the other hand, the ghost propagator is defined in Eq. (4.13) in the continuum for q 6= 0 and (k/N) · (x − y) ≡ ∑µ kµ (x − y)µ /Nµ . The corresponding form of the ghost propagator on the lattice is described in Eq. (4.13). We recall also that momentum q is already defined in Eq. (4.7). We show in Fig. 5.4 the multiplicatively renormalized propagators DL (q) and DT (q) as functions of the three-momentum (q ≡ |~q|, q4 = 0) for β = 6.337, obtained with Nσ = 48 and different Nτ , i.e. for temperature values varying from T = 0.65 Tc up to T ≃ 3 Tc . For details we refer to the upper section of Table 5.1. The renormalization condition is chosen such that DL,T take their tree level values at the subtraction point q = µ. We choose µ = 5 GeV in order √ to be close to the perturbative range and still reasonably away from our lattice cutoff (qmax = 2 3/a ≃ 12.4 GeV). We observe from Fig. 5.4 that the temperature dependence of both DL and DT becomes weaker with increasing momentum. This weakening proceeds faster for DT than for DL . The ultraviolet regions of DT and DL turn out to be “phase-insensitive”. This affect the values of the Z-factor which do not really react to the temperatures. This observation was also reported in [50]. More precisely, while the temperature changes from its minimal value to our maximal one, the change of DT is less than 5% for q > 2.2 GeV, while for DL this is guaranteed for q > 2.7 GeV. For T< ∼ Tc DL shows a comparatively weak temperature dependence also at small momenta. This changes drastically as soon as T > ∼ Tc . In contrast to that DT (q) changes monotonously with T 1 This means the phases of the corresponding P OLYAKOV loop averages to fall into the interval (−π /3, π /3] 68 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD 100 100 DT [GeV −2 ] 10 1 Tc Tc Tc Tc Tc Tc Tc Tc 0.65 0.74 0.86 0.99 1.20 1.48 1.98 2.97 10 DL [GeV −2 ] 0.65 0.74 0.86 0.99 1.20 1.48 1.98 2.97 0.1 1 Tc Tc Tc Tc Tc Tc Tc Tc 0.1 0.01 0.01 0 0.5 1 1.5 2 2.5 q [GeV] 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 q [GeV] FIGURE 5.4: Temperature dependence of the longitudinal (l.h.s.) and the transverse (r.h.s.) gluon propagator for β = 6.337 and a spatial lattice size Nσ = 48. in the infrared region. This can be seen in more detail from Fig. 5.5. There we show the temperature dependence of DL (q) (left panel) as well as of DT (q) (right panel) for six selected momenta in the range up to 1.6 GeV. One can see that DL at fixed momentum shows strong variations in the neighborhood of Tc . It is rising with T below Tc and sharply drops around Tc . This behavior looks most pronounced for zero momentum and gets progressively weaker at higher momenta. For the lowest momenta we observe maxima at T = 0.86 Tc . It remains open, whether the maxima are shifted away from the transition temperature with increasing volume. Similar observations have been recently reported [149] for the case of SU(2) gauge theory where the maximum of DL (0) tend to move away from the transition with decreasing lattice spacing. In any case, our data confirms that the infrared part of DL (p) is strongly sensitive to the temperature phase transition [43, 50]. It may serve to construct some kind of order parameters characterizing the onset of the phase transition, as we will propose below. In contrast to that, DT is ever decreasing and varying smoothly across Tc , showing no visible response to the phase transition at all momenta. We go on with the investigation of the gluon propagator fitting our data with the so called G RIBOV-S TINGL fit formula originally proposed in [26, 150]. This fit formula looks like D(q) = c (1 + d q2n ) . (q2 + r2 )2 + b2 (5.6) This formula was the subject of previous investigations [151] and recently also in [149]. Our momentum range of interest for the G RIBOV-S TINGL interpolation is [0.6 : 8.0] GeV. This range is of a particular interest for the DSE where a a working fit function for the gluon propagator is important to be provided. In fact, our fit function with the corresponding fitting parameters play the role of input data for the DSE. Moreover, this fit function might be compared to the DSE Sec. 5.3. 9 35 (0,0,0,0) (1,0,0,0) (1,1,0,0) (1,1,1,0) (2,1,1,0) (2,2,1,0) 7 6 5 (0,0,0,0) (1,0,0,0) (1,1,0,0) (1,1,1,0) (2,1,1,0) (2,2,1,0) 30 25 DL [GeV −2 ] 8 DT [GeV −2 ] 69 Results on the gluon and ghost propagators at T > 0 4 3 20 15 10 2 5 1 0 0 0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 T /Tc 2 2.5 3 3.5 T /Tc FIGURE 5.5: The longitudinal propagator, DL , (l.h.s.) and the transverse one, DT , (r.h.s.) vs. temperature for a few low momenta, the latter represented as (k1 , k2 , k3 , k4 ). β = 6.337 and Nσ = 48. Parameters T /Tc Nτ 0.65 18 0.74 16 0.86 14 0.99 12 1.20 10 1.48 8 1.98 6 2.97 4 r2 (GeV2 ) 0.372(29) 0.296(22) 0.257(22) 0.359(30) 1.029(41) 1.547(47) 2.455(75) 5.327(159) b(GeV2 ) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 DL fits d(GeV−2 ) 0.192(8) 0.206(7) 0.221(8) 0.209(10) 0.155(6) 0.118(4) 0.086(4) 0.045(2) c(GeV2 ) 4.29(17) 4.11(13) 3.70(13) 3.89(16) 5.43(21) 7.12(24) 9.55(37) 17.15(73) χd2 f 1.49 1.40 1.57 1.83 1.27 1.06 1.35 0.51 r2 (GeV2 ) 0.751(24) 0.756(20) 0.847(22) 0.869(26) 0.951(25) 0.886(138) 0.856(109) 0.927(126) b(GeV2 ) 0.0 0.0 0.0 0.0 0.0 0.810(167) 1.398(62) 2.559(33) DT fits d(GeV−2 ) 0.153(4) 0.161(3) 0.152(4) 0.157(4) 0.147(4) 0.146(11) 0.133(8) 0.100(6) c(GeV2 ) 5.40(14) 5.31(11) 5.50(12) 5.45(14) 5.56(13) 5.70(42) 6.15(34) 7.58(41) χd2 f 1.17 0.99 1.09 1.44 1.17 1.46 0.93 1.01 TABLE 5.2: Results from fits with Eq. (5.6) (n = 1) for DL (l.h.s.) and DT (r.h.s.) corresponding to the Monte Carlo data shown in Eq. (5.4) (β = 6.337, Nσ = 48). The fit range is [0.6 : 8.0] GeV. The values in parentheses provide the fit errors. The boldface printed b-values indicate that they are fixed to zero. predictions in order to have a control over the truncations often used to solve these latter. Expected logarithmic corrections needed for the ultraviolet limit have been neglected here (for a thorough discussion see [113]). We put throughout n = 1. In a first attempt we have left b varying. We obtained values compatible with b = 0 except for DT (q) at the highest three temperature values inspected. Therefore, in all other cases we have repeated the fits with fixed b = 0 and obtained χd2 f -values reasonably below 2.0. The fit parameters can be found in Table 5.2.2 5.3.2 Improving the sensitivity around Tc Hereafter we perform an original study connected to the sensitivity of the observables around the phase transition. As we have seen in the former section DL shows a sensitivity when passing 2 Note that for b = 0 Eq. (5.6) is equivalent to the interpolation formula D(q) = γ (q2 +δ 2 ) β + (q2 +δ 2 )2 . 70 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD 1.4 1.2 1 (1,0,0,0) (1,1,0,0) (1,1,1,0) (2,1,1,0) (2,2,1,0) 1.2 1 0.8 0.8 χ α 1.4 (1,0,0,0) (1,1,0,0) (1,1,1,0) (2,1,1,0) (2,2,1,0) 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 0.5 1 1.5 2 2.5 3 T /Tc 3.5 0.5 1 1.5 2 2.5 3 3.5 T /Tc FIGURE 5.6: Temperature behavior of the ratios χ (Eq. (5.7), left panel) and α (Eq. (5.9), right panel) at low momenta as given in the legend, for a spatial lattice size Nσ = 48 and β = 6.337. through the phase transition temperature Tc . This fact motivated us to construct out of DL additional observables which might strongly react to the phase transition. The original hope was that these observables might be useful for full QCD with NF = 2. Nevertheless, the fact that the nature of the phase transitions in the pure gauge QCD and the full QCD are quite different plays an important role on the efficiency of such ‘improved‘ observables. In fact, it is well know that quenched QCD demonstrates a first phase transition while in full QCD one expects a crossover. Still, we have tried to form “observables” constructed from the gluon propagator which can serve as “order parameters” for the deconfinement transition. We tried many functional combinations, and we picked up some of the most interesting for our purposes. First, we plot the ratio χ = [DL (0, T ) − DL (q, T )]/DL (0, T ) (5.7) as a function of T /Tc in the left panel of Fig. 5.6. The first observation is that all the curves labeled by the momentum 4-tuples in the legend show approximate plateau below Tc . Then, passing the phase transition they suddenly fall off with slopes becoming slightly smaller with increasing momentum, but still with visible temperature sensitivity. This means that χ can be used as an indicator for the deconfinement transition3 . Moreover, the transition can be traced even at rather high momentum. This was not so clear from the lift hand side of Fig. 5.5, where the behavior of DL at higher momenta looks rather smooth. A non-trivial consequence from the behavior of χ, at least in the interval 0.65 Tc < T < Tc , is a factorization conjecture as DL (q; T ) ≃ A(q) · B(T ) . 3 at least for pure gauge theory. (5.8) Sec. 5.3. Results on the gluon and ghost propagators at T > 0 71 Then, as long as the temperature T varies in the given interval, the change of DL can be described by a momentum independent rescaling. This is a rather nontrivial property from which further conclusions can be drawn. For example, in the interpolation formula (Eq. (5.6)) above, we should find the mass parameter r2 and the parameter d to be (approximately) temperature independent as long as T < Tc . This constancy of these parameters is well-established from our fit table (the left panel of Table 5.2). Another interesting construction we tried along this study is the ratio α= DL (0, T ) − DL (q, T ) , DL (0, Tmin ) − DL (q, Tmin ) Tmin = 0.65 Tc , (5.9) which according to the factorization in Eq. (5.8) should be approximately momentum independent. Indeed, this can be seen from the right panel of Fig. 5.6. Moreover, α(q, T ) should resemble qualitatively the temperature dependence of DL at q = 0. Above Tc , however, α falls off reaching very small values at higher temperatures (around 2 Tc ). Therefore, we conclude that both quantities χ and α behave as indicators for the finite-temperature transition. It remains to be seen, whether they also map out the (pseudo)critical behavior in unquenched QCD. Let us note that our volumes are not large enough to study the infrared asymptotic behavior. Moreover, at the lowest momenta we expect systematic deviations related to finite-size effects, lattice artifacts, and G RIBOV copy effects. This concerns also the parameters χ and α because of their dependence on the value DL (q = 0). The systematic effects will be discussed in Section 5.3.3, in order to identify the momentum range, where they play only a negligible role, i.e. to define a momentum range free of systematic effects. Besides the observables α and χ we studied other interesting observables showed in Fig. 5.7. These new quantities are combinations of our DT and DL . These quantities exhibit the relative difference between DT and DL in a sense to enhance the critical behavior around Tc . We call these two quantities ψ and θ which look like in equations as ψ = 1 − DT /DL (5.10) θ = (DL − DT )/(DL + DT ) (5.11) and From Fig. 5.7 we observe that ψ takes rise below T c with the temperature reaching some maximum around Tc . This rise of values translate the difference in values between DT and DL as one might see in Eq. (5.10). Thus, This difference takes its maximum around Tc showing that DL is higher than DT for all momenta. The highest difference occurs in the case of the zero momenta. After such a rise below Tc one observe a dramatic fall down right after Tc and for all momenta. The values of ψ for T > Tc are negative demonstrating that DT is smaller that DL in this regime. Now, concerning the so called ‘asymmetry‘ order parameter in Eq. (5.11). We observe nearly 72 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD 1 0.8 (0,0,0,0) (1,0,0,0) (1,1,0,0) (1,1,1,0) (2,1,1,0) (2,2,1,0) 0.8 0.6 0.4 0.4 θ ψ 0.2 (0,0,0,0) (1,0,0,0) (1,1,0,0) (1,1,1,0) (2,1,1,0) (2,2,1,0) 0.6 0 -0.2 0.2 0 -0.4 -0.2 -0.6 -0.8 -0.4 0.5 1 1.5 2 T /Tc 2.5 3 3.5 0.5 1 1.5 2 2.5 3 3.5 T /Tc FIGURE 5.7: Additional improved observables reacting to the phase transition. ψ defined as 1 − DT /DL (l.h.s.) and the so called ’asymmetry’ order parameter θ = (DL − DT )/(DL + DT ) (r.h.s.). All data are obtained at β = 6.337 on a lattice with spatial size Nσ = 48. the same scenario as for ψ. This means, we see a rise for all momenta below Tc and fall down above reaching then negative values. However, all momenta are behaving nearly with the same strength and the errors computed are quite large. Concluding one should say that even if ψ and θ are quite good quantities to determine Tc they take negative values. In other words, the drawback is that these quantities do not take zero values above Tc as an ideal order parameter would do. Therefore, ψ and θ should not be taken as order parameter for pure QCD. To summarize, our gluon propagator results show agreements with findings in other recent investigations [42, 152, 43, 50, 149]. We observed that the strongest response to the phase transition occur in the gluonic chromoelectric sector (the longitudinal propagator) rather than in the gluonic chromomagnetic one (the transverse propagator). Our results are more focused on the mid-range of momenta ([0.6 : 8.0] GeV), namely around 1 GeV, which are interesting for the DSE as input data. Regarding the investigation of the ghost propagator at finite temperature we computed the ghost propagator according to Eq. (4.13), restricting it for simplicity to the diagonal threemomenta and vanishing M ATSUBARA frequency, kµ = (k, k, k, 0) with k = 1, . . . , 7. The data are again normalized at µ = 5 GeV, such that the ghost dressing function equals unity at q = µ. The result for the latter function is displayed in Fig. 5.8. In comparison with the gluon propagator we see the ghost propagator to change relatively weakly with the temperature (we are using here the logarithmic scale instead of the linear one as for the gluon results). This is in agreement with the observation in [42]. An increase becomes visible at temperature values T > 1.4 Tc for the lowest momenta studied (see Fig. 5.8). The relative insensitivity with respect to the temperature is the reason why we will not further consider the ghost propagator in what follows. Sec. 5.3. 73 Results on the gluon and ghost propagators at T > 0 0.65 0.74 0.86 0.99 1.20 1.48 1.98 2.97 1 Tc Tc Tc Tc Tc Tc Tc Tc 1.3 J(q, T )/J(q, Tmin ) G(q)[GeV −2 ] 10 0.1 1 1.3 0.6 1 1.4 1.8 2.2 2.6 3 0.6 1 1.4 1.8 2.2 2.6 3 0.6 1 1.4 1.8 2.2 2.6 3 1 1.3 1 0.01 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 q [GeV] T /Tc FIGURE 5.8: The momentum dependence of the ghost propagator G(q) in physical units for different temperatures (l.h.s.). The renormalized ghost dressing function J(q, T ) for various temperature values (r.h.s.) and its dependence on the temperature shown for the fixed diagonal 3-momenta ((k, k, k, 0), k = 1, 2, 3) and normalized with J(q, Tmin ) for Tmin = 0.65Tc (r.h.s.). The lowest panel shows the lowest momentum. All data are obtained at β = 6.337 on a lattice with spatial size Nσ = 48. 5.3.3 Study of the systematic effects Within this section we concentrate on the study of the systematic effects exclusively on the gluon propagator. We start investigating the role of the P OLYAKOV sector choice on the values of the components of the gluon propagators, namely: DT and DL . Remember, we have already discussed this point at the beginning of Section 5.3.1. We said in brief that this P OLYAKOV sector dependence holds for higher couplings. In fact, this original study motivated us to concentrate only on the physical sector of the P OLYAKOV loops. Next, we move to a systematic study of the finite volume effects above and below Tc . Lastly, the G RIBOV problem is explored on the lattice giving more insight into the problematic of the choice of gauge copies for the momenta range [0.6 : 8.0] GeV. 5.3.4 The P OLYAKOV sector effects The P OLYAKOV loop, as discussed before, represents one of the most important observables in QCD at finite temperature. The P OLYAKOV loop is defined by Eq. (4.1) as the trace of the ordered product of gauge link variables in time direction. As the P OLYAKOV loop (at each spatial) point is a translational invariant quantity one needs also to average it over space indices as defined in Eq. (5.4). This latter complex number quantity might be written for each configuration as L = Re(L) + i Im(L) (5.12) with the phase φ = Im(L) Re(L) . Therefore, each gauge configuration map uniquely to some value of φ . The P OLYAKOV values displayed in the complex plane lie in one of three sectors delimited 74 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD by the center elements of the SU(3) group. The physical sector is defined only for phases φ satisfying −π/3 < φ < π/3. (5.13) The other sectors, namely the second and the third one are considered unphysical. What we did in this work was to compare the effect on the gluon propagator on configurations lying to different P OLYAKOV sectors. Here in Fig. 5.9 we compare two sectors, namely the physical sector to the second sector (with π/3 < φ < π). We use in this case data at Tc with nσ = 48. Moreover, the data are presented in physical units. We observe on the right hand side the dramatic change in values for DL for momenta below 1.8 GeV. This change rises while moving to lower momenta. On the contrary, DT looks not so effected by the P OLYAKOV sector change for all momenta. The strong jump in values for DL might be explained by the fact that DL is a function of the temporal gauge links U4 as well as the P OLYAKOV loop itself. Therefore, any jump from sector to another one might affect the values of DL as it is the case for P OLYAKOV loop. We conclude that the role of the choice of P OLYAKOV loop sector is important to get reasonable propagator data. We have seen that the most dramatic reaction happens to the longitudinal propagator DL , and DT seems not to be sensible. Still, it is not clear that such difference might also happen in the (de)confined phase of QCD especially in the continuum limit, i.e. higher β . This should be an interesting topic to be investigated in the future. Therefore, we decided for data production always to stick to the physical sector of the P OLYAKOV loop. That is, we took in consideration only configurations lying in the physical sector. In practice, after producing configurations randomly according to the Monte-Carlo process we evaluated each corresponding P OLYAKOV loop sector. In case the sector is not physical we apply an appropriate rotation exp(i θ ) to the gauge links to come back to the physical sector. Therefore, in our pure gauge investigations we computed our gluon propagators only on configurations in the physical sector4 . 5.3.5 Finite volume effects In order to assess finite-volume effects is to compare the data shown before in Fig. 5.4 with the ones obtained on even larger spatial volumes while keeping fixed the coupling (at β = 6.337) and two temperature values, T = 0.86 Tc (confinement) and T = 1.2 Tc (deconfinement), respectively. Within this study the linear spatial extent varies from 48a = 2.64 fm to 64a = 3.52 fm (see also the middle section in Table 5.1). In Fig. 5.10 and Fig. 5.11 we show the corresponding plots for DL and DT , respectively. In all four cases we observe the effects to be small for momenta above 0.6 GeV.5 For lower momenta, 4 This 5 is possible because Z(3) is still a global symmetry of the pure gauge action. This is not anymore the case in the fermionic case. Below Tc the transverse propagator changes by less than 12 %, the longitudinal one by less than 5 %. Above Tc the transverse propagator varies by less than 8 % and the longitudinal one by less than 11 %. Sec. 5.3. 75 Results on the gluon and ghost propagators at T > 0 1000 10 Second Polyakov sector Real Polyakov sector Second Polyakov sector Real Polyakov sector DT [GeV −2 ] DL [GeV −2 ] 100 10 1 1 0.1 0.1 0 0.5 1 1.5 2 2.5 3 0 q [GeV] 0.5 1 1.5 2 2.5 3 q [GeV] FIGURE 5.9: Comparative study of behavior of DL (l.h.s.) and DT (r.h.s.) in physical units evaluated on the physical (real) and the second P OLYAKOV sectors. These data correspond to critical temperature Tc . especially at zero momentum, systematic deviations become more visible. With increasing volume the infrared values of DL seem to rise, whereas for DT the opposite is the case. This behavior has already been reported for pure gauge theories in [41, 44] for SU(2) and in [50] for SU(3), respectively. 5.3.6 The G RIBOV ambiguity investigated The present study of the G RIBOV copies was designed to determine the effect of the choice of the gauge copies to compute gauge-dependent quantities as the gluon propagators. To study G RIBOV copy effects we compare “first”, i.e. randomly occurring copies (fc) with “best” copies (bc) 6 . The copies in question were produced as follows: we searched for copies within all 33 = 27 Z(3) sectors characterized by the phase of the spatial P OLYAKOV loops, i.e. P OLYAKOV loops in one of the three spatial directions. For this purpose the Z(3) flipping operations [34, 50] were carried out on all link variables Ux,i (i = 1, 2, 3) attached and orthogonal to a 3D hyperplane with fixed xi by multiplying them with exp {±2πi/3}. Such global flips are equivalent to non-periodic gauge transformations and do not change the pure gauge action. For the 4th direction, we stick to the sector with |arg P| < π/3 which provides maximal values of the functional Eq. (3.28) at the β -values considered in this section [50]. Thus, the flip operations combine for each lattice field configuration the 27 distinct gauge orbits of strictly periodic gauge transformations into one larger gauge orbit. The number of copies actually considered in each of the 27 sectors depends on the rate of convergence (with increasing number of investigated copies) of the propagator values assigned to the best copy, in particular at zero momentum. From our experience with SU(3) theory [50] 6 This strategy is called sometimes in the literature the fc-bc method 76 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD 40 35 4.5 4 25 20 15 3.5 3 2.5 2 10 1.5 5 0 -0.2 643 × 10 563 × 10 483 × 10 5 DL [GeV −2 ] 30 DL [GeV −2 ] 5.5 643 × 14 563 × 14 483 × 14 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 -0.2 1.6 0 0.2 0.4 q [GeV] 0.6 0.8 1 1.2 1.4 1.6 q [GeV] FIGURE 5.10: Finite-size effect study for DL at β = 6.337. l.h.s.: T = 0.86 Tc , r.h.s.: T = 1.20 Tc . 7 6 4.5 4 4 3 3.5 3 2.5 2 2 1.5 1 0 -0.2 643 × 10 563 × 10 483 × 10 5 DT [GeV −2 ] 5 DT [GeV −2 ] 5.5 643 × 14 563 × 14 483 × 14 1 0 0.2 0.4 0.6 0.8 q [GeV] 1 1.2 1.4 1.6 0.5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 q [GeV] FIGURE 5.11: Same as in Fig. 5.10 but for DT . we expect that the effect of considering gauge copies in different flip-sectors is more important than probing additional gauge copies in each sector. For this reason and to save CPU time we have considered one gauge copy for every Z(3)-sector; therefore, in total ncopy = 27 gauge copies for every configuration. To each copy the simulated annealing algorithm with consecutive over-relaxation was applied in order to fix the gauge. We take the copy with maximal value of the functional Eq. (3.28) as our best realization of the global maximum and denote it as “bc”. The parameters of the SA algorithm in the study of G RIBOV copies were slightly different from those described above in Section 5.1.3. 2000 SA combined simulation sweeps with a ratio 11:1 between micro-canonical and heat bath sweeps were applied starting with Tsa = 0.5 and ending at Tsa = 0.0033. Since this procedure is quite CPU time consuming we restricted this investigation to coarser Sec. 5.3. 35 fc bc −2 ] 25 20 DL [GeV −2 ] 30 DL [GeV 77 Results on the gluon and ghost propagators at T > 0 15 10 5 0 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.2 fc bc 0 0.2 0.4 q [GeV] 0.6 0.8 1 1.2 1.4 1.6 q [GeV] fc bc −2 ] 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 -0.2 DT [GeV DT [GeV −2 ] FIGURE 5.12: Comparison of the bc with the fc G RIBOV copy result for the longitudinal propagator DL (unrenormalized) (l.h.s.: T = 0.86 Tc , r.h.s.: T = 1.20 Tc ). 0 0.2 0.4 0.6 0.8 q [GeV] 1 1.2 1.4 1.6 4.5 4 3.5 3 2.5 2 1.5 1 0.5 -0.2 fc bc 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 q [GeV] FIGURE 5.13: Same as in Fig. 5.12 but for the transverse propagator DT (l.h.s.: T = 0.86 Tc , r.h.s.: T = 1.20 Tc ). lattices 6 × 283 and 8 × 283 with larger lattice spacing keeping fixed the temperature to both values T = 0.86 Tc and T = 1.20 Tc , respectively. Moreover, the physical 3D volume (2.64 fm)3 were also approximately fixed. In Fig. 5.12 and Fig. 5.13 we compare bc with fc results for the gluon propagator components DL and DT , respectively. As one can see, DL is almost insensitive to the choice of G RIBOV copies (at least, for the comparatively small values of Nτ we consider). This observation has been already reported in [44] for the SU(2) case and in [50] for the SU(3) case. On the contrary, the transverse propagator is strongly affected in the infrared region. This behavior is independent of the temperature. Moreover, we see that the transverse gluon propagator values in the infrared become lowered for bc compared with fc results. These observations resemble those made already in [44] and [50]. To complement our results we have also studied the ratios DL ( f c)/DL (bc) and DT ( f c)/DT (bc). The result is shown in Fig. 5.14. We observe that for DL ratios fall down very 78 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD 1.5 1.5 DT DL DT DL 1.4 D(f c)/D(bc) D(f c)/D(bc) 1.4 1.3 1.2 1.1 1 1.3 1.2 1.1 1 0.9 0.9 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 q [GeV] 1 1.5 2 2.5 3 3.5 4 q [GeV] FIGURE 5.14: The ratios DL ( f c)/DL (bc) and DT ( f c)/DT (bc) as a function of the physical momenta q[GeV ]. These data were produced for β = 5.994 and (nσ ,nτ )=(28,6). These data correspond to the temperatures 1.20 Tc (r.h.s.) and 0.86 Tc (l.h.s.). quickly for non-zero momenta. This behavior is the same for the deconfined (1.20 Tc ) and the confined (0.86 Tc ) phases. These effects are getting smaller with higher momenta. Therefore, except for the zero momenta where we see a deviation of 20 % in the ratio higher momenta might be neglected. Regarding DT ratios one sees that deviations in ratios reach 40 % in deconfined phase and 35 % in the confined one. However, we observe that above Tc points are not joining the horizontal unity line as fast as the DL case. That is, DT is a bit more affected by G RIBOV effects for the low momenta. Still, we point out that a roughly stable behavior with small fluctuations is reached for p > 800 MeV . This looks to be the case for the deconfined and the confined phase as well. The main conclusion of this section is that G RIBOV copy effects may be neglected for all nonzero momenta in the case of the longitudinal propagator (at least, for comparatively small values of Nτ ), and for momenta above 800 MeV in the case of the transverse propagator. The momentum range where the last statement is true might depend on the temperature. 5.3.7 Scaling effects study and the continuum limit Throughout the present section we show results of the gluon propagator regarding scaling properties and the continuum limit in pure gauge theory. Our strategy is to perform this systematic study moving to smaller values of the lattice spacing keeping the values of other parameters fixed, i.e. studying the gluon propagator following the limit: lima→0 limVphys =fixed limT =fixed . This order of limits is crucial and must not be inverted. Hence, we exploited in this investigation data already presented in Section 5.3.5 at the two fixed temperatures 0.86 Tc and 1.20 Tc , and also fixed physical volume to (2.7 fm)3 . Therefore, in order to check the scaling properties we have used the same reference values for the temperature below and above Tc as discussed before (i.e., 0.86 Tc and 1.20 Tc ). We kept Sec. 5.3. T /Tc 0.86 0.86 0.86 0.86 1.20 1.20 1.20 1.20 79 Results on the gluon and ghost propagators at T > 0 Parameters β Nσ 5.972 28 6.230 42 6.337 48 6.440 56 5.994 28 6.180 38 6.337 48 6.490 58 Nτ 8 12 14 16 6 8 10 12 Z-factors Z̃T Z̃L 1.43 1.43 1.45 1.47 1.48 1.53 1.64 1.66 1.46 1.46 1.52 1.52 1.62 1.63 1.62 1.65 r2 (GeV2 ) 0.317(20) 0.254(9) 0.262(12) 0.256(7) 0.995(37) 0.985(20) 0.960(19) 1.018(18) DL fits d(GeV−2 ) 0.138(24) 0.224(7) 0.224(11) 0.220(6) 0.153(10) 0.163(6) 0.180(7) 0.162(5) c(GeV2 ) 4.67(26) 3.90(8) 3.80(12) 3.86(7) 5.46(24) 5.34(13) 4.96(13) 5.27(11) χd2 f 0.30 0.44 0.42 0.24 0.80 0.28 0.22 0.06 r2 (GeV2 ) 0.810(23) 0.835(16) 0.867(18) 0.880(15) 0.894(26) 0.924(22) 0.982(27) 0.963(19) DT fits d(GeV−2 ) 0.148(7) 0.151(5) 0.142(6) 0.143(4) 0.144(7) 0.142(6) 0.133(8) 0.140(5) c(GeV2 ) 5.49(17) 5.69(12) 5.62(14) 5.65(11) 5.55(18) 5.71(16) 5.87(21) 5.77(13) χd2 f 1.19 0.52 0.14 0.36 1.10 0.57 0.59 0.45 TABLE 5.3: Left panel: Renormalization factors Z̃T,L of the renormalized propagators DT,L (q, µ) according to Eq. (5.14). The renormalization point is µ = 5 GeV. Right panels: Fit parameters and χd2 f for fits of DL (l.h.s.) and DT (r.h.s.) using the generic fit function D(q2 ) acc. to Eq. (5.6), but with b = 0.The fit range is restricted to [0.6 : 3.0] GeV. The fit errors are indicated in parentheses. also the spatial volume fixed at (2.7 fm)3 , and compared the renormalized propagators at four different values for the lattice spacing a(β ) (see Table 5.1). Our results are displayed for the momentum range up to 1.5 GeV in Fig. 5.15 for DL and in Fig. 5.16 for DT , respectively. Gauge fixing has been carried out as originally described in Section 5.1.3. We provide the renormalization factors for DL,T (q, µ) ≡ Z̃L,T (a, µ) Dbare L,T (q, a) (5.14) in the left panel of Table 5.3. As expected the Z-factors of DL and DT approximately agree. From Fig. 5.15 and Fig. 5.16 we see that the scaling violations happen to be reasonably small for momenta above 0.8 GeV. This shows that our choice of a = 0.055 fm for β = 6.337 was already close to the continuum limit. To determine the a-dependence at five particular physical momenta p we need interpolations of the momentum dependence in between the data points. For the fit within the interval 0.6 GeV ≤ q ≤ 3.0 GeV we have used again Eq. (5.6) with parameter b fixed to zero. The values of the fit parameters are displayed in the right hand panels of Table 5.3. In all cases we find χ 2 -values per degree of freedom around or below unity. The propagators, interpolated to the set of selected momentum values, are shown in Fig. 5.17 for DL and in Fig. 5.18 for DT , respectively, as functions of the lattice spacing a. We show them together with the respective fit curves D(a; p) = D0 + B · a2 (5.15) assuming only O(a2 ) lattice artifacts. The corresponding fit results are collected in Table 5.4. The respective fit parameters D0 represent the continuum limit values of the propagators at the preselected momenta. Our lattice propagator data obtained for β = 6.337 as discussed in Section 5.3.1 can now be 80 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD Parameters T /Tc p(GeV ) 0.86 0.70 0.86 0.85 0.86 1.00 0.86 1.20 0.86 1.40 1.20 0.70 1.20 0.85 1.20 1.00 1.20 1.20 1.20 1.40 DL fits DT fits −2 B D0 (GeV ) B D0 (GeV −2 ) -1.3(28.1) 7.68(16) 32.3(20.0) 3.20(11) 13.5(14.5) 4.63(8) 19.5(14.0) 2.42(8) 12.3(7.9) 2.95(4) 11.7(9.8) 1.83(5) 7.0(4.1) 1.75(2) 5.9(6.4) 1.27(4) 3.0(2.6) 1.12(1) 3.2(4.4) 0.90(2) 23.1(9.3) 2.48(5) 30.7(11.0) 2.84(6) 15.8(6.5) 1.93(4) 18.5(7.4) 2.19(4) 11.8(4.7) 1.49(2) 10.8(4.8) 1.68(2) 7.4(3.3) 1.07(2) 5.3(3.0) 1.19(2) 4.8(2.2) 0.77(1) 2.4(1.8) 0.86(1) TABLE 5.4: Results of the fits for DL (l.h.s.) and DT (r.h.s.) as a function of the lattice spacing a using the fit function D(a; p) acc. to Eq. (5.15). The errors of the fit parameters are given in parentheses. χd2 f in all cases is close or well below unity. See also Fig. 5.17 and Fig. 5.18. compared with the values extrapolated to the continuum limit. This is shown in Fig. 5.19. In more detail, we can compare the continuum extrapolated values at some lower momentum – say at q = 0.70 GeV – with those obtained from a(β = 6.337) = 0.055 fm and interpolated to the same momentum. Then we find deviations being smaller than 4 %. Thus, we are really justified to say that the results obtained for β = 6.337 in the given momentum range are already very close to the continuum limit. Additionally, the continuum limit extrapolated propagators can be easily fitted with Eq. (5.6). The results are shown in Fig. 5.20. We conclude that for the higher β -values and the momentum range considered in this thesis we are close to the continuum limit. Moreover, systematic effects as there are finite-volume and G RIBOV copy effects seem to be negligible for momenta above 0.8 GeV. 81 Results on the gluon and ghost propagators at T > 0 40 35 30 25 20 15 10 5 0 6 283 × 8, β = 5.972 423 × 12, β = 6.230 483 × 14, β = 6.337 563 × 16, β = 6.440 283 × 6, β = 5.994 383 × 8, β = 6.180 483 × 10, β = 6.337 583 × 12, β = 6.490 −2 ] 5 DL [GeV DL [GeV −2 ] Sec. 5.3. 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 q [GeV] 0.6 0.8 1 1.2 1.4 q [GeV] FIGURE 5.15: The longitudinal propagator DL , renormalized at µ = 5 GeV, obtained for fixed physical volume and temperature but varying a = a(β ). l.h.s.: T = 0.86 Tc , r.h.s.: T = 1.20 Tc . 7 4 −2 ] 5 283 × 6, β = 5.994 383 × 8, β = 6.180 483 × 10, β = 6.337 583 × 12, β = 6.490 5 DT [GeV −2 ] 6 DT [GeV 6 283 × 8, β = 5.972 423 × 12, β = 6.230 483 × 14, β = 6.337 563 × 16, β = 6.440 3 2 4 3 2 1 1 0 0 0 0.2 0.4 0.6 0.8 q [GeV] 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 q [GeV] FIGURE 5.16: Same as in Fig. 5.15 but for the transverse propagator DT . l.h.s.: T = 0.86 Tc , r.h.s.: T = 1.20 Tc 82 CHAPTER 5. RESULTS IN THE PURE GAUGE SECTOR OF QCD 6 14 p=0.70 GeV p=0.85 GeV p=1.00 GeV p=1.20 GeV p=1.40 GeV −2 ] 8 p=0.70 GeV p=0.85 GeV p=1.00 GeV p=1.20 GeV p=1.40 GeV 5 DL [GeV −2 ] 10 DL [GeV 12 6 4 4 3 2 1 2 0 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.02 0.04 a [fm] 0.06 0.08 0.1 0.12 a [fm] FIGURE 5.17: DL vs. lattice spacing a for a set of different preselected momenta p. l.h.s. T = 0.86 Tc ; r.h.s. T = 1.20 Tc . 7 7 p=0.70 GeV p=0.85 GeV p=1.00 GeV p=1.20 GeV p=1.40 GeV 4 −2 ] 5 p=0.70 GeV p=0.85 GeV p=1.00 GeV p=1.20 GeV p=1.40 GeV 6 DT [GeV DT [GeV −2 ] 6 3 2 1 5 4 3 2 1 0 0 0 0.02 0.04 0.06 a [fm] 0.08 0.1 0.12 0 0.02 0.04 0.06 0.08 0.1 a [fm] FIGURE 5.18: Same as in Fig. 5.17 but for DT . l.h.s. T = 0.86 Tc ; r.h.s. T = 1.20 Tc . 0.12 83 Results on the gluon and ghost propagators at T > 0 9 8 7 6 5 4 3 2 1 0 −2 ] a=0.055 fm, T/Tc = 1.20 a=0.055 fm, T/Tc = 0.86 a→0 fm, T/Tc = 1.20 a→0 fm, T/Tc = 0.86 DT [GeV DL [GeV −2 ] Sec. 5.3. 0.6 0.8 1 1.2 1.4 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 1.6 a=0.055 fm, T/Tc = 1.20 a=0.055 fm, T/Tc = 0.86 a→0 fm, T/Tc = 1.20 a→0 fm, T/Tc = 0.86 0.6 0.8 1 q [GeV] 1.2 1.4 1.6 q [GeV] 9 8 7 6 5 4 3 2 1 0 −2 ] T/Tc = 1.20 T/Tc = 0.86 DT [GeV DL [GeV −2 ] FIGURE 5.19: Comparison of the renormalized propagators DL (q) (l.h.s.) and DT (q) (r.h.s.) obtained from the Monte Carlo simulation at β = 6.337 with some continuum limit extrapolated values. 0.4 0.6 0.8 1 q [GeV] 1.2 1.4 1.6 9 8 7 6 5 4 3 2 1 0 T/Tc = 1.20 T/Tc = 0.86 0.4 0.6 0.8 1 1.2 1.4 1.6 q [GeV] FIGURE 5.20: Continuum extrapolated values of DL (q) (l.h.s.) and DT (q) (r.h.s.) together with their respective interpolation curves for two temperature values. CHAPTER 6 Results for full QCD W investigate the temperature behavior of the L ANDAU gauge gluon and ghost propagators in the QCD sector of two flavors (NF = 2) maximally twisted mass fermions. We focus on the set of pion masses, 300 MeV << mπ << 500 MeV reaching the smallest lattice spacing of around 0.06 f m . We study the dependence of the gluon propagator as a function of the temperature. We show that at lower momenta the gluon propagator behaves smoothly in the crossover region. Still, the longitudinal component DL seems to react stronger as the transversal part DT specially for zero momenta. The ghost propagator on the other hand demonstrates only a very weak temperature dependence. A good parametrization to our gluon data was also undertaken thanks to the G RIBOV-S TINGL formula for the momenta range [0.4, 3.0] GeV. In fact, this latter range is important for DSE whose different solutions deviate from each other around 1 GeV. In general, our data may serve as input for general continuum functional methods. During our present work we relied on the thermalized configurations provided by the tmfT collaboration. These configurations were generated on a four-dimensional lattice of spatial size of Nσ = 32 and a temporal one of Nτ = 12. Here, one considers QCD with a mass-degenerate doublet of twisted mass fermions, cf. the review by [121]. The corresponding gauge action is a combination of a tree-level S YMANZIK improved gauge action and the twisted-mass action for the fermionic part. We refer to Chap. 3 for a precise description of such actions. Our results are presented in [52]. E . 6.1 Lattice setting and parameters As already said, we consider the four-dimensional periodic lattice of spatial size of Nσ = 32 and a temporal one of Nτ = 12. Configurations were thermalized and provided to us by the tmfT collaboration. We had the task to gauge fix the configuration to the L ANDAU gauge using simulated annealing followed by over-relaxation as we did for the pure gauge case. The configurations are 85 86 CHAPTER 6. RESULTS FOR FULL QCD provided at the maximal twist by tunning the hopping parameter to its critical value kc . Therefore, at maximal twist we get an automatic a-improved fermion formulation [121]. Our hopping parameters of interest are based on the values of β provided by the European Twisted Mass Collaboration (ETMC) [122]. Furthermore, kc for intermediate lattice spacing a(β ) values were obtained thanks to interpolations as practized in [123]. As usual at finite temperature QCD, the imaginary-time extent corresponds to the inverse temperature T −1 = Nτ a. To be able to set the physical scale for each β we interpolated the data provided by the ETMC collaboration [122] at the β values of 3.90, 4.05, 4.2. This allowed us to map each β value to some lattice spacing a in F ERMI for example, and so to be able to construct the momenta and the propagators in physical units. We are reaching during the present investigations lattice spacings a < 0.09 f m. The values of our parameters were already investigated thanks to the chiral condensate and the Polyakov loop and as well as their susceptibilities within the tmfT collaboration [153, 154, 123, 124], and a very smooth behavior was found. Furthermore, one observes signals for a a breakdown of chiral symmetry and a deconfinement transition at slightly different temperatures Tc = Tχ and Tdeconf , respectively, in agreement with the observation reported in [108]. Our runs parameters are listed on Table 6.1. In this table we show the β values together with the corresponding lattice spacings, the temperature and the independent number of configurations for the three sets of pion masses. In order to reduce systematic hyper-cubic momentum selection effects, we have applied for most of our computations the so-called cylinder cut [113]. Therefore, we took in considerations only diagonal and first off-diagonal momenta for the gluon propagator and strictly diagonal momenta for the ghost momenta. Moreover, only modes with zero M ATSUBARA frequencies are studied, i. e. k4 = 0. Now, according to refs. [123, 124], we display in Table 6.2 the (pseudo-) critical couplings βc and the corresponding temperatures Tχ and Tdeconf for the three pion mass values as obtained 2 and from the behavior of from fits around the respective maxima of the chiral susceptibility σψψ the (renormalized) Polyakov loop hRe(L)iR , respectively 6.2 Results on the gluon and ghost propagators 6.2.1 Fitting the bare gluon and ghost propagators The temperatures of interest in this work correspond to the temperatures range covering temperatures where the chiral restoration and deconfinement are expected to happen. These two letter phenomena occur at two different temperatures denoted respectively by Tχ and Tdeconf . However, the values of these temperatures within the lattice community are still under discussion. As an example, for QCD with 2+1 flavors authors in [155] show a temperature separation of about 20 − 30 MeV between the deconfining and the chiral transition temperatures. This latter result conflicts with data provided by others [156] who claim both temperatures to coincide. Sec. 6.2. 87 Results on the gluon and ghost propagators mπ [MeV] β a [fm] T [MeV] 316(16) 3.8400 8.77(47)·10−2 187 316(16) 3.8800 8.25(22)·10−2 199 316(16) 3.9300 7.65(13)·10−2 215 316(16) 3.9525 7.39(12)·10−2 222 316(16) 3.9600 7.31(12)·10−2 225 316(16) 3.9675 7.22(11)·10−2 228 316(16) 3.9750 7.14(11)·10−2 230 316(16) 3.9900 6.98(11)·10−2 235 398(16) 3.8600 8.51(32)·10−2 193 398(16) 3.8800 8.25(22)·10−2 199 398(16) 3.9300 7.65(13)·10−2 215 398(16) 3.9700 7.20(11)·10−2 228 398(20) 3.9900 6.98(11)·10−2 236 398(16) 4.0050 6.82(10)·10−2 241 398(20) 4.0175 6.69(10)·10−2 246 398(20) 4.0250 6.62(10)·10−2 248 398(20) 4.0400 6.47(10)·10−2 254 469(24) 3.9500 7.42(12)·10−2 222 469(24) 3.9700 7.20(11)·10−2 228 469(24) 3.9900 6.98(11)·10−2 235 469(24) 4.0100 6.77(10)·10−2 243 469(24) 4.0200 6.67(10)·10−2 247 469(24) 4.0300 6.57(10)·10−2 250 469(24) 4.0400 6.47(10)·10−2 254 469(24) 4.0500 6.38(10)·10−2 258 469(24) 4.0700 6.19(12)·10−2 266 r0 /a 4.81 5.17 5.63 5.84 5.91 5.98 6.05 6.19 4.99 5.17 5.63 6.00 6.19 6.34 6.46 6.53 6.68 5.81 6.00 6.19 6.39 6.48 6.58 6.68 6.78 6.98 r0 · T 0.40 0.43 0.47 0.49 0.49 0.50 0.50 0.52 0.42 0.43 0.47 0.50 0.52 0.53 0.54 0.54 0.56 0.48 0.50 0.52 0.53 0.54 0.55 0.56 0.56 0.58 ncon f 293 299 255 273 151 250 113 290 159 173 209 198 156 150 271 226 113 146 348 120 210 250 256 152 150 200 κc a · µ0 0.162731 0.00391 0.161457 0.00360 0.159998 0.00346 0.159385 0.00335 0.159187 0.00331 0.158991 0.00328 0.158798 0.00325 0.158421 0.00319 0.162081 0.00617 0.161457 0.00600 0.159998 0.00561 0.158927 0.00531 0.158421 0.00517 0.158053 0.00506 0.157755 0.00498 0.157579 0.00493 0.157235 0.00483 0.159452 0.00779 0.158926 0.00752 0.158421 0.00738 0.157933 0.00718 0.157696 0.00708 0.157463 0.00699 0.157235 0.00689 0.157010 0.00680 0.156573 0.00662 Z̃T Z̃L Z̃J 0.6380(80) 0.6264(108) 0.66862(32) 0.6208(34) 0.6139(50) 0.66939(10) 0.6117(70) 0.6116(103) 0.67264(14) 0.6156(59) 0.6122(84) 0.67252(15) 0.6151(71) 0.6101(102) — 0.6120(63) 0.6079(114) 0.67388(14) 0.6146(77) 0.5982(126) — 0.6092(71) 0.6118(97) 0.67536(14) 0.6191(64) 0.6147(98) 0.66730(21) 0.6202(54) 0.6192(76) — 0.6076(59) 0.6080(84) 0.67005(23) 0.6087(59) 0.6119(89) — 0.6075(139) 0.6090(231) 0.67293(13) 0.6156(87) 0.6112(115) 0.6784(60) 0.6036(78) 0.6063(101) 0.67504(16) 0.6001(64) 0.6005(94) 0.67517(11) 0.6029(119) 0.6167(178) 0.67519(26) 0.6121(55) 0.6020(82) 0.67142(16) 0.6116(80) 0.6024(110) 0.67128(14) 0.6098(70) 0.6041(102) 0.67271(22) 0.6086(54) 0.6093(73) 0.67322(12) 0.5947(54) 0.5927(77) 0.67147(14) 0.6013(72) 0.6017(101) 0.67388(14) 0.6033(80) 0.6028(121) 0.67353(16) 0.5971(62) 0.6072(91) 0.67485(17) 0.5972(143) 0.6119(195) 0.67829(32) TABLE 6.1: The pion masses, the values of the inverse bare coupling β , the lattice spacing a in fm, the temperature T in MeV, the chirally extrapolated Sommer scale r0 [147] in lattice units and r0 T , and the number ncon f of configurations are shown for the ensembles. The spatial Nσ = 32 and temporal Nτ = 12 extents are the same for all ensembles. The number ncopy of gauge copies is fixed to 1. In the middle subtable the critical hopping parameter κc and the bare twisted mass aµ0 (in units of the lattice spacing) are also shown. The renormalization factors for the transverse and longitudinal gluon as well as for the ghost dressing function obtained for the renormalization scale µ = 2.5 GeV are given in the rightmost subtable, denoted as Z̃T , Z̃L and Z̃J , respectively. 88 CHAPTER 6. RESULTS FOR FULL QCD Label mπ [MeV] 2 βc from σψψ Tχ [MeV] βc from hRe(L)iR Tdeconf [MeV] A12 316(16) 3.89(3) 202(7) - B12 398(20) 3.93(2) 217(5) 4.027(14) 249(5) C12 469(24) 3.97(3) 229(5) 4.050(15) 258(5) TABLE 6.2: Extracted (pseudo-) critical couplings βc and corresponding temperatures for the ensembles A12, B12, and C12 (see revised version of [123]) corresponding to three different pion mass values mπ and a time-like lattice extent Nτ = 12 . Investigations for the case NF = 2 using improved W ILSON fermions [157] are also supporting a coincidence of both temperatures within errorbars. Moreover, the value of transition temperature is also another problematic issue. For example, the B ROOKHAVEN /B IELEFELD collaboration [158] using staggered fermions gets for transition temperature Tc = 196(3) MeV, which is much higher than the transition temperatures found by the W UPPERTAL group [155] for the deconfining and chiral transitions - Tdeconf = 170(7) MeV, and Tχ = 146(5) MeV, respectively. This discrepancy might be resolved thanks to the use of finer lattices [155]. Thereby, one understand the importance to check the consistency of these results preferably using different lattice discretizations, and to compare between them. Therefore, many efforts are devoted to investigate different discretizations to understand the nature of the finite temperature phase transition as in [156] (domain wall), [154] (twisted mass) and [159] (improved W ILSON fermions). We provide hereafter gluon and ghost propagator data using the twisted-mass discretization which might be confronted for example to [50] using improved W ILSON fermions. Quite recently, in [160] quarks flavors effects on the ghost and gluon propagators have been studied for the cases NF = 2 and NF = 2 + 1 and for mass range of 270 to 510 MeV using twisted mass fermions. In this last reference no dependence on the temperature for the gluon and ghost propagators was found. Moreover, the authors observed a decrease of the gluon propagator with increasing the number of flavors while the ghost propagator is slightly enhanced. We present in Fig. 6.1 first results for the bare, i.e. unrenormalized dressing functions for a few β (or equivalently temperatures) for the pion masses mπ = 469, 398 and 316 MeV . The unrenormalized transverse ZT and longitudinal ZL gluon dressing functions together with the unrenormalized ghost dressing function J as a function of the physical momentum q for a few temperatures indicated in the legends by their β -values. We agreed to take the renormalization point for this fermionic study µ = 2.5 GeV. In order to find the renormalized dressing function one needs to apply the following relation ren (q, µ) ≡ Z̃T,L (µ)ZT,L (q) ZT,L J ren (q, µ) ≡ Z̃J (µ)J(q) (6.1) Sec. 6.2. 89 Results on the gluon and ghost propagators ren (µ, µ) = J ren (µ, µ) = 1 and are shown in Table 6.1. with the Z̃-factors defined such that ZT,L A further observation is that the values of Z̃T and Z̃L are quite close to each other independent of the temperatures considered. This difference in values does not exceed 5% for all pion masses. Therefore, we conclude that the ultraviolet part of our gluon propagator is not affected with crossing any pseudo-critical temperature, and remains not phase sensitive. The aim is to present data set to be used as an input for DS or FRG equations within the momentum region 0.4 GeV ≥ q ≥ 3.0 GeV. We present these data in terms of fitting formulas for the bare gluon and ghost dressing functions. At a first glance one observes from Fig. 6.1 that the the behavior of ZL to be quite different from the behavior of ZT , whatever mπ values are considered. The transverse dressing function ZT shows less response to the temperature while the curves describing ZL (q) fan out for momenta below the renormalization scale µ = 2.5 GeV according to the β values (or temperatures). This observation was already made us previously in the case of pure gauge theory [49]. Still in this latter case the strong temperature response of DL in comparison to the present fermion case is due to the existence of a confirmed first order phase transition in the pure gauge sector of QCD. As we did before for the case of pure gauge theory (see [49]) we fit here again the gluon dressing function with the G RIBOV-S TINGL formula [26, 150] used in Refs. [151, 149] and derived in the so-called “Refined G RIBOV-Z WANZIGER” approach [161, 162]. We remind the reader that this fit function looks like Z f it (q) = q2 c (1 + d q2n ) . (q2 + r2 )2 + b2 (6.2) 2 Our fit results (with excellent χdo f values within the fitting range [0.4 GeV, 3.0 GeV]) for all available temperature values are displayed in Table 6.3. We have made many tests, and found out that this formula works nicely in the given range already without the b2 term and for a fixed exponent n = 1. However, it might happen that moving to the infrared (smaller momenta) this formula would generate a non-zero b2 -term. Therefore, in this case a pair of complex-conjugate (complex masses) would arise. For the ultraviolet region, and especially above 3 GeV, we had also encountered problems to describe the data with this fit formula as logarithmic corrections (important at this regime) are not taken into account. We have also tried to parametrize the ghost dressing function J using another type of fit function, namely J f it (q) =  f2 q2 k + h q2 . q2 + m2gh (6.3) Here, the fit parameter mgh plays a mass-like role. We examined different situations keeping this parameter as a free parameter. However, we got results consistent with mgh = 0. Therefore, we dropped this infrared mass parameter from our fit procedure. Our fit results are provided for the range [0.4 GeV, 4.0 GeV] and shown in Table 6.4. Because of small statistical errors on 90 CHAPTER 6. RESULTS FOR FULL QCD 3.8400 3.9300 3.9675 3.9900 3.8400 3.9300 3.9675 3.9900 2 3.8400 3.9300 3.9675 3.9900 2 ZT ZL J(q) 2 1 1 1 0 1 2 3 4 5 0 1 q[GeV] 2 3 4 5 1 q[GeV] 2 3.8600 3.9300 4.0050 4.0400 3.8600 3.9300 4.0050 4.0400 2 2 J(q) ZT 1 4 3.8600 3.9300 4.0250 4.0400 ZL 2 3 q[GeV] 1 1 0 1 2 3 4 5 0 1 q[GeV] 2 3 4 1 5 q[GeV] 2 3.95 4.01 4.04 4.07 3.95 4.01 4.04 4.07 2 3 4 q[GeV] 3.95 4.01 4.04 4.07 2 ZT ZL J(q) 2 1 1 1 0 1 2 q[GeV] 3 4 5 0 1 2 q[GeV] 3 4 5 1 2 3 4 q[GeV] FIGURE 6.1: The unrenormalized transverse gluon ZT (left panel), longitudinal gluon ZL (middle panel) dressing functions and the unrenormalized ghost dressing function J (right panel) as functions of the momentum q [GeV] for different (inverse) coupling values β (or temperatures) as given in the legend. The corresponding pion mass values (from top to bottom panels) are mπ ≃ 316, 398 and 469 MeV. Sec. 6.2. 91 Results on the gluon and ghost propagators Parameters T [MeV] 187 199 215 222 225 228 230 235 Parameters T [MeV] 193 199 215 228 236 241 246 248 254 Parameters T [MeV] 222 228 235 243 247 250 254 258 266 c/a2 1.334(132) 1.183(59) 1.032(101) 1.049(93) 0.932(103) 1.023(108) 0.938(119) 0.914(98) c/a2 1.271(116) 1.218(83) 0.982(91) 1.034(86) 1.171(265) 1.006(145) 0.922(133) 0.845(94) 0.824(173) c/a2 0.952(78) 0.993(100) 0.883(96) 0.998(82) 1.010(80) 0.900(87) 0.870(103) 0.826(89) 0.971(250) DL fits d/a2 0.744(138) 0.872(79) 1.013(188) 0.941(159) 1.148(233) 0.979(206) 1.165(277) 1.143(224) DL fits d/a2 0.799(143) 0.808(107) 1.092(177) 0.928(160) 0.699(430) 0.924(281) 1.087(297) 1.278(270) 1.252(502) DL fits d/a2 1.127(169) 1.003(199) 1.203(236) 0.945(157) 0.918(156) 1.108(223) 1.131(275) 1.251(260) 0.894(559) ar 0.415(14) 0.390(7) 0.370(13) 0.370(11) 0.355(14) 0.369(13) 0.359(15) 0.358(13) 2 χdo f 0.19 0.16 0.12 0.16 0.13 0.27 0.09 0.38 c/a2 1.868(142) 1.729(60) 1.371(104) 1.218(76) 1.208(88) 1.256(83) 1.099(93) 0.982(70) ar 0.401(13) 0.391(9) 0.356(12) 0.366(10) 0.383(42) 0.366(19) 0.356(18) 0.346(13) 0.348(24) 2 χdo f 0.19 0.11 0.11 0.67 0.21 0.15 0.12 0.11 0.08 c/a2 1.811(109) 1.637(88) 1.389(84) 1.151(71) 1.035(171) 1.092(104) 1.035(87) 1.020(78) 0.837(111) ar 0.348(10) 0.351(12) 0.343(13) 0.368(10) 0.356(9) 0.349(11) 0.340(14) 0.344(13) 0.371(32) 2 χdo f 0.35 0.32 0.19 0.23 1.08 0.67 0.25 0.25 0.01 c/a2 1.338(80) 1.290(104) 1.126(91) 1.009(61) 1.027(67) 1.096(79) 0.941(84) 1.053(69) 0.792(130) DT fits d/a2 0.420(78) 0.463(40) 0.647(111) 0.757(96) 0.744(122) 0.682(107) 0.852(156) 1.026(143) DT fits d/a2 0.459(66) 0.523(63) 0.651(86) 0.819(110) 0.965(310) 0.784(162) 0.891(164) 0.904(144) 1.237(329) DT fits d/a2 0.645(89) 0.648(127) 0.797(139) 0.937(115) 0.946(124) 0.775(135) 1.018(191) 0.781(126) 1.291(405) ar 0.510(12) 0.486(5) 0.431(10) 0.411(8) 0.405(10) 0.410(9) 0.387(11) 0.369(9) 2 χdo f 0.13 0.66 0.36 0.22 0.05 0.10 0.07 0.38 ar 0.501(9) 0.477(8) 0.437(8) 0.397(8) 0.383(21) 0.381(12) 0.373(10) 0.368(9) 0.340(15) 2 χdo f 0.32 0.66 0.37 0.48 0.25 0.25 0.44 0.21 0.12 ar 0.427(8) 0.416(11) 0.389(10) 0.371(7) 0.375(8) 0.380(9) 0.359(11) 0.367(8) 0.328(34) 2 χdo f 0.63 0.14 0.28 0.32 0.07 0.48 0.22 0.67 0.15 TABLE 6.3: Results from fits with the Gribov-Stingl formula Eq. (6.2) for the unrenormalized ZT (right table) and ZL (left table) dressing functions. The fit range is [0.4 : 3.0] GeV. The values in parentheses indicate the fit errors estimated with the bootstrap method. The parameters b and n were fixed to b = 0 and n = 1, respectively. The pion mass values are mπ = 316(16) MeV (upper), mπ = 398(20) MeV (middle) and mπ = 469(24) MeV (bottom subtables), respectively. 92 CHAPTER 6. RESULTS FOR FULL QCD mπ (MeV) 316(16) 316(16) 316(16) 316(16) 316(16) 316(16) 398(16) 398(16) 398(20) 398(20) 398(20) 398(20) 398(20) 469(24) 469(24) 469(24) 469(24) 469(24) 469(24) 469(24) 469(24) 469(24) β 3.8400 3.8800 3.9300 3.9525 3.9675 3.9900 3.8600 3.9300 3.9900 4.0050 4.0175 4.0250 4.0400 3.9500 3.9700 3.9900 4.0100 4.0200 4.0300 4.0400 4.0500 4.0700 a2 f 2 0.4580(17) 0.41822(7) 0.37046(9) 0.35672(9) 0.34636(7) 0.33093(8) 0.4464(21) 0.3802(19) 0.3380(07) 0.441(84) 0.3189(09) 0.3155(08) 0.3135(21) 0.3621(09) 0.3519(07) 0.3405(12) 0.3295(07) 0.3270(08) 0.3130(08) 0.3100(11) 0.3082(11) 0.2913(20) h/a2 1.0916(61) 1.0904(19) 1.1355(39) 1.1387(36) 1.1501(33) 1.1571(30) 1.0419(40) 1.0962(53) 1.1466(25) 0.90(14) 1.1623(31) 1.1630(25) 1.2200(90) 1.1325(38) 1.1426(31) 1.1555(45) 1.1607(24) 1.1557(27) 1.1538(26) 1.2015(55) 1.2109(51) 1.2156(79) k 0.5111(78) 0.4950(23) 0.5438(55) 0.5462(52) 0.5642(54) 0.5736(56) 0.4444(35) 0.4945(58) 0.5612(39) 0.41(06) 0.5834(60) 0.5853(52) 0.696(21) 0.5397(53) 0.5554(48) 0.5787(81) 0.5873(48) 0.5778(51) 0.5660(48) 0.648(12) 0.677(12) 0.682(19) 2 χdo f 0.69 8.59 2.29 2.40 5.20 7.01 31.1 21.4 24.0 0.21 6.4 17.4 0.93 2.3 4.2 2.3 13.1 7.0 7.9 2.3 0.94 0.39 TABLE 6.4: Fit results for the unrenormalized ghost dressing function with the fitting function according to Eq. (6.3). The momentum fitting ranges are [0.4 : 4.0] GeV. 2 values are not optimal. Moving to lower temperatures even makes these our ghost data our χdo f values worser, and even including the mass-like term would not improve the situation. Still, the fitting curves do not deviate too much from the data points reaching a maximum deviation of 5%. 6.2.2 The T dependence of the gluon and ghost propagators In order to identify the dependency of the gluon and ghost data as a function of the temperature, we present in Fig. 6.2 the ratios of the renormalized dressing functions or propagators ren RT,L (q, T ) = Dren T,L (q, T )/DT,L (q, Tmin ), ren RG (q, T ) = G ren (q, T )/G (q, Tmin ). (6.4) (6.5) Sec. 6.2. Results on the gluon and ghost propagators 93 Within Fig. 6.2 we show ratios as functions of the temperature T for 6 fixed (interpolated) momentum values q 6= 0. To make the temperature effects more visible we normalized them with respect to the lowest temperature values Tmin available for the given pion masses as described in Eq. (6.4). We observe a monotonous decrease for the values of RL (q, T ) with the temperature. The gradient of this decrease is stronger for smaller momenta. This behavior is also seen through the crossover region. On the other hand, RT (q, T ) shows a slight increase within the same range. Regarding the ghost ratio RG (q, T ) shows a small rise at small momenta, specially at at T ≃ Tdeconf . This might be explained as an artifact of the fit function, which does not work for this momenta range. Furthermore, we show the ratio of the renormalized transverse gluon propagator RT at zero momentum as a function of the temperature at the three pion mass values in the upper row of Fig. 6.3. We see a clear rise towards Tdeconf for the middle mass, whereas for the other mass values there are only weak indications for such a behavior. We have drawn in the lower row the data obtained for the inverse renormalized longitudinal propagator DL at zero momentum versus temperature. This quantity can be identified as a quantity proportional to the square of infrared gluon screening mass. This quantity rises as expected for all the three sets of masses having an inflexion temperature point within the crossover region. Therefore, D−1 L (0) might play an important role to indicate some temperature reaction in the crossover region. Yet, the zero momentum data we are providing here are certainly subject to different factors as finite size effects and Gribov effects. Moreover, we think that increasing our statistics is necessary to precise our conclusions. We summarize our results saying that in all these three pion mass cases, DL react stronger than DT especially for momenta below 1.5 MeV . In fact, we presented a computation of Landau gauge gluon and ghost propagators in the range 0.4 GeV to 3.0 GeV within lattice QCD with N f = 2 flavors, see also our paper [52]. We were able to cover the whole crossover temperature range for our three charged pion masses in the range from 300 MeV up to 500 MeV thanks to configurations provided by the tmfT collaboration. We provide fit results which turned out optimal for the longitudinal as well as transversal gluon dressing function but somewhat suboptimal for the ghost dressing function. Our goal is to provide helpful input data to the DS and FRG equations to study the behavior of the hadronic matter. The longitudinal propagator seems to react stronger crossing the transition region as the transverse does for non-zero momenta. We have also presented separately the zero momenta data. We showed the zero momenta transverse propagator which reacts to the crossover region. Moreover, D−1 L (0) (∝(square) electric screening mass) was also computed. We observed typical behavior for this latter giving indications where the crossover happens for the three pion masses. Still, our results need to be sharpened using higher statistics. 94 CHAPTER 6. RESULTS FOR FULL QCD RT (p, T ) RL (p, T ) RG (p, T ) 0.5 0.7 0.9 1.2 1.4 1.2 1.0 0.8 0.6 Tχ Tχ 180 200 220 T [MeV] 240 180 Tχ 200 220 T [MeV] 240 180 200 220 T [MeV] 240 0.5 0.7 0.9 1.2 1.4 1.2 1.0 0.8 0.6 Tχ 190 Tχ Tdeconf 210 230 250 T [MeV] 190 Tdeconf 210 230 250 T [MeV] Tχ 190 Tdeconf 210 230 250 T [MeV] 0.5 0.7 0.9 1.2 1.4 1.2 1.0 0.8 0.6 Tχ Tdeconf 220 T 240 260 [MeV] Tχ Tdeconf 220 T 240 260 [MeV] Tχ Tdeconf 220 T 240 260 [MeV] FIGURE 6.2: Ratios RT , RL and RG for the renormalized transverse Dren T (left panel), longituren ren dinal DL (middle panel) and ghost G (right panel) propagators, respectively, as functions of the temperature T at a few non-zero momentum values p (indicated in units of [GeV]. The corresponding pion masses (from top to bottom) are mπ ≃ 316, 398 and 469 MeV. The vertical bands indicate the chiral and deconfinement pseudo-critical temperatures with their uncertainties (see Table 6.2). Sec. 6.2. 95 Results on the gluon and ghost propagators RT (p = 0, T ) mπ ≈ 316 MeV mπ ≈ 398 MeV mπ ≈ 469 MeV 1.20 1.00 Tχ Tχ Tdeconf Tχ Tdeconf 180 200 220 240 260 280 180 200 220 240 260 280 180 200 220 240 260 280 T [MeV] T [MeV] T [MeV] DL−1 (p = 0) [GeV2 ] 0.24 0.22 mπ ≈ 316 MeV mπ ≈ 398 MeV mπ ≈ 469 MeV 0.20 0.18 0.16 0.14 0.12 0.10 0.08 Tχ Tχ Tdeconf Tχ Tdeconf 180 200 220 240 260 280 180 200 220 240 260 280 180 200 220 240 260 280 T [MeV] T [MeV] T [MeV] FIGURE 6.3: The upper row shows the ratio RT at zero momentum for the three pion mass values indicated. The lower panels show the inverse renormalized longitudinal gluon propagator D−1 L (0). CHAPTER 7 Alternative study for the L ANDAU gauge fixing D this chapter we present an original study of the gluon propagator D(q) at zero temperature for the gauge group SU(3). In fact, we explore here the influence of the G RIBOV problem on D(q), and especially around the 1 GeV region as well as the zero-momentum values D(0). As discussed before in Section 3.4.2 the gauge fixing to the L ANDAU gauge traditionally practiced by maximizing the gauge functional leaves the G RIBOV copies problem open. We believe that such problem might be the origin to the fact that lattice results so far are supporting a decoupling behavior of the propagators contradicting the K UGO -O JIMA confinement criteria, see Section 2.1.4. On the other hand, we know that the DSE is providing both the decoupling solution and the scaling one which are quite different solutions around the momentum sensible region of 1 GeV (and also below) [89]. These two types of DSE solutions arise due to different boundary conditions on the ghost propagator at zero momentum. We believe that such boundary conditions might suppress the optimal influence of the G RIBOV effects. That is, in order to judge with DSE might happen one needs somehow to get rid of the G RIBOV ambiguity, and try to define the gauge uniquely. In this respect LQCD might judge which solution happens in an effective way. In fact, our new ingredient here is to avoid the G RIBOV problem using a new criteria, namely considering gauge copies with minimal FADDEEV-P OPOV operator eigenvalues λmin . That is, this choice condition defines uniquely the gauge. We expect that a free-G RIBOV copies gauge condition should be the key to compare lattice propagators to the DSE solutions. We concentrate here on the gluon rather on the ghost propagator as we already know that ghost propagator gets more singular for small λmin thanks to its spectral representation [137, 138]. URING 97 98 CHAPTER 7. ALTERNATIVE STUDY FOR THE LANDAU GAUGE FIXING 7.1 Correlation between gauge functional and λmin In order to study the SU(3) gluon propagator D(q) in pure gauge theory we hoped to understand the correlation between gauge functional and λmin values. That is, we wanted first to understand how moving to smaller λmin would effect in some sense the gauge functional values. Therefore, the maximum gauge functional value mostly used to fix the gauge might be here seen to not correspond to a gauge copy with the smallest λmin . In [137, 138] it has been already observed that gauge copies with smallest λmin are not in general the best copies bc with the highest gauge functional. However, it has also been observed that moving to larger volumes would shift the FADDEEV-P OPOV eigenvalues to zero. This latter observation is in agreement with Z WANZIGER conjecture [28]. We consider in the present SU(3) exploratory study a lattice volume of 164 with the inverse coupling of β = 6.0. Here, we generate a number of configuration and gauge copies equal respectively to Nconf = 34 and Ncopy = 50. We study in Fig. 7.1 the variations of the normalized gauge functional F, namely (Fmax − F)/Fmax , with the smallest FP eigenvalues λmin (for fixed configuration). The tendency is that moving to smaller λmin one finds only a few number of gauge copies with higher (Fmax − F)/Fmax values. This means that moving to the (unknown) absolute minima of F (being part of the fundamental modular region (FMR)) one shifts away from the G RIBOV horizon (characterized by λ = 0). In general, one concludes from Fig. 7.1 that no correlation is observed. This is in agreement with results from [137, 138] where best copies (in the gauge functional sense) do not correspond necessarily to the lowest λ . In fact, this is a fortunate situation since results regarding the gluon propagator might in principle provide independent results according to the criteria: highest F and λmin . In the next section let us compare the gluon propagator results measured according to different criteria, namely: highest F vs. smallest λmin gauge copies. This study makes it apparent whether the G RIBOV would effect D(q) especially for the low-lying λmin regime. Note, that a typical solution of the gluon and ghost propagator have been mainly supported by the lattice namely called the decoupling solution [59, 60, 61]. These solutions show for D(q) the realization of a plateau in the deep infrared. These solutions are not in agreement with the KOGU -O JIMA confinement criterion, but still considered as one of the mathematical solutions of DSE besides the scaling solutions, see Section 2.1.4 for more discussion. 7.2 The gluon propagator and its zero-momentum value D(0) As said in the former section, in order to perform our previous study one needs to generate a couple of gauge copies for a fixed configuration. Next, one measures on each gauge copy its corresponding FP eigenvalues. Our new criteria is to choose the gauge copy with the smallest eigenvalue, namely λmin . Therefore, one may compute the gluon propagator on such gauge copies, and even compare the same propagator to other different criteria as maximizing the gauge functional adopted so far. More interesting for us is the comparison between the lowest λmin and Sec. 7.2. 99 The gluon propagator and its zero-momentum value D(0) 0.0012 (Fmax-Fi)/Fmax 0.001 0.0008 0.0006 0.0004 0.0002 0 0 0.005 0.01 0.015 0.02 0.025 λmin FIGURE 7.1: Correlation study between the normalized gauge functional F and the λmin . Fmax corresponds to the gauge copies with the highest F for a fixed configuration. Here we produce Nconf = 34 and Ncopy = 50 for a lattice volume of 164 and β = 6.0. the highest F. Let us recall that maximizing the gauge functional is a standard way to fix the gauge. This standard procedure was also used by us in order to fix the gauge copies to the L ANDAU gauge thanks to iterative methods as simulated annealing for example. In the upcoming results we focus exclusively on the SU(3) gluon propagator in the pure gauge theory. To perform the present study we increase the number of gauge copies up to Ncopy = 100 in order to maximize the probability to find gauge copies with small FP eigenvalues. We observe from Fig. 7.2 the situation is not trivial for producing gauge copies with smaller λmin . Moreover, one sees that the first gauge copies generated for a fixed configuration is nearly never the copy holding the lowest λmin , and therefore a lot of gauge copies need to be produced in order to have a chance to minimize λmin . Here, we study unfortunately only Ncopy = 100 which might be not sufficient for such study. Moreover, in principle, we need to reach the momentum plateau (in the infrared) to observe the influence of FP eigenvalues. That is, moving to higher volumes and therefore even more smaller λmin . Here we fix the volume only to 164 and the (inverse) coupling β is fixed to 6.00. Therefore our gluon propagator results might be afflicted by finite volume effects obviously, and not really reach the deep infrared where the G RIBOV effects dominate. Actually, we observe in Fig. 7.3 that comparing the (bare) gluon propagator measured on different type of gauge copies, namely gauge copies with the lowest λmin vs. the ones with highest gauge functional F, show only small differences for our momentum range. Therefore, we conclude that the G RIBOV effects are absent. This is due in fact to the small volume we used which does not enable to reach small momenta where G RIBOV effects manifest. In parallel, following the strategy as us, the authors in [163] did the same study with the gauge group SU(2). This investigation is less expensive than our SU(3) study, and hence the volume of 564 is reached using β = 2.3. There 100 CHAPTER 7. ALTERNATIVE STUDY FOR THE LANDAU GAUGE FIXING min λmin fc a.λmin 0.01 0.001 0 5 10 15 20 25 30 35 configurations no. FIGURE 7.2: The lowest FP eigenvalues λmin as a function of the number of configurations. The lattice volume and β correspond to 164 and 6.00 respectively. The number of configurations Nconf = 34, and the number of gauge copies Ncopy = 100. it is observed that the D(q) takes different values according to the different criteria (minimal FP vs. maximum gauge functional) around the region of 1 GeV . Moreover, to get such effects the authors generated at least 210 gauge copies and 60 thermalized configurations to access very small λmin . Back to our SU(3) results we conclude that due small volume and not sufficient statistics one does not see real effects for the considered momenta range. Moreover, we see also in Fig. 7.4 that zero-momentum gluon propagator values D(0) show no correlation with λmin . This means that D(0) might be not affected when moving to smaller λmin , and no conclusion might be drawn at least for the present lattice parameters. Sec. 7.2. 101 The gluon propagator and its zero-momentum value D(0) 90 Highest gauge functional Lowest λmin 80 70 D/a2 60 50 40 30 20 10 0 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 a.q FIGURE 7.3: The bare gluon propagator D/a2 as a function of the momenta a · q in units of the lattice spacing. The parameters are the same as in Fig. 7.2. 120 D(0) 100 80 60 40 0 0.005 0.01 0.015 0.02 0.025 λmin FIGURE 7.4: A scatter plot showing the zero-momentum gluon propagator value D(0) as a function of λmin . The lattice parameters are the same as in Fig. 7.2. CHAPTER 8 Conclusion In this thesis we dealt with different aspects of the SU(3) L ANDAU gauge gluon and ghost propagators within lattice QCD at finite temperature. Our present work was threefold: First of all, we investigate the pure gauge sector of QCD, also called the quenched approximation of QCD, see also our paper [49]. Within this sector we computed the gluon and ghost propagators for different temperatures covering the first order phase transition (from 0.5 Tc to 3 Tc ). Our critical (inverse) coupling in this case was fixed to βc = 6.337 corresponding to Nτ = 12. We got clear signal for a temperature phase order transition thanks to the P OLYAKOV loop (and its susceptibility) and the longitudinal part of the gluon propagator DL . The transverse part DT of the gluon propagator shows a weak temperature dependence. We improved the sensitivity around Tc even for higher momenta thanks to functional combinations of DL . These latter new constructs might play the role of ‘order parameters‘ at least for quenched QCD. Our pure gauge study concentrate on the momenta range [0.6,8.0] GeV, and not intend the reach the infrared region. In fact, our goal is to provide data as input for the DS equations for momenta around the sensible range of 1 GeV. For that, we extracted the continuum limit of our data after investigating finite size and G RIBOV effects. We show that these latter are small at least for our limited momentum range. We believe that choosing a higher critical beta (corresponding to a = 0.055 fm) and a moderate 3d volume ((2.64 fm)3 ) favored us to get close the continuum limit. On the other hand the ghost propagator shows weak temperature dependence as expected. A second important part of our work was to study full QCD with NF = 2 maximally twisted fermions, see [52]. Thanks to the tmfT collaboration we had access to configurations corresponding to three values of (charged) pion masses, namely: 316, 398 and 469 MeV. These configurations were generated using the twisted mass action providing an automatic O(a) improvement. For this study we concentrate on the momenta range [0.4,3.0] GeV. We computed the gluon and ghost propagators, and show their temperature dependence. The strong temperature response to the crossover region belongs to the (electric sector) DL rather than (magnetic sector) DT . We show indications for a crosover reaction thanks to DL (for non zero momentum) and DT at zero momentum. Still, one needs to be careful and considers more statistics in order to draw 103 104 CHAPTER 8. CONCLUSION a definitive conclusion. To interpolate we fitted successfully our gluon data with the S TINGL G RIBOV formula. Regarding the ghost propagator the fitting results were not optimal. We have also considered D−1 L (0) (proportional to the electric screening mass) for our pion mass sets. This observable is an interesting order parameter showing where the crossover might happen. In our case, this observable shows indications for transition for our masses at different temperatures scales as expected. However, higher statistics are wished to sharpen the region where theory undergoes a crossover. Our goal from this fermionic study is to provide valuable data to the continuum functional methods, and also to give some indications for a crossover around the critical temperature region. Finally, we experimented a new method to study the influence of the G RIBOV problem especially on the gluon propagator for the gauge group SU(3). In fact, we tried to understand how strong is the influence of the choice of the gauge copies on the behavior of the gluon propagator. Below the sensible region of 1 GeV it is already known that DSE provides two different solutions corresponding to different boundary conditions. It is the role of LQCD to support or to reject one of these solution. We believe that G RIBOV ambiguity is an obstacle to clarify the support of LQCD to some well defined solution of DSE. Therefore, we adopted here a new criteria, to fix the gauge uniquely, and therefore avoid the G RIBOV problem. Namely, we take only gauge copies as close as possible to the G RIBOV horizon, i. e. gauge copies with the smallest FADDEEV-P OPOV (FP) operator eigenvalues. In principle, the gluon propagator computed on these latter gauge copies might show deviations to the one obtained from standard maximal gauge functional procedure. However, we observe small deviations (within errorbars) because of the moderate lattice volume used, and eventually the reduced number of gauge copies and configurations. Appendix 1 A note on the over-relaxation method The method known as over-relaxation (OR) is an algorithm aiming to accelerate convergence of the gauge-fixing process. This method might be explained as an iterative process where the update on the gauge transformation matrices gx is expressed as: gnew = gch gold , (1) where gch is the change in the update step. Therefore, in principle choosing gold = gx , gold = gsolution and starting from a cold start for x example, i. e. gold = I, one finds . gch = gnew = gsolution x (2) Thus, over-relaxation is implemented by applying iteratively gch , or equivalently to make the following replacement gsolution → (gsolution )α , 1 < α < 2. x x (3) In fact, the relaxation method corresponds to setting α = 1. During our gauge fixing process we have used OR after the so-called simulated annealing method[126, 127, 128, 31, 32]. The combination of these two methods improves the gauge-fixing significantly getting then gauge transformations close to the global maxima of the gauge functional in the L ANDAU gauge. We fixed the maximum number of OR iterations to Nitmax = 80000. This number was never reached but still as expected the number of iterations took higher values around the critical temperature Tc . 105 106 ANHANG . APPENDIX 2 The G ELL -M ANN matrices The generators for the group SU(3) in the standard representation are given by 1 Ta = λa , 2 where the eight namely  0 λ1 = 1 0  0 λ4 = 0 1  0 λ7 = 0 0 (4) G ELL -M ANN λa Matrices are 3 × 3 generalizations of the PAULI matrices,     1 1 0 0 −i 0 0 0 , λ2 =  i 0 0 , λ3 = 0 0 0 0 0 0 0     0 1 0 0 −i 0 0 0 , λ5 = 0 0 0  , λ6 = 0 0 0 0 i 0 0    0 0 1 0 0 1 0 −i , λ8 = √ 0 1 0  . 3 0 0 −2 i 0  0 0 −1 0 , 0 0  0 0 0 1 , 1 0 (5) 3 The gamma matrices On the lattice, one mostly uses the Euclidean gamma matrices γµ , with µ = 1, 2, 3, 4. This latters matrices may be related to the M INKOWSKI matrices γµM with µ = 0, 1, 2, 3. The latter matrices satisfy {γµM , γνM } = 2 gµ,ν I, (6) where gµ,ν denotes the metric defined as gµ,ν = diag(1, −1, −1, −1) and I is the 4 × 4 unit matrix. Following these conventions one defines the Euclidean gamma matrices γµ as γ1 = −iγ1M , γ2 = −iγ2M , γ3 = −iγ3M , γ4 = γ0M . 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DYSON -S CHWINGER equation for the ghost-gluon vertex. . . . . . . . . . DYSON -S CHWINGER equation for the ghost propagator. . . . . . . . . . . The Columbia phase diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 21 22 22 37 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 Interpolation to our critical beta βc . . . . . . . . . . . . . . . . . . . . Study of the bare DT and DL with respect to momenta preselection. . . TheP OLYAKOV loop and its susceptibility. . . . . . . . . . . . . . . . DL and DT as functions of T . . . . . . . . . . . . . . . . . . . . . . . T dependence of DL and DT for a lower momenta. . . . . . . . . . . . T dependence of χ and α. . . . . . . . . . . . . . . . . . . . . . . . T dependence of ψ and θ . . . . . . . . . . . . . . . . . . . . . . . . T dependence of the ghost propagator. . . . . . . . . . . . . . . . . . P OLYAKOV sectors effects on DL and DT . . . . . . . . . . . . . . . . Finite-size effect study for DL . . . . . . . . . . . . . . . . . . . . . . Finite-size effect study for DT . . . . . . . . . . . . . . . . . . . . . . G RIBOV copies effect study for DL . . . . . . . . . . . . . . . . . . . G RIBOV copies effect study for DT . . . . . . . . . . . . . . . . . . . G RIBOV copies effect study for DL ( f c)/DL (bc) and DT ( f c)/DT (bc). DL as a function of the a = a(β ). . . . . . . . . . . . . . . . . . . . . DT as a function of the a = a(β ). . . . . . . . . . . . . . . . . . . . . Extrapolation of DL to zero a. . . . . . . . . . . . . . . . . . . . . . Extrapolation of DT to zero a. . . . . . . . . . . . . . . . . . . . . . Extrapolated DL and DT vs. β = 6.337 data. . . . . . . . . . . . . . . Extrapolated DT and DL together with the interpolated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 64 66 68 69 70 72 73 75 76 76 77 77 78 81 81 82 82 83 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 123 124 LIST OF FIGURES 6.1 6.2 6.3 The bare dressing function ZT and ZL . . . . . . . . . . . . . . . . . . . . . . . T dependence of the rations RT , RL and RG . . . . . . . . . . . . . . . . . . . . The ratio RT at p = 0 and D−1 L (0). . . . . . . . . . . . . . . . . . . . . . . . . 7.1 7.2 7.3 7.4 Correlation of gauge functional and λmin . . . . Lowest λmin vs. Nconf . . . . . . . . . . . . . . Comparing of D/a2 for different gauge criteria. Scatter plot for D(0) vs. λmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 94 95 . 99 . 100 . 101 . 101 LIST OF TABLES 2.1 Basic properties of quarks and gluons. . . . . . . . . . . . . . . . . . . . . . . 10 5.1 5.2 5.3 5.4 Lattice parameters for the pure gauge QCD simulations runs. Fit results using the G RIBOV-S TINGL formula. . . . . . . . Renormalization factors Z̃T,L and fit results. . . . . . . . . . Fit results for DL and DT as a function of a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 69 79 80 6.1 6.2 6.3 6.4 Full QCD study parameters. . . . . . . . . . . . Pseudo critical couplings for A12, B12, and C12. Fit results for ZT and ZL . . . . . . . . . . . . . . Fit results for the bare ghost dressing function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 88 91 92 . . . . . . . . . . . . . . . . . . . . . . . . 125 Selbständigkeitserklärung Ich erkläre, dass ich die vorliegende Arbeit selbständig und nur unter Verwendung der angegebenen Literatur und Hilfsmittel angefertigt habe. Rafik Aouane Berlin, den 20.12.2012 127