physics department
- university of Pisa
School of Graduate Studies in Basic Sciences“Galileo Galilei”
XXX Cycle
PhD Thesis:
“The Roberge-Weiss QCD transition
and Debye color screening”
Michele Andreoli
supervised by Prof. Massimo D’Elia
iii
Dedicato a coloro che mi hanno spinto in questa tardiva
avventura: i miei figli Remo, Chiara e Francesca, e mia moglie Tina, per aver sopportato per tre anni un padre e un marito
mentalmente assente.
Except where otherwise indicated, this PhD thesis is my own original work. (C) 2017 Michele Andreoli
Ringraziamenti
Ringrazio il gruppo di fisici teorici conosciuti al Dipartimento di Fisica dell’Università di Pisa quali Claudio Bonati, Francesco Negro, Michele Mesiti, e Andrea Rucci, con i quali ho spesso discusso questioni specifiche, Giulio Pelaggi ed
Elena Viggiani, per l’amabile compagnia, ma soprattutto il mio supervisore, il professor Massimo D’Elia, per tutte le cose che mi ha pazientemente insegnato e per
Disclaimer
This document, including all graphics, data & text processing, is realized using Free Software only, as Linux Ubuntu, LATEX, LyX, Bash, GNU C, Python, Numpy and
Introduction
Understanding the behaviour and properties of strongly interacting matter is one of the most challenging tasks of High-Energy Physics.Due to the non-abelian nature of Quantum Chromo-Dynamics (QCD), the theory that describes the strong force, analytic or perturbative treatment of many physically interesting situations is not possible. Therefore, the interaction of experimental results, numerical investigations, and progress in analytical methods are the cornerstone of research aimed at under-standing this type of interaction. The famous Eightfold Way by Gell-Mann has laid the foundation for the modern description of hadronic matter, interpreted as quark composite particles. Within the Standard Particle Physics Model, nowadays com-monly accepted, strong force is established between particles carrying color charge, through the exchange of gluons, which are also colored.
One of the most important issues of the QCD is understanding the confinement mechanism: in vacuum and at low temperatures, quarks and gluons are confined to strongly bound states. At higher temperatures, in the order of 1012K, it is ex-pected that hadronic matter will undergo a transition to the Quark Gluon Plasma phase (QGP), where quarks and gluons may exist as unbound colored states, in the thermal medium. This new form of matter is of particular interest to the physics of the Early Universe and has been studied in several heavy-ion collision experi-ments, where QGP status is expected to form. Unfortunately, there is no "exact" analytical formalism for this kind of phenomenon and we must resort to effective QCD Lagrangians models. Many of them have been proposed, leading to a wealth of different predictions to be verified. Lattice simulations provide an extraordinary “first principle” tool for the theoretical QCD vacuum analysis and the deconfinement transition, as they are able to explore conditions in which analytical or perturbative methods fail.
This thesis is focused on one of the typical phenomena of the plasma phase: the color screening effect and the possible influence of a non zero baryon density. As it is known by Electromagnetism, in the presence of a charged plasma, the electric field is shielded with a characteristic length scale rD. The inverse of rD is the Debye
mass screening mD. It represents the effective mass of the photon in a static thermal
background. In the case of EM, these masses are of the order of e·T , where “e” is the charge of the electron, so they increase linearly with the temperature. Our aim is to see whether similar behaviour can be confirmed in non-Abelian plasma, that is, in plasma where interaction mediators, gluons, also interact with each other. In the literature, there are many works about screening masses in QCD, but simulations are generally made at zero chemical potential, or under unphysical conditions (number
of flavors, etc).
In this work Debye’s masses are extracted from the Polyakov loop correlators
hL(0)L?(r)i, calculated at imaginary chemical potential, the only case in which the Monte Carlo Method is still applicable without modification, and then compared to those that can be obtained from the Thermal Field Theory, via analytical continuation. The thing is made even more attractive by the peculiar periodicity that emerges in the phase diagram of QCD at imaginary chemical potential, where the so-called Roberge-Weiss transition comes into play; a phenomenon already under deep investigation, for the fundamental interest in the role of center symmetry Z3. By decomposing
the Polyakov loop in real and imaginary part, we can get particular combinations of correlators having the magnetic and electric field quantum numbers under charge conjugation and Euclidean time inversion. The problem is that this separation of the magnetic and electrical contibution depends on the chemical potential µ. The basic idea is that it is possible, by diagonalizing the 2×2 correlations matrix, to separate the two types of masses in this case as well.
This thesis is organized as follows: in Chapter I, QCD is presented, in the for-mulation based on Feynman integral, with its basic symmetries and some important properties such as asymptotic freedom and confinement. In Chapter II the lattice for-mulation, the action SQCD discretized, and the possible improvements are discussed.
Chapter III describes the numerical methods used, such as the Monte Carlo Method, etc, as well as the important issue related to the positivity of the Boltzmann factor at non-zero chemical potential (sign problem). Chapter IV discusses the Roberge-Weiss symmetry and Chapter V the phase transition in QCD. Chapter VI sets out the methods for calculating the correlators in the various channels, diagonalizing the correlation matrix. Chapter VII is a preliminary study of the correlators at the Roberge-Weiss transition µ/T =iπ and, finally, in the Chapter VIII the results of the simulations with different chemical potentials are presented.
***
La comprensione del comportamento e delle proprietà della materia fortemente interagente è uno dei compiti piu impegnativi della Fisica alle Alte Energie.
A causa della natura non abeliana della chromo-dinamica quantistica (QCD), la teoria che descrive la forza forte, un trattamento analitico o perturbativo di molte situazioni fisica-mente interessanti non è possibile. Pertanto l’interazione dei risultati sperimentali, le indagini numeriche e il progresso nei metodi analitici costituiscono una pietra angolare della ricerca dedicata alla comprensione di questo tipo di interazioni.
La famosa Eightfold Way di Gell-Mann ha posto le basi per la moderna descrizione della materia adronica, interpretata come particelle composite costituite da quark. All’interno del
vii
Modello Standard della Fisica delle Particelle, oggi comunemente accettato, la forza forte si stabilisce tra particelle che trasportano carica di colore, attraverso lo scambio di gluoni, anch’essi colorati.
Uno dei problemi più importanti della QCD è la comprensione del meccanismo di confi-namento: nel vuoto e a basse temperature, quarks e gluoni sono confinati in stati fortemente legati. A temperature piu alte, dell’ordine di 1012K, si prevede che la materia adronica subisca una transizione al Quark Gluon Plasma (QGP), in cui quarks e gluoni possono esistere sin-golarmente in diversi stati di colorazione, nel mezzo termico. Questa nuova forma di materia è di particolare interesse per la fisica dell’Early Universe ed è stata studiata in diversi esperi-menti di collisione di ioni pesanti, in cui è prevista la formazione di stato di QGP in equilibrio termico.
Purtroppo non esiste un formalismo analitico “esatto” per questo fenomeno e si deve ricorrere all’uso di modelli basati su Lagrangiane QCD efficaci di vario tipo. Sono stati proposti molti modelli, che portano ad una ricchezza di previsioni diverse e da verificare. Le simulazioni numeriche su reticolo forniscono uno strumento straordinario per l’indagine teorica del vuoto QCD e la transizione di confinamento / deconfinamento dai principi primi, in quanto sono in grado di esplorare condizioni in cui i metodi analitici o perturbativi falliscono. Questa tesi è focalizzata su uno dei tipici fenomeni della fase plasmatica: l’effetto di screen-ing di colore e l’eventuale influenza di una densità barionica non nulla sul fenomeno stesso. Com’è noto dall’Elettromagnetismo, nel plasma carico il campo elettrico viene schermato con una scala di lunghezza caratteristica rD. L’inverso di rD è la massa di screening Debye mD.
Essa rappresenta la massa efficace del fotone in un background termico statico. Nel caso EM, queste masse sono dell’ordine di e·T, dove e è la carica dell’elettrone, per cui aumentano linearmente con la temperatura.
Il nostro scopo è vedere se un comportamento simile può essere osservato anche nel plasma non abeliano, cioè nel plasma in cui i mediatori dell’interazione, i gluoni, interagiscono anche tra loro e, eventualmente, con quali differenze al variare del potenziale chimico.
In letteratura esistono molti lavori sul calcolo delle masse di screening in QCD, ma gen-eralmente a potenzile chimico nullo. In questo lavoro i valori delle masse di Debye vengono estratti dai correlatori dei loop di Polyakov hL(0)L?(r)i, calcolati a potenziale chimico im-maginario, l’unico caso in cui il Metodo di Montecarlo è ancora applicabile senza modifiche, per poi confrontarle con quelle che si possono ottenere dalla Thermal Field Theory, via esten-sione analitica.
La cosa è resa ancor più attraente dalla peculiare periodicità che emerge nel diagramma di fase della QCD a potenziale chimico immaginario µ, dove entra in gioco la cosiddetta transizione di Roberge-Weiss, già ampiamente sotto indagine per il fondamentale interesse legato al ruolo della simmetria centrale Z3. Mediante la scomposizione del loop di Polyakov in
parte reale e parte immaginaria, si ricavano particolari combinazioni di correlatori che hanno, sotto coniugazione di carica e inversione temporale euclidea, i numeri quantici del campo magnetico e del campo elettrico. Il problema è che questa separazione in parte magnetica e
parte elettrica dipende dal valore del potenziale chimico µ. L’idea di base è che sia possibile, mediante diagonalizzazione della matrice 2×2 delle correlazioni, separare i contributi dei due tipi di masse anche in questo caso e seguirne l’andamento in funzione del potenziale chimico. Questa tesi è organizzata come segue: nel Capitolo I viene presentata la QCD come teoria di Gauge, nella formulazione basata sugli integrali di Feynman, le sue simmetrie e alcune proprietà importanti come asymptotic freedom e confinement. Nel Capito II la formulazione su reticolo, l’azione SQCD discretizzata, e i possibili improvements. Nel capitolo III si descrivono
i metodi numerici usati, quali Metodo di Montecarlo, etc, nonchè le questioni legate alla positività o meno del fattore di statistico di Boltzmann (sign problem) a potenziale chimico non nullo. Nel Capitolo IV si discute della simmetria di Roberge-Weiss e nel capitolo V della transizione di fase in QCD. Nel capitolo VI si stabiliscono i metodi per calcolare i correlatori nei vari canali e diagonalizzare la matrice di correlazione. Nel Capitolo VII c’è uno studio preliminare sul comportamento dei correlatori alla transizione di Roberge-Weiss µ/T = iπ, mentre nel Capitolo VIII vengono presentati i risultati delle simulazioni a potenziale chimico variabile lungo la linea a Fisica Costante.
Contents
1 Quantum Chromodynamics 1
1.1 Continuum Formulation of QCD . . . 1
1.1.1 Gauge transforms and Parallel transports . . . 2
1.1.2 Path integral quantization . . . 3
1.2 QCD at Finite Temperature and Density . . . 4
1.2.1 Temperature and Z partition sum . . . 4
1.2.2 Implementing the chemical potential in gauge theories . . . 6
1.2.3 Canonical formalism . . . 7
1.3 Symmetries of QCD . . . 8
1.3.1 Chiral symmetry . . . 8
1.3.2 Center Symmetry and Pokyakov loop . . . 9
1.4 Quarks Confinement . . . 12
1.4.1 Asymptotic Freedom . . . 12
1.4.2 Confinement and Quark Free Energy . . . 12
1.4.3 Quark-antiquark potential . . . 14
2 Lattice QCD basics 17 2.1 Links and Plaquettes . . . 17
2.2 The pure-gauge Action . . . 19
2.3 Fermions on the Lattice . . . 20
2.3.1 Wilson fermions . . . 20
2.3.2 Fermion determinant . . . 22
2.3.3 Staggered formulation . . . 24
2.4 Continuum Limit . . . 24
2.4.1 Tuning the temperature . . . 25
2.5 Action improvements . . . 26
2.6 Final QCD Lagrangian . . . 27
3 Computational aspects of Lattice simulations 29 3.1 The RHMC algorithm . . . 29
3.2 Importance sampling . . . 29
3.2.1 Rational approximations . . . 30
3.2.2 Preconditioning . . . 31
3.2.3 Hybrid MonteCarlo . . . 32
3.2.4 Energy violation and Metropolis acceptance . . . 35
3.3 The sign problem . . . 35
3.3.1 The sign of the Boltzmann’s factor . . . 35
3.3.2 Possible approaches . . . 36
4 The Roberge-Weiss Symmetry 39 4.1 Center Symmetry in the Continuum . . . 39
4.2 Center trasformation properties of the fermion determinant . . . 41
4.3 The Charge symmetry . . . 41
4.4 Center Symmetry and observables . . . 43
4.5 Polyakov loop Distribution and Roberge-Weiss Transition Lines . . . 44
5 The QCD Phase transition 49 5.1 Transition order in SU(N) and effective potentials . . . 50
5.2 Detecting the transition . . . 52
5.2.1 Order parameter and Susceptibility . . . 53
5.2.2 Finite Size Scaling and Universality . . . 55
5.2.3 Susceptibilities from Lattice . . . 56
5.2.3.1 Baryon density . . . 56
5.2.3.2 Chiral condensate . . . 58
5.3 Density of states and histograms . . . 59
5.3.1 The role of B4, the Binder’s cumulant . . . 61
5.4 Mass dependence and Columbia plot . . . 62
6 Debye color screening 67 6.1 Propagators and screening masses . . . 67
6.2 Perturbative regime and temperature dependence . . . 69
6.3 Electric and Magnetic channels . . . 70
6.3.1 Notations . . . 70
6.3.2 Physical intepretation and quantum numbers . . . 72
6.4 Computing correlators using FFT . . . 73
6.5 Diagonalization of the correlation matrix . . . 74
6.6 The Generalized Eigenvalue method . . . 75
6.7 Rotation angle and chemical potential . . . 79
7 Color screening at the Roberge-Weiss transition 83 7.1 Simulation Setup . . . 83
7.2 Discussions . . . 85
8 Numerical simulations 91 8.1 Simulation Setup . . . 91
CONTENTS xi
8.3 Mass ratios . . . 99
8.4 Comparison with other methods . . . 99
8.5 Final comments . . . 100
9 Conclusions and Outlook 105 A Appendices 107 A.1 Quark Free Energy . . . 107
A.2 Euclidean formulation . . . 108
A.3 Debye screening in QED . . . 110
A.4 The string model of the phase transition . . . 111
A.5 Baryon charges and chemical potential . . . 111
A.6 Integrated time correlation . . . 113
A.7 The jackknife method . . . 114
A.8 Histogram Reweighting . . . 116
A.9 FFT Alternative methods . . . 117
A.10 Multi-histograms Methods . . . 118
A.11 Cumulants expansion . . . 120
A.12 Scaling properties on the lattice . . . 122
A.13 Statistical Mechanics reminder . . . 124
A.14 Thermodynamics quantities from Lattice . . . 126
List of Figures
1.1 Finite temperature is introduced constraining the theory on a torus,
with compactified radius 1/T. . . 5
1.2 Chiral condensate as function of β = 6/g2. (Simulation parameters: Nf =4, mqa=0.09, lattice 123×4). . . 9
1.3 Roots of unity in the complex plane (example for z6 =1) . . . 9
1.4 Paths on the lattice: trivial loops and Polyakov loops. . . 11
1.5 Diagram of h|P|i at the transition. Here β = 6/g2 e amq = 0.09 and Nf =4 (reweighted data). . . 14
1.6 Polyakov loops ad distance r . . . 14
1.7 Cornell quark-antiquark potential V(r) . . . 15
2.1 The lattice grid: links, fields and plaquettes . . . 18
2.2 Plaquettes . . . 19
2.3 Lattice paths in the hopping expansion . . . 23
4.1 The quantity µβ in the complex plane . . . 42
4.2 Roberge-Weiss boundaries (the vertical lines) and center sectors orien-tation (the arrows) . . . 44
4.3 In this figure (from left to right, top to bottom) βµ is gradually changed between tra 0 e 2πi (the blue arrow), at very low T. The Polyakov loop distribution, as complex number, is symmetryc and substantially insensitive to the chemical potential (Simulation data: Nf =2, β=2.0) 46 4.4 Polyakov loop scatter plot: In this plot che chemical potential is fixed to µβ = iπ, and the inverse coupling β is varied in the critical zone, around βc ≈ 5.56 (Simulation parameters: Nf = 8, amq = 1.0,L = 123×4) . . . 46
4.5 This set of diagrams are th analogue of fig (4.3), but at high temper-ature (in this simulation Nf = 12 , amq = 0.15, β = 7.0, L = 323×4, 10000 samples) . . . 47
5.1 Schematic behaviour of the Polyakov loophLifor a) second order tran-sition b) first order (dashed: smooth crossover) . . . 51
5.2 Pressure p for a large set of β =6/g2 ( Simulation parameters: amq = 0.15, Nf = 12, lattices: 163×4, 15000 samples. The pseudocritical point in βc=4.56) . . . 53
5.3 Left side: Example of fit of the susceptibility peaks versus Ns. Right side:
An example of collapse plot for a 2th order transition (Nf =8) . . . 56
5.4 Left panel: baryon numberhNi. Right panel: The peak ofhδ|N|2i. Sim-ulation parameters: Nf = 4, amq = 0.2, and imaginary chemical po-tential µ/T =π(multihistogram reweighting ) . . . 57
5.5 The peak of the chiral subsceptibity at the transition point ( amq=0.09, Nf =4, lattices: {123, 163, 203}x4 (multihistogram reweighted data) . . . . 58
5.6 1th order transition (plaquette histogram). Simulations data: Nf = 8, mqa =0.1, lattice 28x4, 15000 trajectories. . . 60
5.7 2th order transition. p(ImP) histogram). Simulations data: Nf = 4, mqa =0.5, lattice 243x4, 15000 trajectories. . . 61
5.8 Binder cumulant B4. In the phase transition, the unique peak of the probability distribution of P(|Im(P)|)at|Im(P)| = 0 (ordered phase) split in two peaks as the system passes to the disordered phase, and the B4passes from 3 to 1, as in figure. (Simulation parameters: Nf =4, mqa =0.09, 2800 measures, with data reweighting) . . . 63
5.9 Mass dependence of TRW . . . 63
5.10 Tricritical points . . . 64
5.11 Columbia plot: quark mass dependence of thermal transition . . . 65
6.1 From top left: (a) Θ(r)rotation angle for µ/T = 0 and µ/T = iπ; (b) KXY(r), mixed correlator (c) Magnetic and Electric Correlators diag/undiag comparison (Simulation data: L = 323×8, µ/T = iπ, β = 3.94, T ≈ 300 K ) . . . 76
6.2 Diagonalized correlators λ(r, r0), for the magnetic and electric sec-tors. In the two diagrams we compare λ(M)(r, r0) and λ(E)(r, r0) ob-tained with the two methods: the standard one and GEVP (General-ized Eigenvalue Problem), with r0·T= {0, 0.2, 0.4}. (Simulation data: L=323×8, µ/T=i·0.3, β=3.94, T ≈300 K ) . . . 78
6.3 Diagonalized correlators λ(r, r0). The same setup as in fig (6.2), but with larger r0: r0·T ={1.0, 1.2}. . . 79
6.4 Symmetry axis. . . 80
6.5 The Polyakov loop distribution and the imaginary chemical potential (the arrow) corresponding to angles 22, 120, 240 (degree). . . 81
6.6 Rotations angle θ(r)at various µ/T=i{120, 240}◦(Simulation details: Nf =12, L=123×4, β=7.0) . . . 82
7.1 Diagonalization process: (a) rotation angleΘ(r)(in degree) (b) diag/nodiag comparison, for β=5.672 . . . 84
LIST OF FIGURES xv
7.2 Susceptibily of|Im(P)|, and FSS analysis (multi-histogram data reweighted),
for the case Nf =8 . . . 85
7.3 Interquark potential V(r) as function of rT, for various T/Tc below the RW phase transition. The χ2/do f of the fit is reported in the small labels. . . 86
7.4 σ(T)/Tc2, for various T/Tcbelow the RW phase transition. The χ2/do f of the fit is reported in the small labels. . . 87
7.5 KM(r)as function of rT and T/Tc, above the RW transition. . . 87
7.6 KE(r), as function of rT and T/Tc, above the RW transition. . . 88
7.7 Fits results of K(r) with ansatz (7.3): Magnetic and elettric screening mass. The χ2/do f of the fit is reported in the small labels. . . 88
7.8 Fits results of K(r)with ansatz (7.3): exponents ηM,E(T)of the Coulomb-like part of the potential, above βc. Line η=0 corresponds to the mean field theory. The χ2/do f of the fit is reported in the small labels. . . 89
8.1 Side (a): diagonalization angle θ(r), as function of the distance r·T. Side (b): comparison between diagonalized/undiagonalized correla-tors, for βG=3.94 and µ/Tπ=1.0 . . . 94
8.2 Magnetic diagonal correlator KM(r) as function of rT, for several TπµI and β (log scale). . . 96
8.3 Electric correlators KE(r) as function rT, for several TπµI and β. The errors are considerably larger than the magnetic case, so the range of r is also reduced (log scale). . . 97
8.4 Illustrating the magnetic dominance mE >mM: KE(r)drops faster that KM(r)(log scale). . . 98
8.5 Screening masses mE,M/T comparation, as function of TπµI 2 and β. The value µI Tπ = 13, although included in the plot, was not used for fit purpose. . . 101
8.6 Diagonalized mM, mE, and not diagonalized mm, me masses, for the magnetic (M) and electric (E) channel. . . 102
8.7 αM,E coefficients, as function of T, and their ratio αE/αM. . . 103
8.8 γM,E coefficients, as function of T . . . 103
8.9 mE/mM as function of(T, µ). . . 104
8.10 mE/mM as function of(µ, T). . . 104
A.1 Fig (a): heavy quark free energy∆F/T analytic continuation, as func-tion of µR/πT (see eq (A.3)) Fig (b): fitting of the the inverse coefficient 1/ef(T) =a(T−b)c, as function of T (see eq (A.4) . . . 109
A.2 |Im(P)|: Integrated time correlation. The number of truly indepen-dent lattice measurements, scales as 1/τint. Near the critical point,
this causes an huge loss of accuracy (the critical slowling down phe-nomenon) (Simulation parameters: Nf = 8, lattice {12, 16, 20}3×4,
Chapter 1
Quantum Chromodynamics
Quantum ChromoDynamics (QCD) is the established theory of strong force. It describes the fundamental interaction of elementary particles named quarks and gluons.
“In this chapter, the functional formulation of Quantum Chromodynamics is briefly in-troduced. First, the introduction of a finite temperature in the mathematical framework of the theory is discussed, and other aspects related to the introduction of chemical potential and its effects on the spectrum. Then some basic symmetries of the QCD are discussed, such as gauge invariance, Chiral Symmetry and Central Symmetry and its order parameter (the Polyakov loop). Finally, we discuss the most important open question for the QCD, namely the problem of confinement: in particular, from the concept of asymptotic freedom, it is seen the bond between free energy and Polyakov loop, and how the latter can be used to intercept (at least in the case of pure gauge) the phase transition”.
1.1
Continuum Formulation of QCD
QCD is a local SU(Nc)(Nc =3)gauge theory, with fermions ψ(x)in the fundamental
representation, and gluons Aµ(x)in the adjoint representation:
(
Aµ(x) ≡∑aTaAµa(t, x), a =1 . . . Nc2−1
ψ(x) ≡ψi(color),α(spin), f(f lavor)(x)
(1.1) Ta are the generators of the group, Nc×Nc hermitian matrices acting on the color
indices with the following standard propreties:
[Ta, Tb] =i fabcTc, Tr[TaTb] =
1
2δab (1.2)
Gluons and fermions are coupled via the Yang-Mills mechanism, i.e through the covariant derivative
Dµ =∂µ−ig·Aµ. (1.3)
In view of numerical treatment of the theory, it is usefull to analytically continue the time component of the 4-vectors to purely imaginary values:
dt→ −idτ, d4x → −id4xE, dt2−dx2→ −(dτ2+dxE2) (1.4) In Euclidean space the action is mapped to its Euclidean version as (A.2)
iS→ −SE (1.5) where SE = ˆ d4xE " 1 2Tr FµνFµν+ Nf
∑
f=1 ¯ ψf γµDµ+mf ψf # (1.6) Since we will work almost solely in Euclidean space-time, we skip the subscript E in the following.The gluon sector contains only the field strength tensor Fµν =∂µAν−∂νAµ−ig[Aµ, Aν] = i g[Dµ, Dν] =
∑
a TaF a µν(t, x), (1.7) with: Fµνa =∂µAaν−∂νAaµ+g·f abcAb µ·A c ν (1.8)The fermion sector is bilinear in ψ(x)and linear in Aµ(x). After variation in the
fermion fields ψf, it produces the quark Dirac equation of motion
∂S ∂ ¯ψf
= D/+mf
ψf =0 (1.9)
Each term in (1.6) describes the interaction between quarks and gluons, as well as nonlinear three and four gluon-gluon interactions.
Except for the number of flavors Nf and their masses mf, that are free parameters,
the structure of the QCD Lagrangian is completely fixed by the local symmetry and the Lorentz invariance. [22]
1.1.1 Gauge transforms and Parallel transports
Gauge theories are characterized by invariance under local G-transformations, where G is an arbitrary Lie group.
§1. CONTINUUM FORMULATION OF QCD 3
The local nature of the symmetry implies that we cannot form expressions such as ¯ψyψx, because these involves quantities ad different points x, y, with incompatible
trasformation laws: we should “parallel transport” the vector ψx from x to y. In this
wayΓy,xψxtransforms as ψyand ¯ψyΓyxψxis gauge invariant.
A parallel transport Γ[C] along a path C is an element of the group G. As basic requirement (continuity) it should become the identity when C reduces to a single point.
An infinitesimal transport xµ → xµ+eµ is defined through the Lie derivative of
the generic field ψ:
Dµψx = lim eµ→0 Γ−eψx+e−ψx eµ (1.10) so Γe≈e−e·(D−∂)≈ eigeµAµ (1.11)
For a finite transport Γ[C]: [61, 28]
Γ[C]≡ Peig´CAµ(x)·dxµ (1.12)
Generally, Γ[C]depends on the curve C: parallel translations along two different paths, connecting the same points x and y , will usually not give the same result because the commutator [Dµ, Dν] is generally not null, even for an abelian theory
such as QED. As a result,Γ[C]may differ from the identity even if C is closed. Γ itself is not gauge invariant. For a path C between two points xi, xf, it
trans-forms as
Γ0[C] =G(x
f)Γ[C]G−1(xi) (1.13)
From the requirement (1.13), we have the transformation rules: D0µ= GDµG−1, igA
0
µ =G(∂µ−ig·Aµ)G−1 (1.14)
So, parametrizing G(xµ) =e−i∑aωa(x)·Ta, with ωa(x)∈R continuous and
differen-tiables scalar functions, we have: ψ0 = Gψ A0µ = GAG†− 1 igG∂µG† Fµν0 = GFµνG−1 (1.15)
1.1.2 Path integral quantization
The quantization in the continuum is realized using Feynman’s path integrals.
with the weight e−S(ψ,Aµ). So, if O is a generic observable, we have hOi = ´ DψDψ¯DAOe−SG[A]e−∑f ´ ψ†fMfψf d4xO ´ DψDψ¯DA e−SG[A]e−∑f ´ ψ†fMfψf d4x (1.16) where Mf =mf +D/ (1.17)
is the Dirac’s fermion operator.
DψDψ¯ andDAµare integration measures in the configuration spaces: DA=
∏
x∏
µ,a dAaµ(x), Dψ=∏
x i,α, f∏
dψi,α, f(x) (1.18) dAaµ(x) is the Haar measure over the SU(3) group manifold. This is the
gauge-invariant measure induced by the standard metric ds2 = cost·Tr[dU†dU] over the
group, where U∈SU(3).
Each gauge field contribute to sum an infinite number of copies related by gauge transformations, leading to potential divergences. To avoid such multiple counting, we need to fix the gauge picking up one representative among the copies, for exam-ple, using the Faddeev-Popov’s method1) .
However, that gauge-fixing is not necessary if functional integrals are carried out numerically over compact spaces, as in Lattice QCD. [68]
1.2
QCD at Finite Temperature and Density
1.2.1 Temperature and Z partition sum
The starting point for the analysis of the equilibrium thermodynamics, is the statistic sum Z(V, T): Z(V, T) =Tr[e−1TH] =
∑
φ hφ|e− 1 TH|φi (1.19)On the other hand, in the path integrals formulation quantum amplitudes like
1)Briefly, in this method the gauge fixing condition F[A] =0 is added in the integral as
decomposi-tion of the unity:´ dFδ(F) =1. Parametrizing the Lie’s group with continous functions ωa(x), we can
write´ dFδ(F) =´ ∏adωadet[δωδF]δ(F(ω))were Jab= [δωδF]abis the jacobian of the transformation. The
delta-fuction can be approximated as exp(−1
2ξF2)with ξ→0 and det[J] =
´
∏adcad¯cae− ¯cJc, where the
§2. QCD AT FINITE TEMPERATURE AND DENSITY 5
Figure 1.1: Finite temperature is introduced constraining the theory on a torus, with com-pactified radius 1/T.
hφ|e−τ H|φiare represented by Feynman’s integrals: hφ|e−τ H|φi ∼ ˆ φ(0,x)=±φ(τ,x) Dφ e− ´τ 0 dτd3xLE[φ] (1.20)
where LE is the euclidean lagrangian density and φ(τ, x) is a generic field, sub-jected to periodic (bosons) or anti-periodic (fermions) boundary conditions
φ(0, x) =±φ(τ, x) (1.21) The sum itself∑φ is replaced with an integration over the B.C. field φ(0, x).
So, the simplest prescription is to put τ = T1 in the path integral (1.20) , writing: [46, 68] Z(V, T) = ˆ Dφ(0, x) ˆ φ(0,x)=±φ(T1,x)D φ e− ´1/T 0 dτd3xLE[φ] (1.22)
Integration in Dφ(0, x) leads, usually, to an overall multiplicative constant (and formally infinite), usually omitted.
In the standard treatment, the QCD vacuum is approximated with a box of vol-ume V, in thermal contact with a heat bath at temperature T.
In addition to its dependence on volume V and temperature T, the partition function Z also depends on a set of chemical potentials (µf) and on the quark masses
mf, for f =1, .., Nf different quark flavors, that enters in the fermion matrix.
As final result for the grand canonical QCD partition function ZGC(T, V, µf, mf, . . .)
we obtains: ZGC(T, V, µf, mf, . . .) = ˆ B.C.D ψDψ¯DA e−SG[A]e− ´1 T 0 dτd3x LQCD[ψ,Aµ] (1.23)
The only, but fundamental, difference compared to the T=0 case (infinite time extension) is due to the compactness of the time direction:
Aµ(τ+
1
T, x) =Aµ(τ, x), ψ(τ+ 1
This has effects on how the fields are expanded in Fourier’s serie: Aµ(τ, x) = ˆ d4p (2π)4A˜µ(p)eip·x, ψ(τ, x) = ˆ d4p (2π)4ψ˜(p)e ip·x (1.25)
with p·x = pµ·xµ = ωτ+px(euclidean metric), because now the spectrum is
discretized (Matsubara frequencies): ωn = π T−1(2n) =2πn·T (bosons) ωn = π T−1(2n+1) = (2n+1)·πT (f ermions)
Integration over the momentum k0 is replaced by sums over the Matsubara
fre-quencies k0=2πn/β, n∈Z, so the basic recipe to change integrals in sums is:
ˆ dω 2π d3p (2π)3 ⇒
∑
n 1 T−1 !∑
k 1 L3 ! = T V∑
n,k 1.2.2 Implementing the chemical potential in gauge theoriesIn the “naive” formulation, the chemical potential is added in the QCD Lagragian with the rule [1]
m→m−µγ0 (1.26)
If we apply this prescription to the fermion hamiltonian ˆ
H=γ0(−i6∂·γ+m) (1.27)
we get the standard term−µ ˆN, where ˆN = ψ†ψis the baryon number operator, as well the right sign in the Boltzmann factor :
Tr[e−β ˆH]→Tr[e−β ˆH+µ ˆN] (1.28)
With this substitution, the Dirac operator becomes:
M = (∂µ−igAµ)γµ−µfγ0+mf (1.29)
Unfortunately, a term like µψ†ψin the Lagrangian leads to renormalization prob-lems.
Anyway, analyzing the form of the Dirac operator, (1.29), having µ and A4 the
same factor(γ0), one see that µ enters in the M operator like the 4-th component of
the gauge field Aµ:
§2. QCD AT FINITE TEMPERATURE AND DENSITY 7
This suggest that chemical potential should be implemented adding on the sys-tem an external U(1) gauge field A4 =−giµ. [2]
With this prescription, infinitesimal transportΓ in the time direction picks-up the additional factor
Γ(eµ) =eigeµAµ =eµe4 (1.31)
any other unchanged. [20, 68].
This observation will be essential in the lattice formulation of the theory. 1.2.3 Canonical formalism
At finite baryon density, we can define two kind of partition functions: the “canoni-cal” and the “gran canoni“canoni-cal” one.
In the “gran canonical” partition function ZGC(µ, T, V), the sum over the stats|νi is unrestricted, and Nν can fluctuate.
Grouping the terms with the same baryon number Nν, we have
ZGC(µ, T, V) =
∑
ν e−βEν+βµNν = (1.32) =∑
N eβµN∑
Nv=N e−βEν =∑
N eβµNZ C(N, T, V) (1.33) where ZC(N, T, V) =∑
ν: Nν=N e−βEν(Nν) (1.34)is the “canonic” partition function, with the sum restricted to the ensemble at fixed number Nv =N 2).
In the thermodynamic limit V → ∞ fluctuations are negligible, and the two de-scriptions are equivalent.
The relation (1.32) can be formally inverted by the help of the Cauchy’s integral formula ZC(N) = ˛ dζ 2πi ZGC(ζ) ζN+1 (1.35)
where ζ =eβµ(and dζ =ζd(βµ)) is the fugacity, and also written a:
ZC(N) =
˛
d(βµ)
2πi ZGC(µ)·e
−βµ·N (1.36)
Here the integration is along a path in the βµ complex plane, parallel to the imaginary axis βµ=iθ, and closing to the infinity.
2)Formal definitions are: Z
GC(µ, T, V) =Tr e−β( ˆH−µ ˆN)and ZC(N, T, V) =Tr e−β ˆHδ(Nˆ −N) .
This relation shows that ZGC(µ)and ZC(N)are connected by a Laplace trasform,
thus they share the same informations. [12]
1.3
Symmetries of QCD
1.3.1 Chiral symmetry If in the Lagrangian
LQCD =ψ(D/+m)ψ, m≡diag(m1, m2, . . . mNf) (1.37)
the mass term is absent, QCD is symmetric under independents unitary transfor-mations of the left (L) and right (R) components of the 4D-spinor ψ:
L= 1−γ5
2 ψ, R=
1+γ5
2 ψ (1.38)
Indeed, while the kynetic term keeps L and R decoupled
ψ /Dψ= L /DL+R /DR, (1.39) in the mass term they are mixed:
ψψ=ψ†γ0ψ=LR+RL (1.40) and the symmetry is explicitly broken.
In this case, the lagrangian is invariant only under the vectorial subgroup
SU(Nc)V⊂SU(Nc)L×SU(Nc)R (1.41)
where L ed R transforms with the same unitary matrix.
Anyway, the mass of the Goldstone’s bosons associated to this symmetry break-down (the π mesons) is small, and this shows that the chiral symmetry remains an approximated symmetry also in the real theory, where m is small, but not zero.
Chiral symmetry implies that the number of particles of the two kinds of chirality (left and right) 3) are equal. In this case, a good order parameter for the chiral
symmetry is the quantity
hψψi = hLR+RLi = −T V
∂ln Z
∂m (1.42)
3)Chirality is the value of γ 5
§3. SYMMETRIES OF QCD 9 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 β 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 h ¯ ψψi Nf= 4 123× 4, m=0.09 163× 4, m=0.09 203× 4, m=0.09
Figure 1.2: Chiral condensate as function of β = 6/g2. (Simulation parameters: Nf = 4,
mqa=0.09, lattice 123×4).
called chiral condensate.
Under γ5the L component changes sign:
γ5 : L→ −L and γ5 : R→R, (1.43)
so ψψ is odd
γ5: ψψ → −ψψ (1.44)
As consequence, in a chiral-symmetric vacuum hψψi = 0, and a finite not null value signals the formation of the quark-antiquark condensate and the breaking of the chiral symmetry (see fig (1.2))
1.3.2 Center Symmetry and Pokyakov loop
Figure 1.3:Roots of unity in the complex plane
(example for z6=1)
The SU(N) theory has an interesting property that plays an important role for the QCD phase transition: symme-try under the center transformations.
The “center” of the group G is the set of elements z ∈ G (a subgroup) that commute with every element of the group[z, G] =0.
For SU(N), the elements having this property are of the form z·IN, with z∈
C, such that t zz? = 1 and det(z·I) =
nothing more than the set of N-roots of the unit ZN ={ei2π
n
N, n=0, . . . N−1}. (1.45)
We consider now periodic gauge transforms, but with an additional factor z∈ ZN
on the last time slice (also called “large”gauge trasforms) [47] :
G(τ+β, x) =z·G(τ, x), z ∈Z3 (1.46)
The bosons fields changes with the law: ˜
Fµν =GFµνG−1, A˜µ= GAG†−
1 igG∂µG
† (1.47)
z commutes with each group generators [z, Ta] = 0. In addition zz? = 1, so it
completely cancels out from the fields, which remain periodic in τ. The same thing happens for gauge action SG[A] =
´
d4x12 Tr FµνFµν and
integra-tion measureDA4)
This implies the important result: QCD without quarks possesses an exact central symmetry at the action level.
On the contrary, since ˜ψ= Gψ is made by an anti-periodic (ψ)and non-periodic (G) factors, the transformed field ˜ψ is subject to boundary condition incompatible with the Fermi-Dirac statistics, whenever is z6=1:
˜
ψ(β) =−z·ψ˜(0) (1.48) The presence of dynamical quarks breaks explicitly the symmetry.
Typical quantities that are affected by this transformation are parallel transports Γ[C] along paths C wrapping n-times around the compactified time direction. (see fig. (1.4)). They are topologically non trivial, i.e. not reducible at one point with continuous transformations.
The Polyakov loop Ωab(x) = Peig´ 1 T 0 dτ A4(τ,x) ab , a, b=1 . . . Nc (1.49)
is the simplest example of this kind of quantity with the winding number n =±1. Consider now the averaging value of the gauge invariant quantity Tr[Ω] over
4)According to the definition, the Haar measure depends only on the product∏|∆λ|2, where λ are
§3. SYMMETRIES OF QCD 11
Figure 1.4: Paths on the lattice: trivial loops and Polyakov loops.
physical volume P= lim V→∞ 1 V Nc ˆ d3xTr Peig´ 1 T 0 dτ A4(τ,x) , (1.50)
Under Z3 , crossing the temporal boundary τ = T1 (see fig. (1.4)), the Polyakov
loop picks a factor of z:
P(z) =zP, z=ek2πi3 ∈ Z(3) (1.51)
Indicating withhiAthe average over the gauge ensamble{A}, we have
hPiA= hP(z)iA(z)= zhPiA(z) =zhPiA (1.52)
The first equality holds, because it is a simple changing of integration variables; the second one, is the result of the invariance ofDA and SG[A].
Combining the two, we get
hPi =zhPi (1.53)
whose solution (for z 6= 1) is hPi = 0: in the pure gauge theory with exact Z3
symmetry, the Polyakov’s loop should be zero.
Consequently, P acts as an order parameter: a finite expectation value hPi 6= 0 signals a spontaneous breaking of the Center Symmetry. [See 33, 30, 27]
1.4
Quarks Confinement
1.4.1 Asymptotic Freedom
Renormalization Group Theory shows that the 2nd-order loop expansion of the GellMann-Low beta-function βGL(µ) = ∂∂gln(Rµ) in terms of the renormalized coupling gR is
∂gR
∂ln(µ) = −b0g
3
R−b1g5R+. . . (1.54)
where b0 and b1 are renormalization-scheme independent parameters, and µ is
the momentum scale:
( b0 = 16π12 11−23Nf b1 = 16π12 2 102− 38 3Nf (1.55)
Solving this equation and introducing the integration constant ΛQCD, we have:
µ(gR) =ΛQCD b0g2R
+b1/2b02e+ 1
2b0g2R (1.56)
This relations imply a scale dependence of the the QCD coupling: when b0 > 0,
i.e. 11− 2
3Nf >0, we have gR →0 when µ →∞ : the effective interaction decreases
as momentum scale is increased (asymptotic freedom), making perturbative methods appliable.
1.4.2 Confinement and Quark Free Energy
At ordinary temperatures quarks appair permanently confined into hadrons.
There are, however, many indications that the strongly interacting matter at high temperatures behaves fundamentally different from the low temperature case.
As demonstrated by asymptotic freedom, the gauge coupling decreases when the temperature rises. This leads to the prediction of a new state of matter, the quark gluon plasma (QGP), in which the basic constituents of QCD, quark and gluon, should propagate almost freely.
At a certain point, for a fiven finite temperature Tc (about 150 MeV), as shown in
the lattice theories, a transition happens between the two phases.
From the physical point of view, confinement could be measured by the free energy∆F required to add a source quark q in the system and is given by the average of the Polyakov loop
e−1T∆Fq = 1
Nch
§4. QUARKS CONFINEMENT 13
To see that, we start solving the Dirac’s equation for a massless particle a rest in
(D/−ig /A) ψ=0 (1.58) In the static limit ∂
∂x →0, only the time derivative ∂4 are left, and we have
(∂4−igA4) ψ(τ) =0 (1.59) which, for a quark with color a, has solution:
ψa(τ, x) = h
Peig´dτ A4(τ,x)i
ab ψ
b(0, x) (1.60)
The relation (1.60) shows that the quantum amplitute for the process|ψ, ai → |ψ, bi, in the classical limit, is:
hψˆb
τ(x)ψˆ
†a
0 (x)i ∼Ωba(x) (1.61)
where Ωba(x)are element matrix of is the Polyakov loop (see eq: (1.49)).
The free Gibbs energy increment∆Fq5) due to the quark source
e−T1∆Fq = 1 ZNa
∑
a ha|e−τ H|ai = 1 ZNc∑
a hψˆa 0(x)e−τ Hψˆa0(x)i = = 1 ZNc ˆ DψDψe¯ −τ H "∑
a ψ†a0 (x)ψaτ(x) # = 1 Nch Tr[Ω(x)]iwhere we used the (euclidean) Heisenberg time evolution for the fields: e−τ H ˆ
ψ0a(x) =ψˆτa(x)e−τ H (1.62) In the pure gauge QCD, this results in a confinement criterion: ifhPi =0, adding a source quark q in the system costs an infinite amount of energy ∆Fq = ∞, and
quarks are confined.
So, at least for a purely gluonic SU(N) gauge theory, the phase transition could be characterized by the breaking of a global Z(N)symmetry above Tc:
deconfinement⇔Z3symmetry breaking ⇔ hPi 6=0 (1.63)
On the other hand, in full QCD with dynamical fermions, this simple confinement
5)The meaning of a term such g¸ dτ A(
τ, x) in the exponent can be understood in another way:
in the minkowskian metric, the action for a particle of “charge” g and current jµ = (ρ, ρv) is S =
−´dx4gAµjµ. If the particle is at rest (v = 0) we have S = −
´
dtgA0(t, x), where gA0(t, x) is the
potential energy of a static “charge g” in x. After the Wick euclidean rotation (dt=−idτ, A0=−iA4),
5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 β 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 h| P |i Nf= 4 123× 4, m=0.09 163× 4, m=0.09 203× 4, m=0.09
Figure 1.5: Diagram of h|P|i at the transition. Here β = 6/g2 e amq = 0.09 and Nf = 4
(reweighted data).
criterion fails: the color flux tube emanating from the source q is no longer forced to go to infinite, but can end in a dynamical anti-quark ¯q, creating a q ¯q system with finite∆F.
As result, hPi can be not null also in the confined phase and it is not an order parameter anymore.
Nevertheless, the Polyakov loop is still a good indicator for the deconfinement: the change ofhPifrom small values (low temperature) to large values (high temper-ature), signals the phase transition, and this happens at a well determined T = Tc
(see fig. (1.5)).
The contribution to the heavy quarks free energy related to the introdution of a baryon chemical potential can be extracted from the large distance behaviour of the Polyakov loop .
See Appendix (A.1) for an example of this kind of simulation.
1.4.3 Quark-antiquark potential
Figure 1.6: Polyakov loops ad distance r
While hTr[Ω(x)]i gives the Gibbs free energy ∆F of a single quark, the cor-relator hTr[Ω(x)]·Tr[Ω†(y)]i give the
§4. QUARKS CONFINEMENT 15 quark-antiquark potential Vq¯q(r), e−T1Vq¯q = Zq¯q Z = 1 N2 ch Tr[Ω(x)]·Tr[Ω†(y)]i (1.64) where r= |x−y|, for r→∞.
This can be show as in this way: [68] e−T1Vq¯q = 1 ZN2 c
∑
a,b ha, ¯b|e−τ H|a, ¯bi = 1 ZN2 c∑
a,b hψˆ0†a(x)ψˆb0(y)e−τ Hψˆa0(x)ψ0†b(y)i (1.65)Using again the rule (1.62), we have e−T1Vq¯q = 1 ZN2 c ˆ DψDψ¯
∑
a,b he−τ H ψ0†a(x)ψ0b(y)ψaτ(x)ψ†bτ (y)i = = 1 ZN2 c ˆ DψDψ¯∑
a,b he−τ H ψaτ(x)ψ†a0 (x)·ψ†bτ (y)ψ0b(y)i = = 1 N2 ch∑
a Ωaa(x)·∑
a Ω† bb(y)i = 1 N2 ch Tr[Ω(x)]·Tr[Ω†(y)]iFigure 1.7: Cornell quark-antiquark potential V(r)
Theoretical arguments, based on the strong-coupling limit (g → ∞) and con-firmed by simulations, show that quark-antiquark potential is well interpolated by the Cornell potential (see fig (1.7)):
V(r) =a− b
r +σ·r (1.66)
If the linear term in V(r) is not null, the effective interaction between particles increases with distance, and quarks are permanently confined into hadrons.
When temperature is increased, the value of σ decreases, and becomes zero for some T =Tc (deconfinement).
Chapter 2
Lattice QCD basics
The non-perturbative properties characterizing low energy hadronic physics (confinement, chiral symmetry breaking, etc) are lost when strongly interacting matter is heated or com-pressed, while at low temperature we expects a more rich structure.
It is therefore crucial to have a method for studying QCD in the non-perturbative region. Lattice QCD, the discretized version of Quantum Chromodynamics, is a successful first-principle method that meets the request. It satisfies the basic requirements for a strong force theory, namely gauge invariance and (partially) chiral invariance.
This chapter introduces the basic concept, such U links and plaquettes, using the parallel transport concept as basis. Transport on closed circuits can be used to calculate Yang-Mills’s action in the elementary volume of the lattice (Wilson). Then we discuss briefly the intro-duction of fermions, in the staggered formulation (Kogut-Susskind), and other questions such continuum limit and action improvements.
2.1
Links and Plaquettes
In the lattice formulation, field theory is defined in a gauge-invariant way on a dis-crete space-time domain
Λ=aZ4= {x|xµ
a ∈Z} (2.1)
, an hypercubic collection of Ns3×Nτ lattice “sites”, labeled with x ≡ (τ, x). The
minimum λ on the lattice is 2a, so p∈ [−π/a, π/a], the so called Brillouin zone. This serves at least two purposes: a) to provide an ultra-violet cut-off for the theory, restricting highest momentum to π/a (a being the lattice spacing) and b) to evaluate the path integrals in the Euclidean formulation stochastically, using Monte-carlo methods.
After discretizing, all physical units disappear from the theory: chemical po-tentials and quark masses are expressed in units of the lattice spacing, ˜µf = µfa,
˜
mf =mfa.
Figure 2.1: The lattice grid: links, fields and plaquettes
On the lattice the funda-mental degrees of freedom are fermion fields ψx ∈ Λ, that
reside on the sites of the lat-tice and carry flavor, color and Dirac indices1), and bosonic, bosonic fields in the form of SU(Nc)matrices Ux,µ.
At least at the naive level, the discretization of the fermion sector is straightforwardly achieved by introducing dimensionless Grassmann valued fermion fields and replacing derivatives by finite central differ-ences, such:
∆xψx ≈
(ψx+a−ψx−a)
2a (2.2)
The discretization of a gauge theory is actually a bit more involved: before sub-tracting two fields defined in different lattice points, we have to parallel-transport them.
The links variables Uµ(x)∈SU(3)exactly describe this parallel transport process:
Γ(x+aµ ⇒x) =P exp ig ˆ x x+µaˆ dxµAµ(x) (2.3) from site x to the neighboring site in the ˆµdirection x+µaˆ . In the limit a→0, it can be approximated by
Uµ(x)≡e−igaµAµ (2.4)
Using Uµ, it is possible to define two kind of derivatives (forward and backward):
(∆(+)
µ ψ = 1a(Uµ(x+a ˆµ)ψx+aµˆ−ψx)
∆(µ−)ψ = 1a(ψx−Uµ†(x−aµˆ)ψx−aµˆ)
(2.5) useful in the following.
§2. THE PURE-GAUGE ACTION 19
2.2
The pure-gauge Action
Consider the trasport around a (minimal) closed path, starting at x and oriented as µν, the “plaquette” (see fig (2.2))
Ux;µν=P exp ig ˛ DxµAµ(x) =Ux,µUx+µˆ,νUx†+ˆν,µU†x,ν (2.6)
This quantity, unlike Tr[U], is not gauge-invariant. Indeed:
Ux;µν0 =G(x)Ux;µνG†(x) (2.7)
The fact that transport along a closed path is different from identity implies that transportΓ[C]depends on the path chosen, as well as on the starting and end points. This difference is basically due to the fact that Lie derivatives at different direc-tions do not commute[Dµ, Dν]6=0, so we expect that
Ux;µν →1 when Fµν =
i
g[Dµ, Dν]→0, (2.8)
eg. when the “curvature” of the configuration space is infinitesimal.
Simbolically, the infinitesimal transport in the µ direction can be written as Ux,µ =
eiagAµ(x), and for the plaquette:
Ux;µν≈e−iagAµ(x)e−iagAν(x+aµˆ)eiagAµ(x+aˆν)eiagAν(x) (2.9)
Figure 2.2: Plaquettes
With the help of the Baker-Campbell-Hausdorff formula 2)and negletting terms
2)eAeB=eA+B+1
of order a3 in the exponent, it can be written as: Ux;µν ≈e−ia
2g·(∂
µAν−∂νAµ−ig[Aµ,Aν]) ≈e−ia2g·Fµν (2.10)
where we recognize Fµν =∂µAν−∂νAµ−ig[Aµ, Aν].
It is easy to see that expanding the U links in power of lattice spacing a for a →0, we have [59] e−iga2Fµν ≈1−iga2F µν− 1 2g 2a4F2 µν+. . . . (2.11)
The discretization of the Yang-Mills part of the action leads to:
Sgauge = ˆ d4x1 2
∑
µ,vTr[F 2 µν]≈a 4∑
µ<ν Tr[Fµν2 ]≈ a4 2 g2∑
µ<ν Tr[1−Ux;µν] ≈a42Nc g2∑
µ<ν [1− 1 Nc Re Tr Ux;µν]where we used the property Tr[Fµν] =∑aFµνa Tr[T
a] =0 .
So, in the simplest formulation, the gauge action can be written as : [59, 68]
Sgauge = βG
∑
x,µ<ν 1−1 3Re Tr Ux;µν , βG = 2Nc g2 (2.12)2.3
Fermions on the Lattice
2.3.1 Wilson fermions
The fermion action is a bilinear form in the fields ψx
Sf ermions=
∑
f ¯ ψfMfψf ≡∑
f ,x,y ¯ ψf(x) Mf xyψf(y) (2.13)with associated matrix
Mf =mf +D/(mf, µf) (2.14)
From the point of view of the numerical scheme, the implementation of D is based on its relation with the finite differences operator∆
∆ f = f(x+a)− f(x)
a = (
eaD−1
a )f(x) (2.15) Various schemes can be used, truncating the formal expansion of D in powers of
§3. FERMIONS ON THE LATTICE 21 a: D= ln(1+a∆)−1 a =∆− 1 2a∆ 2+ a2 3∆ 3+ (2.16)
or even more sophisticated ones, involving high-order combinations of "forward" and "backward" derivatives (2.5).
The simplest is to take a symmetric combination of these, plus a quadratic term, proportional to r: / DW = 1 2γµ(∆ (+) µ +∆ (−) µ ) +r·a∆ (+) µ ∆ (−) µ (2.17)
The Wilson operator Fourier representation is / DW(p) =iγµ sin(apµ) a +2r· sin2(apµ/2) a (2.18)
The additional term becomes zero in the limit a →0 at fixed momentum p, so it is in effect an "artifact" of the discretization process.
The reason for his introduction to (2.17) is soon explained. [19]
As is well known, the spectrum of the theory is dominated by the propagator (1/D) poles located in the Brillouin zone (BZ) [−π/a, π/a]. They are the zero’s of
/
D(p)in the momentum representation.
In the bosonic case, the kinetic energy contribution φ†φ would be sin
2(ap
µ/2)
a ,
with zeros in apµ= n2π. Excluding p=0, they are all outside BZ.
Different is the case of fermions: here the kinetic term for r = 0 contributes 16 poles pµ ∈ (±πa)⊗4, of which 15 are in the BZ zone. They correspond to elementary
excitations that have no analogue in the continuous (fermion doubling problem). The additional term solves the problem: at the boudary of the BZ zone apµ= nπ,
his contribution diverges for a→0.
These modes would be assigned an effective mass O(1a): m(p) =m+ r
a
∑
µ6=0
(1−cos(apµ)) (2.19)
and, at the continuum, they disappear from the spectrum.
Unfortunately, the Dirac’s operator, in Wilson’s formulation, does not anticom-mutes anymore with γ5
{γ5, /DW} 6=0 if r6=0 (2.20)
This implies that, even for zero mass fermions, the kinetic term ¯ψ /DWψexplicitly breaks chiral symmetry, at finite lattice spacing a.
It can be demonstrated (Nilsen-Ninomiya theorem) that it is impossible to construct a theory, bilinear in the fermionic fields, which is local, invariant for translation,
chirally symmetrics and with the correct continuum limit, without giving up any of these properties. [61]
In the case of Wilson’s formulation, one chooses to renounce to chiral symmetry. After a rescaling of the fields, and choosing r = 1, the fermion matrix is usually written as (Mf)xy = δxy−κf ±3
∑
ν=±0 e−aµfδ|v|,0·sign(v)· [(1−γ ν)Uν(x)δx+ˆν,y] (2.21) with κf = 1 2amf+8. [59]Here x and y refer to lattice sites, ˆν is a unit vector on the lattice; aµf and amf are
respectively the chemical potential and the quark mass in lattice units. 2.3.2 Fermion determinant In the definition of Z Z= ˆ DψDψ¯DA e−SQCD[A,ψ] (2.22) with SQCD =Sgauge[A] +
∑
f ˆ ψ†fMf(mf, µf)ψf d4x (2.23) we can carry out the integration over ψf analytically, using the standard identity 3):det A=N
ˆ
DηD¯ηe−∑ijη¯iAijηj If we also assume quark degeneration, we have
Z= ˆ DU e−Sgauge
∏
f det Mf = ˆ DU e−Sgaugedet MNf (2.24)The fermion determinant det M(U)describes the vacuum polarization effects due to the dynamical quarks and makes the effective action highly non-local.
For this reason simulations with dynamical quarks are very resource demanding and the quenched approximation is often employed, where det M(U)is set to 1.
Using the the so called TrLog trick4):
det M=eTr(ln(M)) (2.25)
3)N is an unessential constant, η
iare Grasmanian vectors. 4)Valid for positive definite matrices.
§3. FERMIONS ON THE LATTICE 23
and introducing the notation
h. . .iG =
ˆ
DUe−SG(. . .) (2.26)
we can write Z in the compact form: Z=h(det M(U))iNf
G =heNf Tr (ln(M))
iG (2.27)
Writing M = m+D, the formula (2.27) has an interesting expansion in power of/ /
D/m: the so-called hopping expansion. [27] Indeed, using the Taylor formula
ln(1+x) =
∑
k≥1 (−1)kxk k , (2.28) we have: Z= ˆ DUe−Sgauge eNfTr[ln(m+D/)]i G= ˆ DUe−Se f f[U] (2.29)Figure 2.3: Lattice paths in the hopping expansion
with Se f f[U] =SG[U] +Nf
∑
k≥1 (−1)k Tr[D/ k] k·mk−1 (2.30)This is equivalent to taking the ensemble average with the following probability distribution:
P(U) = e−
Se f f[U]
´
DUe−Se f f[U] (2.31)
Each terms in Se f f[U]has a straightforward interpretation in terms of k-loop on
n→m, while the tracehTr[D/·D/· · · ·D/
| {z }
k times
]iGimplies that n should be equal to m (closed
path).
These k-loops, in contrast to gauge action plaquettes, are non-trivial: they can turn around completely in the direction of time.
At the lowest order, therefore, the hopping expansion contributes to the action with terms proportional to the Polyakov’s loop h· <(L(x)), with h∼κN f, similar to those of an external magnetic field in a spin system.
2.3.3 Staggered formulation
The naive discretization leads to the unwanted “doublers”: 16 spurious poles in the quark propagator at the edges of the Brillouin zone pµ= ±π/a.
In the Kogut-Susskind formulation, the fermionic degrees of freedom are spread-ing, or staggered, over the lattice, reducing the doublers by a factor of 1/4.
The result is that the single flavor is represented taking the 4th root of the quark determinant det[M14]and for Nf flavors:
(det M)stagg =
∏
f
det Mf[U, µf]
1/4
(2.32)
and the Gamma matrices γµ disappear form the fermion matrix, substituted with
simples phases ηµ(x) = (−1)∑ν≤µxµ (Mf)xy = amfδx, y+ 4
∑
ν=1 ην(x) 2 h e−aµfδν,4U x; νδx,y−ˆν (2.33) − e+aµfδν,4U† x−ˆν; νδx,y+ˆν i . (2.34)2.4
Continuum Limit
For lattice QCD to give results compatible with Physics5), a prescription is required in order to make the limit a→0 consinstently. In particular, the limit has to be taken along lines of constant physics, i.e. keeping temperature T = 1
Nτa and mass ratios
fixed. In this case, the limit a→0 is equivalent to taking Nτ →∞.
When studyng Thermodynamics propertries, such the phase transition, we are interested in changing T with continuity, just one value of Nτ. In this case, the lattice
5)In particular, the lattice spacing a should be small enough to catch small (a 1/Λ) and large
§4. CONTINUUM LIMIT 25
spacing a is varied tuning the bare couping constant g, but keeping the renormalized coupling gR unchanged. The precise relation a(g)is dictated by RG laws.
Assuming the functional form for gR
gR =gR(g(a), ln(aµ)) (2.35) we have ∂gR ∂ln(a) = ∂gR ∂g0 ∂g ∂ln(a)+ ∂gR ∂ln(µ) =0 (2.36) where ∂g
∂ln(a) is the lattice analog of the Gellman-Low function.
Solving for it, we have:
∂g ∂ln(a) =− ∂gR ∂ln(µ) ∂gR ∂g (2.37) The series of gR(g)starts with
gR = g+c·g3+· · · , (2.38) so ∂gR ∂g =1+O(g 2) (2.39) Substituting (2.39) in (2.37), we obtain: ∂g ∂ln(a) =b0g 3+b 1g5+. . . (2.40)
with the same coefficients (apart from the sign) which appair in (1.54). The integration is straightforward, leading to
a(g) = 1 ΛL b0g2 −b1/2b20 e− 1 2b0g2 (2.41)
wereΛLis a new integration constant6). (see for example [49, 51])
2.4.1 Tuning the temperature
Temperature in Lattice QCD is fixed by the lattice extension in the compactified time direction τ = Nτa (see fig (1.1)):
T= 1
Nτa
(2.42)
6)It is possible to show that the two constants,Λ
QCDandΛL, are related by a multiplicative factor.
There are two different ways to vary T on the lattice.
Following the definition (2.42), one way is to change Nτ. But Nτ is an integer and
can be changed only in discrete steps.
Another way, is to keep fixed Nτ and change the lattice spacing a. However, after
the lattice discretization, the parameter a does not appear anymore in the theory and must be controlled through its relation a=a(g)with the bare coupling g.
Using (2.41) and the definition of T, eq.(2.42), in the leading order of the weak coupling expansion (i.e. with b1 =0)we have:
T(g)∼ ΛL
Nτ
e
1
2b0g2 (2.43)
For QCD (Nf = 3) b0 is positive (as opposite of QED): this implies that the high
temperature regime (T→∞)is mapped to the weak-coupling regime (g→0). One should note that decreasing a, not only increase T, but also decrease the physical volume V =a4·Ns3Nτ, breaking the thermodinamic condition V1/3 T.
2.5
Action improvements
Typically, the lattice action discretization errors areO(a2)for the gauge andO(a)for
fermions fields.
Progress has also been made in recent years in the construction of better dis-cretization schemes for the QCD Lagrangian, which significantly reduced the sys-temic errors and cut-off effects introduced by finite size of the grid.
These discretization errors can be removed by the Symanzik improvement pro-gram that relies on higher order difference scheme, i.e. considering combination of link variables that extend beyond the elementary square ([See,for example 16]): the generalized plaquettes Wi; µνn×m, i.e. trace of the n×m loop constructed from the gauge links along the directions µ, ν departing from the i site.
Calculation of quantities like correlators, for example, can be done much more reliably when the operators are less contaminated by high order fluctuations.
The smeared links Uµare constructed substituing the original link with a smeared
version, obtained reducing the mixings of states with the high frequency modes. Smearing is usually done replacing the original link with a gauge-covariant aver-age of different paths connecting the same two points, then projecting to SU(3)(APE smearing).
The projection may be viewed as an arbitrary trick to remain in the group, making the method less palatable from the theoretical point of view.
But the main problem is another: modern Montecarlo are based on the molecular dynamics, a technique that requires the computation of the “molecular force” (3.26),
